CConservative Updating
Matthew Kovach ∗ February 2, 2021
Abstract:
This paper provides a behavioral analysis of conservatism in beliefs. I introducea new axiom, Dynamic Conservatism, that relaxes Dynamic Consistency when informationand prior beliefs “conflict.” When the agent is a subjective expected utility maximizer, Dy-namic Conservatism implies that conditional beliefs are a convex combination of the priorand the Bayesian posterior. Conservatism may result in belief dynamics consistent with con-firmation bias, representativeness, and the good news-bad news effect, suggesting a deeperbehavioral connection between these biases. An index of conservatism and a notion of com-parative conservatism are characterized. Finally, I extend conservatism to the case of anagent with incomplete preferences that admit a multiple priors representation.
Keywords:
Conservative updating, prior-bias, non-Bayesian updating, confirmation bias,representativeness, good news-bad news effect, multiple priors.
JEL:
D01, D81, D9. ∗ Department of Economics, Virginia Tech. E-mail: [email protected]. I would like to thank Aurélien Bail-lon, Jonathan Chapman, Federico Echenique, Bart Lipman, Pietro Ortoleva, Kota Saito, Marciano Siniscalchi,Gerelt Tserenjigmid, and Leeat Yariv for helpful feedback. This paper was previously titled
Sticky Beliefs: ACharacterization of Conservative Updating and is based on chapter 3 of my dissertation at Caltech. a r X i v : . [ ec on . T H ] J a n Introduction
Many papers in both economics and psychology have documented biases in belief updat-ing (see, for instance, Camerer (1995), Kahneman and Tversky (1972), El-Gamal and Grether(1995)). A recurrent finding is that people tend to exhibit conservatism (Phillips and Edwards(1966), Edwards (1982), Aydogan et al. (2017), and Möbius et al. (2014)). A conservative up-dater only partially incorporates new information into her beliefs; hence she puts too muchweight on her prior. Despite its prevalence, conservatism has yet to be behaviorally founded.I introduce a novel behavioral postulate, Dynamic Conservatism, to capture conservatismwithin the framework of conditional preferences over acts (see Savage (1954) and Anscombeand Aumann (1963)). This axiom weakens Dynamic Consistency to accommodate “prefer-ence stickiness.” Put loosely, Dynamic Conservatism allows for violations of Dynamic Con-sistency only when the initial preference and the information are in conflict. Further, I showthat conservative preferences are consistent with several well-known biases, including con-firmation bias, the representativeness heuristic, and the good news-bad news effect.For a deeper intuition behind Dynamic Conservatism, consider the following hypothet-ical of an agent’s reaction to information regarding climate change. Suppose this agent ini-tially favors the use of coal for power, yet she concedes that using alternative energy is betterif climate change is occurring. If she obtains information regarding the scientific consensusthat climate change is occurring, would she now support using alternative energy over coal?If she is dynamically consistent (i.e., Bayesian), then the answer is yes because she has al-ready conceded that alternative energy is better in that contingency. If she is conservative,she may still prefer coal. When there is a conflict between information (climate change news)and an agent’s initial preference (use coal), conservatism may result in a violation of DynamicConsistency. Dynamic Conservatism allows for such violations.Dynamic Conservatism restricts the agent so that she may violate Dynamic Consistencyonly in situations in which there is a conflict between the information and her initial prefer-ence. To illustrate, consider how our agent would feel if she had initially preferred alternativeenergy to coal. Now that the evidence for climate change is consistent with her initial pref-erence, she must continue to prefer alternative energy. That is, no matter how conservativeshe is, it would be absurd for her to (i) initially support alternative energy, (ii) acknowledgethat alternative energy is better if climate change is occurring, and then (iii) state that shesupports coal upon receipt of information regarding the consensus that climate change isoccurring. Dynamic Conservatism rules out reversals of this form.Suppose the agent is a subjective expected utility maximizer (SEU) and her prior beliefs2re given by a probability distribution µ . In this case, Dynamic Conservatism characterizesa subjective expected utility agent whose posterior beliefs after A , denoted µ A , are a convexcombination of the prior and the Bayesian posterior: µ A = δ ( A ) µ + (1 − δ ( A )) B ( µ, A ) (1)where δ ( A ) ∈ [0 , and B ( µ, A ) is the Bayesian update of µ given A . When δ ( A ) > , theagent is reluctant to move away from her initial beliefs, and therefore she is conservative.When δ ( A ) = 0 , she is Bayesian; when δ ( A ) = 1 , she is unresponsive to the information.Consequently, δ ( A ) can be interpreted as her degree of conservatism (at A ). In general, acollection of conditional preferences admits a conservative subjective expected utility representation when posterior beliefs are given by Equation 1 for each event.An important feature of the representation in Equation 1 is the dependence of the de-gree of conservatism, δ ( A ) , on the realized event: bias may be source-dependent. Becauseof source dependence, conservatism is able to capture belief dynamics consistent with con-firmation bias (Nickerson (1998)), the representativeness heuristic (Kahneman and Tversky(1972)), and the good news-bad news effect (Eil and Rao (2011)). Although these biases havebeen thought of as distinct aspects of behavior, this finding suggests that there may be a corebehavioral aspect of conservatism that unifies these biases.I then provide an analysis of various aspects related to an agent’s degree of conservatism.Intuitively, δ ( A ) captures how conservative an agent is (at A ). When δ ( A ) is event indepen-dent, δ ( A ) = δ for some δ ∈ [0 , , then the agent is described by a single behavioral param-eter and δ is an index of conservatism. I establish a behavioral characterization of this casethrough Weak Consequentialism, a novel weakening of Consequentialism. I then provide asimple definition of comparative conservatism that rests upon the comparison of a binaryact with a constant outcome. This provides an efficient method for ordering people by eachone’s degree of conservatism.I close by exploring the implications of conservatism when agents do not have precisebeliefs. To do so, I consider a collection of possibly incomplete conditional preferences andimpose an adapted version of Dynamic Conservatism on these preferences, which I call Un-ambiguous Dynamic Conservatism. In this setting, the agent has a set of possible beliefs andprefers an act, f , to another, g , when f provides (weakly) greater expected utility than g for every possible belief. Unambiguous Dynamic Conservatism characterizes a conceptually Relatedly, − δ ( A ) might be thought of as a measure of her confidence in the new information. Thisinterpretation will be useful for the discussion of source dependence in section 4. The setup and model are presented in section 2. Behavioral foundations are presented in sec-tion 3. The connection between conservatism and other belief biases is discussed in section 4.The analysis of degrees of conservatism is discussed in section 5. I present the extension tomultiple priors and incomplete preferences in section 6. The remainder of this section dis-cusses related literature.While numerous experimental papers suggest that people do not update their beliefs ina Bayesian manner, relatively few provide an axiomatic analysis of non-Bayesian updating.Epstein (2006) provided one of the first axiomatic analysis of non-Bayesian updating. Heutilized a setup of preferences over menus and modeled non-Bayesian updating as a temp-tation (á la Gul and Pesendorfer (2001)) to modify prior beliefs in response to an interimsignal. This was extended to an infinite-horizon model by Epstein et al. (2008). Because ofthe menu-preference setting, beliefs are typically dependent on both the information andthe option set. More recently, Ortoleva (2012) utilized the conditional preference approachto introduce the Hypothesis Testing Representation. In his model, a decision maker maybe surprised by low probability events and, in response, adopt a new prior before applyingBayes’ rule. This representation captures “over-reaction" to information and is conceptuallydistinct from conservatism. The notion of conservative updating I characterize results in violations of Dynamic Con-sistency and Consequentialism. While violations of these properties are well documented,the causes of these violations are still not fully understood. Additionally, violations of Con-sequentialism are relatively understudied compared to violations of Dynamic Consistency.Violations of Dynamic Consistency and Consequentialism were documented by Dominiaket al. (2012) in an Ellsberg-style experiment. While these violations were more frequentamong ambiguity averse subjects, they also occur among ambiguity neutral subjects. More Additionally, such a decision maker always satisfies Consequentialism, which is typically violated by con-servative updating. Further, all subjects report a "loss of confidence" in their choices after information, with the greatest loss
There is a (nonempty) finite set S of states of the world, a collection of events given by analgebra Σ over S , and a (nonempty) set of consequences X . Let F denote the set of functions f : S → X , referred to as an act. Following a standard abuse of notation, for any x ∈ X ,by x ∈ F I mean the constant act that returns x in every state. Lastly, for any f, g ∈ F and for any A ∈ Σ , let f Ag denote the act that returns f ( s ) when s ∈ A and returns g ( s ) when s ∈ A c ≡ S \ A . Following the literature, I assume that X is a convex subset of a vectorspace. Thus, mixed acts can be defined point-wise, so that for every f, g ∈ F and λ ∈ [0 , ,by λf + (1 − λ ) g I mean the act that returns λf ( s ) + (1 − λ ) g ( s ) for each s ∈ S .I assume that the agent has preferences over F conditional on her information. Formally,the agent has a collection of preference relations, { (cid:37) A } A ∈ Σ over F , where (cid:37) A are her pref-erences after observing A . Let (cid:31) A and ∼ A represent the asymmetric and symmetric parts of (cid:37) A . The case when the agent has no information is represented by (cid:37) S := (cid:37) . For an event A ,say that A is (cid:37) -null (or simply null) if f Ag ∼ g for any f, g ∈ F . Otherwise, A is non-null.Let Σ + ⊂ Σ denote the collection of non-null events.Let ∆( S ) denote the set of probability measures over S . Typical elements, µ, π ∈ ∆( S ) are called beliefs. For any probability µ and non-null event A ∈ Σ + , define the Bayesianupdate of µ given A by B ( µ, A )( B ) = µ ( B ∩ A ) µ ( A ) for B ∈ Σ . Finally, for any two utility functions of confidence occurring among subjects who violated both Dynamic Consistency and Consequentialism. Thisis consistent with the interpretation of − δ ( A ) as the degree of confidence in the information, with lowerconfidence leading to more violations. It is not difficult to extend to an infinite state space by assuming Σ is a σ -algebra and F is the set offinite-valued Σ -measurable functions. For instance, X may be an interval of monetary prizes or, as in the classic Anscombe and Aumann (1963)setting, a set of lotteries over some prize space. , v : X → R , say u ≈ v if u is a positive affine transformation of v . Definition 1.
Say that a collection of preferences { (cid:37) A } A ∈ Σ admits a conservative subjec-tive expected utility (conservative SEU) representation if there are a non-constant utilityfunction u : X → R , a prior probability µ ∈ ∆( S ) , and a function δ : Σ + → [0 , such thatfor all A ∈ Σ + , f (cid:37) A g ⇐⇒ (cid:88) s ∈ S u ( f ( s )) µ A ( s ) ≥ (cid:88) s ∈ S u ( f ( s )) µ A ( s ) and µ A = δ ( A ) µ + (1 − δ ( A )) B ( µ, A ) . Example 1.
There are two payoff states, P = { R, B } and two signals Θ = { r, b } . Let S = P × Θ and, slightly abusing notation, let r = { ( R, r ) , ( B, r ) } and b = { ( R, b ) , ( B, b ) } denote the events corresponding to an r or b signal, respectively. The prior µ ∈ ∆( S ) isgiven in Table 1 below. µ r bR / / B / / Table 1: Prior over S = P × Θ .This example has a natural interpretation as a standard signaling experiment with priorover P given by µ P = ( , ) , and (asymmetric) signal accuracy σ ( r | R ) = , σ ( b | B ) = .If the agent admits a conservative SEU representation, posterior beliefs after r and b ,written in terms of the marginal belief on R , are µ r ( R ) = δ ( r ) 58 + (1 − δ ( r )) 45 µ b ( R ) = δ ( b ) 58 + (1 − δ ( b )) 13 . One feature of the representation is that δ is event (i.e., signal) dependent, allowing for arich variety of other “biases” to emerge. For instance, when δ ( r ) < δ ( b ) , the agent’s beliefsexhibit confirmation bias in addition to conservatism. That is, because her conservatism biasis greater when she receives a signal that is counter to her “prior hypothesis” (i.e., µ ( R ) > > µ ( B ) ) she is more reluctant to incorporate “disconfirming information." This and otherbiases are further discussed in section 4. 6 Behavioral Foundations
The first axiom imposes a subjective expected utility (SEU) representation at each informa-tion set. The conditions for this are well-established in the literature.
Axiom 1 (Conditional SEU) . For each A ∈ Σ , (cid:37) A admits a non-degenerate subjective ex-pected utility representation.Before introducing the conservatism axiom, for comparison, I first state the classic axiomsof Dynamic Consistency and Consequentialism. An excellent discussion of both axioms isprovided in Ghirardato (2002), and so I will only briefly discuss them. Axiom 2 (Dynamic Consistency) . For any A ∈ Σ , A non-null, and for all f, g ∈ F , f Ag (cid:37) g = ⇒ f (cid:37) A g. Axiom 3 (Consequentialism) . For any A ∈ Σ and for all f, g ∈ F , f ( s ) = g ( s ) for all s ∈ A = ⇒ f ∼ A g. In essence, f Ag (cid:37) g reveals that the decision maker believes she would abandon g for f if A occurred. Dynamic Consistency states that if this is the case, then f must be preferredto g after being told A has occurred. Consequentialism states that whenever two acts areidentical within A , the agent must be indifferent between them after A . It is well knownthat Dynamic Consistency and Consequentialism, when combined with Conditional SEU,are necessary and sufficient for Bayesian updating (Ghirardato, 2002). Hence, these must berelaxed to allow for conservatism.To illustrate how these axioms are relaxed, recall the introductory example on the useof alternative energy versus coal. The act f is “use alternative energy,” g is “use coal,” andthe event A is “climate change is occurring.” The agent concedes that alternative energy isbetter if climate change is occurring. Depending on the agent’s initial preference between f and g , there are two relevant cases: Case 1: f (cid:37) g and f Ag (cid:37) g, Case 2: g (cid:31) f and f Ag (cid:37) g. In case 1, the preference for the fixed action f (over g ) is consistent with the preference forthe contingent action f Ag (over g ). In case 2, the preferences are in conflict. Dynamic Con-7istency requires the agent to conclude that f is better than g in the conditional preferencein both cases. However, intuition suggests that a conservative agent may violate DynamicConsistency in case 2. Indeed, she does so because she does not fully believe A has occurred.Similarly, a conservative agent would never violate Dynamic Consistency in case 1 since sheprefers f to g irrespective of A ’s occurrence. The following axiom takes this intuition as thedefining behavior of conservatism. Axiom 4 (Dynamic Conservatism) . For any A ∈ Σ + and for all f, g ∈ F ,(i) f (cid:37) g (ii) f Ag (cid:37) g (cid:41) = ⇒ f (cid:37) A g. Further, if both (i) and (ii) are strict, then f (cid:31) A g .Dynamic Conservatism requires that if the agent (i) prefers f to g ex-ante and (ii) fore-casts that she would abandon g for f if A occurred, f Ag (cid:37) g , then she must prefer f to g conditional on A . Therefore, while Dynamic Conservatism allows for some violations of Dy-namic Consistency (e.g., the climate change example), it restricts violations to cases wherethe agent cannot be completely sure that she is making the right decision. Hence, we canthink of Dynamic Conservatism as allowing for a form of "stickiness" or skepticism aboutnew information. Seen through this lens, Dynamic Conservatism can be viewed as a cau-tious response when the reliability of information is (subjectively) uncertain. Consequently,Dynamic Conservatism permits an agent to state f Ag (cid:37) g and g (cid:31) A f . Theorem 1.
The following are equivalent:(i) The collection { (cid:37) A } A ∈ Σ satisfies Conditional SEU and Dynamic Conservatism;(ii) The collection { (cid:37) A } A ∈ Σ admits a conservative subjective expected utility representation.Further, if ( u, µ, δ ) and ( u (cid:48) , µ (cid:48) , δ (cid:48) ) both represent { (cid:37) A } A ∈ Σ , then (i) u (cid:48) is a positive affine trans-formation of u , (ii) µ (cid:48) = µ , and (iii) δ (cid:48) ( A ) = δ ( A ) for all A such that A, A c ∈ Σ + . Theorem 1 shows that when conditional preferences admit a SEU representation, Dy-namic Conservatism is the precise behavioral content of conservative updating. Further, the Another way of viewing Dynamic Conservatism is through a two-self interpretation. Condition (i) capturesa self that does not learn, while condition (ii) captures a Bayesian self. Dynamic Conservatism states thatwhen the selves agree, then the agent’s behavior necessarily reflects this agreement. When they disagree, thenDynamic Conservatism says nothing. The representation result, however, shows that the agent’s behavior mustbe governed by a “compromise” belief. δ ( A ) , is uniquely pinned down at essentially every A . Example 1 (Continued) . To further illustrate this result, recall our example with two pay-off states P = { R, B } and two signals Θ = { r, b } . As illustrated in Figure 1, DynamicConservatism ensures that the lower contour set of the conditional indifference curve pass-ing through f must contain the intersection of the lower contour sets determined by (i) theinitial preference (e.g., µ ) and (ii) the preference that corresponds to performing Bayesianupdating B ( µ, r ) . This intersection is shaded in blue. In essence, conservatism pulls theconditional indifference curve to to be more aligned with the initial preference. u ( R ) u ( B ) f µ B ( µ, r ) Figure 1: Indifference curves after the agent receives an r signal. The (orange) dashed linecorresponds to an indifference curve passing through f for some δ ( r ) ∈ (0 , . Remark 1.
It is easy to see from Figure 1 that over-inference or “prior-neglect” may be capturedby rotating the (orange) indifference curve away from µ and beyond B ( µ, r ) (e.g., intuitively,this is captured by δ ( r ) < ). This would imply the existence of an act g , such that f (cid:37) g , f r g (cid:37) g but g (cid:31) r f , violating Dynamic Conservatism. However, over-inference is difficultto capture in the conditional preference framework with a general state space. This is because δ ( A ) < results in negative values for the probabilities of certain states. Thus, the behavioralfoundations of such behavior remain an open question. Dynamic Conservatism may be extended to any event A ∈ Σ , rather than merely the non-null events, Σ + . When A is null, f Ag ∼ g for any f, g . It then follows from Dynamic Conservatism that (cid:37) = (cid:37) A . Thus,conservative agents react to null events by completely ignoring them. Source Dependence and Belief Biases
When beliefs (or attitudes) about uncertainty vary with the source of that uncertainty, thenbeliefs (or attitudes) are said to be source-dependent. This section demonstrates that source-dependent conservatism captures belief dynamics consistent with confirmation bias, repre-sentativeness, and the good news-bad news effect. Source dependence might be related tofamiliarity with a source. An implication of this notion is that there may be increased skep-ticism about new sources, which is conceptually similar to the notion of source-dependentambiguity attitudes (Abdellaoui et al. (2011) and Chew and Sagi (2008)). Alternatively, sourcedependence might be related to message complexity, whereby subjectively simpler messagesare accepted more readily. Confirmation bias (see Nickerson (1998) for an excellent review) refers to a tendency to ac-cept information that supports already believed hypotheses and to downplay conflicting in-formation. Confirmation bias emerges from conservatism bias when the weight attached tothe prior is greater for “disconfirming" news than for “confirming" news.
Example 1 (Continued) . There are two payoff states, P = { R, B } , and two signals, Θ = { r, b } , and the prior µ is given in Table 1 below. µ r bR / / B / / If the agent admits a conservative SEU representation, posterior beliefs after r and b ,written in terms of the marginal belief on R , are µ r ( R ) = δ ( r ) 58 + (1 − δ ( r )) 45 µ b ( R ) = δ ( b ) 58 + (1 − δ ( b )) 13 . When δ ( r ) < δ ( b ) , the agent’s beliefs exhibit confirmation bias in addition to conser-vatism; when she receives a signal that is counter to her “prior hypothesis” (i.e., µ ( R ) > This reasoning is consistent with theories in psychology that posit that conservatism results from thedifficulty of aggregating sources of information (Slovic and Lichtenstein, 1971) and noisy recollection (Hilbert,2012). > µ ( B ) ), she is more reluctant to incorporate “disconfirming information." I provide abehavioral characterization of confirmation bias in Proposition 2. The representativeness heuristic (Kahneman and Tversky, 1972) refers to a tendency to re-act more strongly to signals that are representative of, or similar to, an underlying state.Consider a setup like the confirmation bias example, but suppose the agent has a similar-ity relation (cid:68) where ( A, a ) (cid:68) ( B, b ) denotes a is more representative of A than b is of B . If ( A, a ) (cid:68) ( B, b ) implies δ ( E a ) ≤ δ ( E b ) , then a form of the representativeness heuristic ispresent. Beliefs depend on both the objective information provided by the signal and thedegree to which the agent perceives that signal as representative of the payoff-relevant vari-ables. When subjects react differently to “types” of news, they exhibit the good news-bad newseffect (Eil and Rao (2011), Möbius et al. (2014), and Charness and Dave (2017)). Source-dependent conservatism may also allow for this effect. Informally, a decision maker reactsmore to good news when her degree of bias is smaller. Similar to the representativeness ex-ample, suppose E a (cid:68) E b denotes E a is better news than E b . Then asymmetric updating occurswhen E a (cid:68) E b implies δ ( E a ) ≤ δ ( E b ) . The reverse behavior, stronger reaction to bad news,is captured by reversing the inequality. Theorem 1 allows for the degree of conservatism to depend on the information received.In many settings, this is useful as it allows for source-dependent reactions to news. How-ever, it is often convenient to fully describe behavior with a single parameter, or an index ofconservatism. Further, a constant degree of conservatism simplifies the task of identifyingbehavioral parameters from data and increases the predictive power of the model. I showthat constant conservatism is characterized by a consistency condition linking conditionalpreferences across information sets that may be viewed as a weak form of Consequentialism.11 xiom 5 (Weak Consequentialism) . For any
A, B, C ∈ Σ + with C ∩ ( A ∪ B ) = ∅ and forall f, y, z ∈ F , f Cy (cid:37) A z ⇐⇒ f Cy (cid:37) B z. To see how this is a weak form of Consequentialism, suppose C ∩ ( A ∪ B ) = ∅ andthat Consequentialism holds. Consider f Cy for an arbitrary f . Since for all s ∈ A ∪ B , f Cy ( s ) = f Cy ( s ) , it follows from Consequentialism that both f Cy ∼ A y and f Cy ∼ B y .Thus, under Consequentialism, f is irrelevant, and under ordinal preference consistency,Weak Consequentialism always holds. On the other hand, Weak Consequentialism does notimpose indifference between f Cy and y but requires a consistent relative preference; if f Cy is preferred to z after A , then it is also preferred after B . Proposition 1.
Suppose the collection { (cid:37) A } A ∈ Σ admits a Conservative SEU representation ( u, µ, δ ) . Then the following are equivalent:(i) The collection { (cid:37) A } A ∈ Σ satisfies Weak Consequentialism.(ii) There is a unique δ ∈ [0 , such that δ ( A ) = δ for all A such that A and S \ A arenon-null. A key strength of this result is that now δ may serve as a simple index of conservatismbias and may be elicited with only a few questions. By weakening Weak Consequentialism to depend on the ex-ante likelihood of events, a be-havioral characterization of conservative preferences that are consistent with confirmationbias may be obtained. To do so, I first define a qualitative likelihood ordering over events.
Definition 2.
For any
A, B ∈ Σ + , say that A is more likely than B , denoted A ≥ l B iffor all x, y ∈ X , x (cid:37) y implies xAy (cid:37) xBy .Recall that Weak Consequentialism ensures a constant δ by precisely calibrating theagent’s willingness to bet on C across A and B . Note that this is independent of the ex-anterelative likelihood of A or B . Confirmation bias, on the other hand, suggests that subjectivelymore likely events are incorporated more accurately. In other words, because B is viewed asless likely than A , the agent is more willing to bet on C after B than after A . This is capturedby the following behavioral axiom. 12 xiom 6 (Generalized Confirmation Bias) . For any
A, B, C ∈ Σ + with C ∩ ( A ∪ B ) = ∅ and for all x, y, z ∈ F if x (cid:31) y and A ≥ l B , then xCy (cid:37) B z = ⇒ xCy (cid:37) A z. Proposition 2.
Suppose the collection { (cid:37) A } A ∈ Σ admits a Conservative SEU representation ( u, µ, δ ) . Then the following are equivalent:(i) The collection { (cid:37) A } A ∈ Σ satisfies Generalized Confirmation Bias.(ii) µ ( A ) ≥ µ ( B ) if and only if δ ( A ) ≤ δ ( B ) . Intuitively, a more conservative agent is less responsive to information, which can be cap-tured in the representation by a larger weight on the prior belief. Analogously, one person ismore conservative than another if he places a larger weight on his prior belief than she doeson her prior belief. This can be formally defined in terms of preferences over binary acts.
Definition 3.
Say that { (cid:37) A } A ∈ Σ is more conservative than { (cid:37) A } A ∈ Σ if for all A and all x, y , y ∈ X satisfying (i) x (cid:31) i y i and (ii) xAy (cid:37) z ⇐⇒ xAy (cid:37) z for all z ∈ XxAy (cid:37) A z = ⇒ xAy (cid:37) A z. If both agents are Bayesian, then xAy ∼ iA x for any x, y ∈ X . A conservative agent,however, may worry about the low payoff y on A c . The more conservative the agent, thelower his certainty equivalent for a bet on A . Consequently, a more conservative agentplaces a lower value on xAy than a less conservative agent. Hence, when agent (he) ismore conservative than agent (she), his certainty equivalent is lower than hers, and sowhenever he prefers to bet on A , so must she. Importantly, this definition dos not requirethe agents to have the same initial beliefs, but only the same tastes over constant acts. Proposition 3.
Suppose ( { (cid:37) iA } A ∈ Σ ) i =1 , admit Conservative SEU representations ( u i , µ i , δ i ) i =1 , where u ≈ u and Σ = Σ . Then the following are equivalent:(i) { (cid:37) A } A ∈ Σ is more conservative than { (cid:37) A } A ∈ Σ .(ii) δ ( A ) ≥ δ ( A ) for every A such that A and S \ A are non-null. By allowing y and y to differ, we account for the different prior beliefs about the ex-ante likelihood of A .Accordingly, we may take y = y if and only if µ ( A ) = µ ( A ) . A ). Under the assumption of constant conservatism(i.e., Weak Consequentialism holds), a single elicitation of a (conditional) certainty equivalentsuffices to order all subjects. Agents may struggle to come up with a single, probabilistic belief; they perceive a situationto be ambiguous. Under ambiguity, we often suppose the agent has multiple beliefs. Tostudy conservatism with multiple beliefs, I suppose the agent has a collection of (incomplete)preferences, each of which admits a multiple-prior “unanimity” representation à la Bewley(2002).Similar settings have been used to study objective versus subjective rationality (Gilboaet al., 2010), distinguish indecisiveness in tastes versus beliefs (Ok et al., 2012), and differ-entiate ambiguity perception and attitude (Ghirardato et al., 2004). Unlike in Gilboa et al.(2010), I am focused purely on the evolution of beliefs and so do not consider the secondary“subjective” preference relation. Because of this my results hold irrespective of the agent’sambiguity attitude.
Definition 4.
A preference relation (cid:37) admits a multi-prior expected utility representationif there are a utility u : X → R and a nonempty, closed, convex set of beliefs M ⊆ ∆( S ) such that for all acts f, g ∈ F , f (cid:37) g ⇐⇒ (cid:88) s ∈ S u ( f ( s )) µ ( s ) ≥ (cid:88) s ∈ S u ( g ( s )) µ ( s ) for every µ ∈ M . (2)Suppose the agent has a collection of possibly incomplete preferences { (cid:37) ∗ A } A ∈ Σ definedover F . For an event A , say that A is unambiguously non-null if for all x, y such that x (cid:31) ∗ y , xAy (cid:31) ∗ y . Let Σ ∗ + denote the set of unambiguously non-null events. For any set ofprobabilities M and any A ∈ Σ ∗ + , the set obtained when each belief in M is updated byBayes’ rule is denoted B ( M , A ) = { π ∈ ∆( S ) | π = B ( µ, A ) for some µ ∈ M} . Lastly, forany set Y ⊆ ∆( S ) , let conv ( Y ) denote the convex hull of Y . See Gilboa and Marinacci (2011) for an excellent summary of the literature. xiom 7 (Conditional Multi-prior Expected Utility) . For each A ∈ Σ ∗ + , (cid:37) ∗ A admits a non-degenerate multi-prior expected utility representation. That is, there exists a pair ( u A , M A ) that represents (cid:37) ∗ A as in Equation 2.As in the case of subjective expected utility, the conditions for such a representation arewell-known in the literature and so are not re-stated.The following axiom, Unambiguous Dynamic Conservatism, is precisely Dynamic Con-servatism applied to the the unambiguous preferences. Axiom 8 (Unambiguous Dynamic Conservatism) . For any A ∈ Σ ∗ + , and for all f, g ∈ F f (cid:37) ∗ gf Ag (cid:37) ∗ g (cid:41) = ⇒ f (cid:37) ∗ A g. Further, if both (i) and (ii) are strict, then f (cid:31) ∗ A g .The following theorem is the direct counterpart of Theorem 1 for multiple priors. Theorem 2.
The following are equivalent:(i) The collection { (cid:37) ∗ A , } A ∈ Σ satisfies Conditional Multi-prior Expected Utility and Unam-biguous Dynamic Conservatism.(ii) There is a non-constant utility function u : X → R such that u A = u for every A ∈ Σ ,and for each A ∈ Σ ∗ + , M A ⊆ conv ( M ∪ B ( M , A )) . (3) In this case, we say the agent admits a conservative multi-prior representation . As before, Unambiguous Dynamic Conservatism ensures that risk attitudes are unchangedby the information (i.e., u is independent of A ). However, Theorem 2 is considerably moregeneral than Theorem 1, which corresponds to the special case where (cid:37) ∗ and (cid:37) ∗ A are com-plete. An important point of departure is that Theorem 2 allows for an agent to expand her setof beliefs. For instance, this occurs when (cid:37) ∗ is complete but (cid:37) ∗ A is not. At the same time, myresult complements results by Ghirardato et al. (2008) and Faro and Lefort (2019), who foundthat imposing dynamic consistency on the unambiguous preference ensures prior-by-priorupdating (e.g., M A = B ( M , A ) ). 15 xample 2 (Single prior-multiple weights) . Suppose M = { µ } and for each A ∈ Σ ∗ + thereexists a closed, convex set of weights W A ⊂ [0 , such that M A = { π ∈ ∆( S ) | π = δ A µ + (1 − δ A ) B ( µ, A ) and δ A ∈ W A } . In this case, the agent is SEU before information. However, because she is uncertain about thereliability of the information, her set beliefs expands. Her posterior beliefs are constructedvia a set of “confidence weights” that she places on the news. In fact, it is straightforward toshow that this example corresponds to precisely the case in which the initial preference isSEU.
Corollary 1.
Suppose { (cid:37) ∗ A } A ∈ Σ satisfies Conditional Multi-prior Expected Utility and Unam-biguous Dynamic Conservatism and that the initial preference, (cid:37) ∗ , is complete. Then for every A ∈ Σ ∗ + there exists a closed, convex set of weights W A ⊂ [0 , such that M A = { π ∈ ∆( S ) | π = δ A µ + (1 − δ A ) B ( µ, A ) and δ A ∈ W A } . Example 3 (Multiple priors-single weight) . Suppose for some map δ : Σ ∗ + → [0 , , M A = δ ( A ) M + (1 − δ ( A )) B ( M , A ) . In this case, the agent has multiple beliefs, but she has a well-defined opinion of the newssource. Thus she uses a single parameter to shift her beliefs after each event A . Example 3 (Continued) . Suppose there are two payoff states, P = { R, B } , two signals Θ = { r, b } , and M = conv ( { µ, µ (cid:48) } ) , which are shown in Table 2 below. µ r bR /
10 2 / B /
10 3 / µ (cid:48) r bR /
10 3 / B /
10 2 / Table 2: Priors µ and µ (cid:48) .Both µ, µ (cid:48) generate the same marginals over P : µ ( R ) = µ (cid:48) ( R ) = 3 / . The distinctionbetween µ and µ (cid:48) is in how they treat signals. Under µ signals are informative, while under µ (cid:48) they are not. If the agent admits a conservative multi-prior representation, posterior beliefsafter r and b , written in terms of the belief over payoff-states P , are M r = conv (cid:18)(cid:26)(cid:18) − δ ( r )5 , δ ( r )5 (cid:19) , (cid:18) , (cid:19)(cid:27)(cid:19) b = conv (cid:18)(cid:26)(cid:18) δ ( b )5 , − δ ( b )5 (cid:19) , (cid:18) , (cid:19)(cid:27)(cid:19) . These belief sets are illustrated in Figure 2 in terms of the marginal belief on R . Notice . δ ( θ ) = 1 [ [ M b δ ( b ) = 0 δ ( b ) > M r δ ( r ) = 0 δ ( r ) > ]] Figure 2: Posterior probabilities for payoff state R .that R is ambiguous after the signal, save for the extreme case δ ( θ ) = 1 , and the range is in-creasing as δ ( θ ) decreases. Consequently, the conditional certainty equivalent for a bet on R (e.g., a monetary act paying $ x if R and $0 otherwise) depends on the both the agent’s degreeof conservatism and her ambiguity attitude. In the case of ambiguity aversion (e.g., maxminexpected utility (MEU) representation (Gilboa and Schmeidler, 1989)), the agent may exhibitan “all news is bad news” effect. For more flexible models, such as the α -maxmin model, thisneed not be the case. The certainty equivalents for a bet on R can be seen in Table 3. In thetable, I vary δ ( · ) and ambiguity attitude; -maxmin (e.g., Gilboa and Schmeidler (1989)) is themost ambiguity averse and -maxmin is maximally ambiguity seeking. Certainty equivalentsfor intermediate values of α may be directly calculated from these extreme cases. δ ( r ) \ α . x . x / . x . x . x . x δ ( b ) \ α . x . x / . x . x . x . x Table 3: Certainty equivalents for the monetary bet paying x -utils in payoff state R , oth-erwise, for an α -MEU agent. The left panel reports values after r , while the right reportsvalues after b . A ProofsA.1 Preliminary Results
Consider the following two properties for a binary relation (cid:37) on F . By adapting axioms introduced by Gilboa et al. (2010), or more recently by Frick et al. (2020), one maycharacterize an ( α -)maxmin representation consistent with { (cid:37) ∗ A } . This additional structure on the agent’sambiguity attitude generates sharper predictions regarding her willingness to bet on various events. -Completeness: For any x, y ∈ F , either x (cid:37) y of y (cid:37) x . Monotonicity: If f ( s ) (cid:37) g ( s ) for all s ∈ S , then f (cid:37) g . Lemma 1.
Consider a collection of preferences { (cid:37) A } A ∈ Σ such that (i) (cid:37) satisfies C-Completenessand Monotonicity and (ii) the collection satisfies Dynamic Conservatism. Then for all A ∈ Σ such that A is non-null, and any x, y ∈ X , x (cid:37) y ⇐⇒ x (cid:37) A y. Proof.
Since (cid:37) is complete for constant acts, suppose x (cid:37) y . By Monotonicity of (cid:37) this isequivalent to xAy (cid:37) y for all A . Then by Dynamic Conservatism, x (cid:37) A y . Suppose that x (cid:37) A y but y (cid:31) x . Then it follows from Monotonicity and the fact that A is non-null that yAx (cid:31) x . From Dynamic Conservatism it follows that y (cid:31) A x , a contradiction. Hence x (cid:37) y . Lemma 2.
Suppose (cid:37) ∗ admits a multi-prior expected utility representation ( u, M ) . For each A ∈ Σ and all f, g ∈ F , f Ag (cid:37) ∗ g ⇐⇒ f Ah (cid:37) ∗ gAh for all h ∈ F . Proof.
Let A be any event and let f, g, h be any acts in F . f Ag (cid:37) ∗ g ⇔ (cid:88) s ∈ A u ( f ( s )) µ ( s ) + (cid:88) s ∈ S \ A u ( g ( s )) µ ( s ) ≥ (cid:88) s ∈ A u ( g ( s )) µ ( s ) + (cid:88) s ∈ S \ A u ( g ( s )) µ ( s ) for all µ ∈ M⇔ (cid:88) s ∈ A u ( f ( s )) µ ( s ) ≥ (cid:88) s ∈ A u ( g ( s )) µ ( s ) for all µ ∈ M⇔ (cid:88) s ∈ A u ( f ( s )) µ ( s ) + (cid:88) s ∈ S \ A u ( h ( s )) µ ( s ) ≥ (cid:88) s ∈ A u ( g ( s )) µ ( s ) + (cid:88) s ∈ S \ A u ( h ( s )) µ ( s ) for all µ ∈ M⇔ f Ah (cid:37) ∗ gAh This lemma only relies on A being non-null, and it in fact holds for more general state spaces. Lemma 3.
Suppose (cid:37) ∗ admits a multi-prior expected utility representation ( u, M ) . For each A ∈ Σ ∗ + and all f, g ∈ F , say f (cid:68) ∗ A g if f Ah (cid:37) ∗ gAh for some h . Then (cid:68) A admits a multi-priorexpected utility representation ( u, B ( M , A )) . roof. By Lemma 2, (cid:68) ∗ A does not depend on the choice of h . It then follows that f (cid:68) ∗ A g ifand only if for every µ ∈ M (cid:88) s ∈ A u ( f ( s )) µ ( s ) + (cid:88) s ∈ S \ A u ( h ( s )) µ ( s ) ≥ (cid:88) s ∈ A u ( g ( s )) µ ( s ) + (cid:88) s ∈ S \ A u ( h ( s )) µ ( s ) ⇐⇒ µ ( A ) (cid:88) s ∈ A u ( f ( s )) µ ( s ) ≥ µ ( A ) (cid:88) s ∈ A u ( g ( s )) µ ( s ) ⇐⇒ (cid:88) s ∈ A u ( f ( s )) π ( s ) ≥ (cid:88) s ∈ A u ( g ( s )) π ( s ) for all π ∈ B ( M , A ) , where B ( M , A ) = { π ∈ ∆( S ) | π = B ( µ, A ) for some µ ∈ M} .Note that when (cid:37) ∗ is complete, then we have the case of subjective expected utility and M and B ( M , A ) are singleton sets. A.2 Proof of Theorem 1
Proof.
Necessity is clear so only sufficiency is shown. By Conditional SEU, there exists a ( u A , µ A ) for each A ∈ Σ that represents (cid:37) A . Further, by Dynamic Conservatism, preferencessatisfy ordinal preference consistency (see Lemma 1): x (cid:37) y if and only if x (cid:37) A y . Hencewe may assume u = u A for all A . Further, as X is convex, it is without loss to suppose [ − , ⊂ u ( X ) , as u .For each A ∈ Σ + , define the binary relation (cid:68) A on F by f (cid:68) A g if and only if f Ag (cid:37) g .Then by Lemma 3, (cid:68) A has an expected utility representation ( u, B ( µ, A )) .Next, define the set D A := { π ∈ ∆( S ) | δµ + (1 − δ ) B ( µ, A ) for δ ∈ [0 , } . By DynamicConservatism, it follows that µ A ∈ D A . Suppose not, then as D A and { µ A } are closed, convexsets, there exists a separating hyperplane a ∈ R | S | so that µ A · a > ˆ µ · a for all ˆ µ ∈ D A . Let ¯ z = max ˆ µ ∈ D A ˆ µ · a and let ¯ a = a − ¯ z (1 , . . . , . Then µ A · ¯ a > ≥ ˆ µ · ¯ a for all ˆ µ ∈ D A . (4)We may suppose that ¯ a ∈ [ − , | S | , since we can always multiply both sides of (4) by (cid:15) > .Further, there are acts f, g such that u ( g ( s )) − u ( f ( s )) = ¯ a ( s ) for every s ∈ S . Consequently, (cid:88) s ∈ S µ A ( s ) u ( g ( s )) > (cid:88) s ∈ S µ A ( s ) u ( f ( s )) (5)19nd (cid:88) s ∈ S ˆ µ ( s ) u ( f ( s )) ≥ (cid:88) s ∈ S ˆ µ ( s ) u ( g ( s )) for all ˆ µ ∈ D A . (6)By construction, µ, B ( µ, A ) ∈ D A and so by (6) it follows that f Ag (cid:37) g , f (cid:37) g . However, by(5) g (cid:31) A f , which contradicts Dynamic Conservatism. Hence µ A ∈ D A . As A was arbitrary,the preceding argument applies to any non-null A . It is standard to show that u is unique upto a positive, affine transformation and, since u ( x ) > u ( y ) for some x, y ∈ X , that µ and µ A are also unique. Given uniqueness of µ and µ A , it is trivial that there is a unique δ ( A ) thatsatisfies Equation 1 whenever µ ( A ) < . When µ ( A ) = 1 (i.e., A c is null), µ = B ( µ, A ) = µ A and (cid:37) = (cid:37) A . When A, A c are both non-null, define δ : Σ + → [0 , as the unique solution to µ A = δ ( A ) µ + (1 − δ ( A )) B ( µ, A ) . When A c is null, define δ ( A ) arbitrarily. A.3 Proof of Proposition 1
Proof.
Theorem 1 shows that for each A , there is a δ ( A ) ∈ [0 , satisfying the representation.Suppose Weak Consequentialism holds. It is sufficient to show that for any pair of non-nullevents A, B ∈ Σ , such that both µ ( A ) < and µ ( B ) < , δ ( B ) = δ ( A ) . Case 1: ( µ ( A ∪ B ) < ). Fix any non-null C satisfying C ∩ ( A ∪ B ) = ∅ and choose x, y, z such that xCy ∼ A z . By Weak Consequentialism, it follows that xCy ∼ B z . Sincepreferences admit a conservative SEU representation, it follows that µ A ( C ) u ( x ) + (1 − µ A ( C )) u ( y ) = u ( z ) , (7) µ B ( C ) u ( x ) + (1 − µ B ( C )) u ( y ) = u ( z ) . (8)Since x, y are arbitrary, it is without loss to suppose that u ( x ) > u ( y ) . Then, combining(7) and (8), it is clear that µ A ( C ) = µ B ( C ) . Since C ∩ ( A ∪ B ) , it follows that B ( µ, A )( C ) =0 = B ( µ, B )( C ) , and hence µ A ( C ) = δ ( A ) µ ( C ) and µ B ( C ) = δ ( B ) µ ( C ) . Hence equality istrue if and only if δ ( A ) = δ ( B ) . Case 2: ( µ ( A ∪ B ) = 1 ). Suppose A ∩ B (cid:54) = ∅ . Then notice that A, A ∩ B ( B, A ∩ B )satisfy the conditions of Case 1. It follows then that δ ( A ) = δ ( A ∩ B ) = δ ( B ) . Now suppose A ∩ B = ∅ . Since there are at least three non-null events, without loss there exists somenon-null set A (cid:48) such that A (cid:48) ⊂ A and µ ( A (cid:48) ) < µ ( A ) . Then we may again apply Case 1 to Alternatively, there exists A (cid:48) such that A ⊂ A (cid:48) and µ ( A ) < µ ( A (cid:48) ) , but this is just a relabeling. Note that , A (cid:48) , A \ A (cid:48) , showing that δ ( A ) = δ ( A (cid:48) ) = δ ( A \ A (cid:48) ) . Further, since A ∩ B = ∅ , it followsthat µ ( A (cid:48) ∪ B ) < , and so δ ( A (cid:48) ) = δ ( B ) . A.4 Proof of Proposition 2
Proof.
Let A ≥ l B and fix any non-null C satisfying C ∩ ( A ∪ B ) = ∅ . Choose any x, y, z such that x (cid:31) y and suppose xCy ∼ B z . It follows that µ B ( C ) u ( x ) + (1 − µ B ( C )) u ( y ) ≥ µ A ( C ) u ( x ) + (1 − µ A ( C )) u ( y ) . (9)Since u ( x ) − u ( y ) > , simplifying the above yields µ B ( C ) ≥ µ A ( C ) . The result then followssimply from the fact that µ B ( C ) = δ ( B ) µ ( C ) and µ A ( C ) = δ ( A ) µ ( C ) . The reverse directionis routine. A.5 Proof of Proposition 3
Proof.
Suppose { (cid:37) iA } A ∈ Σ admit representations ( u i , µ i , δ i ) i =1 , where u ≈ u . Then withoutloss u = u = u . Suppose agent is more conservative than agent . Note that A and A c are (cid:37) i -non-null if and only if µ i ( A ) ∈ (0 , . Pick some x, y , y satisfying the conditions ofDefinition 3; then u ( x ) µ ( A ) + u ( y )(1 − µ i ( A )) = u ( x ) µ ( A ) + u ( y )(1 − µ ( A )) . Case 1: ( µ ( A ) = µ ( A ) ). It is without loss to suppose y = y = y for some y andthat u ( x ) = 0 . Further, we may ignore the dependence of µ on i . If { (cid:37) A } A ∈ Σ is moreconservative than { (cid:37) A } A ∈ Σ , it follows that δ ( A )(1 − µ ( A )) u ( y ) ≤ δ ( A )(1 − µ ( A )) u ( y ) ,or δ ( A ) ≥ δ ( A ) . The reverse direction is similar. Case 2: ( µ ( A ) (cid:54) = µ ( A ) ). Note that µ ( A ) − µ ( A ) = (1 − µ ( A )) − (1 − µ ( A )) (cid:54) = 0 . (10)By hypothesis, u ( x ) µ ( A ) + u ( y )(1 − µ i ( A )) = u ( x ) µ ( A ) + u ( y )(1 − µ ( A )) from which,when combined with (10), it follows that (1 − µ ( A ))[ u ( y ) − u ( x )] = (1 − µ ( A ))[ u ( y ) − u ( x )] . (11)From { (cid:37) A } A ∈ Σ is more conservative than { (cid:37) A } A ∈ Σ , we conclude that V A ( xAy ) = δ ( A )[ µ ( A ) u ( x ) + (1 − µ ( A )) u ( y )] + (1 − δ ( A )) u ( x ) such a pair of nested events must exist for at least one of A or B . δ ( A )[ µ ( A ) u ( x ) + (1 − µ ( A )) u ( y )] + (1 − δ ( A )) u ( x ) = V A ( xAy ) . Simplifying the above expression yields δ ( A )(1 − µ ( A )) u ( y ) − δ ( A )(1 − µ ( A )) u ( x ) ≤ δ ( A )(1 − µ ( A )) u ( y ) − δ ( A )(1 − µ ( A )) u ( x ) , which directly implies δ ( A )(1 − µ ( A ))[ u ( y ) − u ( x )] ≤ δ ( A )(1 − µ ( A ))[ u ( y ) − u ( x )] . (12)The result that δ ( A ) ≥ δ ( A ) then follows by combining (12) with (11) and the facts that u ( y i ) − u ( x ) < and < µ i ( A ) < for i = 1 , . A.6 Proof of Theorem 2
Proof.
The proof of this theorem is similar to the proof of Theorem 1. First, for each A ∈ Σ ,there exists a pair ( u A , M A ) such that (cid:37) ∗ A is represented by Equation 2. For any A ∈ Σ ∗ + ,define the binary relation (cid:68) ∗ A on F by f (cid:68) ∗ A g if and only if f Ag (cid:37) ∗ g . Then by Lemma 3, (cid:68) ∗ A has a multi-prior expected utility representation ( u, B ( M , A )) .As in the proof of Theorem 1, since (cid:37) ∗ A is complete on constant acts for every A it followsfrom Lemma 1 that for any A ∈ Σ ∗ + , x (cid:37) y if and only if x (cid:68) ∗ A y if and only if x (cid:37) A y . Henceit is without loss to suppose that u = u A .Next, let D A := conv ( M , B ( M , A )) . Since M is a closed subset of ∆( S ) , it is compact.Further since A is unambiguously non-null, B ( M , A ) is closed and hence also compact.Further, they are both convex. Then D A is also compact and convex.Now, suppose for contradiction that M A ⊆ D A is false. Then there exists some ˜ µ A ∈M A \ D A . Following an argument similar to a Theorem 1, there exists a separating hy-perplane a ∈ R | S | so that ˜ µ A · a > ˆ µ · a for all ˆ µ ∈ D A . Let ¯ z = max ˆ µ ∈ D A ˆ µ · a and let ¯ a = a − ¯ z (1 , . . . , . Then ˜ µ A · ¯ a > ≥ ˆ µ · ¯ a for all ˆ µ ∈ D A . (13)We may suppose that ¯ a ∈ [ − , | S | , since we can always multiply both sides of (13) by (cid:15) > .22urther, there are acts f, g such that u ( g ( s )) − u ( f ( s )) = ¯ a ( s ) for every s ∈ S . Consequently, (cid:88) s ∈ S ˜ µ A ( s ) u ( g ( s )) > (cid:88) s ∈ S ˜ µ A ( s ) u ( f ( s )) (14)and (cid:88) s ∈ S ˆ µ ( s ) u ( f ( s )) ≥ (cid:88) s ∈ S ˆ µ ( s ) u ( g ( s )) for all ˆ µ ∈ D A . (15)By construction, M , B ( M , A ) ⊂ D A and so by (15) it follows that f Ah (cid:37) ∗ gAh and f (cid:37) ∗ g .However, by (14) f (cid:54) (cid:37) ∗ A g , which contradicts Unambiguous Dynamic Conservatism. Hence M A ⊆ D A . References
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