DDynamic Random Sub jective Expected Utility
Jetlir Duraj ∗ Abstract
Dynamic Random Subjective Expected Utility (DR-SEU) allows to model choicedata observed from an agent or a population of agents whose beliefs about objectivepayoff-relevant states and tastes can both evolve stochastically. Our observable, theaugmented Stochastic Choice Function (aSCF) allows, in contrast to previous workin decision theory, for a direct test of whether the agents’ beliefs reflect the true data-generating process conditional on their private information as well as identificationof the possibly incorrect beliefs. We give an axiomatic characterization of whenan agent satisfies the model, both in a static (R-SEU) as well as in a dynamicsetting (DR-SEU). We look at the case when the agent has correct beliefs about theevolution of objective states, as well as at the case when her beliefs are incorrectbut unforeseen contingencies are impossible.We also distinguish in some detail two sub-variants of the dynamic model whichcoincide in the static setting: Evolving SEU, where a sophisticated agent’s utilityevolves according to a Bellman equation and Gradual Learning, where the agent islearning about her taste over time. We prove easy and natural comparative staticresults on the degree of belief incorrectness as well as on the speed of learning abouttaste.Auxiliary results contained in the online appendix extend previous decision the-ory work in the menu choice and stochastic choice literature from a technical as wellas a conceptual perspective.
The study of stochastic choice has found renewed popularity in economics. Along with aconsiderable amount of research on static stochastic choice models, several recent workshave pioneered foundational work into dynamic stochastic choice models. In a dynamicsetting the agent solves a dynamic decision problem and learns as time passes abouteither the environment she is facing or her own evolution of preferences or both. In manyapplications an analyst only observes choices of an agent as well as some (possibly public)signals about payoff-relevant objective states. He doesn’t have information about thestochastic process of the preferences of the agent (the private information of the agent). ∗ [email protected], Acknowledgments: I am indebted to Drew Fudenberg and Tomasz Strzaleckifor their continuous encouragement and support in this project. I thank Jerry Green, Kevin He, EricMaskin and Nicola Rosaia for numerous comments during different stages of this project. I also thankArjada Bardhi, Krishna Dasaratha, Ryota Iijima, Jonathan Libgober, Jay Lu, Maria Voronina and theaudience of Games and Markets at Harvard for their helpful comments. Any errors are mine. [Fudenberg, Strzalecki ’15], [Frick, Iijima, Strzalecki ’17], [Steiner, Stewart, Matejka ’17] to name afew. a r X i v : . [ ec on . T H ] A ug n this paper we consider such a general environment: there are payoff-relevant ob-jective states every period, an agent has every period standard subjective expected utility (SEU) preferences, comprised of beliefs about the objective, payoff-relevant state as wellas a Bernoulli utility over a set of prizes. The subjective state of an agent in each periodconsists of her realized SEU. We assume that these follow an exogenously given stochasticprocess which is well-known to the agent (albeit unknown to the analyst). We assumethe agent can’t influence the given stochastic process and allow for both stochastic tastesand stochastic beliefs. In many real life examples this two-fold randomness is present, e.g.investment and saving behavior may depend both on exogenous, objective randomnesssuch as market conditions as well as on the stochastic evolution of the risk aversion of theagent.We assume that after each history of choices and realizations of the objective states theanalyst observes limiting frequencies of the choice of the agent in decision problems/menusof the current period as well as the realization of the objective states in the current period.The data also reflect variation of the decision problems/menus. Thus, the observable is inevery period, after each history of choices and objective states, a probability distribution over choices from a menu in the current period and over realizations of the objectivestate. Many situations in real life deliver such data, from employment situations in thelabor market, consumption and investment decisions, to educational choices of students,loan practices, etc. Our focus is axiomatic throughout. Under the assumption that the distribution ofthe private information of the agent doesn’t depend on the decision problem she facesand that the analyst has access to a rich observable featuring variation in the decisionproblems, we give conditions on the observable which allow the analyst to uncover thedistribution of the private information of the agent regardless of its arbitrariness. Underthese conditions the analyst can also study whether the agent’s beliefs when makingchoices reflect the correct data-generating process conditional on her private informationand whenever that is not the case he can identify the biases conditional on the agent’sprivate information. While the study of misspecified learning is not new, this is the firstwork, to the best of knowledge, where there are no a priori assumptions on the originof the misspecification. The misspecified beliefs may be because of misspecified priors,because of imprecise observation of private signals by the agent or because conditional onher private information the agent has some arbitrary behavioral biases in beliefs. The model we consider is still falsifiable as we require the agent to be Bayesian withrespect to the stochastic process describing the evolution of her private information, eventhough she may be non-Bayesian with respect to the true data generating process ofthe objective states. Moreover, we don’t allow any misspecified learner to receive hard evidence about misspecification, such as the occurrence of an unforeseen contingency.Thus, in this paper the agent is able to explain any observed string of objective states within her model, even though as time passes her beliefs might diverge more and more Our identification results are valid under more general conditions – see Remark 1 in section 2. This type of data also allows an alternative heterogeneous population interpretation: there is apopulation of agents facing similar choice situations. The analyst observes in many instances the choiceof an agent as well as the realization of some payoff-relevant objective state. We focus on the single-agentcase in the exposition, but intuitions and results can be readily translated. E.g. this model allows for the case of confirmatory bias studied in [Rabin, Schrag ’99] where anagent may misread signals in a way favorable to her current hypothesis. The agent in their model is notBayesian with respect to the correct prior but is so within her model . The richer observable allows comparative static results about the degree of biasednessof beliefs. We show how an analyst can use the data to construct a precise estimator of theextent of the belief biasedness of the agent and how he can compare different agents usingthis estimator. Moreover, since our model allows for both stochastic taste and beliefs, weshow what an analyst can say about the relative speed with which two different agentslearn their taste, given their respective datasets.This paper is most related to [Lu ’16] – who studies the same static model but withunobservable objective states, and [Frick, Iijima, Strzalecki ’17] – who study a fully non-parametric dynamic model as here but without payoff-relevant objective states. Relat-edly, [Dillenberger et al ’14] study the ex-ante menu preference of the agent modeled by[Lu ’16]. Among other things we extend their work to allow for stochastic taste. Con-ceptually the paper is also related to [Lu ’17] who shows how a combination of ex-antepreference over acts and post-signal random choice can overcome the classical issue ofidentification in the Expected
State-dependent
Utility model. Our model illustrates thestrong identification properties of random choice data for the case of state-independentutilities in a rich dynamic environment allowing for stochastic taste. Finally, the observ-able in this paper can be interpreted as a likelihood function of a dynamic choice modelin the spirit of [Rust ’87] and the literature that it inspired. Whereas that literature hasfocused on identification and inference of controlled stochastic processes, this paper offersan axiomatic treatment of such likelihood functions for choice behavior in a general setup with both observable and unobservable states.In the following we explain in detail the organization of the paper mentioning itscontribution at each step.Section 2 focuses on the static model. For each decision problem A an analyst observesthe frequency of an agent’s choice and the realization of a payoff-relevant objective state s (we say agent picks act f from menu A and objective state s is realized with a certainprobability ρ ( f, A, s )). We call this observable an augmented stochastic choice function(aSCF) . We show how the analyst can identify from this observable the space of thesubjective states of the agent. We call this the revealed subjective support of the data.We impose axioms similar to the ones in [Lu ’16] to ensure that the revealed subjectivesupport consists of SEUs that are identified by a belief q about the realization of s as wellas a Bernoulli utility u . Furthermore, we show how the analyst can use the concept of therevealed subjective support to test whether the agent is using the correct data-generatingprocess of objective states, conditional on her private information. This corresponds tothe classical statistical concept of well-calibrated beliefs originating in [Dawid ’82] butnow in a general setting which allows for stochastic taste. Intuitively, an agent has correct interim beliefs only if the observed frequency of the realization of s conditional onobserving f chosen from A is a mixture of beliefs in the subjective support of the datawhich can rationalize the choice of f from A . Whenever this condition fails the analyst The time horizon is assumed to be finite. Thus the agent cannot resort to statistical tests of arbitraryaccuracy to determine that her beliefs might indeed be misspecified. Our proofs modify and extend the proofs of [Lu ’16] and [Frick, Iijima, Strzalecki ’17] in multipledirections as well as extending several other models in the literature. E.g. we extend [Ahn, Sarver ’13]to include objective states and stochastic beliefs. Details are in the online appendix. See [Rust ’94] and [Aricidiacono, Ellickson ’11] for surveys on the dynamic discrete choice literature. The last section of [Lu ’16] also studies the property of well-calibrated beliefs but in a setting of s is always in the support ofher belief q .Section 3 introduces the dynamic model. The observable is now a history-dependentaSCF : for every history h t − occurring with positive probability, the analyst observesfrequencies of the choice in a subsequent decision problem A t together with the realizationof the objective state in the respective period (we say agent picks f t from menu A t afterhistory h t − and objective state s t is realized with probability ρ t ( f t , A t , s t | h t − )). Historieshave empirical content, as they help the analyst identify the serial correlation in the privateinformation of the agent, i.e. in her tastes and beliefs.We assume these history-dependent aSCFs satisfy the assumptions of the static model.In contrast to the static case there is now limited observability : not every menu is observ-able after every history. This is a similar observability problem as in [Frick, Iijima, Strzalecki ’17]and technically its solution in this paper adapts theirs to our more general setting withpayoff-relevant states. It relies in identifying two classes of histories which reveal the sameprivate information. Whenever the observable satisfies the history-dependent version ofthe static model and the two history equivalence properties the analyst can identify thestochastic evolution of the private information of the agent as well as the true data-generating process of the objective states. This is the DR-SEU model, the namesake ofthe paper.After establishing the main characterization result we focus on two special cases ofDR-SEU whose static versions are indistinguishable: Evolving SEU, where the evolutionof agent’s Bernoulli utility is given through a Bellman equation and its specialization,Gradual Learning, where the agent is learning about a fixed but unknown taste. Ad-ditionally, and because we need it for the dynamic characterization results, we describewhen a menu preference comes from an agent who is subjectively learning both aboutobjective states and about her Bernoulli utility/taste through a new axiom called
WeakDominance . Intuitively, such an agent would always prefer to exchange any menu of acts A for a menu ¯ A which allows her to pick any of the prizes occurring in A with positiveprobability irrespective of the realization of the uncertainty she’s facing ex-ante.Section 4 leverages the characterization theorems to prove comparative statics results.In a setting of non-stochastic taste we address the question of how an analyst can compareagents with respect to their biasedness of beliefs. Namely, given a commonly observablecharacteristic, e.g. gender, race or letter grades, if the analyst fixes a direction of biasedbeliefs for every characteristic, he can tell from stochastic choice data when an agent ismore biased than another agent. Intuitively, the choice data give evidence that the morebiased agent values menus uniformly more differently to a fictitious unbiased agent thanthe less biased agent. Finally, in the special case of the Gradual Learning representation,we show how an analyst may distinguish when an agent’s uncertainty for taste fullyresolves and how the analyst may compare different agents with respect to the speedof learning their taste. Intuitively, agent 2 learns her taste more slowly than agent 1whenever the data suggests that agent 2 satisfies Weak Dominance whenever agent 1 non-stochastic taste. The two equivalence properties are called
Contraction history independence and
Linear history in-dependence . Consider an employer at a job fair looking at applications for a job vacancy. The jobconsists of performing a task, after the job fair is concluded, whose outcome has twopotential values coming from S = { g, b } ( g stands for good and b for bad ). We assumethat whether g or b is realized depends on both the ability of the employee as well asother randomness outside of the control of the employee.During the job fair, in the first period of the model ( t = 0) some characteristic s ∈ S = { s (cid:48) , s (cid:48)(cid:48) } of the applicant is revealed to the employer, say ethnicity, gender,education level, etc. We assume the distribution of s over S is known to the employer.This may be justified e.g. if the data about the prevalence of the characteristic s in thepopulation of the applicants at the job fair is public. In the second period ( t = 1) theemployer has beliefs about the outcome of the task, conditional on the revealed character-istic s . These are coded by (ˆ q , ˆ q ) = (ˆ q ( g | s (cid:48) ) , ˆ q ( g | s (cid:48)(cid:48) )) ∈ (0 , . These can potentiallybe different from the true data generating process which here for simplicity is given by q ( g | s (cid:48) ) = q ( g | s (cid:48)(cid:48) ) = . Assume here for simplicity that the analyst knows this data-generating process. In our example we say that the employer has incorrect beliefs if thefollowing holds. Incorrect Beliefs: > ˆ q ( g | s (cid:48) ) > > ˆ q ( g | s (cid:48)(cid:48) ) > We assume in the following that the objective state s (task outcome) is also observableto the analyst after the choice of the employer.Given the observed characteristic s the employer can choose in t = 1 whether to hirethe candidate (formally, act h s : S → R ) or not hire (act nh s : S → R ). In the case of nothiring, the utility of the employer is always zero u s ( nh s ( s )) = 0 for all s ∈ S , s ∈ S .In the case of hiring the employer has (possibly) stochastic utility u s : R → R whichsatisfies u s ( h s ( g )) = g s , u s ( h s ( b )) = b s with g s > > b s almost surely . Stochastic utility conditional on the realization of s is meant to capture the possibilitythat the utility of a successful task for the employer may depend on the specific task to Many situations have the same structure: lending activity of a bank, university applications, etc. Other assumptions are possible. These here are for definiteness.
5e solved, here assumed unobservable to the analyst, besides on the characteristic s ofthe employee. It may also happen due to other characteristics of the candidate besides s which are unobservable to the analyst but relevant to the employer. Finally, we assumethat whenever the employer is indifferent between hiring and not hiring a candidate heuses an unbiased coin to break ties.Besides biases in beliefs we allow for the possibility that the employer cares about therealization of s as well. We require for the random variables g i , b i , i = 1 , C ) g s (cid:48) ≥ g s (cid:48)(cid:48) > > b s (cid:48) ≥ b s (cid:48)(cid:48) almost surely . A successful task benefits the employer more – and a failed one hurts him less – if it isthe deed of an agent of characteristic s (cid:48) rather than s (cid:48)(cid:48) . That is, the employer incursuniformly lower payoffs from s (cid:48)(cid:48) for each outcome.We say that the employer cares about s if the following holds. Preference for s (cid:48) : g s (cid:48) > g s (cid:48)(cid:48) > > b s (cid:48) > b s (cid:48)(cid:48) almost surely.Here we ask for the ‘extreme’ inequalities in condition ( C ) to hold strictly almost surely. Assume now that an analyst has frequency data on both hiring decisions at the jobfair and on the outcome of the task, even though she may not observe the precise typeof the task in every instance. Thus for all s = s (cid:48) , s (cid:48)(cid:48) and s = g, b the analyst observesthe limiting frequency that candidate s is hired, and that state s is realized, denotedby ρ s ( h s , { h s , nh s } , s ). This paper gives conditions on stochastic choice data whichallows the following.- As a first step the analyst can confirm that the true data-generating process isunbiased, i.e. that q ( g | s (cid:48) ) = q ( g | s (cid:48)(cid:48) ) = holds. This corresponds to the constraint ρ s (cid:48) ( f s (cid:48) , { f s (cid:48) , h s (cid:48) } , g ) = ρ s (cid:48)(cid:48) ( f s (cid:48)(cid:48) , { f s (cid:48)(cid:48) , h s (cid:48)(cid:48) } , g ) = 12 . - The analyst can also discern from stochastic choice data whether there is bias in be-liefs, whether the employer cares about the realization of s or whether both are occurringsimultaneously.Namely, whenever the employer is unbiased in beliefs and doesn’t care about the real-ization of s per se he chooses to hire either candidate with the same positive probability.This corresponds to the constraint (cid:88) s ρ s (cid:48) ( f s (cid:48) , { f s (cid:48) , h s (cid:48) } , s ) = (cid:88) s ρ s (cid:48)(cid:48) ( f s (cid:48)(cid:48) , { f s (cid:48)(cid:48) , h s (cid:48)(cid:48) } , s ) . Whenever there is either bias in beliefs or the employer has preference for s (cid:48) he hirescandidate s (cid:48) with strictly higher probability than candidate s (cid:48)(cid:48) . (cid:88) s ρ s (cid:48) ( f s (cid:48) , { f s (cid:48) , h s (cid:48) } , s ) > (cid:88) s ρ s (cid:48)(cid:48) ( f s (cid:48)(cid:48) , { f s (cid:48)(cid:48) , h s (cid:48)(cid:48) } , s ) (1) The employer may have lexicographic preferences; she cares about s first and foremost but given s also takes into account other unobservable features of the candidate. Just as for beliefs other assumptions are here possible. and has preference for s (cid:48) , all elseequal he hires candidate s (cid:48) with a (weakly) higher probability than in the case of eitherbias in beliefs only or preference for s (cid:48) only. This corresponds to a larger gap in (1).This example shows that stochastic choice data coming from standard subjective ex-pected utility (SEU) maximizers can be used to identify biases, whenever the analyst getsinformation for the realization of the objective state (here whether the task is successfulor not). As we show, stochastic choice data allow comparisons of different employers interms of their biases in much more complicated examples than the current one. Consider an undergraduate student who adheres to subjective expected utility (SEU) andhas beliefs about the final outcome in the job market once she graduates. This outcomecomes from a finite objective state space, say, S = { job in finance, job in tech industry, job in government, graduate school, start-up } . At the beginning of the undergraduate education the student is also learning about hertaste regarding possible careers and so has stochastic tastes ˜ v , ˜ v , . . . , ˜ v τ about the finaloutcome. At the end of some student-specific year τ ≥
1, learning about taste ceases: thestudent has a fixed Bernoulli utility v about the final outcome S even though her beliefs q t about the final outcome in S remain stochastic throughout the whole higher educationexperience.Formally, let school years be encoded by t ∈ { , , . . . , T } . Let s t be a period- t signalabout final outcome coming from a finite space of objective signals S t . These can begrades or feedback from faculty, experiences in internships, etc. Let acts (decisions ofa student) correspond to jobs/projects/classes she engages with in each year and menus A t be finite collections of such acts the student can choose from in each education year.Denote the set of menus available in period t by A t . Given a realized signal s t each act f t in period t delivers a lottery over pairs consisting of an instantaneous prize from a finiteset of prizes Z and a continuation decision problem A t +1 from A t +1 . The realization ofthe continuation problem A t +1 corresponds to jobs/internships/classes possibly availableto the student, after she has taken a current class corresponding to the act f t . Say thatan act f t is constant , if the lottery over pairs of current prize and continuation decisiondoesn’t depend on the realization of the signal s t , i.e. it is the same for all s t in S t . E.g. aconstant act is a summer job a student may take only due to financial reasons and whichdoesn’t enhance her intellectual skills in the job market for any possible career.The analyst observes past choices of, say act f l chosen from menu A l as well as therealization of signal s l ∈ S l , and for the current period t ∈ { , . . . , T } she observes after thehistory h t − = ( f , A , s ; . . . ; f t − , A t − , s t − ) the frequencies of triples ( f t , A t , s t ). Thesehistory-dependent frequencies, denoted by ρ t ( f t , A t , s t | h t − ), are to be interpreted as afterhistory h t − student chose f t when facing A t and the objective signal s t was realized. If the history-dependent preference of the student over menus/decision problems from A t , t = 0 , . . . , T were observable, it is intuitive to expect it satisfies the following proper-ties. There is no continuation problem in period t = T . Preference for Flexibility:
Every year the student prefers menus which are largerrather than subsets thereof. That is, B t ∈ A t is less valuable than A t ∈ A t if B t ⊂ A t . This is because a strict subset offers less option value for a SEU agentthan a full menu.B. Weak Dominance for t ≤ τ : At τ = 0, say, she prefers to replace a single act f whose utility depends on the realization of the signal s with a menu of constant acts¯ A = { f ( s ) : s ∈ S } offering the same outcomes (lotteries over Z × A ) as every s − dependent outcome of f . This is because menu ¯ A offers insurance against herstochastic taste in t = 1. Intuitively, summer jobs where the student doesn’t learnnew specialized skills for the job market may be more valuable to a student whois still unsure of her taste about different careers than committing to an internshipwhose outcome is highly dependent on what she learns about her career taste at theend of the current period.C. Strong Dominance for t > τ : From the end of period τ on, whenever the act f t +1 delivers weakly better utility for each realization of the signal in period t + 1 than g t +1 , from the perspective of the end of year t , the menu { f t +1 , g t +1 } is as good as { f t +1 } . This unambiguous comparison of continuation problems in the end of year t becomes possible because at the end of period τ the career tastes of the student havestabilized and are deterministic. Given a fixed taste about distinct careers she isable to at least determine when an act is uniformly more valuable than another, nomatter the realization of the objective signal in the current period t .We show how the properties A-C can be derived from ex-post stochastic choice from menuswithout knowing anything about the preference over menus of the student. Moreover, ourmethods allow the analyst to also determine the speed with which an agent, such as thestudent in this example, learns her final taste v (e.g. to determine the τ of the student).For example, if the act f is taking an internship which requires substantial investment inlearning new skills in a very specific field like finance, i.e. an act whose outcome is highlydependent on s as well as the realization of the future taste ˜ v , we should expect an agentwho knows by the end of period t = 0 that her taste is so that she likes to get a job infinance, to prefer committing to f at the end of t = 0. This should be especially the caseif the alternative is to face a menu which offers acts whose outcomes don’t depend muchon s or the realization of ˜ v such as helping out with grading an undergrad class, takingup a summer job in the library, etc, even though they might be as financially profitableas picking the internship in finance f .Finally, given richness of the data, our characterization results show how an analyst isable to compare different agents according to their speed of learning about taste in similarsituations. The names are justified: after formally introducing the technical set up and the axioms in the mainbody of the paper we show in online appendix section 5 that under Preference for Flexibility, StrongDominance implies Weak Dominance but not the other way around. Intuitively in our example, student 1 learns her taste faster than student 2, if stochastic choice datagive evidence that student 1 satisfies Strong Dominance whenever student 2 does. Static Random Sub jective Expected Utility withobservable ob jective states
In this section we introduce and characterize the static model. This is the crucial buildingblock of the dynamic model of section 3.
Set up in the static model.
Let Z be a prize space assumed to have a separable,metric topological structure. Let S be a finite set of objective states and F the set ofAnscombe-Aumann acts (AA acts) with a typical element given by f : S → ∆( Z ) where∆( Z ) denotes the space of simple lotteries over prizes in Z . Finally, denote by A thecollection of finite, nonempty subsets of F . A typical element in A is called a menu anddenoted by a capital letter, e.g. A ∈ A . A is equipped with the Hausdorff topology.Given a belief of the agent over S , i.e. an element q from ∆( S ) and an Expected Utilityfunction u : ∆( Z ) → R to evaluate simple lotteries, we say the agent satisfies SubjectiveExpected Utility (SEU) with beliefs q and taste u if the utility of an act f is given by q · ( u ◦ f ) := (cid:80) s ∈ S q ( s ) u ( f ( s )). Define N ( A, f ) = { ( q, u ) ∈ ∆( S ) × R X : q · ( u ◦ f ) ≥ q · ( u ◦ g ) , g ∈ A } . This is the set of SEUs which can rationalize the choice of f from menu A .Denote N + ( A, f ) the respective subset of N ( A, f ) where f is not tied to other actsfrom A .Moreover, define M ( A ; u, q ) = { f ∈ A : q · ( u ◦ f ) ≥ q · ( u ◦ g ) , g ∈ A } . This is the set of maximizers when the agent’s belief about objective state of the world is q and her Bernoulli utility is u .The timeline of the one-period model is the following. s realized,payoff f(s)Agentpicks f outof ASEU (q,u)realized passive learning s realized,payoff f(s)Agentpicks f outof Aposteriorbelief qrealizedactive learningAgent picks info structure μ t Figure 1: Timeline for the static setting.Let µ be a probability measure over ∆( S ) × R X , equipped with the sigma-algebra F generated by sets of the form N + ( A, f ) , N ( A, f ) or alternatively with the Borel sigma-algebra of ∆( S ) × R X . The wording objective means that the state s is verifiable by both agent and analyst after it occurs. A lottery is called simple if only finitely many prizes can happen with positive probability. ∆( Z ) isequipped with the topology of weak convergence of probability measures. The set of acts F is equippedwith the product-topology over ∆( Z ) S . In the following we often identify the EU-functional u : ∆( Z ) → R with its Bernoulli utility from R Z . This is constructed as a product sigma-Algebra of the respective Borel sigma-Algebra of weak con-vergence on ∆( S ) and the Borel one from R X (the latter again a product sigma-Algebra).
9e say that µ is regular if µ ( N + ( A, f )) = µ ( N ( A, f )) for any A ∈ A , f ∈ F . In thispaper regular measures µ have the following form: whenever there are ties, i.e. M ( A ; u, q )is not a singleton for some A and SEU pair ( q, u ) the agent randomly picks an auxiliarySEU pair ( p, v ) such that M ( M ( A ; u, q ); p, v ) is a singleton. Observable in the static model.
We assume the analyst observes an augmentedstochastic choice function defined as in part 1) of the following definition.
Definition 1.
1) An augmented stochastic choice function (aSCF) is a map ρ : F × A × S → [0 , with the properties(a) (cid:88) s (cid:88) f ∈ A ρ ( f, A, s ) = 1 , ∀ A ∈ A . (b) ρ ( s ) := (cid:88) f ∈ A ρ ( f, A, s ) = (cid:88) f ∈ B ρ ( f, B, s ) > , ∀ A, B ∈ A , s ∈ S.
2) A stochastic choice function (SCF) ζ is a map ζ : F × A→ [0 , with the property (cid:88) f ∈ A ζ ( f, A ) = 1 , ∀ A ∈ A The second requirement in the definition of aSCF makes sure that we can define theobserved frequency of objective state s independently of the decision problem the agentis facing. This says that objective uncertainty is fully exogenous and independent fromthe problem the agent is facing in addition to being outside the influence of the analyst.Formally, it allows the definition of ρ ( s ) = (cid:80) f ∈ A ρ ( f, A, s ) for any A and s ∈ S , i.e. theprobability of observing s in the data.For a given aSCF ρ we denote in the following by ¯ ρ the SCF derived from summingeach ρ ( f, A, s ) across states. Formally,¯ ρ ( f, A ) := (cid:88) s ∈ S ρ ( f, A, s ) , f ∈ A, A ∈ A . Discussion of the Observable.
Assuming that the data of the analyst comes in theform of aSCFs characterizes an analyst with superior information compared to the setup of [Lu ’16]. In many realistic situations this is a viable assumption: loan performancedata, how students perform in school or how an employee performs in some task is oftenobservable to an outside analyst. The interested reader can peruse the proofs in Section 1 in the online appendix for the mathematicaldetails. This tie-breaking rule is special and it will be reflected in the properties of the data in the form ofa specific axiom: Extremeness-type of Axioms (see next subsections) imply tie-breaking through SEUs. Section 2 of the online appendix considers extensively the case when the observable corresponds toSCF, that is the realization of s is not observable by the analyst. For the static setting the whole theory,up to explicit modeling of the tie-breaking is contained in [Lu ’16], whereas the dynamic version of hismodel can be derived easily using the approach of [Frick, Iijima, Strzalecki ’17]. See the online appendixfor more details. Remark 1.
The observable in Definition 1 has more general applicability, e.g. it can beused even if there is partial observability of s as long as there is full identification in theaggregate .In more detail, assume the analyst observes a signal y ∈ Y about the true realization ofthe objective state s ∈ S instead of its realization. If ˆ µ ( y | s ) gives the (menu-independent)conditional probability of observing signal y when the realized state is s the assumption ofaSCFs as observable is valid for the analysis if the following two conditions hold:- ˆ µ is known by the analyst,- The matrix (ˆ µ ( y | s )) y ∈ Y,s ∈ S is quadratic and has full rank. We now introduce the Random Subjective Expected Utility representation for an aSCF ρ we are after. An agent has private information about both beliefs over the realizationof the objective state s as well as her taste u ∈ R Z . The analyst observes only aggregatefrequencies of choice data and realizations of the objective state from the same agent inmany choice instances or similar aggregate data choices from a population of agents. Definition 2.
A Random SEU representation (R-SEU) of the aSCF ρ is a tuple (Ω , F ∗ , µ, ( q, u, s ) , (ˆ q, ˆ u )) such thatA. (Ω , F ∗ , µ ) is a probability space with finite Ω ,B. ( q, u, s ) : Ω → ∆( S ) × R Z × S is an injective map, has non-constant SEU ( q ( ω ) , u ( ω )) and s ( ω ) ∈ supp ( q ( ω )) for all ω ∈ Ω .C. EitherC1. The representation has correct interim beliefs (cib): µ ( s ∈ ·| q ) = q ( · ) or otherwiseC2. The representation has no unforeseen contingencies (nuc): supp ( µ ( s ∈ ·| q, u )) ⊂ supp ( q ( · )) .D. the ( q, u ) -measurable tiebreaking process (ˆ q, ˆ u ) : Ω → R Z is regular and for all f ∈ A , ρ ( f, A, s ) = µ ( C ( f, A, s )) . Here, C is defined as C ( f, A, s ) = { ω ∈ Ω : f ∈ M ( M ( A, q ( ω ) , u ( ω )) , ˆ q ( ω ) , ˆ u ( ω )) , s ( ω ) = s } .
11n the following ω are called states of the world. C ( f, A, s ) denotes then the collectionof states of the world where the agent chooses f from A and the objective state s isrealized.Before continuing, we note down the true data-generating process (DGP) derived fromthe representation. Definition 3.
For an aSCF ρ that satisfies a R-SEU representation define the DGP, a ∆( S ) -valued random variable ¯ q : Ω → ∆( S ) as ¯ q ( ω )( · ) = µ ( s ∈ ·| q, u )( ω ) . Then the property of correct interim beliefs (cib) can be writtenas ¯ q = q whereas that of unforeseen contingencies (nuc) is written as supp (¯ q ) ⊂ supp ( q ) . For this subsection only, we look at an agent whose preference (cid:23) over acts is continuous but otherwise arbitrary (i.e. not necessarily SEU) and introduce a concept which is helpfulin the characterization results of this paper in addition to having general applicabilityoutside of this model as well. If the only fact the analyst knows about the stochasticchoice of an agent is that it comes from a continuous preference, the sets N ( f, A ) can bewritten as N ( f, A ) = {(cid:23) continuous preference over F : f (cid:23) g, g ∈ A } . Say that the stochastic choice data of an agent satisfies a
Random Utility Model if thestochasticity in choice follows from the randomness of her preference. Formally, we defineas follows.
Definition 4 (Random Utility Model) . Say that a SCF ζ on F satisfies a Random UtilityModel (RUM) if there exists a regular probability measure µ over continuous preferencesover F so that for every A ∈ A and f ∈ A we have ζ ( f, A ) = µ ( N ( f, A )) . The randomness in preferences may originate from her stochastic perceptions of thedecision environment she faces, for example in the special case of SEUs her beliefs maybe stochastic. In the case of SEUs randomness can also come from stochastic tastes.Alternatively, a RUM may be interpreted as representing data from a population ofheterogeneous agents who have deterministic preferences. The following definition showshow to identify from data the collection of preferences underlying a RUM.
Definition 5.
For a SCF ζ which satisfies a RUM let RSSupp ( ζ ) , the revealed subjectivesupport of ζ , be defined through RSSupp ( ζ ) = {(cid:23) over F : ∀ A ∈ A , f ∈ A, if (cid:23) ∈ N ( A, f ) then there exists ( f n , A n ) → ( f, A ) with ζ ( f n , A n ) > } . Here convergence ( f n , A n ) → ( f, A ) is in the product topology of F × A . (cid:23) is in the revealed subjective support of ζ if every choicethat can be rationalized by (cid:23) appears in the data encoded by ζ , up to tie-breaking. If the RUM has support on SEUs, the definition ‘picks out’ the SEUs in the supportof µ from Definition 4 up to positive affine transformations of the respective Bernoulliutilities. Aside.
Another compact and suggestive way to write down the revealed subjectivesupport of a SCF ζ is as follows.For a continuous preference (cid:23) over F denote the set of choices it can rationalize as R (cid:23) , that is R (cid:23) = { ( f, A ) ∈ F × A : (cid:23) ∈ N ( f, A ) } . This is the set of choice data that are consistent with maximization of (cid:23) .The set of choices explained by the data represented by some SCF ζ is N ( ζ ) = { ( f, A ) : f ∈ A, ∃ ( f n , A n ) → ( f, A ) with ζ ( f n , A n ) > n } . Then
RSSupp ( ¯ ρ ) can be characterized as follows. RSSupp ( ζ ) = {(cid:23) : R (cid:23) ⊂ N ( ζ ) } . The following axiomatization of aSCFs is based on previous results about the axiomati-zation of SCFs in [Lu ’16] and [Ahn, Sarver ’13].Axioms 0-1 till 0-5 below are adaptations to our setting of aSCFs of the standardaxioms from Theorem S.1 of [Lu ’16]. They imply that an aSCF comes from an underlyingRUM whose revealed subjective support contains only SEUs. Axiom 0-6 is adapted from[Ahn, Sarver ’13] and ensures that there can only occur finitely many such SEUs.
Standard Axioms in statewise form.
For all s ∈ S it holds Axiom 0-1: Statewise Monotonicity. ρ ( f, A, s ) ≥ ρ ( f, B, s ) for A ⊂ B . Axiom 0-2: Statewise Linearity. ρ ( λf + (1 − λ ) g, λA + (1 − λ ) { g } , s ) = ρ ( f, A, s )for any A ∈ A , g ∈ F and λ ∈ (0 , Axiom 0-3: Statewise Extremeness. ρ ( ext ( A ) , A, s ) = 1 for all A ∈ A . Axiom 0-4: Statewise Continuity.
A (cid:51) A (cid:55)→ ρ ( · , A | s ) is continuous. In more detail: (cid:23) occurs in the data if for every choice pair ( f, A ) either (1) ρ ( f, A ) > and (cid:23) ∈ N ( A, f ) or if (2) ρ ( f, A ) = 0 and (cid:23) ∈ N ( A, f ) then ρ ( f, A ) = 0 only happens due to tie-breaking. Note that F has a mixture structure in the usual way. In particular, one can form conv ( A ), theconvex hull of A for any menu A . Then ext ( A ) is identified with the set of extremum points of conv ( A ). The image of the mapping is the space of simple lotteries on F , equipped with the topology of weakconvergence of probability measures. xiom 0-5: State Independence. To explain this axiom we first introduce someterminology: a menu A is called constant if it contains only constant acts. Given a menu A and a state r ∈ S let A ( r ) = { f ( r ) : f ∈ A } be the constant menu containing alllotteries from acts in A which happen at state r .Then State Independence says: Suppose f ( s ) = f ( s ) , A ( s ) = A ( s ) and A i ( s ) = { f ( s ) } , s (cid:54) = s i , i = 1 ,
2. Then ρ ( f, A , s ) = ρ ( f, A ∪ A , s ).Intuitively, if an act f yields the same payoff in states s and s , payoffs of menu A in s are the same as those of menu A in s and acts in A i only differ in s i then theprobability of choosing f in A is the same as choosing f in A ∪ A , unless the realizationof the Bernoulli utility of the agent depends on whether s or s is realized. Axiom 0-6: Statewise Finiteness.
There is
K > A ∈ A , there is B ⊂ A with | B | ≤ K independent of s such that for every f ∈ A \ B there are sequences f n → m f and B n → m B with ρ ( f n , { f n } ∪ B n , s ) = 0.To state the axiom of correct beliefs we define for a SEU pair ( q, u ) where p is thebelief of the agent and u her Bernoulli utility as π q ( p, u ) = p. That is, the projection tothe belief used from the agent. Furthermore, in the following ρ ( s | f, A ) is the conditionalprobability of observing the realization of the objective state s in the data conditional onthe agent choosing f from menu A . Axiom 0-7: Correct Interim Beliefs (CIB).
For all f ∈ F and A ∈ A with ¯ ρ ( f, A ) > ρ ( ·| f, A ) ∈ π q ( conv ( N ( f, A ) ∩ RSSupp ( ¯ ρ ))) = conv ( π q ( N ( f, A ) ∩ RSSupp ( ¯ ρ ))) . (2)The axiom says that the DGP of the objective state s conditional on observed choice( f, A ) is a mixture of beliefs which correspond to some SEU that fulfill two naturalconditions simultaneously : 1) the SEU is contained in the revealed subjective support ofthe data and 2) the SEU rationalizes the choice f from A .Incorrect beliefs can arise due to different reasons: the agent may observe objectivesignals with noise, she may have a misspecified prior or otherwise have subjectively biasedbeliefs even though they average out to the correct prior. We exclude in this paper thecase when incorrect beliefs originate from non-Bayesian updating with respect to any prior.In contrast to section 6 of [Lu ’16] here the analyst gets information about the real-ization of the objective state and can glean out the true DGP from data. This allowsher to make a direct comparison between the true DGP and the beliefs of the agent. Section 7 of [Lu ’16] constructs a test of CIB based on test acts. His methods requirenon-stochastic taste whereas our axiom is robust to stochasticity of tastes.Now we present a relaxation of the Correct Interim Beliefs Axiom which allows forincorrect beliefs but so that the incorrectness remains undetected by the agent ex-post.This is inconsequential in a static setting but has repercussions in the dynamic setting ofSection 3 where we study an agent who passively learns about objective states as well asher taste in every period. Moreover, in the dynamic model in Section 3 we assume that the agent is sophisticated and thusour model doesn’t allow any prospective overconfindence/underconfidence as in [Lu ’16]. xiom 0-7’: No Unforeseen Contingencies (NUC) For all f ∈ F and A ∈ A with¯ ρ ( f, A ) > supp ( ρ ( ·| f, A )) ⊂ (cid:91) { supp ( q ) : q ∈ π q ( N ( A, f ) ∩ RSSupp ( ¯ ρ )) } . Our first main result gives the axiomatization of aSCFs in a static setting.
Theorem 0.
The aSCF ρ on A admits a R-SEU representation with CIB satisfied ifand only if it satisfies Axioms 0-1 till 0-7. It admits a R-SEU representation with NUCsatisfied if and only if it satisfies Axioms 0-1 till 0-6 together with Axiom 0-7’. In the following whenever for an aSCF ρ the Axioms 0-1- till 0-6 together with 0-7’are satisfied, we say Axiom 0 is satisfied for ρ . We consider here the special case of Theorem 0 where all possible Bernoulli utilities in therepresentation are equal up to positive affine transformations of each other. This impliesthat stochasticity in choice only comes from randomness in beliefs.To facilitate analysis, we require the existence of a best constant act. This requirementis easily expressed in terms of stochastic choice.
Axiom: Existence of a constant best act.
There exists a constant act ¯ f ∈ F suchthat for every act f ∈ F it holds f (cid:54) = ¯ f = ⇒ ρ ( f, { f, ¯ f } ) = 0 . The existence of a best constant act is assured for example if Z consists of monetaryprizes and the preferences of the agent over money are strictly increasing. Whenever thisAxiom is satisfied, it becomes easier to eschew tie-breaking considerations when writingdown other Axioms on data.The axiom on data which ensures that the agent has a deterministic taste is thefollowing. Axiom: C-Determinism*.
For any menu A consisting of constant acts it holds truelim a → ρ (cid:0) af + (1 − a ) ¯ f ; A \ { f } ∪ { af + (1 − a ) ¯ f } (cid:1) ∈ { , } . This says that except for possible stochastic tie-breaking, constant acts are chosendeterministically. On the other hand, if taste is stochastic then choice from constantmenus should be stochastic, even after taking into account possible stochastic tie-breaking.Given this intuition the following characterization result is not surprising.
Proposition 1 (Informational Representation for aSCFs) . Assume that an aSCF ρ hasa R-SEU representation with regular measure µ . Assume that there exists a constant bestact.Then the following are equivalent. This is an adaptation of the
C-Determinism
Axiom from the [Lu ’16] who doesn’t consider tie-breaking explicitly as we do. . For all ( q, u ) , ( p, v ) ∈ RSSupp ( ¯ ρ ) u is a positive affine transformation of v .B. ρ satisfies C-Determinism*. This section is devoted to the dynamic model. We introduce the general representationand two interesting specializations of it. After that, we give axioms for all three repre-sentations.
Set up in the dynamic model.
Let Z be a finite prize space, ∞ > T ≥ t = 0 , . . . , T let S t be finite spaces of objective states . The objective states evolveaccording to a DGP which cannot be influenced by the agent (passive learner situation).Define recursively the spaces of consequences for every period as follows. Let X T = Z and the set of acts F T with a typical element f T : S T → ∆( Z ). Let A T be the collection offinite sets from F T . Then continue inductively by defining X t = Z × A t +1 , where A t +1 isthe collection of finite menus from F t +1 . F t is then the set of acts f t : S t → ∆( X t ). Thus, an act f t at time t < T gives for each possible objective state s t a lotteryover current consumption and a continuation decision problem/menu. We denote f At themarginal act on menus A t +1 and f Zt the marginal act on Z induced by f t .We assume in each period ( q t , u t ) is private information of the agent whereas therealization of s t is observed by both the agent and the analyst. Thus stochasticity inchoice comes from the information asymmetry between the agent and the analyst inthe single-agent interpretation, whereas in the population interpretation the analyst isobserving dynamic data from a population of SEU agents whose preference characteristicsare unknown.Visually the timeline is depicted in Figure 2. t t+1 S t realized (z t , A t+1 )realizedChoice of f t from A t (q t ,u t ) realized Figure 2: Timeline for the dynamic setting.
The observable in the dynamic setting.
The analyst observes histories with a typi-cal element h t as well as history-dependent aSCFs ρ t ( ·| h t − ). The collection of the former isdenoted by H t whereas of the latter simply by ρ and called a dynamic augmented stochas-tic choice function (dynamic aSCF) . These are described recursively as follows. For t = 0the analyst observes an aSCF ρ as in Definition 1. The set H collects all histories h =( f , A , s ) ∈ F ×A × S such that ρ ( h ) >
0. For h ∈ H denote A ( h ) := supp ( f A ) the Furthermore we denote in the following by A ct the collection of period − t menus consisting of constantacts. − h with positive probability. The construction is contin-ued recursively: for any history h t ∈ H t there is an aSCF ρ t +1 ( ·| h t ) which can be used todefine the set of possible continuation menus A t +1 ( h t ). The set of period − ( t + 1) historiesis then H t +1 := { ( h t , f t +1 , A t +1 , s t +1 ) : A t +1 ∈ A t +1 ( h t ) , ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ) > } .In simple words: histories are finite sequences of triplets ( f i , A i , s i ) with the interpre-tation that the data shows that with positive probability f i is chosen from menu A i and s i is the realized objective state in period i . Moreover, a history can only happen if theelements ( f i , A i , s i ) of its sequence happen successively with positive probability startingfrom the ‘oldest’ one ( f , A , s ) to the most recent.The data reflects limited observability in the sense that ρ t is defined only conditionalon histories which happen with positive probability in the data. We show below how thiscan be overcome. We first define properties shared by all representations. The focus is on having propertieswhich are tractable but still allow for a general enough representation.
Say that the triple ( F t , q t , u t , s t ) ≤ t ≤ T is simple w.r.t. the probability space (Ω , F ∗ , µ )if A. each F t is generated by a finite partition such that µ ( F t ( ω )) > ω ∈ Ω. Here F t ( ω ) is the partition cell of F t which contains ω .B. the map ( q t , u t , s t ) : Ω → ∆( S t ) × R X t × S t has non-constant SEU ( q t ( ω ) , u t ( ω )) forall ω and is adapted to the filtration F t , t ≤ T . Moreover, whenever ω (cid:48) (cid:54)∈ F t ( ω ) itholds ( q t ( ω ) , u t ( ω ) , s t ( ω )) (cid:54) = ( q t ( ω (cid:48) ) , u t ( ω (cid:48) ) , s t ( ω (cid:48) )).The tiebreakers (ˆ q t , ˆ u t ) ≤ t ≤ T are regular and preference-based , i.e.A. µ ( ω ∈ Ω : | M ( A t , ˆ q t , ˆ u t ) | = 1) = 1 for all A t ∈ A t .B. conditional on F T ( ω ) the sequence (ˆ q , ˆ u ) , . . . , (ˆ q T , ˆ u T ) is independent andC. µ ((ˆ q t , ˆ u t ) ∈ ·|F T ( ω )) = µ ((ˆ q t , ˆ u t ) ∈ ·| q l ( ω ) , u l ( ω ) , l ≤ t ) for all t .Simplicity and regularity are necessary for a parsimonious representation, whereas the preference-based condition incorporated in C. ensures that the tie-breaking of the agentdepends only on her realized SEU in the period at hand (and through it also on pasthistory) but not on the realization of the objective state in the current period.We define for a triple ( f k , A k , s k ) the set C ( f k , A k , s k ) = { ω ∈ Ω : f k ∈ M ( M ( A k , q k ( ω ) , u k ( ω )) , ˆ q k ( ω ) , ˆ u k ( ω )) , s k ( ω ) = s k } . These are the states of the world which rationalize the observable ( f k , A k , s k ) in period k . Similarly one defines for a history h t = ( A , f , s ; . . . ; A t , f t , s t ) the set of states of theworld which rationalize the occurrence of the history. C ( h t ) = ∩ l ≤ t C ( A l , f l , s l ) . w.r.t. stands for with respect to . .1.2 The general representation. We are now ready to write down the most general representation of a dynamic aSCF. Itdoesn’t impose any functional restrictions on the Bernoulli utilities of the agents and onlya minimal restriction on the evolution of beliefs.
Definition 6.
A Dynamic Random SEU representation (DR-SEU) of the dynamic aSCF ρ is a tuple (Ω , F ∗ , µ, ( F t , ( q t , u t ) , s t , (ˆ q t , ˆ u t )) ≤ t ≤ T ) such thatA. (Ω , F ∗ , µ ) is a finitely additive probability space,B. the filtration ( F t ) ⊂ F ∗ and the F t − adapted process ( q t , u t , s t ) : Ω → ∆( S t ) × R X t × S t is simple,C. the F ∗ -measurable tiebreaking process (ˆ q t , ˆ u t ) : Ω → R X t is regular and preference-based and for all f t ∈ A t , h t − ∈ H t − ( A t ) , ρ t ( f t , A t , s t | h t − ) = µ ( C ( f t , A t , s t ) | C ( h t − )) . D. EitherD.1. The representation has correct interim beliefs (CIB): µ ( s t ∈ ·| q t ) = q t ( · ) for all t ∈ { , . . . , T } ,or otherwiseD.2. The representation has no unforeseen contingencies (NUC): supp ( µ ( s t ∈ ·| q t , u t )) ⊂ supp ( q t ( · )) . Some explanations are in order. History h t − happens with the probability µ ( C ( h t − )):the state of the world has to be so that for each l ≤ t the realized subjective state/SEU( q l , u l ) picks f l from A l , f l survives any possible tie-breaking and finally, in period l theobjective state s l is realized.Conditional on C ( h t − ) occurring, f t is chosen from A t only if the realized subjectivestate in period t given by the pair ( q t , u t ) is so that a SEU-maximizing choice from A t is f t and f t survives any possible tie-breaking.Note that the stochastic process of the objective and subjective states is unconstrained,except for the condition D: the agent uses the correct data-generating process conditionalon her private information (correct interim beliefs) or otherwise she respects the require-ment of (no unforeseen contingencies) , i.e. the agent never gets hard evidence that herbelief process is misspecified. The only other requirement embodied in the definition isthat the agent uses Bayes rule to update her beliefs. As noted before, the general representation doesn’t include any behavioral restrictionson the evolution of the beliefs and tastes of the agent besides the SEU assumptions andthat the agent remains Bayesian after every history with respect to her beliefs about thefuture evolution of tastes and objective states. In particular, her beliefs about the futureSEU realizations may be incorrect. In this subsection we exclude this possibility.18 volving SEU.
This specialization of DR-SEU captures a dynamically sophisticated agent who correctly takes into account the evolution of her future SEU preferences. There is an F t − adapted process of random EU-functionals v t , t = 0 , . . . , T , the felicityfunctions, over instantaneous consumption lotteries l ∈ ∆( Z ) and a discount factor δ > u T = v T and u t for t ≤ T is given by the following Bellman equation. u t ( f t ( s t )) = v t ( f Zt ( s t )) + δ E A t +1 ∼ f At ( s t ) ,q t +1 · u t +1 (cid:20) max f t +1 ∈ A t +1 ( q t +1 · u t +1 )( f t +1 ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (3)Here the conditional expectation E [ ·|F t ] takes into account the randomness coming fromthe lottery f At ( s t ) of the continuation problem as well as from the uncertainty about theSEU of the agent in period t + 1. The agent makes the correct inference about the futureSEU q t +1 · u t +1 , given her current information in F t . Definition 7.
An Evolving SEU representation of the dynamic aSCF ρ is a tuple (Ω , F ∗ , µ, ( F t , q t , u t , s t ) ≤ t ≤ T ) such thatA. (Ω , F ∗ , µ, ( F t , q t , u t , s t ) ≤ t ≤ T ) is a DR-SEU representation.B. (3) holds true for the stochastic process of Bernoulli utilities u t , t = 0 , . . . , T . If we assume there is only one period ( T = 0) then Evolving SEU collapses to thestatic model of section 2. The same holds trivially true for the following special case ofEvolving SEU. Gradual Learning.
This is a specialization of the Evolving SEU representation whichcaptures an agent who is learning about her taste. This results in a martingale conditionon the evolution of the felicities v t , t = 0 , . . . , T . Definition 8.
A Gradual Learning (GL-SEU) representation of the dynamic augmentedstochastic choice rule ρ is a tuple (Ω , F ∗ , µ, ( F t , q t , u t , s t ) ≤ t ≤ T ) such thatA. (Ω , F ∗ , µ, ( F t , q t , u t , s t ) ≤ t ≤ T ) is a Evolving-SEU representation.B. There exists an EU-function v for lotteries in ∆( Z ) such that for all t = 0 , . . . , T itholds v t = E [ v |F t ] . (4)As we show in the following subsection dynamic stochastic choice data are enough todistinguish the two special cases Evolving SEU and Gradual Learning even though thetwo models coincide in the static setting. The first axiomatization concerns the most general representation. This model of sophisticated behavior still doesn’t encompass all possible sophisticated behav-iors allowed by the general DR-SEU representation – see Example 3 concerning [Epstein ’06] and[Epstein et al ’08] in subsection 3.2. [Frick, Iijima, Strzalecki ’17] showed the same insight in a setting of lotteries and without objectivepayoff-relevant states. .2.1 Axioms for DR-SEU Axioms for the general representation in Definition 6 can be classified in two groups. Thefirst group identifies two types of observationally equivalent histories. The second groupcomprises requiring Axiom 0 from the static setting after each history together with atechnical axiom of history continuity . Overcoming limited observability.
Similar to [Frick, Iijima, Strzalecki ’17] we char-acterize histories which are equivalent with respect to the information they reveal throughtwo axioms: Contraction History Independence and Linear History Independence. Thisallows to overcome the limited observability problem.Given a history h t − = ( A , f , s ; . . . , A t − , f t − , s t − ) let ( h t − − k , ( A (cid:48) k , f (cid:48) k , s (cid:48) k )) be thehistory of the form ( A , f , s ; . . . ; A (cid:48) k , f (cid:48) k , s (cid:48) k ; . . . ; A t − , f t − , s t − ). That is, the history ischanged only in period k . Definition 9.
We say that g t − ∈ H k − is contraction equivalent to h t − if for some k wehave g t − = ( h t − − k , ( B k , f k , s k )) where A k ⊂ B k and ρ k ( f k , A k , s k | h k − ) = ρ k ( f k , B k , s k | h k − ) . That is, when expanding the set of opportunities at a period k but otherwise holdingthe history h t − intact, the same stochastic choice results in the period of the expansion. Axiom 1: Contraction History Independence (CHI)
For all t ≤ T, if g t − ∈H t − ( A t ) is contraction equivalent to h t − ∈ H t − ( A t ) then for all s t ∈ S t ρ t ( · , A t , s t | h t − ) = ρ t ( · , A t , s t | g t − ) . Intuitively, if the distribution of the preferences is stable, two contraction equivalenthistories should give the same stochastic choice in the future as well, all else equal. This isbecause in the Definition 9 above, elements from B k \ A k were not attractive to any SEUin the underlying distribution of preferences which has induced either of the histories h t − and g t − , and given the stability of the underlying distribution of preferences the contentof private information revealed from the two histories h t − and g t − is the same. Thisimplies that the continuation stochastic choice should be the same.The other class of equivalent histories is the following. Definition 10.
A finite set of histories G t − ⊂ H t − is linearly equivalent to h t − =( A , f , s ; . . . , A t − , f t − , s t − ) if G t − = { ( h t − − k , ( λA k + (1 − λ ) B k , λf k + (1 − λ ) g k , s k )) : g k ∈ B k } . That is, a history is changed only at a single period by having the revealed choice f k from A k mixed with all possible choices g k from a menu B k .One can calculate from the history-dependent aSCF, the probability choices condi-tional on a set of histories G t − by the formula ρ ( f t , A t , s t | G t − ) = (cid:88) g t − ∈ G t − ρ t ( f t , A t , s t | g t − ) · ρ ( g t − ) (cid:80) h t − ∈ G t − ρ ( h t − ) . xiom 2: Linear History Independence (LHI) For all t ≤ T if G t − ⊂ H t − ( A t )is linearly equivalent to h t − ∈ H t − ( A t ), then ρ t ( f t , A t , s t | h t − ) = ρ t ( f t , A t , s t | G t − ).Intuitively, if we have a set of histories G t − linearly equivalent to history h t − with themixing happening in period k , because of SEU-properties, f k is optimal from A k if andonly if a mixture of the type λf k + (1 − λ ) g k with some g k is optimal from the mixedmenu λ { f k } + (1 − λ ) B k . Therefore, the mixing doesn’t reveal anything new regardingthe private information of the agent and so continuation stochastic choice should be thesame.Now let Axioms 1 and 2 hold for the observable and assume the menu A t is not possiblewith positive probability after history h t − . Define ρ h t − ( f t , A t , s t ) := ρ t ( f t , A t , s t | λh t − + (1 − λ ) d t − ) , for some history d t − = ( g k , { g k } , s k ) ≤ k ≤ t − which leads to menu A t with probability one.LHI ensures that the construction is well-defined and coincides with ρ t ( f t , A t , s t | h t − )whenever A t ∈ A t ( h t − ). Note here that histories of the type d t − don’t reveal anythingabout the private information of the agent. They should be interpreted as tools for theanalyst to obtain variation in the data, much needed for identification of the underlyingparameters. History-Dependent R-SEU and History Continuity.
We model agents who inevery period are SEU but have private information about their preferences. Therefore,the data need to satisfy Axiom 0 from the static setting. This is the content of the nextAxiom.
Axiom 3: R-SEU in every period
For all t ≤ T and h t − , each of the history-dependent aSCFs ρ t ( ·| h t − ) satisfies Axiom 0 from the static setting, i.e. it has a R-SEUrepresentation.The last axiom needed to characterize DR-SEU is a technical form of Continuity.The following definition gives our concept of continuity for histories and is adapted from[Frick, Iijima, Strzalecki ’17]. Definition 11.
1) For a sequence of acts f n say that f n converges in mixture to the act f , written as f n → m f , if there exists h ∈ F and α n → with f n = α n h + (1 − α n ) f .2) For a sequence of menus ( B n ) n ⊂ A say that B n converges in mixture to the act f ,written B n → m f , if there exists B ∈ A and α n with B n = α n B + (1 − α n ) { f } .3) For a sequence of menus ( A n ) n ⊂ A say that A n converges in mixture to the menu A , written A n → m A , if for each f ∈ A there is a sequence ( B nf ) n ⊂ A such that B nf → m { f } and A n = ∪ f ∈ A B nf . We next define menus and histories without ties, a concept we also come across later.
Definition 12.
For any ≤ t ≤ T and h t − ∈ H t − the set of period t − menus withoutties conditional on h t − is denoted by A ∗ t ( h t − ) and consists of all A t ∈ A t such that forany f t ∈ A t and any sequences f nt → m f t , s t ∈ S t and B nt → m A t \ { f t } we have lim n ρ t ( f nt , B nt ∪ { f nt } , s t | h t − ) = ρ t ( f t , A t , s t ) . For t = 0 we write A ∗ := A ∗ ( h t − ) . The set of period t histories without ties is H ∗ t := { h t = ( A , f , s ; . . . ; A t , f t , s t ) ∈ H t : A k ∈ A ∗ t ( h k − ) , for all k ≤ t } . A t without ties is so that no matter the SEU of the agent, she neverneeds to perform tie-breaking. Therefore the menu can be perturbed in any direction andthe probabilities of observing the perturbed act f nt chosen from the perturbed menu B nt converge to the probability of observing f t chosen from A t . A history without ties is sothat every menu occurring in it is without ties.The technical Continuity axiom reads then as follows. Axiom 4: History Continuity
For all t ≤ T, A t , f t and h t − ∈ H t − , ρ t ( f t , A t , s t | h t − ) ∈ co { lim n ρ t +1 ( f t +1 , A t − , s t | h t − ,n ) : h t,n → m h t , h t − ,n ∈ H ∗ t − } . Whenever a history h t − is perturbed slightly, the change is in choices and decisionproblems as the objective states s k , k ≤ t − f t chosen from A t as well as s t realized should change continuously with the history. Theorem 1.
For a dynamic aSCF ρ the Axioms 1-4 are equivalent to the existence of aDR-SEU representation. If we add Existence of a Best Act and C-Determinism* from subsection 2.3.1 to Axiom0, we get a characterization of the special case of DR-SEU representation where the agentknows her Bernoulli utility u t for certain in every period. That is, she is learning onlyabout the objective states. Proposition 2 (Informational Representation for aSCFs) . Assume that a dynamic aSCF ρ has a DR-SEU representation with regular measure µ . Assume that there exists aconstant best prize.Then the following are equivalent after every history h t observed with positive proba-bility.A. For all ( q t , u t ) , ( p t , v t ) ∈ RSSupp ( ¯ ρ t ( ·| h t )) u is a positive affine transformation of v t .B. ρ t ( ·| h t ) satisfies C-Determinism*. Stochastic choice coupled with the SEUassumption imposes enough structure on data to allow the identification of a history-dependent preference relation (cid:23) h t on acts. Intuitively, if the ‘tail’ of the history h t is( f t , A t , s t ), the SEU draw ( q t , u t ) in period t has to rationalize the choice of f t from A t .For every pair of acts g t , r t we can then define g t (cid:23) h t r t if g t is weakly better than r t forevery possible draw of SEU from N ( f t , A t ) that happens with positive probability underthe respective DR-SEU representation. Note that this implies that (cid:23) h t is potentiallyincomplete. The following definition adds tie-breaking considerations to the intuition wejust explained. 22 efinition 13. For each t ≤ T − and h t = ( h t − , A t , f t , s t ) ∈ H t we define the relation (cid:23) h t on F t as follows: For any g t , g (cid:48) t ∈ F t we have g t (cid:23) h t r t if there exist sequences in F t with g nt → m g t and r nt → m r t such that ρ t (cid:18) f t + 12 r nt , A t + 12 { g nt , r nt } , s t (cid:12)(cid:12)(cid:12)(cid:12) h t − (cid:19) = 0 , for all n. Finally, let ∼ h t , (cid:31) h t be the indifference and strict part of (cid:23) h t . Because of Axiom 0 in DR-SEU, specifically the no unforeseen contingencies (NUC) assumption, the preference (cid:23) h t doesn’t depend on the realization of the period − t objectivestate s t as long as that state has positive probability under h t − .We now put the additional axioms characterizing Evolving SEU on (cid:23) h t . Axiom 4: Separability.
For all t ≤ T − , g t , r t ∈ F t we have g t ∼ h t r t whenever g At ( s t ) = d r At ( s t ) and g Zt ( s t ) = d r Zt ( s t ) for all s t ∈ S t .This says that whenever the marginal distributions over the current prize lottery andcontinuation menu of two acts after a history h t are the same then the two acts areindifferent under the revealed preference after the history. It ensures that Bernoulli utility u t has the form u t ( z t , A t +1 ) = v t ( z t ) + δV t ( A t +1 ) . (5)Axiom 4 allows the definition of a history-dependent menu preference over continuationmenus. Definition 14.
Fix a z t ∈ Z . Take a h t ∈ H t and define an ex-post menu preference (cid:23) h t over A t +1 by A t +1 (cid:23) h t B t +1 , if δ ( z t ,A t +1 ) (cid:23) h t δ ( z t ,B t +1 ) . We now add other menu preference axioms to shape the menu preference V from (5)into the form needed for (3). The next three Axioms are standard. Axiom 5: Monotonicity.
Whenever A t +1 ⊆ B t +1 it holds B t +1 (cid:23) h t A t +1 . Axiom 6: Indifference to Timing.
For any A t +1 , B t +1 and α ∈ (0 ,
1) we have αA t +1 + (1 − α ) B t +1 ∼ h t αA t +1 + (1 − α ) B t +1 . Axiom 7: Menu Non-Degeneracy.
There exists A t +1 , B t +1 such that δ ( z t ,B t +1 ) (cid:23) h t δ ( z t ,A t +1 ) for all z t .Before stating the next axiom, we introduce an operation on menus which produces forevery menu a constant menu containing all the lotteries in its acts. Formally, in a settingwith AA-acts from F for a menu A ⊂ F define the menu of constant acts from ¯ A as follows.¯ A = { g ∈ F : g constant act with g ( s ) = f ( s (cid:48) ) for some f ∈ A, s, s (cid:48) ∈ S } . The following axiom ensures that the menu preference (cid:23) h t of Definition 14 can be rep-resented by Expected Utility preferences with stochastic but state-independent Bernoulliutilities. 23 xiom 8: Weak Dominance. For any A t +1 ∈ A t +1 it holds ¯ A t +1 (cid:23) h t A t +1 .Intuitively, from the perspective of the end of period t and compared to the menu A t +1 ,the menu ¯ A t +1 offers insurance w.r.t. the stochasticity of both beliefs and tastes as ex-postin t + 1 the agent can choose her best lottery from any act in A t +1 whereas in A t +1 whichlottery the agent ultimately faces depends on the realization of the objective state s t +1 . Menu Finiteness (technical).
Next we define what it means for a menu preferenceto be finite. This is a technical property we need for tractability.
Definition 15.
For (cid:23) a menu preference over some set of prizes X say that it satisfies Finiteness if there exists K ∈ N such that for menu A there exists B ⊂ A with | B | ≤ K and so that B ∼ A . Axiom 9: Finiteness of Menu preference
For all h t ∈ H t , the menu preference on A t +1 derived from (cid:23) h t satisfies Finiteness as in Definition 15.Finally, we add the sophistication axiom which ensures that the agent correctly predictsher future beliefs and tastes. Intuitively, if enlarging the menu A t +1 to B t +1 is valuable forthe agent just after the realization of history h t and her beliefs about the future evolutionof her preferences are correct, this is because there are possible draws of SEUs in period t + 1 for which elements in B t +1 \ A t +1 are optimal. This should be then reflected in the h t − dependent stochastic choice from B t +1 . Axiom 10: Sophistication
For all t ≤ T − h t ∈ H t and A t +1 ⊂ B t +1 ∈ A ∗ t +1 ( h t ),the following are equivalentA. ρ t +1 ( f t +1 , B t +1 , s t +1 | h t ) > f t +1 ∈ B t +1 \ A t +1 and some s t +1 ∈ S t +1 .B. B t +1 (cid:31) h t A t +1 . Theorem 2.
For a dynamic aSCF ρ satisfying a DR-SEU representation the Axioms4-10 are equivalent to the existence of an Evolving SEU representation. Next, we note down a special cases of the Evolving SEU representation which canbe used to model data from a population of agents with deterministic but heterogeneoustastes who are learning about payoff-relevant objective states. Thus, uncertainty abouttaste resolves in the first period, i.e. after an agent from the population is ‘drawn’, butthere is persistent uncertainty about payoff-relevant objective states.
Example 3: Stochastic taste only in period zero.
If we replace Axiom 8 with thefollowing Strong Dominance axiom then we get a version of Evolving SEU, where tastesare stochastic only in t = 0 and the profile of future tastes is completely determined afterevery period-0 history. This is what [Dillenberger et al ’14] call Dominance in their main theorem. xiom 8’: Strong Dominance For all 0 ≤ t ≤ T − h t ∈ H t we have:If f t +1 ∈ A t +1 and { f t +1 ( s t +1 ) }(cid:23) h t { g t +1 ( s t +1 ) } for all s t +1 ∈ S t +1 then A t +1 ∼ h t A t +1 ∪ { g t +1 } .Intuitively, if the Bernoulli utility is deterministic and if an act is better than anotheruniformly across all states, adding the dominated act to a menu which contains thedominating act doesn’t make the menu more valuable. Proposition 3.
For a dynamic aSCF ρ satisfying a DR-SEU representation the Axioms4-7,8’,9 and 10 are equivalent to the existence of an Evolving SEU representation wherestochasticity of tastes is resolved at the end of period . Finally, we note a special case of DR-SEU involving a sophisticated agent but whichdoesn’t have an Evolving SEU representation.
Example 4. [Epstein ’06] and [Epstein et al ’08] consider a sophisticated agent whoexperiences temptation in beliefs and therefore updates her beliefs about objective statesin a subjective way not necessarily conforming to Bayesian updating with respect to thetrue data-generating process. The ex-post choice versions of these models are special casesof DR-SEU and satisfy C-Determinism*, but they violate Axiom 5 (Monotonicity), whichis necessary for an Evolving SEU representation. Gradual Learning imposes additional restrictions on the evolution of Bernoulli utilities ofan Evolving SEU representation: the agent is learning about a fixed taste.To explain the three additional Axioms which lead to the Gradual Learning represen-tation we introduce some notation.For some t ≤ T − l t , . . . , l T ∈ ∆( Z ) of consumption lotteries, letthe stream of lotteries ( l t , . . . , l T ) ∈ ∆( X t ) ⊂ F t be the period- t lottery that at every period τ ≥ t yields consumption according to l τ . Formally, for any consumption lottery l ∈ ∆( Z )and menu of constant acts A t +1 ∈ A ct +1 define ( l, A t +1 ) ∈ ∆( X t +1 ) to be the lotterywhich has stochastic consumption now and fixed continuation with probability one. Then ( l t , . . . , l T ) = ( l t , A t +1 ) ∈ ∆( X t ) is defined recursively from period T backwards by A T = { l T } ∈ A T and A s = { ( l s , A s +1 ) } ∈ A s for all s = t + 1 , . . . , T −
1. We write( l t , . . . , l τ , m, . . . , m ) if l t +1 = · · · = l T for some m ∈ ∆( Z ) and τ ≥ t . Axiom 11: Stationary Preference over Lotteries [FIS].
For all t ≤ T − , l, m, n ∈ ∆( Z ) and h t we have( l, n, . . . , n ) (cid:31) h t ( m, n, . . . , n ) if and only if ( n, l, . . . , n ) (cid:31) h t ( n, m, n, . . . , n ) . Intuitively, if and only if the felicity v t today is just the average of the future felicity v t +1 tomorrow, it holds true from today’s perspective that postponing the choice between twolotteries by a period results in the same ranking as for the case that the choice is madeimmediately. The model in [Epstein et al ’08] features infinite horizon so the statement above holds for its finitehorizon version. This is similar to the definition in section 4.3 of [Frick, Iijima, Strzalecki ’17]. l, m ∈ ∆( Z ),we say they are h t -non-indifferent if ( l, n . . . , n ) (cid:54)∼ h t ( m, n, . . . , n ) for some n ∈ ∆( Z ).Moreover, to avoid tautologies we require a non-degeneracy condition. Condition 1: Consumption Non-degeneracy
For all t ≤ T − h t , there exists h t − non-indifferent l, m ∈ ∆( Z ). Axiom 12: Constant Intertemporal Trade-off [FIS].
For all t, τ ≤ T − , if l, m are h t − non-indifferent and ˆ l, ˆ m are g τ -non-indifferent, then for all α ∈ [0 ,
1] and n ∈ ∆( Z ):( l, m, n, . . . , n ) ∼ h t ( αl + (1 − α ) m, αl + (1 − α ) m, n, . . . , n ) ⇐⇒ (ˆ l, ˆ m, n, . . . , n ) ∼ g τ ( α ˆ l + (1 − α ) ˆ m, α ˆ l + (1 − α ) ˆ m, n, . . . , n ) . This ensures that the discounting factor δ from the Evolving SEU representation is unique.Finally, we note down the classical axiom which gives δ < Axiom 13: Impatience [FIS].
For all t ≤ T − , h t and l, m, n ∈ ∆( Z ),if ( l, n, . . . , n ) (cid:31) h t ( m, n, . . . , n ), then ( l, m, n, . . . , n ) (cid:31) h t ( m, l, n, . . . , n ).The characterization result for Gradual Learning is then as follows. Theorem 3.
Assume the aSCF ρ satisfies an Evolving SEU model and assume Condition1 is satisfied. Then Axioms 11-13 are equivalent to the existence of a Gradual Learningrepresentation for ρ . The following Proposition proved in Section 4 of the online appendix shows that allthree representations are unique up to positive affine transformations of the Bernoulliutilities the agent uses to evaluate lotteries over the respective consequence spaces X t as well as up to relabeling of the states of the world ω and of the objective states s t .The characterization of uniqueness is a prerequisite for the comparative static exercisesof Section 4. The results mirror closely the identification in [Frick, Iijima, Strzalecki ’17]adapted to our more general setting with agents who hold (possibly incorrect) beliefsabout payoff-relevant states. Proposition 4.
1) Suppose that a dynamic aSCF ρ admits two DR-SEU representations (Ω , F ∗ , µ, ( F t , ( q t , u t ) , s t , (ˆ q t , ˆ u t )) ≤ t ≤ T ) and (Ω (cid:48) , F (cid:48)∗ , µ (cid:48) , ( F (cid:48) t , ( q (cid:48) t , u (cid:48) t ) , s (cid:48) t , (ˆ q (cid:48) t , ˆ u (cid:48) t )) ≤ t ≤ T ) .Then there exists a bijection φ t : F t →F (cid:48) t and F t -measurable functions α t : Ω → R ++ and β t : Ω → R such that for all ω ∈ Ω :(i) µ ( F ( ω )) = µ (cid:48) ( φ ( F ( ω ))) and µ ( F t ( ω ) |F t − ( ω )) = µ (cid:48) ( φ t ( F t ( ω )) | φ t ( F t − ( ω ))) if t ≥ ;(ii) q (cid:48) t ≡ q t for all t ≥ , u t ( ω ) = α t ( ω ) u (cid:48) t ( ω (cid:48) ) + β t ( ω ) whenever ω (cid:48) ∈ φ t ( F t ( ω )) ; iii) µ ((ˆ q t , ˆ u t ) ∈ B t ( ω ) |F t ( ω )) = µ (cid:48) ((ˆ q (cid:48) t , ˆ u (cid:48) t ) ∈ φ t ( B t ( ω )) |F (cid:48) t ( φ t ( ω ))) for any B t ( ω ) = { ( p t , v t ) ∈ ∆( S t ) × R X t : f t ∈ M ( M ( A t , ( q t ( ω ) , q t ( ω )) , p t , v t )) } for some f t ∈ A t ,A t ∈ A t .2) If ρ admits two Evolving-SEU representations then in addition to (i)-(iii) above wehave(iv) α t ( ω ) = α ( ω ) (cid:16) ˆ δδ (cid:17) t , for all ω ∈ Ω and t ≥ ;(v) v t ( ω ) = α t ( ω ) v (cid:48) t ( ω (cid:48) ) + γ t ( ω ) whenever ω (cid:48) ∈ φ t ( F t ( ω )) , where γ T ( ω ) = β T ( ω ) and γ t ( ω ) = β t ( ω ) − δ E [ β t +1 |F t ( ω )] if t ≤ T − .3) If ρ has two Gradual Learning Representations and satisfies Condition 1, then inaddition to (i)-(v) the following holds(vi) δ = δ (cid:48) (vii) β t ( ω ) = − δ T − t +1 − δ E [ β T |F t ( ω )] .
1) shows that agent’s choices uniquely identify the evolution of her private informationin both relevant dimensions: tastes and beliefs. The lack of identification for the Bernoulliutility functions u t is unavoidable. Intuitively, when one rescales the Bernoulli utilitiesby a factor which depends only on information up to time t , the sets of maximal elements M ( A t ; q t , u t ) don’t change.2) shows that the Evolving SEU model allows for stronger identification of the Bernoulliutilities. The scaling factor of Bernoulli utilities needs to be measurable with respect tothe information available at t = 0. This is because in the Evolving SEU model the utilityof the continuation problem enters cardinally into the overall utility of choosing an actfrom a menu. One can then use the same information, namely that available in period t = 0, to build a measuring rod with which utilities can be compared across periods.Obviously, the scaling factor α t still depends on the state of the world ω . In a populationinterpretation of the observable aSCF this means that different agents may use differentinformation available at t = 0 to compare utils intertemporally.3) shows that the Gradual Learning model improves on the identification propertiesof the Evolving SEU model because the discount factor is identified uniquely. This is aconsequence of the Constant Intertemporal Trade-Off Axiom. Under that Axiom any pos-sible scaling of the Bernoulli utilities in addition to depending on time t = 0 informationonly, has to be constant over time. This section offers simple comparative statics results under varying assumptions about therepresentations of the observable aSCF. The characterizations are simple because aSCFsrepresent very rich data sources. 27 .1 A measure of belief biasedness
If the analyst doesn’t observe anything about the realization of objective states, it isimpossible to discuss correctness of beliefs of the agents. Most of the canonical modelsof behavior based only on menu choice as an observable, as in [Dillenberger et al ’14] and[Krishna, Sadowski ’14] and many others, as well as models of stochastic choice withoutobservable objective states as in [Lu ’16] cannot address questions of belief biasedness. Inthis part we illustrate what is possible if the observable of the analyst consists of aSCFs.For simplicity we assume there are best and worst prizes which coincide for all agentsconsidered: that is, constant acts f , ¯ f such that for every aSCF ρ considered it holds:for every f (cid:54) = f we have ¯ ρ ( f, { f , f } ) = 1 and for every f (cid:54) = ¯ f we have ¯ ρ ( f, { ¯ f, f } ) = 0 . Moreover, for simplicity we assume the agents have the same non-stochastic taste u andfocus on comparative statics related to beliefs. We assume there is an underlying state of the world ω coming from a finite set Ω.For example in Example 2 ω may encode gender or ethnicity. An analyst observes twoagents i = 1 , s ∈ S . A state of the world ω goes hand in hand with a set of beliefs about the possiblerealizations of s for each agent and a true data-generating-process (DGP). The analystobserves the aSCFs of the agents which are assumed to have the following form. ρ i ( f, A, s ) = (cid:88) ω ∈ Ω µ ( ω, s ) τ iq i ( ω ) ,u ( f, A ) , i = 1 , . (6)Here µ ∈ ∆( ω × S ) and the tie-breakers τ iq i ( ω ) ,u depend only on the realized SEU ( q i ( ω ) , u )of agent i .We assume µ is either known by the analyst (e.g. an experiment in a lab) or theanalyst gleans it from the data ρ i using Theorem 0.Now assume the analyst fixes a direction q ( ω ) ∈ ∆( S ) for possible biases for every ω ∈ Ω and is interested in finding out how biased, if at all, the beliefs of the agents are inthe direction { q ( ω ) } ω ∈ Ω . The analyst might think that a possible bias for ω correspondsto some ‘extreme’ q ( ω ) (cid:54) = µ ( ·| ω ). A natural way in terms of the aSCF to say that an agent is biased in the direction { q ( ω ) } ω ∈ Ω and that, say, agent 1 has uniformly less biased beliefs than agent 2 is to requirethe following in terms of the representation. Definition 16.
1) Agent i’s beliefs are biased toward the direction q := { q ( ω ) } ω ∈ Ω if andonly if there exists a vector of weights { a ( ω ) } ω ∈ Ω ∈ [0 , Ω such that the following holds Formally speaking all aSCF/SCF-s in this subsection satisfy C-determinism* – choice is stochasticbecause beliefs of an agent are stochastic, besides possible randomness coming from tie-breaking. In thissetting all the machinery of [Lu ’16], esp. the related test acts can be used (see online appendix). Theconditions on the SCFs which imply that the taste of distinct agents are the same are available uponrequest. For example, if Ω encodes gender and the true DGP is that µ ( ·| ω ) is independent of ω , a possibleextreme bias might be to assume that for ω = male , q ( ω ) is ‘tilted’ towards more favorable realizationsof the objective state s whereas for ω = f emale , q ( ω ) is ‘tilted’ towards more unfavorable realizations ofthe objective state s . As Example 1 illustrates, this might be the case with employment data dependingon the vocation and job properties. i ( ω ) = a i ( ω ) q ( ω ) + (1 − a i ( ω )) µ ( ·| ω ) for some a ( ω ) ∈ [0 , .
2) Agent 1’s beliefs are uniformly less biased toward q than agent 2’s beliefs if and onlyif it holds for every ω ∈ Ω that ≤ a ( ω ) ≤ a ( ω ) ≤ . Figure 3 helps describe the definition. q (ω)q (ω) s s s μ( • |ω) •• •• q(ω) Figure 3: In state ω agent 1 has beliefs more aligned to true DGP than agent 2.The associated menu preference approach from [Lu ’16] provides a way to identify theweights of the bias in some direction q . Definition 17 ([Lu ’16]) . Given ¯ ρ , let the associated menu preference (cid:23) ¯ ρ be given by theutility function on menus V ¯ ρ : A→ [0 , with V ¯ ρ ( A ) = (cid:90) ¯ ρ ( A, A ∪ { αf + (1 − α ) ¯ f } ) da. For a fixed weight in α ∈ [0 ,
1] the value ¯ ρ ( A, A ∪ { αf + (1 − α ) ¯ f } ) gives the probabilitythat an element of A beats the act αf + (1 − α ) ¯ f , that is, the probability that the agentprefers items out of the menu A instead of the test act with weight α on the worst prize.Intuitively speaking, a menu is more valuable in the associated menu preference of a SCFif in the aggregate its elements are more preferred than test acts αf + (1 − α ) ¯ f . [Lu ’16]shows that, up to tie-breaking considerations, every stochastic choice function as ¯ ρ can becharacterized through its associated menu preference V ¯ ρ . Thus, except for tie-breaking, ¯ ρ contains no more information about the agent than V ¯ ρ does.Given the direction of bias q define for every weight of biases a : Ω → [0 ,
1] the associatedmenu preference where the agent gives weight a ( ω ) to the belief q ( ω ) whenever the stateof the world Ω is realized. V a ( A ) = (cid:90) Ω max f ∈ A [ a ( ω ) q ( ω ) + (1 − a ( ω )) µ ( ·| ω )] · ( u ◦ f ) µ ( dω ) . This gives a map ψ q : [0 , Ω →{ menu preferences } . Intuitively, one can interpret anyelement a ∈ [0 , Ω as a vector of degrees of biasedness towards q . The image of this map can naturally be identified with value functions of menu preferences. ψ q comes directly from the data: the aSCF-s ρ i , i = 1 , µ ( ·| ω ) (or the analyst knows this already) and the analyst picks the biasvector q . Once can show that once a bias direction q is fixed, every weight vector a definesa unique menu preference V a .This allows the following characterization of the degree of belief-biasedness in direction q in terms of observables/data. Here, recall that the induced menu preference from thestochastic choice function ¯ ρ is also completely constructed from stochastic choice data. Proposition 5.
Assume that the two aSCF ρ i , i = 1 , are as in (6) and consider a vectorof biases q ∈ ∆( S ) Ω . It holds:A. Agent i ’s beliefs are uniformly biased toward the direction q with degree a ∈ [0 , Ω if and only if ψ − q ( V ¯ ρ i ) = a, i.e. if and only if a is the image under ψ q of the menu preference induced fromstochastic choice.B. Agent 1’s beliefs are uniformly less biased toward the direction q than agent 2’sbeliefs if and only if ψ − q ( V ¯ ρ ) ≤ ψ − q ( V ¯ ρ ) . Note that by varying q , an analyst can use the induced menu preference of ¯ ρ i (fromDefinition 17) to identify the actual bias direction of an agent whenever her aSCF doesn’tsatisfy the Axiom of Correct Interim Beliefs from Definition 2. Example 1 continued.
In the context of Example 1 from the Introduction, subsection1.1.1 this Proposition states that stochastic choice data are enough for the analyst toidentify the incorrect beliefs ˆ q i , i = 1 ,
2. Namely, assume directions for the biases q ( s (cid:48) ) =(1 ,
0) and q ( s (cid:48)(cid:48) ) = (0 , s = s (cid:48) will always deliver outcome s = g and a candidate s = s (cid:48)(cid:48) will always deliveroutcome s = b . The Proposition delivers then a ( s (cid:48) ) = 2ˆ q − a ( s (cid:48)(cid:48) ) = 1 − q sothat whenever a : S → [0 ,
1] is identified from data the analyst can recover the incorrectbeliefs ˆ q i , i = 1 , a ∈ [0 , Ω on biases is to require instead a uniformweight a ∈ [0 ,
1] on biases which is independent of the realization of the characteristic ω .The conditions on the induced menu preferences identifying the bias a are then simplerthan in Proposition 16. Nevertheless, in applications, the bias weights will usually differaccording to the realization of the characteristic ω . For example, one might expect in somecases the agent to use the correct conditional DGP µ ( ·| ω ) and in other cases of realized ω -s to use a very biased belief much closer to an ‘extreme’ q ( ω ) (cid:54) = µ ( ·| ω ). Therefore, herewe have focused on the concept of Definition 16 which allows for this additional flexibility. Defining the menu preference of an unbiased agent (a counterfactual) and of a fully biased agent,the condition of biasedness is that the induced menu preference of the agent is a convex combination ofthe menu preferences of the unbiased and fully biased agent and that a corresponds to the weight on thebiased agent. .2 The speed of learning about taste In this subsection we consider agents in a dynamic setting ( T ≥
1) whose stochastic choicedata satisfy the Gradual Learning model and discuss measures across agents of the speedof learning about taste. We assume for all agents considered in this subsection that attime t = 0 their taste is not deterministic. Formally, we require the following conditionson any aSCF of this section. Assumptions
For all aSCFs in this subsection it holds true:A. ρ satisfies a Gradual Learning (GL) representation with T ≥ v t , t ∈ { , . . . , T } .B. ¯ ρ doesn’t satisfy C-Determinism*.B. ascertains that there is non-trivial learning about taste for an agent. On the otherhand, due to Sophistication (assumed as part of A.), if an agent learns her future tasteat the end of a period t , her taste remains deterministic in all future periods.Recall that the preferences (cid:23) h t on continuation menus A t +1 for some history h t ∈ H t from Definition 14 are derived solely from stochastic choice data. If for an agent heruncertainty about future taste is resolved after a history h t the derived menu preferenceon A t +1 derived from (cid:23) h t will satisfy Strong Dominance. On the other hand, StrongDominance will be violated for (cid:23) h t whenever an agent’s uncertainty about future tastedoesn’t get resolved after history h t . The same holds if instead of looking at whetherStrong Dominance is satisfied we look at whether C-Determinism* is satisfied.This suggests a simple way to define the speed of learning about taste of an agent whosatisfies a Gradual Learning model as well as an equally simple way to rank such agentsaccording to their speed of learning about taste. Definition 18.
1) Say that an agent learns her future taste after history h t if her derivedmenu preference on A t +1 from (cid:23) h t satisfies Strong Dominance or equivalently, if ρ t +1 ( ·| h t ) satisfies C-Determinism*.
2) Say that an agent becomes certain of her future taste at time t if she learns her futuretaste after every history h t ∈ H t .3) Say that agent 1 learns her taste faster than agent 2 if the following implication holdstrue for every t ≤ T − :agent 2 becomes certain of her taste at t = ⇒ agent 1 becomes certain of her taste at t. The characterization of these concepts in terms of the GL representation (Definition8) is as follows.
Proposition 6.
1) Suppose an agent has a GL representation with probability space (Ω , F ∗ , µ ) . Then an agent learns her future taste after history h t if and only if condi-tional on C ( h t ) her felicity is deterministic, i.e. v t +1 is a constant function on C ( h t ) . Equivalence holds under the assumption that the data satisfy the GL representation. ) Suppose an agent has a GL representation with underlying probability space (Ω , F ∗ , µ ) .An agent becomes certain of her future taste at time t if and only if her felicity at time t is independent of the state of the world ω , i.e. v t +1 is a constant function on all of Ω .
3) Suppose two agents i = 1 , have GL representations with underlying probability space (Ω , F ∗ , µ ) but otherwise may have different filtrations {F it } t ≤ T and different evolution ofSEUs { ( q it , u it ) } t ≤ T for i = 1 , . Then agent 1 learns her taste faster than agent 2 if andonly if the following implication holds true for every t ≤ T − : v t +1 is a constant function on all of Ω = ⇒ v t +1 is a constant function on all of Ω . Example 4.
Assume that we have two investors i = 1 , the same market condi-tions whose CARA Bernoulli utility over monetary outcomes has the form x (cid:55)→ − e − γ i x where γ i is random according to a discrete distribution taking positive values from a finiteset Γ ⊂ [1 , + ∞ ). In every period each investor decides whether to invest in a risky project f , whose outcome is strongly dependent on market conditions (objective state s t ∈ R + drawn anew each period) through f ( s t ) ∼ √ s t × U nif orm {− , } + s t or to pick invest-ments h ( α ) whose s t -independent outcome satisfies h ( α ) ∼ √ α × U nif orm {− , } + α .Then according to the above Proposition an analyst has two ways of telling who of the twoinvestors has learned her parameter γ i the earliest. If she only has data on choices frommenus containing only acts of the type h ( α ) she finds the first time when the choice ofeach investor on such menus becomes deterministic. If she only has data of choice amongmenus, an indicator that investor 1 learned her preference parameter earlier is that shestarts preferring menus where f is present to menus where f isn’t present earlier in timethan investor 2 does. We have introduced a dynamic stochastic choice model general enough to encompasssituations where a subjective expected utility agent has both stochastic taste as wellas stochastic beliefs about the realization of objective payoff-relevant states. Under theassumption that the analyst has access to data which reveal the agent’s history-dependentchoices as well as the sequence of realizations of objective states we have characterizedaxiomatically the case when the analyst can uncover the otherwise arbitrary evolution ofthe private information of the agent.The assumed richness of the data allows the analyst to test whether the agent is usingcorrectly specified beliefs about objective states conditional on her private information andif not, to determine the bias of the agent as well as to compare different agents accordingto their biasedness of beliefs. We have also characterized special cases of the generalrepresentation, Evolving SEU and Gradual Learning, which would have been otherwiseindistinguishable in the static setting. Finally, in the case of Gradual Learning, we haveshown how an analyst is able to detect from data that the agent has stopped learningabout her taste and that therefore the randomness in choice only comes from randomnessin beliefs.Information acquisition is outside the scope of this model and constitutes the naturalnext step in research. E.g. we shouldn’t expect the student in Example 2 not to try32nd actively learn early about her final job market outcome. So it natural to expectIndifference to Timing to be violated; if an agent tries to actively learn about futuretastes by spending resources after history h t we should expect her to satisfy instead theweaker condition:if A t +1 ∼ h t +1 B t +1 then αA t +1 + (1 − α ) B t +1 (cid:22) h t A t +1 . That is, since contingent planning costs utility, the agent is averse to it whenever she isex-ante indifferent between two decision problems. Introducing information acquisitionin this framework would also allow a better study of misspecified learning.Other directions to pursue are as follows. We haven’t considered consumption depen-dence as [Frick, Iijima, Strzalecki ’17] do in their DREU model of stochastic taste only. Developing ‘systems’ of DR-SEUs coming from agents in strategic situations is also leftfor future research, as is characterizing meaningful relaxations of the Sophistication as-sumption in the Evolving SEU model.Finally, on another perspective, this paper is about identification and not inference. Inapplications data sets are naturally finite. We leave for future research characterizationsof stochastic dynamic behavior when data sets are finite.
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The Appendix is organized as follows. Appendix A is devoted to the proof of Theorem 0.Appendix B describes the Ahn-Sarver representations in the dynamic setting. These aremore convenient for proofs and their equivalence to the Filtration-based representationsfrom the main text of the paper is proved in the online appendix. Appendix C proves theexistence of so-called separating histories . These are an essential tool in the proof of themain characterization theorems. Most of Appendix D is devoted to the proof of Theorem1, the rest of it to the proofs of Section 4. The proof of the rest of the characterizationtheorems is in the online appendix. Besides the rest of the auxiliary results, the latteralso contains most of the technical work needed to extend the menu choice literatureto the setting of SEUs, add explicit tie-breaking to [Lu ’16] and beliefs about objectivepayoff-relevant states to [Ahn, Sarver ’13].
A Random Sub jective Expected Utility with observ-able ob jective states (AS-version)
A.1 Separation property for acts - static setting
We prove a separation property for menus of acts, similar to Lemma 1 in [Ahn, Sarver ’13](separation property for lotteries).We start with a trivial remark which will be used extensively in the following.
Remark 2.
1) A SEU preference encoded by ( q, u ) is constant (i.e. consists of onlyindifferences) if and only if u is constant.2) Two SEU representations ( q, u ) and ( q (cid:48) , u (cid:48) ) represent the same SEU preference ifand only if q = q (cid:48) and u ≈ u (cid:48) . The separation property for acts is as follows.
Lemma 1 (Separation property in the AA setting) . Let Z (cid:48) be any set (possibly infinite)and let { ( q k , u k ) : k = 1 , . . . , K } ⊂ ∆( S ) × R Z (cid:48) be a set of pairwise distinct SEU repre-sentations s.t. u k is non-constant for all k = 1 , . . . , K . Then there is a collection of acts { f k : k = 1 , . . . , K } ⊂ F s.t. q k · u k ( f k ) > q k · u k ( f l ) for any distinct l, k ∈ { , . . . , K } . Proof.
We divide the proof in three steps.
Step 1.
Assume first that u k (cid:54)≈ u l for all l (cid:54) = k and that Z (cid:48) is finite. Then we are in thesetting of Lemma 13 from [Frick, Iijima, Strzalecki ’17] and can use a menu of constantacts to realize the separation property required. Step 2.
Assume now that u k ≈ u l for all l (cid:54) = k and that Z (cid:48) is finite. W.l.o.g. wecan assume that u k = u l = u and that im ( u ) = [0 , q k (cid:54) = q l for all l (cid:54) = k . It is enough in this case to solve the following problem:(P) For all k find p k ∈ ∆( S ) s.t. q k · p k > q k · p l , l (cid:54) = k. Now we are again in the setting of Lemma 13 in [Frick, Iijima, Strzalecki ’17], if we takeas Bernoulli utilities the q k -s. Formally, it follows q k (cid:54)≈ q l whenever S has more thanone element as one can check using uniqueness result in the classical vNM Theorem.35hus, Lemma Lemma 13 in [Frick, Iijima, Strzalecki ’17] gives probability distributions p k , k = 1 , . . . K satisfying (P). Now, we can easily construct the acts needed by the formula u ( f k ( s )) = p k ( s ) , s ∈ S, k = 1 , . . . , K . Note that this trick works because ∆( S ) ⊂ [0 , S . Step 3.
Assume now that we are in the general case ( q k , u k ) (cid:54)≈ ( q l , u l ) , l (cid:54) = k . Thereexists a finite Z ⊂ Z (cid:48) s.t. all u k are non-constant in ∆( Z ). We are going to choose acts f : S → ∆( Z ). Assume w.l.o.g. that for all k we have im ( u k ) ⊆ [0 , u k in classes r = 1 , . . . R ≤ K s.t. if l, k are so that u k ≈ u l they belong to thesame class. Within the same class, normalize the Bernoulli utilities to be equal. Thus,we can rewrite the SEU preferences given as { ( q rl , u r ) : r = 1 , . . . R, l = 1 . . . , K r } . Now pick constant acts h r , r = 1 . . . , R as in Step 1 with u r ( h r ) > u r ( h r (cid:48) ) , r (cid:54) = r (cid:48) . Pickalso within each group r ∈ { , . . . , R } acts f rl , l = 1 , . . . , K r with image in ∆( Z ) s.t. q rl · u r ( f rl ) > q rl · u r ( f rl (cid:48) ) , l (cid:54) = l (cid:48) . We claim that the separating acts we are after can betaken of the form λf rl + (1 − λ ) h r , r = 1 , . . . , R ; l = 1 , . . . , K r whenever λ > λ ∈ (0 ,
1) with( P q rl · u r ( λf rl + (1 − λ ) h r ) > q rl · u r ( λf r (cid:48) l (cid:48) + (1 − λ ) h (cid:48) r ) , whenever ( r, l ) (cid:54) = ( r (cid:48) , l (cid:48) ) . Consider first the case r = r (cid:48) . Then l (cid:54) = l (cid:48) and (P1) is true for all λ by linearity of theBernoulli functions and the choice of f rl .Consider then the case r (cid:54) = r (cid:48) . Given that u r ( h r ) > u r ( h r (cid:48) ) and the linearity of theBernoulli functions, for a fixed pair of tuples ( r, l ) (cid:54) = ( r (cid:48) , l (cid:48) ) (P1) becomes true whenever λ is small enough for that pair. This gives a positive upper bound on λ . Since the numberof pairs ( r, l ) is finite, overall there exists a 1 > λ > r, l ) (cid:54) = ( r (cid:48) , l (cid:48) ). A.2 Proof for the Axiomatization of aSCFs (AS-version)
Pick an element y ∗ ∈ X and set U = { u ∈ R X : u ( y ∗ ) = 0 } .We first define the AS-version (Ahn-Sarver version) of the representation. Definition 19.
1) Let ρ be an aSCF for acts in F over ∆( X ) where X is a separablemetric space and S , the set of objective states is finite.We say that ρ admits an AS-version R-SEU representation if there is a triple ( SubS, µ, { (( q, u ) , τ q,u ) : ( q, u ) ∈ SubS } ) such thatA. SubS is a finite subjective state space of distinct and non-constant SEUs and µ is aprobability measure on SubS × S .B. For each ( q, u ) ∈ SubS the tie-breaking rule τ q,u is a regular sigma-additive probabil-ity measure on ∆( S ) × U endowed with the respective product Borel sigma-Algebra. . For all f ∈ F , A ∈ A and s ∈ S we have ρ ( f, A, s ) = (cid:88) ( q,u ) ∈ SubS µ ( q, u, s ) τ q,u ( f, A ) , (7) where τ q,u ( f, A ) := τ q,u ( { ( p, w ) ∈ ∆( S ) × U : f ∈ M ( M ( A ; u, q ); w, p ) } ) .2) We say that the AS-version R-SEU representation has no unforeseen contingencies if supp ( µ ( ·| q, u )) ⊆ supp ( q ) for all ( q, u ) ∈ SubS .3) We say that the AS-version R-SEU representation has correct interim beliefs if µ ( ·| q, u ) = q ( · ) for all ( q, u ) ∈ SubS . The next Theorem gives the axiomatization of aSCFs which have an AS-version R-SEUrepresentation.
Theorem 4.
The aSCF ρ on A admits an AS-version R-SEU representation if and onlyif it satisfiesA. Statewise MonotonicityB. Statewise LinearityC. Statewise ExtremenessD. Statewise ContinuityE. Statewise State IndependenceF. Statewise FinitenessMoreover, it additionally has a No Unforeseen Contingencies representation if and onlyif it additionally satisfies No Unforeseen Contingencies. Finally, it has a
Correct InterimBeliefs representation if and only if it additionally satisfies Correct Interim Beliefs.
Proof of Theorem 4. Necessity.
Checking this is routine. In particular, one checkseasily that
RSSupp ( ¯ ρ ) = supp ( µ ). Sufficiency.
We prove this in several steps.
Step 1.
We construct the SCFs ¯ ρ from ρ as well as ρ ( · , ·| s ) for all s ∈ S . Due to theaxioms on ρ all of ¯ ρ as well as ρ ( · , ·| s ) , s ∈ S satisfy all axioms from Theorem 1 in theonline appendix. In particular, we have the following representations: for all f ∈ A, A ∈ A ¯ ρ ( f, A ) = (cid:88) ( q,u ) ∈ SubS ψ ( q, u ) τ q,u ( f, A ) (8)and ρ ( f, A | s ) = (cid:88) ( q,u ) ∈ SubS ( s ) ψ s ( q, u ) τ sq,u ( f, A ) . (9)with appropriate probability measures ψ and ψ s on finite sets of SEUs. Step 2.
Due to simple probability accounting it holds37 ρ ( f, A ) = (cid:88) s ∈ S ρ ( f, A | s ) ρ ( s ) . (10)If it were true that supp ( ψ s ) (cid:54)⊆ supp ( ψ ) for some s ∈ S then by use of separating menusas constructed in Lemma 1 one could come to a contradiction to (10). The same kindof contradiction argument and use of Lemma 1 leads to exclusion of the case supp ( ψ ) \∪ s ∈ S supp ( ψ s ) (cid:54) = ∅ . In all we have established supp ( ψ ) = ∪ s ∈ S supp ( ψ s ) . In particular, we can extend w.l.o.g. ψ s for all s to all of supp ( ψ ) by setting it to zerooutside of supp ( ψ s ). Step 3.
By a similar mixing argument as in Proposition 2 in the online appendix (seestep 3 there) one can easily show that whenever ( q, u ) ∈ supp ( ψ ) ∩ supp ( ψ s ) we have τ sq,u = τ q,u . In particular, we can write the representations for ρ ( · , ·| s ) as ρ ( f, A | s ) = (cid:88) ( q,u ) ∈ SubS ψ s ( q, u ) τ q,u ( f, A ) . (11)By plugging (11) in (10), rearranging and using the uniqueness result for the AS-representationof ρ from Proposition 2 in the online appendix we get ψ ( q, u ) = (cid:88) s ∈ S ψ s ( q, u ) ρ ( s ) , ( q, u ) ∈ supp ( ψ ) . (12)By setting µ ( q, u, s ) = ψ s ( q, u ) ρ ( s ) we define a probability measure over SubS × S whose marginal over SubS is full support and which satisfies (7).
Step 4.
Take a separating menu ¯ A = { f ( q, u ) : ( q, u ) ∈ supp ( ψ ) } for supp ( ψ ). Weshow that the following property (P) gives us the representation for correct interim beliefs.( P ) ρ ( ·| f ( q, u ) , ¯ A ) = q ( · ) , ( q, u ) ∈ supp ( ψ ) . Claim. (P) implies the representation with correct interim beliefs.
Proof of Claim.
For the menu ¯ A and each ( q, u ) ∈ supp ( ψ ) we have ψ s ( q, u ) = ρ ( f ( q, u ) , ¯ A | s ) = ρ ( f ( q, u ) , ¯ A, s ) ρ ( s ) = ρ ( s | f ( q, u ) , ¯ A ) ¯ ρ ( f ( q, u ) , ¯ A ) ρ ( s ) = q ( s ) ψ ( q, u ) ρ ( s ) . (13)Here, only in the last equality we have used (P) and the definition and representation of¯ ρ from Theorem 1 in the online appendix. We write this as the identity(!) ρ ( s ) ψ s ( q, u ) = q ( s ) ψ ( q, u ) . Summing (!) w.r.t. ( q, u ) we get the identity (!!) ρ ( s ) = (cid:80) ( q,u ) ∈ SubS ψ ( q, u ) q ( s ) forall s ∈ S and thus a unique solution for ψ s in (13). It is then trivial to see that therepresentation holds because of (11) and (!!).38 tep 5. In this step we show that (P) is implied by Correct Interim Beliefs.Denote in general for each q ∈ ∆( S ) such that ( q, u ) ∈ supp ( µ ) for some u ρ ( ·| f ( q, u ) , ¯ A ) =ˆ q ( q, u )( · ).Suppose by contradiction that there exists some ( q, u ) ∈ supp ( µ ) with ρ ( ·| f ( q, u ) , ¯ A ) (cid:54) = q ( · ). If it holds for some ˆ u that (ˆ q ( q, u ) , ˆ u ) ∈ supp ( µ ) = RSSupp ( ¯ ρ ) then we know that(ˆ q ( q, u ) , ˆ u ) (cid:54)∈ N ( ¯ A, f ( q, u )) ∩ RSSupp ( ¯ ρ ) = { ( q, u ) } as ¯ A is separating for RSSupp ( ¯ ρ ) andˆ q ( q, u ) (cid:54) = q , which implies ( q, u ) (cid:54)≈ (ˆ q ( q, u ) , ˆ u ). But clearly | N ( ¯ A, f ( q, u )) ∩ RSSupp ( ¯ ρ ) | = |{ ( q, u ) }| = 1.Overall it follows that Correlated Interim Belief axiom is violated at the choice data( f ( q, u ) , ¯ A ). Step 6.
We show that the following property (P!) gives us the representation for nounforeseen contingencies.( P !) supp ( ρ ( ·| f ( q, u ) , ¯ A )) ⊂ supp ( q ( · )) , ( q, u ) ∈ supp ( ψ ) . Claim. (P!) implies the representation with unforeseen contingencies.We look at (13), but leave out the final equality. The Claim follows immediately.
Step 7.
In this step we show that (P!) is implied by No Unforeseen Contingencies.Suppose by contradiction that there exists some ( q, u ) ∈ supp ( ψ ) = RSSupp ( ¯ ρ ) with supp ( ρ ( ·| f ( q, u ) , ¯ A )) (cid:54)⊆ supp ( q ( · )). Pick again a separating menu for RSSupp ( ¯ ρ ) andnote that | N ( ¯ A, f ( q, u )) ∩ RSSupp ( ¯ ρ ) | = |{ ( q, u ) }| = 1. Overall it follows that the NoUnforeseen Contingencies axiom is violated for the choice data ( f ( q, u ) , ¯ A ).We note down uniqueness. Proposition 7.
The AS-version REU-representation for an aSCF ρ is essentially uniquein the sense that for each two representations the only degree of freedom is positive affinetransformations of the Bernoulli utilities of elements in the support of the measures overSEUs. Proof.
For the case of CIB this follows directly from Proposition 2 in the online appendixapplied to the SCF corresponding to the aSCF.For the case of NUC, if there are two different representations for ρ with respectivemeasures µ, µ (cid:48) it follows from Proposition 2 in the online appendix that the marginalsare equal: (cid:80) s µ ( q, u, s ) = (cid:80) s µ (cid:48) ( q, u, s ) for all ( q, u, s ). In particular, up to equivalenceclasses of positive affine transformations of the Bernoulli utility functions the support ofthese two marginals in ∆( S ) × R X is equal for the two measures. Assume then w.l.o.g.the same normalization for both supports. Taking now a separating menu ¯ A for the SEUsin the support of the two measures µ, µ (cid:48) , we have from the representation property that ρ ( f ( q, u ) , ¯ A, s ) = µ ( q, u, s ) = µ (cid:48) ( q, u, s ) for all s. This concludes the proof.
Proof for Proposition 1. Sufficiency.
Define the SCF on ∆( X ) by the formula The online appendix shows equivalence between AS-based representations and Filtration-based rep-resentations. Here a slight abuse of notation as we haven’t written down the isomorphism between constant menusof acts and menus of lotteries, but the context gives clarity. ( f, A ) = ρ ( f, A ) , A is menu of constant acts . Note that Theorem 1 in the online appendix gives with some slight abuse of notation τ ( f, A ) = (cid:88) ( q,u ) ∈ SubS for some q µ ( q, u ) τ q,u ( { ( p, w ) ∈ ∆( S ) × U : f ∈ M ( M ( A ; q, u ); p, w ) } ) . Since the beliefs play no role in the decision of the agent (all acts are constant), one canrewrite this as τ ( f, A ) = (cid:88) u ∈ π u ( SubS ) µ ( u ) τ (cid:48) u ( { w ∈ U : f ∈ M ( M ( A ; u ); w ) } ) , where µ ( u ) = (cid:80) q :( q,u ) ∈ SubS µ ( q, u ) > τ (cid:48) u = (cid:80) q :( q,u ) ∈ SubS µ ( q,u ) µ ( u ) τ q,u . Note that τ (cid:48) u is aregular tie-breaker for lotteries. Obviously this gives an S-based REU representation as in Theorem 4 of [Frick, Iijima, Strzalecki ’17].C-Determinism* implies then directly that τ has only one state in the sense of the S-basedrepresentation from [Frick, Iijima, Strzalecki ’17]. In particular, u ≈ v for all u, v ∈ U such that ( q, u ) , ( p, v ) ∈ supp ( µ ) for some q, p ∈ ∆( S ). Necessity.
Consider a menu of constant acts A . Then for all ( q, u ) , ( p, u ) ∈ supp ( µ )we have M ( A ; u, q ) = M ( A ; v, u ) =: M ( A, q ), so that by a small abuse of notation whichuses the fact that the menu A is constant we can write ρ ( f, A ) = (cid:88) ( q,u ) ∈ SubS µ ( q ) τ q ( { w ∈ U : f ∈ M ( M ( A ; u ); w ) } ) . The existence of a best constant act ¯ f means u ( ¯ f ) > u ( f ) whenever f (cid:54) = ¯ f and f alsoconstant.Note now that for each g ∈ A, g (cid:54) = f we have for either u ( af + (1 − a ) ¯ f ) > u ( g ) or u ( af + (1 − a ) ¯ f ) < u ( g ) for all a < a . It follows that ρ ( af + (1 − a ) ¯ f ; A \ { f } ∪ { af + (1 − a ) ¯ f } ) ∈ { , } , for all a < B AS-Based Representations for the dynamic setting
The proofs in this appendix are done in the AS-version of the representations. Here weexplain what these are. The online appendix then establishes the equivalence betweenthe two types of representations. Here, the w breaking ties from M ( A, u ) is drawn as follows: first draw a ( q, u ) where ( q, u ) hasprobability µ ( q,u ) µ ( u ) and then, draw (conditionally independently across the ( q, u )-s) w according to themarginal of τ q,u on U . This works because the tie-breakers are preference-based . Otherwise one arrives easily at a contradiction through separating lotteries to either µ ( u ) > u or to the C-Determinism* Axiom. .1 Dynamic Random Subjective Expected Utility (DR-SEU) Definition 20.
We say that a history-dependent family of aSCF ρ = ( ρ , . . . , ρ T ) has aDR-SEU representation if there exists • a finite objective state space S and a collection of partitions S t , t = 1 . . . , T of S such that S t is a refinement of S t − . • a finite collection of states of the world Θ t , t = 0 , . . . , T (an element is of thetype ( q t , u t , s t ) ∈ ∆( S t ) × R X t × S t ). The sequence Θ t , t ≤ T has a partitionalstructure and there are no repetitions: each element ( q t , u t , s t ) is indexed by thepredecessors ( q , u , s ; . . . ; q t − , u t − , s t − ) . Moreover we have the restriction that s k ∈ supp ( q k ) . • a collection of probability kernels ψ k : Θ k − → ∆(Θ k ) for k = 0 , . . . , T with a typical element in the image written as ψ q k − ,u k − ,s k − k . Inparticular, the probability that ( q k , u k , s k ) is realized after θ k − occurs is ψ θ k − k ( q k , u k , s k ) . • a sequence of tie-breakers: for all t = 0 , . . . , T a regular probability measure τ ( q t ,u t ) over ∆( S t ) × R X t , for all ( q t , u t ) = π qu ( θ t ) for some θ t ∈ Θ t .such that the following two conditions hold. DR-SEU 1 (a) every ( q t , u t ) ∈ π qu (cid:16) supp ( ψ θ t − t ) (cid:17) represents a non-constant SEU preference.(b) supp ( ψ θ t − t ) ∩ supp ( ψ θ (cid:48) t − t ) = ∅ whenever θ t − (cid:54) = θ (cid:48) t − , both in Θ t − . (c) ∪ θ t − supp ( ψ θ t − t ) = Θ t .(d) either (correct interim beliefs) The kernels ψ satisfy ψ θ k − k ( s k | q k , u k ) = q k ( s k ) or otherwise (no unforeseen contingencies) supp (cid:16) ψ θ k − k ( ·| q k , u k ) (cid:17) ⊂ supp ( q k ) . DR-SEU 2
The SCF ρ t after a history h t − = ( A , f , s ; . . . , A t − , f t − , s t − ) is given by ρ t ( s t , f t , A t | h t − ) = (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:104)(cid:81) t − k =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) (cid:105) · ψ θt − t ( θ t ) τ πqu ( θt ) ( f t ,A t ) (cid:80) πs ( θ ,...,θt − s ,...,st − (cid:104)(cid:81) t − k =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) (cid:105) . This means that there can be repetitions in terms of the SEUs ( q t , u t ) but whenever this happens adifferent s t is realized. With the obvious conventions for k = 0. This implies, that whenever π s ( θ t − ) = π s ( θ t − ) and two elements θ t ∈ supp ( ψ θ (cid:48) t − t ) , θ (cid:48) t ∈ supp ( ψ θ (cid:48) t − t )with π qs ( θ t ) = π qs ( θ (cid:48) t ) we must have u t (cid:54) = u (cid:48) t . .2 Evolving Subjective Utility (Evolving SEU) The Evolving Subjective Expected Utility representation is a special case of DR-SEU.In the pre-choice situation in period t when the agent knows ( q t , u t ) = π qu ( θ t ) andsatisfies the Evolving SEU representation she evaluates acts according to the followingSEU functional E q t [ u t ( f t )] = E s t ∼ q t [ u t ( f t ( s t ))] = E s t ∼ q t [ v t ( f Zt ( s t ))] + δV π qu ( θ t ) t ( f At ) . (14)Here V π qu ( θ t ) t ( f At ) is defined in two steps. First we define V θ t t ( A t +1 ) = (cid:90) max f t +1 ∈ A t +1 E q t +1 [ u t +1 ( f t +1 )] dψ θ t t +1 ( q t +1 , u t +1 ) . (15)This gives the value of a menu when the agent knows the menu, but not the SEU withwhich it will evaluate the acts. This is the situation just after ( z t , A t +1 ) is known to theagent at the end of period t .A moment before, i.e. when the agent doesn’t know s t yet the value of f At is given by V π qu ( θ t ) t ( f At ) := (cid:88) s t (cid:88) A t +1 ∈ suppf At ( s t ) q t ( s t ) f At ( s t )( A t +1 ) V θ t t ( A t +1 ) =: (cid:88) s t q t ( s t ) V θ t t ( f At ( s t )) . (16)Note that the uncertainty that is integrated out in (16) is the objective one concerning s t and that we have used equation (15) to define the extension of V θ t t to lotteries overmenus. We can rewrite this in integral form as follows. V π qu ( θ t ) t ( f At ) = (cid:90) max f t +1 ∈ A t +1 E q t +1 [ u t +1 ( f t +1 )] dψ π qu ( θ t ) t +1 ( q t +1 , u t +1 ) , where ψ π qu ( θ t ) t +1 ( q t +1 , u t +1 ) := (cid:80) s t q t ( s t ) ψ θ t t +1 ( q t +1 , u t +1 ) = (cid:80) s t ,s t +1 q t ( s t ) ψ θ t t +1 ( q t +1 , u t +1 , s t +1 ). B.3 Gradual SEU-Learning.
Gradual SEU-learning is the case of Evolving SEU with the additional requirement thather sequence of expected utility functionals from consumption v t , t = 0 , . . . T form aMartingale. In the following we use the projection π v , which for a u t as in (14) gives thecorresponding v t .Normalize v t (¯ p ) = 0 for all t where ¯ p is the uniform lottery over Z . This is possiblebecause Z is assumed to be finite for the dynamic setting. After a θ t = ( q t , u t , s t ) it hasto hold for the sequence π v ( θ t ) from the Evolving SEU representation π v ( θ t ) = 1 δ (cid:88) ( q t +1 ,u t +1 ) ∈ π qu (Θ t +1 ) ψ θ t t +1 ( q t +1 , u t +1 ) · π v ( u t +1 ) = 1 δ E [ π v ( θ t +1 ) | θ t ] . (17) I.e. agent is Expected Utility w.r.t. lotteries over menus. Separating histories
We first define histories consistent with a state θ t . Then we define separating histories for a fixed state θ t . The main result of this section establishes the existence of separatinghistories (Lemma 7).Let us assume that we have an aSCF ρ which satisfies DR-SEU 1. We define thepredecessor of a state θ as pred ( θ t ) = ( θ , . . . , θ t − ). Definition 21.
For a state θ t = ( q t , u t , s t ) denote by pred ( θ t ) = ( θ , . . . , θ t − ) the uniquepredecessor of θ t from (cid:81) t − i =0 Θ i . The concept is well-defined because of DR-SEU 1 (a)-(b).
Definition 22.
Given a history h t = ( A , f , s ; A , f , s ; . . . ; A t , f t , s t ) say that θ t isconsistent with h t if for the unique predecessor of θ t , given by ( θ , . . . , θ t − ) we have t (cid:89) k =0 τ π qu ( θ k ) ( f k , A k ) · ψ θ k − k ( θ k ) > . Here we use the convention ψ θ − := ψ . Note that multiple states θ t can be consistent with the same history h t .Define QU θ k ( A k +1 , f k +1 , s k +1 ) = { ( q k +1 , u k +1 ) : ( q k +1 , u k +1 , s k +1 ) ∈ supp ( ψ θ k k +1 ) and f k +1 ∈ M ( A k +1 ; q k +1 , u k +1 ) } . This is the set of SEU-s ( q k +1 , u k +1 ) occurring right after θ k which can rationalize thedata ( A k +1 , f k +1 , s k +1 ).For time t = 0 define QU ( A , f , s ) = { ( q , u ) : ( q , u , s ) ∈ supp ( ψ θ − ) and f ∈ M ( A ; q , u ) } . We prove first the following Lemma.
Lemma 2 (Pendant to Lemma 1 in [Frick, Iijima, Strzalecki ’17]) . Fix any θ t and its pre-decessor ( θ , . . . , θ t − ) . Suppose h t = ( B , g , s ; . . . ; B t , g t , s t ) satisfies QU θ k − ( B k , g k , s k ) = { π qu ( θ k ) } . Then for all k = 0 , . . . , t , only θ k in Θ k can be consistent with h k . Proof.
Fix any l = 0 , . . . , t and consider θ (cid:48) l ∈ Θ l \ { θ l } with pred ( θ (cid:48) l ) = ( θ (cid:48) , . . . , θ (cid:48) l − ).Let k ≤ l be smallest such that θ (cid:48) k (cid:54) = θ k . Then π qu ( θ (cid:48) k ) ∈ π qu ( supp ( ψ θ k − k )). So QU θ k − ( B k , g k , s k ) = { ( q k , u k ) } (which is assumed) implies either (A) ( q k , u k ) (cid:54) = ( q (cid:48) k , u (cid:48) k )or (B) ( q k , u k ) = ( q (cid:48) k , u (cid:48) k ) , s k (cid:54) = s (cid:48) k (otherwise contradiction to θ k (cid:54) = θ (cid:48) k ).In the case of (B) the definition of the QU-sets implies then that s (cid:48) k (cid:54)∈ supp ( q k ), i.e. q (cid:48) k ( s (cid:48) k ) = 0. In the case of (A) the definition of the QU-sets implies g k (cid:54)∈ M ( B k ; q (cid:48) k , u (cid:48) k ),i.e. τ q (cid:48) k ,u (cid:48) k ( g k , B k ) = 0. Overall we have that θ (cid:48) l is not consistent with h l .Next we show that θ l is consistent with h l . Note that from the definition of historiesw.r.t. to some aSCF it follows that ρ ( g l , B l | h l ) >
0. DR-SEU 2 then implies (cid:88) π qu ( θ ,...,θ t ) ∈× i ≤ l SEU i (cid:34) l − (cid:89) k =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) (cid:35) · ψ θ l − l ( θ l ) τ π qu ( θ l ) ( f l , A l ) q l ( s l ) > .
43f it happens that pred ( θ l ) (cid:54) = ( θ , . . . , θ l − ) then (cid:104)(cid:81) l − k =0 ψ θ k − k ( θ k ) (cid:105) · ψ θ l − l ( θ l ) = 0 just bythe definition of DR-SEU 1. If otherwise pred ( θ l ) = ( θ , . . . , θ l − ) but θ l (cid:54) = θ (cid:48) l then weshowed above that (cid:104)(cid:81) l − k =0 q k ( π s ( θ k )) τ π qu ( θ k ) ( f k , A k ) (cid:105) · τ π qu ( θ l ) ( f l , A l ) q l ( s l ) = 0. Definition 23.
A separating history for θ t with pred ( θ t ) = ( θ , . . . , θ t − ) is a history h t = ( B , g , s ; . . . ; B t , g t , s t ) ∈ H ∗ t such that QU θ k − ( B k , g k , s k ) = { π qu ( θ k ) } for all k ≤ t .For the case k = 0 we abuse notation and write QU − ( B , g , s ) = QU ( B , g , s ) . Remark 3.
1) Let A t ∈ A t arbitrary. After introducing LHI below, one sees easily,that when mixing a separating history for θ t with a deterministic history such that ithas the same projection on objective states as h t − one can assume that h t − is so that A t ∈ A ∗ t ( h t − ) . In particular separating histories are not unique.2) By definition, θ t is the only state in Θ t − consistent with h t − if h t − is a separatinghistory for θ t − . Write D t − = { d t − ∈ H t − : d t − = ( { f } , f , s ; . . . { f t − } , f t − , s t − ) , f i ∈ F i } , for the set of histories such that the menu is degenerate in each period and look at itssubset DC t − = { d t − ∈ H t − : d t − = ( { h } , h , s ; . . . { h t − } , h t − , s t − ) , h i , i ≤ t − } . The latter consists of deterministic histories where the agent faces only constant acts andthus objective states don’t matter.Note that given a menu A t (cid:54)∈ A t ( h t − ) we can always choose a h t − ∈ DC t − with A t ∈ supp ( h At − ). Then we can define the extended aSCF as follows. Definition 24.
For a history h t − ∈ H t − , A t ∈ A t and s t ∈ S t define ρ h t − t ( · , A t , s t ) = ρ t ( · , A t , s t | λh t − + (1 − λ ) d t − ) , for some λ ∈ (0 , , where d t − ∈ DC t − is so that λh t − + (1 − λ ) d t − ∈ H t − ( A t ) . We prove the extension is well-defined.
Lemma 3.
Suppose that ρ satisfies LHI. Fix t ≥ , A t ∈ A t , h t − = ( A , f , s ; . . . , A t − , f t − , s t − ) ∈H t − and ( λ , . . . , λ t − ) , (ˆ λ , . . . , ˆ λ t − ) ∈ (0 , t .Suppose d t − = ( h , { h } , s ; . . . ; h t − , { h t − } , s t − ) , ˆ d t − = (ˆ h , { ˆ h } , s ; . . . ; ˆ h t − , { ˆ h t − } , s t − ) ∈DC t − ( A t ) . Then we have ρ t ( · , A t , s t | λh t − + (1 − λ ) d t − ) = ρ t ( · , A t , s t | ˆ λ ˆ h t − + (1 − ˆ λ ) ˆ d t − ) . In particular, ρ h t − t is well-defined. For this to hold it suffices that A t ∈ supp ( h At − ). Here the mixture operation for histories is valid for every pair of histories which share the samesub-history of objective states – the mixture operation only acts on the sub-history of acts and menus.Recall that mixture of menus is defined through the Minkowski sum. roof. Let k = max { n = 0 , . . . , t − h n (cid:54) = ˆ h n } .Suppose that k = −
1. This means that d t − = ˆ d t − . If λ i > ˆ λ i for i = 0 , . . . , t − i − th entry of λh t − + (1 − λ ) d t − can be rewritten as an appropriate mixtureof the i − th entry of ˆ λ ˆ h t − + (1 − ˆ λ ) ˆ d t − and ( A i , f i , s i ). If on the other hand λ i ≤ ˆ λ i for i = 0 , . . . , t − i − th entry of λh t − + (1 − λ ) d t − can be rewritten as anappropriate mixture of the i − th entry of ˆ λ ˆ h t − + (1 − ˆ λ ) ˆ d t − and ( A i , f i , s i ). Starting from i = 0 and using LHI and working our way up the index i = 0 , . . . , t − λh t − + (1 − ˆ λ ) ˆ d t − with its correspondingentry from λh t − + (1 − λ ) d t − . This shows the result for the case k = − k ≤ m − ≤ m ≤ t −
1. We show that the claim still holds for k = m . Define the followingobjects. B m = 12 A m + 12 { h m } , ˆ B m = 12 A m + 12 { ˆ h m } , r m = 12 f m + 12 h m , ˆ r m = 12 f m + 12 ˆ h m g n = 12 h n + 12 l n , ˆ g n = 12 ˆ h n + 12 l n , for soon to be specified l n , n = 1 , . . . , t −
1. Namely, define l n recursively so that theysatisfy λ n A n + (1 − λ n ) { g n } , ˆ λ n A n + (1 − ˆ λ n ) { ˆ g n } , A n + 12 { h n } , A n + 12 ˆ { g n } , { g n } ∈ supp ( l An − ) . Finally augment the constant act l m − so that23 B m + 13 { ˆ g m } ,
23 ˆ B m + 13 { g m } , { g m } + 12 { ˆ g m } ∈ supp ( l Am − ) . Denote c t − := ( g n , { g n } , s n ) t − n =0 and ˆ c t − := (ˆ g n , { ˆ g n } , s n ) t − n =0 both in DC t − . Notethat we have λh t − + (1 − λ ) c t − , ˆ λh t − + (1 − ˆ λ )ˆ c t − ∈ H t − ( A t ) by construction. Also,the last entry at which c t − and ˆ c t − differ is m . Thus by repeated application of LHIwe can replace λh t − + (1 − λ ) d t − by λh t − + (1 − λ ) c t − and ˆ λh t − + (1 − ˆ λ ) ˆ d t − byˆ λh t − + (1 − ˆ λ )ˆ c t − . c t − , ˆ c t − and also satisfy the following relations.( a ) : 12 h t − + 12 d t − , h t − + 12 ˆ d t − ∈ H t − ( A t ) , ( b ) : 23 B m + 13 { ˆ h m } , { h m + 12 ˆ H m } ∈ supp ( h Am − ) , ( c ) : 23 ˆ B m + 13 { h m } , { h m + 12 ˆ H m } ∈ supp (ˆ h Am − ) . These imply immediately( d ) : (cid:18) B m + 13 { ˆ g m } , r m + 13 { ˆ g m } (cid:19) = (cid:18)
23 ˆ B m + 13 { g m } ,
23 ˆ r m + 13 { g m } (cid:19) = (cid:18) A m + 23 { h m + 12 ˆ h m } , f m + 23 ( 12 h m + 12 ˆ h m ) (cid:19) . The argument is the same as in the proof of Lemma 15 in [Frick, Iijima, Strzalecki ’17]. It is based onthe fact that when mixing a history h t − with a degenerate history from D t − , then the sets of maximizers N ( A i , f i ) doesn’t change. (cid:18) ( 12 h t − + 12 c t − ) − m , (cid:18) B m + 13 { ˆ g m } , r m + 13 { ˆ g m } , s m (cid:19)(cid:19) and (cid:18) ( 12 h t − + 12 c t − ) − m , (cid:18)
23 ˆ B m + 13 { g m } ,
23 ˆ r m + 13 { g m } , s m (cid:19)(cid:19) are in H t − ( A t ). Moreover, (d) implies that the first history is an entry-wise mixture of h t − with e t − = ( c t − − m , { h m + ˆ h m ) } , h m + ˆ h m , s m ), whereas the second is an entry-wisemixture of ˆ c t − with ˆ e t − = ( ˆ d t − − m , { h m + ˆ h m ) } , h m + ˆ h m , s m ).The base case of the induction ( k = −
1) gives ρ t ( · ; A t , s t | λh t − + (1 − λ ) c t − ) = ρ t ( · ; A t , s t | h t − + 12 t − c t − )and ρ t ( · ; A t , s t | ˆ λh t − + (1 − ˆ λ )ˆ c t − ) = ρ t ( · ; A t , s t | h t − + 12 t − ˆ c t − ) . But note that the entry where e t − , ˆ e t − first differ is strictly less than m . Hence applyingthe inductive hypothesis we have ρ t (cid:18) · ; A t , s t (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ( 12 h t − + 12 c t − ) − m , (cid:18) B m + 13 { ˆ g m } , r m + 13 { ˆ g m } , s m (cid:19)(cid:19)(cid:19) = ρ t (cid:18) · ; A t , s t (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ( 12 h t − + 12 ˆ c t − ) − m , (cid:18)
23 ˆ B m + 13 { g m } ,
23 ˆ r m + 13 { g m } , s m (cid:19)(cid:19)(cid:19) . Combining this together with the implication from the base case we get the result.In the next Lemma we show that the extended aSCF satisfies the formula in DR-SEU2.
Lemma 4.
Suppose that we have an aSCF ρ which has a DR-SEU representation as inDefinition 20 till some period T ∈ N . Then the extended version of ρ as in Definition 24will satisfy DR-SEU 2, i.e. for all t ≤ T, ∀ f (cid:48) t , A (cid:48) t and h t − = ( A , f , s ; . . . ; A t − , f t − , s t − ) and f t , A t we have ρ t ( s t , f t , A t | h t − ) = (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:104)(cid:81) t − k =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) (cid:105) · ψ θt − t ( θ t ) τ πqu ( θt ) ( f t ,A t ) q t ( s t ) (cid:80) πs ( θ ,...,θt − s ,...,st − (cid:81) t − k =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) . Proof. If h t − ∈ H t − ( A t ) then the claim follows directly from DR-SEU2. Assume thusthat h t − (cid:54)∈ H t − ( A t ) and take d t − = ( { h } , h , s ; . . . ; { h t − } , h t − , s t − ) ∈ DC t − with d t − ∈ H t − ( A t ) and compatible with the sub-history of objective states so that accordingto Definition 24 we can define for some λ ∈ (0 , ρ t ( f t , A t , s t | h t − ) := ρ t ( f t , A t , s t | λh t − + (1 − λ ) d t − ) . Note that(1) the formula depends on the menus and acts chosen only through the tiebreakers τ . 462) d t − ∈ D t − implies that for all s ≤ tf s ∈ M ( M ( A s ; q s , u s ) , p s , w s ) ⇐⇒ λf s + (1 − λ ) h s ∈ M ( M ( λA s + (1 − λ ) { h s } ; q s , u s ) , p s , w s ) .
1) and 2) imply immediately that for all s ≤ tτ q s ,u s ( f s , A s ) = τ q s ,u s ( λf s + (1 − λ ) h s , λA s + (1 − λ ) { h s } ) . From here the result follows from applying DR-SEU 2 to the history λh t − +(1 − λ ) d t − .We define Θ( h t − ) ⊂ Θ t − as the set of states θ t − consistent with h t − in the sense ofDefinition 22. Lemma 5. [Pendant to Lemma 14 in [Frick, Iijima, Strzalecki ’17]] Fix t ∈ { , . . . , T } and suppose that we have a DR-SEU representation up to time t . Take any h t − =( A , f , s ; . . . ; ( A t − , f t − , s t − ) ∈ H t − and A t ∈ A t . Then the following are equivalent.A. A t ∈ A ∗ t ( h t − ) .B. For each θ t − ∈ Θ( h t − ) and ( q t , u t ) ∈ π qu (cid:16) supp ( ψ θ t − t ) (cid:17) we have | M ( A t ; q t , u t ) | = 1 . Proof.
From A. to B.:
We prove the contrapositive. Suppose that there is θ t − ∈ Θ( h t − )and ( q t , u t ) ∈ π qu ( supp ( ψ θ t − t )) with | M ( A t ; q t , u t ) | >
1. Pick any f t ∈ M ( A t ; q t , u t ) with τ q t ,u t ( f t , A t ) >
0. Since u t is non-constant by DR-SEU 1, we can find lotteries ∆( X t ) with u t ( r ) < u t (¯ r ). Fix a sequence α n ∈ (0 ,
1) with α n → f nt = α n δ r + (1 − α n ) f t aswell as g nt = α n δ r + (1 − α n ) g t and ¯ g nt = α n δ ¯ r + (1 − α n ) g t for all g t ∈ A t \ { f t } . Let B nt = { g nt : g t ∈ A t \{ f t }} and ¯ B nt = { ¯ g nt : g t ∈ A t \{ f t }} . Finally let B nt = B nt ∪ ¯ B nt . Thenwe have B nt → m A t \ { f t } and f nt → m f t . Furthermore, since | M ( A t ; q t , u t ) | > g t ∈ A t \ { f t } such that q t · u t (¯ g nt ) > q t · u t ( f nt ). This implies τ q t ,u t ( f nt , B nt ∪ { f nt } ) = 0.Furthermore, note that for ( q (cid:48) t , u (cid:48) t ) ∈ π qu (Θ t ) \ { ( q t , u t ) } we always have N ( M ( A t ; q (cid:48) t , u (cid:48) t ); f t ) = N ( M ( B nt ∪ { f nt } ; q (cid:48) t , u (cid:48) t ); f nt ) ⊇ N ( M ( B nt ∪ { f nt } ; q (cid:48) t , u (cid:48) t ); f nt ) , which implies τ q (cid:48) t ,u (cid:48) t ( f t , A t ) ≥ τ q (cid:48) t ,u (cid:48) t ( f nt , B nt ∪{ f nt } ) for all n . Letting pred ( θ t − ) = ( θ , . . . , θ t − )Lemma 4 implies that for all n and all s t ∈ S t ρ t ( f t , A t , s t | h t − ) − ρ t ( f nt , B nt ∪ { f nt } , s t | h t − ) = (cid:80) πs ( θ (cid:48) ,...,θ (cid:48) t )=( s (cid:48) ,...,s (cid:48) t ) (cid:20)(cid:81) t − k =0 ψ θ (cid:48) k − k ( θ (cid:48) k ) τ πqu ( θ (cid:48) k ) ( f k ,A k ) (cid:21) · ψ θ (cid:48) t − t ( θ (cid:48) t ) (cid:16) τ πqu ( θ (cid:48) t ) ( f t ,A t ) − τ πqu ( θ (cid:48) t ) ( f nt ,B nt ∪{ f nt } ) (cid:17)(cid:80) πs ( θ (cid:48) ,...,θ (cid:48) t − s (cid:48) ,...,s (cid:48) t − (cid:81) t − k =0 ψ θ (cid:48) k − k ( θ (cid:48) k ) τ πqu ( θ (cid:48) t ) ( f t ,A t ) ≥ (cid:81) t − k =0 ψ θ k − k ( θ k ) τ π qu ( θ t ) ( f t , A t ) (cid:80) π s ( θ (cid:48) ,...,θ (cid:48) t − )=( s (cid:48) ,...,s (cid:48) t − ) (cid:81) t − k =0 ψ θ (cid:48) k − k ( θ (cid:48) k ) τ π qu ( θ (cid:48) t ) ( f t , A t ) > . The last line doesn’t depend on n so we getlim sup n →∞ ρ t ( f nt , B nt ∪ { f nt } , s t | h t − ) < ρ t ( f t , A t , s t | h t − ) . Note that we need Lemma 4 here because the history h t − is not assured to lead to B nt ∪ { f nt } withpositive probability.
47y Definition 12 we have A t (cid:54)∈ A ∗ t ( h t − ). From B. to A.:
Suppose A t satisfies B. Consider any f t ∈ A t , f nt → m f t , B nt → m A t \ { f t } .Consider a θ t − ∈ Θ( h t − ) and ( q t , u t ) ∈ π qu ( supp ( ψ θ t − t )).By 2. we either have M ( A t ; q t , u t ) = { f t } or f t (cid:54)∈ M ( A t ; q t , u t ). In the former case q t · u t ( f t ) > q t · u t ( g t ) for all A t (cid:51) g t (cid:54) = f t . By linearity we have q t · u t ( f nt ) > q t · u t ( g nt ) forall g nt ∈ B nt for all n large enough.This implies τ q t ,u t ( f t , A t ) = lim n τ q t ,u t ( f nt , B nt ∪ { f nt } ) = 1. In the case that f t (cid:54)∈ M ( A t ; q t , u t ) we have similarly q t · u t ( f t ) < q t · u t ( g t ) for some A t (cid:51) g t (cid:54) = f t . But thenlinearity implies τ q t ,u t ( f t , A t ) = lim n τ q t ,u t ( f nt , B nt ∪ { f nt } ) = 0.Overall, for all θ t − ∈ Θ( h t − ) and ( q t , u t ) ∈ π qu ( supp ( ψ θ t − t )) it holds τ q t ,u t ( f t , A t ) =lim n τ q t ,u t ( f nt , B nt ∪ { f nt } ). By looking at the formula in Lemma 4 we see that this impliesfor all s t ∈ S t and all n large enough ρ t ( f nt , B nt ∪ { f nt } , s t | h t − ) = ρ t ( f t , A t , s t | h t − ) . This finishes the proof.Before continuing, we register the piece of notation for an arbitrary f t ∈ F t : supp Z ( f t ) := ∪ q ∈ supp ( f t ) supp ( q ). Lemma 6. [Pendant to Lemma 17 in [Frick, Iijima, Strzalecki ’17].] Suppose we have aDR-SEU representation till time T . Fix any θ t − ∈ Θ t − , separating history h t − for θ t − and A t ∈ A t . Then there exists a sequence A nt → m A t with A nt ∈ A ∗ t ( h t − ) . Moreover,given a ( q (cid:48) t , u (cid:48) t ) ∈ π qu ( supp ( ψ θ t − t )) and f t ∈ M ( A t ; q t , u t ) we can ensure in this construc-tion that there is f nt ( q (cid:48) t , u (cid:48) t ) ∈ A nt with f nt ( q (cid:48) t , u (cid:48) t ) → m f t such that QU θ t − ( A nt , f nt ( q (cid:48) t , u (cid:48) t ) , s t ) = { ( q (cid:48) t , u (cid:48) t ) } for all s t ∈ supp ( q (cid:48) t ) . Proof.
Let QU ( θ t − ) := π qu ( supp ( ψ θ t − t )). By Definition 20 there exists a finite set Y t ⊆ X t such that (i) for any ( q t , u t ) ∈ QU ( θ t − ), u t is non-constant over Y t ; (ii) forany distinct ( q t , u t ) (cid:54) = ( q (cid:48) t , u (cid:48) t ), both in supp ( ψ θ t − t ), ( q t , u t ) (cid:54) = ( q (cid:48) t , u (cid:48) t ) on F t ( Y t ) and (iii) ∪ f t ∈ A t supp Z ( f t ) ⊆ Y t .By (i) and (ii) and Lemma 1 we can find a separating menu C t = { f t ( q t , u t ) : ( q t , u t ) ∈ QU ( θ t − ) } , i.e. such that for all ( q t , u t ) ∈ QU ( θ t − ) we have M ( C t ; q t , u t ) = { f t ( q t , u t ) } .Pick z ( q t , u t ) ∈ argmax y ∈ Y t u t ( y ) for all ( q t , u t ) ∈ QU ( θ t − ), write by a small abuse ofnotation again z ( q t , u t ) for the constant act paying out z ( q t , u t ) with probability one ateach state of the world and define the constant act b t = | Y t | (cid:80) y ∈ Y t δ y ∈ ∆( Y t ). Again, wedenote by b t with a small abuse of notation the constant act which pays out the lottery b t in each state of the world.By (i) we have q t · u t ( z ( q t , u t )) > q t · u t ( b t ) for all ( q t , u t ) ∈ QU ( θ t − ). If we then defineˆ f t ( q t , u t ) = αf t ( q t , u t ) + (1 − α ) z ( q t , u t ) we still have q t · u t ( ˆ f t ( q t , u t )) > q t · u t ( b t ) if wechoose α ∈ (0 ,
1) small enough. This is because of the ‘finiteness’ of all the data goinginto the problem. Note also, that if we define ˆ C t = { ˆ f t ( q t , u t ) : ( q t , u t ) ∈ QU ( θ t − ) } westill have M ( ˆ C t ; q t , u t ) = { ˆ f t ( q t , u t ) } .Now pick for each ( q t , u t ) ∈ QU ( θ t − ) a f t ( q t , u t ) ∈ M ( A t ; q t , u t ). To also provethe ‘moreover’ part, pick f t ( q t , u t ) as required in the ‘moreover’ part. Fix a sequence (cid:15) n ∈ (0 ,
1) going to zero. For each n and ( q t , u t ) ∈ QU ( θ t − ) := supp ( ψ θ t − t ) let f nt ( q t , u t ) = Note that because of Remark 3 this is w.l.o.g. Recall this denotes the set of acts whose images are contained in ∆( Y t ). − (cid:15) n ) f t ( q t , u t ) + (cid:15) n ˆ f t ( q t , u t ). Moreover, for each g t ∈ A t define g nt = (1 − (cid:15) n ) g t + (cid:15) n b t .Finally, take A nt = { f nt ( q t , u t ) : ( q t , u t ) ∈ QU ( θ t − ) } ∪ { g nt : g t ∈ A t } . Note that A nt → m A t . Finally, note that by construction we have M ( A nt ; q t , u t ) = { f nt ( q t , u t ) } .Since by Remark 3, part 2) θ t − is the only state consistent with h t − Lemma 5 andthe construction here imply A nt ∈ A ∗ t ( h t − ), as required. The last required property, i.e. QU θ t − ( A nt , f nt ( q t , u t ) , s t ) = { ( q t , u t ) } for any s t ∈ supp ( q t ) is true by construction.The next result proves the existence of separating histories. Lemma 7. [Pendant to Lemma 2 in [Frick, Iijima, Strzalecki ’17].] For any θ t ∈ Θ t with pred ( θ t ) = ( θ (cid:48) , . . . , θ (cid:48) t − ) there always exists a separating history. Proof.
By Lemma 1 and DR-SEU 1 we can construct for Θ a menu B = { f π qu ( θ )0 : θ ∈ Θ } ∈ A such that QU ( B , f π qu ( θ )0 , π s ( θ )) = { π qu ( θ ) } for all θ ∈ Θ . Proceeding in-ductively, again using Lemma 1 and DR-SEU 1, we can find a menu B k ( θ k − ) = { f π qu ( θ k ) k : π qu ( θ k ) ∈ π qu ( supp ( ψ θ k − k )) } for all θ k − ∈ Θ k − such that (!) QU θ k − ( B k ( θ k − ) , f π qu ( θ k ) k , π s ( θ k )) = { π qu ( θ k ) } for all π qu ( θ k ) ∈ π qu ( supp ( ψ θ k − k )).Moreover, we can assume that B k +1 ( θ k ) ∈ supp A ( f π qu ( θ k ) k ) for all k = 0 , . . . , t − θ k ∈ Θ k by mixing each f π qu ( θ k ) k with the constant act delivering ( z, B k +1 ( θ k )) for a z ∈ Z fixed throughout. If the mixing puts small enough probability on the constant act inquestion, then (!) is preserved.This implies in particular that h t := ( B , f θ (cid:48) , s (cid:48) ; . . . ; B t ( θ (cid:48) t − ) , f π qu ( θ t ) t , π s ( θ t )) ∈ H t .Moreover, since QU θ (cid:48) k − ( B k ( θ (cid:48) k − ) , f π qu ( θ (cid:48) k ) k , π s ( θ (cid:48) k )) = { π qu ( θ (cid:48) k ) } , it follows by Lemma 2 thatonly the state θ (cid:48) k is consistent with h k for k = 0 , . . . , t . Additionally, by construction forall ( q k , u k ) ∈ π qu ( supp ( ψ θ (cid:48) k − k )) we have M ( B k ( θ (cid:48) k − ); q k , u k ) = { f q k ,u k k } . Hence, by Lemma5 we have B k ( θ (cid:48) k − ) ∈ A ∗ k ( h k − ). Since this holds for all k we have overall h t ∈ H ∗ t . Insummary it follows that h t is a separating history for θ t . D Proof of the main result in the dynamic setting
Here we prove the representation theorem in its AS-version for DR-SEU. The proofs forthe special cases Evolving SEU and Gradual Learning are in the online appendix.
D.1 Proof for DR-SEU
D.1.1 Sufficiency
We proceed by induction on t ≤ T . First consider t = 0. Because of the axioms and X being a separable metric space we have the existence of an AS-version R-SEU rep-resentation for ρ on H . Depending on the version of the representation we are lookingat, i.e. whether CIB or NUC is satisfied, we also have the respective property for therepresentation at time t = 0. Set SEU = { π qu ( θ ) : θ ∈ Θ } .Suppose next that we have the representation for all t (cid:48) ≤ t . We now construct therepresentation for t + 1. 49o this end, pick a subjective state θ t ∈ Θ t and pick an arbitrary separating history h t ( θ t ) for θ t . This exists by Lemma 7. Define ρ θ t t +1 ( · , A t +1 , s t +1 ) = ρ ( · , A t +1 , s t +1 | h t ( θ t )) . Here we use for the right-hand side the extended aSCF, which is well-defined as perLemma 24. As per axioms we get a representation ρ θ t t +1 ( f t +1 , A t +1 , s t +1 ) = (cid:88) ( q t +1 ,u t +1 ) ∈ SEU θtt +1 ψ θ t t +1 ( q t +1 , u t +1 , s t +1 ) τ ( q t +1 ,u t +1 ) ( f t +1 , A t +1 ) . (18)Again, depending on the respective property required by the axioms on beliefs, CIBor NUC, the kernel ψ θ t t +1 satisfies the respectively required property in DR-SEU 1.We set SEU t +1 = (cid:116) θ t SEU θ t t +1 and define Θ t +1 accordingly by the collection of all( q t +1 , u t +1 , s t +1 ) such that ( q t +1 , u t +1 ) ∈ SEU t +1 and s t +1 ∈ supp ( q t +1 ). We extend themeasures ψ θ t t +1 to all of SEU t +1 by setting them to zero outside of SEU θ t t +1 .We see that DR-SEU 1 is satisfied by Definition.With this definition we can rewrite (18) as ρ θ t t +1 ( f t +1 , A t +1 , s t +1 ) = (cid:88) θ t +1 ∈ Θ t +1 ψ θ t t +1 ( θ t +1 ) τ π qu ( θ t +1 ) ( f t +1 , A t +1 ) . Before showing DR-SEU 2, we show that the definition of ρ θ t t +1 doesn’t depend on theparticular separating history for θ t picked in its definition. Lemma 8.
Fix any θ t ∈ Θ t with pred ( θ t ) = ( θ , . . . , θ t − ) . Suppose h t = ( f , A , s ; . . . ; f t , A t , s t ) ∈H t satisfies QU θ k − ( A k , f k , s k ) = { π qu ( θ k ) } for all k = 0 , , . . . , t . Then for any A t +1 ∈A t +1 and s t +1 ∈ S t +1 it holds ρ t +1 ( · , A t +1 , s t +1 | h t ) = ρ θ t t +1 ( · , A t +1 , s t +1 ) . Proof.
Step 1.
Let ˜ h t = ( ˜ f , ˜ A , ˜ s ; . . . ; ˜ f t , ˜ A t , ˜ s t ) ∈ H t denote the separating history for θ t used to define ρ θ t t +1 . We first prove the Lemma under the assumption that h t ∈ H ∗ t , i.e. that h t is itself a separating history for θ t . Note that since h t , ˜ h t ∈ H ∗ t and QU θ k − ( A k , f k , s k ) = QU θ k − ( ˜ A k , ˜ f k , ˜ s k ) = { ( q k , u k ) } Lemma 5 implies that M ( A k , q k , u k ) = { f k } and M ( ˜ A k , q k , u k ) = { ˜ f k } .Pick lotteries ( r , . . . , r t ) ∈ ∆( X ) × · · · × ∆( X t ) such that A t +1 ∈ supp ( r At ) and sothat for all k = 0 , . . . , t − { B k +1 , ˜ B k +1 , B k +1 ∪ ˜ B k +1 } ⊂ supp ( r Ak ) , where B l = A l + { ˜ f l } + { r l } and ˜ B l = ˜ A l + { f l } + { r l } for l = 0 , . . . , t . Here wehave identified lotteries with their respective constant acts. Define also the mixture act g l = f l + ˜ f l + r l .Linearity of SEU functionals implies QU θ k − ( B k , g k , s k ) = QU θ k − ( ˜ B k , g k , ˜ s k ) = QU θ k − ( ˜ B k ∪ B k , g k , ˜ s k ) = { ( q k , u k ) } . The symbol (cid:116) means we join them into a union of disjoint sets, i.e. if a SEU ( q, u ) appears in thesupport of two distinct θ t , θ (cid:48) t then we count it twice. Note that in the last equality it is irrelevant whether we write ˜ s k or s k because of the argument inthe first paragraph of the first step of the proof.
50e also have M ( B k , q k , u k ) = M ( ˜ B k , q k , u k ) = M ( ˜ B k ∪ B k , q k , u k ) = { g k } . This implies that for all k = 0 , . . . , t and ( q (cid:48) k , u (cid:48) k ) ∈ π qu (cid:16) supp ( ψ θ k − k − ) (cid:17) we have τ q (cid:48) k ,u (cid:48) k ( g k , B k ) = τ q (cid:48) k ,u (cid:48) k ( g k , ˜ B k ) = τ q (cid:48) k ,u (cid:48) k ( g k , ˜ B k ∪ B k ) = (cid:40) , if π qu ( θ k ) = π qu ( θ (cid:48) k )0 , if π qu ( θ k ) (cid:54) = π qu ( θ (cid:48) k ) . By DR-SEU 2 of the inductive hypothesis it follows for all k = 0 , . . . , t − ψ θ t − t ( q t , u t , s t ) = ρ t ( g t , ˜ B t , s t | ˜ B , g , s ; . . . ; ˜ B t − , g t − , s t − )= ρ t ( g t , B t , s t | B , g , s , . . . , B t − , g t − , s t − )= ρ t ( g t , ˜ B t ∪ B t , s t | ˜ B , g , s , . . . , ˜ B k ∪ B k , g k , s k , . . . , ˜ B t − ∪ B t − , g t − , s t − )= ρ t ( g t , ˜ B t ∪ B t , s t | B , g , s , . . . , ˜ B k ∪ B k , g k , s k , . . . , ˜ B t − ∪ B t − , g t − , s t − ) . Note that in these relations we could have replaced everywhere s k with ˜ s k , since both arein the support of q k by the definition of the operator QU θ k − .Since all the histories considered above are compatible with A t +1 we apply CHI recur-sively to get ρ t +1 ( · , A t +1 , s t +1 | B , g , s ; . . . ; B t , g t , s t ) = ρ t +1 ( · , A t +1 , s t +1 | ˜ B ∪ B , g , s ; . . . ; ˜ B t ∪ B t , g t , s t )= ρ t +1 ( · , A t +1 , s t +1 | ˜ B , g , s ; . . . ; ˜ B t , g t , s t ) . (19)Here s t +1 ∈ S t +1 is arbitrary. Use LHI and Lemma 3 (well-definiteness of the extendedaSCF) to get ρ t +1 ( · , A t +1 , s t +1 | h t ) = ρ t +1 ( · , A t +1 , s t +1 | B , g , s ; . . . ; B t , g t , s t ) ,ρ t +1 ( · , A t +1 , s t +1 | ˜ h t ) = ρ t +1 ( · , A t +1 , ˜ s t +1 | ˜ B , g , ˜ s ; . . . ; ˜ B t , g t , ˜ s t ) . (20)Finally, we put (19) and (20) together to get ρ t +1 ( · , A t +1 , s t +1 | h t ) = ρ t +1 ( · , A t +1 , s t +1 | ˜ h t ) . This establishes the proof for the case that h t ∈ H ∗ t . Step 2.
Now suppose that h t (cid:54)∈ H t ∗ . Take any sequence of (valid) histories h t,n ∈ H t ∗ with h t,n → m h t with h t,n = ( A n , f n , s n ; . . . ; A nt , f nt , s nt ) for each n . Existence is ensured bythe Axiom of History Continuity. Claim.
For all large n we have QU θ k − ( A nk , f nk , s nk ) = { π qu ( θ k ) } for all k = 0 , . . . , t . Proof of Claim.
Take some subsequence ( h t,n l ) l ≥ of ( h t,n ) n ≥ . We have ρ k ( f n l k , A n l k , s n l k | h k − ,n l ) > k = 0 , . . . , t by the definition of histories. Assume that by DR-SEU 2 for k ≤ t we can find θ (cid:48) t,n l ∈ Θ t with pred ( θ (cid:48) t,n l ) = ( θ (cid:48) ,n l , . . . , θ (cid:48) t − ,n l ) and ( θ (cid:48) ,n l , . . . , θ (cid:48) t,n l ) (cid:54) =( θ ,n l , . . . , θ t,n l ) such that π qu ( θ (cid:48) k,n l ) ∈ QU θ (cid:48) k − ,nl ( f n l k , A n l k , s n l k ) for all k = 0 , . . . , t . Since S ×· · · × S t is finite, by choosing an appropriate subsequence we can assume ( θ (cid:48) ,n l , . . . , θ (cid:48) t,n l ) =( θ (cid:48) , . . . , θ (cid:48) t ) (cid:54) = ( θ , . . . , θ t ) for all l . Pick the smallest k such that θ (cid:48) k (cid:54) = θ k and pick any g k ∈ A k . Since A n l k → m A k we can find g n l k ∈ A n l k with g n l k → m g k . Since we have for51ll l that π qu ( θ (cid:48) k ) ∈ QU θ (cid:48) k − ( f n l k , A n l k , s n l k ), so π qu ( θ (cid:48) k )( f n l k ) ≥ π qu ( θ (cid:48) k )( g n l k ) and thus also π qu ( θ (cid:48) k )( f k ) ≥ π qu ( θ (cid:48) k )( g k ) by linearity of the SEU represented by π qu ( θ (cid:48) k ).Moreover, by choice of k we have π qu ( θ (cid:48) k ) ∈ π qu ( supp ( ψ θ (cid:48) k − k − )) = π qu ( supp ( ψ θ k − k − )). Butthe fact that QU θ k − ( f k , A k , s k ) = { π qu ( θ k ) } implies that π qu ( θ (cid:48) k ) = π qu ( θ k ) for all k . Wehave thus shown that each subsequence ( h t,n l ) l ≥ of ( h t,n ) n ≥ has a subsequence with theproperty required by the claim. A simple argument by contradiction now establishes theclaim. End of Proof of Claim.
The Claim establishes that for all large enough n , h t,n satisfies the assumption of theLemma. Since h t,n ∈ H ∗ t , Step 1 then shows that ρ t +1 ( f t +1 , A t +1 , s t +1 | h t,n ) = ρ θ t t +1 ( f t +1 , A t +1 , s t +1 )for all large enough n and all f t +1 , s t +1 . History Continuity now allows to close the argu-ment and prove that ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ) = ρ θ t t +1 ( f t +1 , A t +1 , s t +1 ) . As a next step we establish that ρ t +1 ( ·| h t ) is a weighted average of the ρ θ t t +1 for θ t consistent with h t . Lemma 9. [Pendant of Lemma 4 in [Frick, Iijima, Strzalecki ’17]] For any f t +1 ∈ A t +1 and h t = ( A , f , s ; . . . ; A t , f t , s t ) ∈ H t ( A t +1 ) we have ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ) = (cid:80) πs ( θ ,θ ,...,θt )=( s ,...,st ) (cid:81) tk =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) · ρ θtt +1 ( f t +1 ,A t +1 ,s t +1 ) (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:81) tk =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) . Proof.
Let ( θ t , . . . , θ mt ) be the set of elements from Θ t that are consistent with history h t , as defined in Definition 22. For each j = 1 , . . . , m let ˆ h t ( j ) = ( B j , f j , s ; . . . ; B jt , f jt , s t )be a separating history for θ jt . Note that such a history exists because under θ jt and itspredecessors the ‘right’ sub-history of objective states ( s , . . . , s t ) has positive probability.We can assume w.l.o.g. that for each k = 1 , . . . , t in all objective states s t − thereis a positive probability (albeit possibly small) for ( z, A k + B jk ) for some z . This canbe achieved by mixing with constant acts. Thus, w.l.o.g. we can ensure that h t ( j ) := h t + ˆ h t ( j ) ∈ H t ( A t +1 ).Note first that it holds for all j = 1 , . . . , mρ ( h t ( j )) = t (cid:89) k =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( f k , A k ) . (21)This follows from the following calculation. ρ ( h t ( j )) = t (cid:89) k =0 ρ k ( 12 f k + 12 f jk ; 12 A k + 12 B jk , s k | h k ( j ))= (cid:88) ( θ (cid:48) ,...,θ (cid:48) t ) t (cid:89) k =0 ψ θ (cid:48) k − k ( θ (cid:48) k ) τ π qu ( θ (cid:48) k ) ( 12 f k + 12 f jk , A k + 12 B jk )= t (cid:89) k =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( 12 f k + 12 f jk , A k + 12 B jk )= t (cid:89) k =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( f k , A k ) . h t ( j ) is a separating historyfor θ jt (see Lemma 5). Since θ jt is consistent with h t it follows ψ θ k − k ( θ k ) · τ π qu ( θ jk ) ( f k , A k ) > k = 0 , . . . , t and therefore also:for every π qu ( θ (cid:48) k ) ∈ π qu ( supp ( ψ θ jk − k )), τ π qu ( θ (cid:48) k ) ( f k + f jk , A k + B k ) > π qu ( θ (cid:48) k ) = π qu ( θ jk ). This yields the third equality above.Define now H t = { h t ( j ) : j = 1 , . . . , m } ⊂ H t ( A t +1 ). By repeated application of LHIwe have that ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ) = ρ t +1 ( f t +1 , A t +1 , s t +1 | H t ) . (22)Moreover, we have that ρ t +1 ( f t +1 , A t +1 , s t +1 | H t ) = (cid:80) mj =1 ρ ( h t ( j )) ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ( j )) (cid:80) mj =1 ρ ( h t ( j ))= (cid:80) mj =1 (cid:81) tk =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( f k , A k ) · ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ( j )) (cid:80) mj =1 (cid:81) tk =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( f k , A k )= (cid:80) mj =1 (cid:81) tk =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( f k , A k ) ρ θ jt t +1 ( f t +1 , A t +1 , s t +1 ) (cid:80) mj =1 (cid:81) tk =0 ψ θ jk − k ( θ jk ) τ π qu ( θ jk ) ( f k , A k )= (cid:80) π s ( θ ,...,θ t )=( s ,...,s t ) (cid:81) tk =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) · ρ θ t t +1 ( f t +1 , A t +1 , s t +1 ) (cid:80) π s ( θ ,...,θ t )=( s ,...,s t ) (cid:81) tk =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) . (23)Here the first equality holds by definition of choice conditional on a set of histories.The second follows from (21). Note that h t ( j ), being a separating history for θ jt andconsistent with h t , implies QU θ jk ( f k + f jk , A k + B jk , s k ) = { π qu ( θ jk ) } for each k . Hence,Lemma 8 implies that ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ( j )) = ρ θ jt t +1 ( f t +1 , A t +1 , s t +1 ). This yields thethird equality.Finally, note that if ( θ , . . . , θ t ) ∈ Θ × · · · × Θ t has ( θ , . . . , θ t ) (cid:54) = ( θ j , . . . , θ jt ) for all j ,then either θ t (cid:54)∈ { θ jt : j = 1 , . . . , m } or θ t = θ jt for some j but pred ( θ jt ) (cid:54) = ( θ , . . . , θ t − ). Ineither case we have (cid:81) tk =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) = 0 by the inductive step up to t . Thisjustifies the last equality in (23).Combining (22) and (23), we obtain the desired conclusion.We show that our construction satisfies DR-SEU2 at step t + 1 as well. We recall therepresentation in (18) and combine it with Lemma 9 to get for any h t = ( A , f , s ; . . . ; A t , f t , s t ) ∈H t ( A t +1 ) ρ t +1 ( f t +1 , A t +1 , s t +1 | h t ) = = (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:81) tk =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) · (cid:16)(cid:80) θt +1 ψ θtt +1 ( θ t +1 ) τ πqu ( θt +1) ( f t +1 ,A t +1 ) (cid:17)(cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:81) tk =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) = (cid:80) π s ( θ ,...,θ t +1 )=( s ,...,s t +1 ) (cid:81) t +1 k =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) (cid:80) π s ( θ ,...,θ t )=( s ,...,s t ) (cid:81) tk =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) . .1.2 Necessity Suppose that ρ admits a DR-SEU representation as in Definition 20. From the represen-tation in DR-SEU 2 and from Lemma 3 we have that for a fixed h t ∈ H t the static aSCFrule ρ t ( ·| h t ) satisfies the static axioms. Claim 1. ρ satisfies CHI. Proof.
Take any h t − = ( h t − − k , ( A k , f k , s k )) and ˆ h t − = ( h t − − k , ( B k , f k , s k )) with A k ⊆ B k and ρ k ( f k , A k , s k | h k − ) = ρ k ( f k , B k , s k | h k − ). From DR-SEU 2 this implies (cid:88) ( θ ,...,θ k ) (cid:32) k (cid:89) l =0 ψ θ l − l ( θ l ) τ π qu ( θ l ) ( f l , A l ) (cid:33) = (cid:88) ( θ ,...,θ k ) (cid:32) k (cid:89) l =0 ψ θ l − l ( θ l ) τ π qu ( θ l ) ( f l , B l ) (cid:33) . (24)It follows from τ π qu ( θ l ) ( f k , A k ) ≤ τ π qu ( θ l ) ( f k , B k ) that equality in (24) can hold if andonly if τ π qu ( θ l ) ( f k , A k ) = τ π qu ( θ l ) ( f k , B k ) whenever θ k is consistent with h k . This impliesimmediately due to DR-SEU 2 that ρ t ( ·| h t − ) = ρ t ( ·| ˆ h t − ) . Claim 2. ρ satisfies LHI. Proof.
Take any A t , s t and h t − = ( A , f , s ; . . . ; A t − , f t − , s t − ) ∈ H t − ( A t ) and H t − ⊆H t − ( A t ) of the form H t − = { h t − − k , ( λA k + (1 − λ ) B k , λf k + (1 − λ ) g k , s k )) : g k ∈ B k } forsome k < t, λ ∈ (0 ,
1) and B k = { g jk : j = 1 , . . . , m } ∈ A k . Let ˜ A k = λA k + (1 − λ ) B k andfor each j = 1 , . . . , m let ˜ f jk = λf k + (1 − λ ) g jk and ˜ h t − ( j ) = ( h t − − k , ( ˜ A k , ˜ f jk , s k )).By DR-SEU 2, for all f t we have ρ t ( f t , A t , s t | h t − ) = (cid:80) π s ( θ ,...,θ t − )=( s ,...,s t − ) (cid:81) tl =0 ψ θ l − l ( θ l ) τ π qu ( θ l ) ( f l , A l ) (cid:80) π s ( θ ,...,θ t − )=( s ,...,s t − ) (cid:81) t − l =0 ψ θ l − l ( θ l ) τ π qu ( θ l ) ( f l , A l ) , and by definition also ρ t ( f t , A t , s t | H t − ) = (cid:80) mj =1 ρ (˜ h t − ( j )) ρ t ( A t , f t , s t | ˜ h t − ( j )) (cid:80) mj =1 ρ (˜ h t − ( j )) . For each j = 1 , . . . , m DR-SEU 2 yields ρ t ( f t , A t , s t | ˜ h t − ( j )) = (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:16)(cid:81) tl =0 ,l (cid:54) = k ψ θl − l ( θ l ) τ πqu ( θl ) ( f l ,A l ) (cid:17) · ψ θk − k ( θ k ) τ πqu ( θk − ( ˜ f jk , ˜ A k ) (cid:80) πs ( θ ,...,θt − s ,...,st − (cid:16)(cid:81) t − l =0 ,l (cid:54) = k ψ θl − l ( θ l ) τ πqu ( θl ) ( f l ,A l ) (cid:17) · ψ θk − k ( θ k ) τ πqu ( θk ) ( ˜ f jk , ˜ A k ) , as well as ρ (˜ h t − ( j )) = t − (cid:89) l =0 ,l (cid:54) = k ρ l ( f l , A l , s l | ˜ h l − ) ρ k ( ˜ f jk , ˜ A k , s k | ˜ h k − )= (cid:88) π s ( θ ,...,θ t − )=( s ,...,s t − ) (cid:32) t − (cid:89) l =0 ,l (cid:54) = k ψ θ l − l ( θ l ) τ π qu ( θ l ) ( f l , A l ) (cid:33) · ψ θ k − k ( θ k ) τ π qu ( θ k ) ( ˜ f jk , ˜ A k ) .
54e put the last three formulas together and rearrange to obtain ρ t ( f t , A t , s t | H t − ) = (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:16)(cid:81) tl =0 ,l (cid:54) = k ψ θl − l ( θ l ) τ πqu ( θl ) ( f l ,A l ) (cid:17) · ψ θk − k ( θ k ) ( (cid:80) mj =1 τ πqu ( θk ) ( ˜ f jk , ˜ A k ) ) (cid:80) πs ( θ ,...,θt − s ,...,st − (cid:16)(cid:81) t − l =0 ,l (cid:54) = k ψ θl − l ( π qu ( θ l )) τ πqu ( θl ) ( f l ,A l ) π q ( θ l )( s l ) (cid:17) · ψ θk − k ( π qu ( θ k )) ( (cid:80) mj =1 τ πqu ( θk ) ( ˜ f jk , ˜ A k ) ) π q ( θ k )( s k ) . But note that for all θ k ∈ Θ k it holds m (cid:88) j =1 τ π qu ( θ k ) ( ˜ f jk , ˜ A k ) = m (cid:88) j =1 τ π qu ( θ k ) (cid:16) ( q (cid:48) , u (cid:48) ) ∈ ∆( S k ) × R X k : f jk ∈ M ( M ( ˜ A k ; π qu ( θ k )); ( q (cid:48) , u (cid:48) )) (cid:17) = (cid:88) g jk ∈ B k τ π qu ( θ k ) (cid:0) ( q (cid:48) , u (cid:48) ) ∈ ∆( S k ) × R X k : f k ∈ M ( M ( A k ; π qu ( θ k )); ( q (cid:48) , u (cid:48) )) , g jk ∈ M ( M ( B k ; π qu ( θ k )); ( q (cid:48) , u (cid:48) )) (cid:1) = τ π qu ( θ k ) (cid:0) ( q (cid:48) , u (cid:48) ) ∈ ∆( S k ) × R X k : f k ∈ M ( M ( A k ; π qu ( θ k )); ( q (cid:48) , u (cid:48) )) (cid:1) = τ π qu ( θ k ) ( f k , A k ) . By plugging this into the formula for ρ t ( f t , A t , s t | H t − ) we see that ρ t ( f t , A t , s t | h t − ) = ρ t ( f t , A t , s t | H t − ) . Claim 3. ρ satisfies History Continuity. Proof.
Fix any ( f t , A t , s t ) and h t − = ( f , A , s ; . . . ; f t − , A t − , s t − ) ∈ h t − . Let Θ t − ( h t − ) ⊆ Θ t − denote the set of period-( t −
1) states that are consistent with h t − . Define ρ θ t − t ( f t , A t , s t ) = (cid:80) θ t ψ θ t − t ( θ t ) τ π qu ( θ t ) ( f t , A t ). By Lemma 4 we have ρ t ( f t , A t , s t | h t − ) = (cid:80) π s ( θ ,...,θ t )=( s ,...,s t ) (cid:81) tk =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) (cid:80) π s ( θ ,...,θ t − )=( s ,...,s t − ) (cid:81) t − k =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f k , A k ) = (cid:80) πs ( θ ,...,θt )=( s ,...,st ) (cid:81) t − k =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) · (cid:80) θt ψ θt − t ( θ t ) τ πqu ( θt ) ( f t ,A t ) (cid:80) πs ( θ ,...,θt − s ,...,st − (cid:81) t − k =0 ψ θk − k ( θ k ) τ πqu ( θk ) ( f k ,A k ) . We see that ρ t ( f t , A t , s t | h t − ) ∈ co { ρ θ t − t ( f t , A t , s t ) : θ t − ∈ Θ t − ( h t − ) } . Fix any θ t − ∈ Θ t − ( h t − ). To prove the claim it suffices to show that ρ θ t − t ( f t , A t , s t ) ∈ { lim n ρ t ( f t , A t , s t | h t − n ) : h t − n → m h t − , h t − n ∈ H ∗ t − } . To this end, let pred ( θ t − ) = ( θ , . . . , θ t − ) and let ¯ h t − = ( B , g , s ; . . . ; B t − , g t − , s t − ) ∈H ∗ t − be a separating history for θ t − . By Lemma 6 for each k = 0 , . . . , t − A nk ∈ A ∗ k (¯ h k − ) and f nk ∈ A nk with f nk → m f k and QU θ k − ( A nk , f nk , s k ) = { π qu ( θ k ) } for all n and all k = 0 , . . . , t −
1. Working backwards from k = t − A nk and f nk with a mixture putting small weight on a constant act yielding ( z, A nk +1 )for some z so as to ensure that A nk +1 ∈ supp A ( f nk ( s k )), irrespective of s k ∈ S k . This canbe done maintaining the previous properties of A nk and f nk .By construction it follows h t − n = ( A n , f n , s ; . . . ; A nt − , f nt − , s t − ) ∈ H ∗ t − ( A t ) and thisis also a separating history for θ t − . 55y Lemma 4 the latter fact implies for each n that ρ t ( f t , A t , s t | h t − n )= (cid:80) θ t (cid:16)(cid:81) t − k =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f nk , A nk ) (cid:17) · ψ θ t − t ( θ t ) τ π qu ( θ t ) ( f t , A t ) (cid:81) t − k =0 ψ θ k − k ( θ k ) τ π qu ( θ k ) ( f nk , A nk )= (cid:88) θ t ψ θ t − t ( θ t ) τ θ t ( f t , A t )= ρ θ t − t ( f t , A t , s t ) . The desired claim follows since h t − n → m h t − . D.2 Proofs for the Comparative Statics part
D.2.1 Proof of Proposition 5
This is a trivial application of Lemma 24 in the online appendix.