MMechanism Design with News Utility ∗ Jetlir Duraj † Abstract
News utility is the idea that the utility of an agent also depends on changes inher beliefs over consumption and money. We introduce news utility into otherwiseclassical static Bayesian mechanism design models. We show that a key role isplayed by the timeline of the mechanism, i.e. whether there are delays between theannouncement stage, the participation stage, the play stage and the realization stageof a mechanism. Depending on the timing, agents with news utility can experiencetwo additional news utility effects: a surprise effect derived from comparing to pre-mechanism beliefs, as well as a realization effect derived from comparing post-playbeliefs with the actual outcome of the mechanism.We look at two distinct mechanism design settings reflecting the two mainstrands of the classical literature. In the first model, a monopolist screens an agentaccording to the magnitude of her loss aversion. In the second model, we considera general multi-agent Bayesian mechanism design setting where the uncertainty ofeach player stems from not knowing the intrinsic types of the other agents. We giveapplications to auctions and public good provision which illustrate how news utilitychanges classical results.For both models we characterize the optimal design of the timeline. A timelinefeaturing no delay between participation and play but a delay in realization is neveroptimal in either model. In the screening model the optimal timeline is one withoutdelays. In auction settings, under fairly natural assumptions the optimal timelinemay have delays between all three stages of the mechanism.
Most situations in practice to which the theory of classical static Bayesian mechanismdesign is applicable can be thought of as consisting of three distinct stages: first, themechanism is announced to the agents and the agents decide whether to participate;second, the agents decide what to play in the mechanism and finally, the mechanismoutcome consisting of a consumption allocation and money transfers to the designer isrealized.Classical models of Bayesian mechanism design generally assume that the agents pos-sess quasi-linear utility and are Expected Utility maximizers. Absent discounting issuesthe analysis is the same in the classical model for the cases where the above mentioned ∗ I am thankful to Drew Fudenberg, Matthew Rabin and Tomasz Strzalecki for continuous support inthis project as well as to Fabian Herweg for introducing me to this topic. I thank Daniel Clark and KevinHe for numerous comments as well as Arjada Bardhi, Krishna Dasaratha, Jerry Green, Annie Liang andEric Maskin for comments during different stages of this project. Any errors are mine. † [email protected] a r X i v : . [ ec on . T H ] A ug tages may happen with delay from each other. This is because the agents in the classicalmodel are time-consistent as well as insensitive to the timing of the realization of uncer-tainty. This paper characterizes mechanism design for agents who violate the last twoassumptions in a specific way: they experience news utility and loss aversion . These twofeatures make the agents sensitive to whether uncertainty is resolved with delay and leadto time-inconsistent behavior. We look at the case of news-utility agents who are sophis-ticated with regard to this time-inconsistency. These assumptions lead to considerabledifferences to classical analysis: besides changes in several key intuitions from classicalsettings, the issue of the optimal design of the timeline of the mechanism becomes salient.More precisely, in this paper we assume agents possess quasi-linear intrinsic utility andadd the innovation that their utilities depend on changes in their beliefs over consumptionand money (henceforth called news utility ). News utility is assumed separable in the goodand money dimension and the comparison of new to old beliefs uses a classical gain-lossfunction featuring loss aversion (see e.g. [Kahneman, Tversky ’79]) – relative to goodnews, utility losses from ‘bad news’ are compounded due to loss aversion. The agents inthis paper are forward-looking with respect to both intrinsic and news utility as well assophisticated about their future behavior. We assume news utility is produced only fromobjective sources, i.e. there is no self-production of news utility and that each agent takesinto account future news utility in expectation in any decision instance. Just as for theclassical part of the utility, the belief over future consumption and money used to weighnews utility is induced by an agent’s play and other random factors in the environment.We consider the consequences of such preferences in two different Bayesian mechanismdesign settings reflecting two of the main strands of the classical literature: monopolisticscreening of a single agent as well as multi-agent mechanism design such as auction orpublic goods settings. In the screening model we assume the uncertainty facing the mo-nopolist concerns a behavioral parameter of the agent: her loss aversion level in the goodand money dimension. We also assume that the agent learns her intrinsic type only uponconsumption. This simple model already matches many situations in the real world wherethe intrinsic value of a consumption good is discovered only upon consumption. In themulti-agent model we assume that all ‘behavioral’ features of the preferences are commonknowledge, the agents know their intrinsic type when presented with the mechanism, andan agent’s uncertainty only comes from not knowing the intrinsic type of the other agents.Therefore, the informational side of the multi-agent model is a straightforward extensionof the classical multi-agent Bayesian mechanism design model with quasilinear utility,whose prime examples in the literature are auction settings or provision of public goods.In stark contrast to the classical setting, with the new preferences it matters whetherthere are delays between stages of a mechanism so that we distinguish three main timelinesfor the analysis. In timeline A, the mechanism is implemented without delay: the announcement of itsexistence, the decision to participate and what to play happen almost concurrently so that Besides strategic use of delays by a designer, exogenously given delays due to technological constraintsbetween stages where uncertainty persists in the agents’ minds are a recurrent feature of life: goods needto be produced, information must travel, etc. Additional timelines are equivalent to the ones presented here under two simplifying assumptions:no-discounting of utility and the the designer cannot randomize. See subsection 1.1 and Proposition 1for more details. Figure 2 from subsection 2.1 and Figure 3 from subsection 3.1 respectively depict indetail the timelines for the two different mechanism design models. surprise effect ) coming fromcomparing the pre-mechanism beliefs in the consumption and money dimensions pinneddown by her outside option with the new (degenerate) beliefs induced by the realizationof the mechanism outcome.In timeline B the participation and play stages of the mechanism happen withoutdelay but the mechanism outcome is realized with delay. Besides the surprise effectthe agent experiences now a second bout of news utility coming from comparing theactual realization of the mechanism outcome with the distribution induced by her playdecision and the environment (dubbed realization effect ). She takes this into account inexpectation at the play stage. This results in lower interim utility because of loss aversion:delaying the outcome after a play decision hurts the agent because bad news hurt morein expectation than good news elate.Finally, in timeline C there is a delay between the participation stage and the playstage, besides the delay between the play stage and the moment the mechanism outcomeis realized. In this case the agent’s time inconsistency becomes observable as differentselves with different objectives decide on participation and play. The play-self doesn’ttake into account the surprise effect of the participation self whereas the latter takes intoaccount the future behavior of the play-self. This wedge between participation and playcan be fruitfully used by the designer in certain situations as she may be able to exploit theplay-self better once she is locked-in after deciding to take part in the mechanism. Thiscomes at a cost though: in optimal mechanisms the participation-self, being sophisticatedand anticipating her future decisions, may need to be subsidized in comparison to theother timelines.Say that a direct mechanism is incentive compatible if revealing own private informa-tion is an equilibrium for the respective timeline of the mechanism. Similar to the classicalsetting, monotonicity conditions are key in characterizing incentive compatibility and theresulting expected transfers. But due to news utility, monotonicity applies to modifiedinterim perceived valuations instead of the interim intrinsic valuations from the classicalsetting. As a consequence, incentive compatible allocation rules only determine perceivedexpected transfers to the designer, up to type-independent constants.Say that an incentive compatible mechanism is individually rational if it gives anequilibrium participation utility higher than the outside option to every agent. Whereasthe same self decides about incentive compatibility and individual rationality wheneverthere is no delay between participation and play decisions (timelines A and B), differentselves decide on them when there is a delay between the two decision stages (timeline C).For allocation rules whose incentive compatibility is unproblematic in the classicalmodel, loss aversion can lead to failure of incentive compatibility whenever it is highenough and the timeline features delays. We illustrate this in the case of the ex-postefficiency rule for public good provision with symmetric agents and private informationconcerning the intrinsic type. We also show how incentive compatibility may be restoredin that setting under certain conditions on the distribution of the intrinsic valuation ofthe public good, whenever the number of agents is high enough. Intuitively, with a largenumber of agents the law of large numbers kicks in and the ex-post efficiency rule impliessmall news utility costs for delays as the probability of provision becomes either very highor very low. Ex-post efficiency means the designer would like to maximize the welfare of the agents under completeinformation.
This subsection explains in detail the preferences and the decision procedure of the agents.Agents experience intrinsic utility from actual consumption and payments as well as newsutility from changes in beliefs.
Intrinsic utility.
The agent derives intrinsic utility from consumption of a profile a of consumption goods coming from a set of physical allocations A which is a closed,connected subset of an Euclidean space R d , d ≥ t . Moreover, the utility of the agent from the pair of consumption goods and monetarypayment ( a, t ) depends also on a parameter which we call her intrinsic type θ and which4omes from a closed interval of R denoted Θ = [ θ, ¯ θ ]. We assume that intrinsic utility foreach agent with type θ is quasilinear and of the form V ( a, θ, t ) = v ( a ) θ − t. (1) v : A→ R + is assumed differentiable. Let in the following ∆( A × Θ × R ) denote the set ofBorel probability distributions over A × Θ × R .We assume the agent conforms to Expected Utility in the intrinsic part of her utility.That is, if the uncertainty over u = ( a, θ, t ) is captured by a distribution G ∈ ∆( A× Θ × R )then the agent experiences expected intrinsic utility of the form E u ∼ G [ V ( u )] . Hereafter, u ∼ G means the random variable u is distributed according to the distribution G . News utility.
Besides intrinsic utility, the agent experiences news utility whenever herbeliefs about the realization of u objectively change from H to some G ∈ ∆( A × Θ × R ).News utility is experienced in two dimensions: in the consumption dimension (upper index g in the following, g stands for good ), and in the monetary dimension (upper index m inthe following). For any G ∈ ∆( A × Θ × R ) denote by G g the distribution of v ( a ) θ inducedby G . This depends only on the marginal distribution of G on A ×
Θ. Moreover, let G m be the marginal distribution of G on the monetary payments. Whenever the agent’sbelief changes from G to H she experiences news utility in the dimension j = g, m givenby N j ( G j | H j ) = µ j (cid:90) ξ j ( c G j ( p ) − c H j ( p )) dp. Here for any real-valued distribution F , c F ( p ) is the p -percentile of F , p ∈ (0 , More-over, ξ j ( y ) = (cid:40) y if y ≥ λ j y if y < Here λ j > j . A negative change causesthe agent to experience a disutility greater in magnitude than the elation caused by apositive change of the same size. µ j > j . Overall, newsutility of the change from H to G is given through the sum of the news utilities of thetwo dimensions. N ( G | H ) = N g ( G g | H g ) + N m ( G m | H m ) . The timelines we have in mind are the following. First, the agent is offered a menu oflotteries C which is a subset of ∆( A × Θ × R ). We assume this is a non-empty compact set For any p ∈ (0 ,
1) the p -percentile c F ( p ) of F is determined by the conditions F ( c F ( p )) ≥ p and F ( c ) < p for any c < c F ( p ). This percentile-per-percentile comparison first appeared in [K¨oszegi, Rabin ’09]. See [Pagel ’17] foralternative specifications of the news utility which can incorporate correlation between dimensions.
5f ∆(
A × Θ × R ) where we have equipped the latter with topology of weak convergenceof probability measures (this fits both mechanism design models below). Then she picksa lottery from C and finally the outcome of the lottery is realized. There may exist delaysbetween the moment the menu C is offered to the agent and the moment she chooses from C as well as the moment the uncertainty from the lottery she picked from C is realized.These delays may be due to technological constraints or they may be introduced throughthe outside party, call it designer, which designs the menu C . We assume that there isno discounting of time whenever a delay is present. The timelines the agent may face depending on the timing of delays are depicted inFigure 1. ADCB
Accept ordecline MChoice from M (if accept)Uncertainty isrealized delay delaydelaydelay
Accept ordecline M Uncertainty isrealizedChoice from M (if accept) Choice from M (if accept)Choice from M (if accept)Accept ordecline MAccept ordecline M Uncertainty isrealizedUncertainty isrealized
Figure 1: Different timelines.In the following we also assume that the agent has an exogenously given belief over
A × Θ × R given by F . This is what she expects to happen if she is not notified of theoption of choosing from C . In mechanism design settings below F is determined by theoutside option of the agent.To calculate the overall utility of an agent we impose the following assumptions onthe agent’s behavior. Assumption 1:
The agent is sophisticated, Bayesian and forward-looking. That is,she takes into account the optimal behavior of future selves, uses Bayes rule to update Some typical examples of technological constraints comprise settings where communication takestime or where delivery of payments/goods or production of a good whose consumption value is uncertaintakes time. In the mechanism design settings we consider not every menu C out of ∆( A × Θ × R ) is feasible. Inparticular, we don’t allow the designer to randomize so that the subjective randomness the agents faceis only due to the environment. It is not hard to introduce discounting to the model but discounting doesn’t yield any additionaldeep insight besides making the model much more cumbersome. Additional timelines featuring delay between the announcement of the existence of C and the momentthe agent is required to decide on whether to accept C or not are equivalent to existing ones under the no-discounting assumption by the same argument as we show for timelines C and D below (see Proposition1). Those timelines would be relevant in a model of ‘deciding when to decide’ which is outside the scopeof this paper. C the agent takes into accountthe actual choice she will make from C . Moreover, at each decision moment after a delayno past intrinsic or news utility is taken into account. We note here that Assumption1 doesn’t necessarily imply a temporal coordination of selves in the sense that the selfpicking from C needs to break ties in favor of the self who decides to accept or reject C . Assumption 2:
News utility is produced only from objective sources, that is, there isno self-production of news utility. In particular, the agent conforms with Expected Utilityat each moment in time with respect to any subjective randomization device. Under this assumption it is unproblematic to assume that the agent doesn’t possess arandomization device when picking from C . Assumption 3:
At any moment in time, if no delay is present there is at most one newnews utility term. It comes from comparing beliefs before an objective source of newswith those after the objective source of news.Assumptions 2 and 3 imply that there are at most two instances of production of newsutility in the above timelines:1) when the menu C is presented to the agent if she decides to accept it (surpriseeffect) , as well as2) when the uncertainty of the lottery she picked from the menu is realized (realizationeffect) .Assume the agent ultimately chooses F ∈ C . Then the surprise effect in timelinesB,C,D corresponds to experiencing N ( F | F ). The realization effect for timelines B,C andD corresponds to experiencing N ( u | F ) whenever u ∈ A × Θ × R is the realization of F .In timeline A the two news utility effects coincide and there is only one news utilityterm comprised of N ( u | F ) whenever u ∈ A × Θ × R is realized. This is because timelineA stands for the case where the decisions of whether to accept C , what lottery to pick outof C and the realization of the resulting uncertainty all happen without delay and almostconcurrently. Assumption 4:
Each agent in a decision moment takes into account future news util-ity terms in expectation by weighting them with the belief induced by her actual decisionbe it a decision on or off -equilibrium path. This implies for an agent in timelines
B, C, D who accepted the menu C , that the expec-tation of her news utility from the realization effect whenever she picks F ∈ C is given by E u ∼ F [ N ( u | F )]. We call such a term an expected news utility term . An idea of coordination among selves to give a past self a higher utility underlies the PPE conceptin [K¨oszegi, Rabin ’06]. The related PPE concept used in [K¨oszegi, Rabin ’06] (see also [K¨oszegi, Rabin ’09]) to model agentswho experience expectation-based loss aversion allows for the possibility of self-production of news utilityand therefore adds an additional constraint to the maximization problem of the agent. This may lead toexistence and characterization problems (see [K¨oszegi ’10]). These are excluded by our assumptions. Just as in the case of Assumption 2 this is in stark difference to the PPE solution concept in[K¨oszegi, Rabin ’06] and [K¨oszegi, Rabin ’09]. ecision procedure of a single agent. In timeline A the expected news utility fromaccepting the menu C and choosing F ∈ C enters the overall decision utility of the agentas E u ∼ F [ N ( u | F )].For the case that the agent doesn’t accept menu C she experiences utility O ( F ) = E u ∼ F [ V ( u )] + E u ∼ F [ N ( u | F )] , regardless of whether the realization of F happens with a delay or not.
13 14
For the casethat F is degenerate, say puts probability one on u , we have O ( F ) = V ( u ).Whenever the agent accepts the menu, she experiences in timeline A the decisionutility max F ∈C E u ∼ F [ V ( u )] + E u ∼ F [ N ( u | F )] . She accepts C if and only if this utility is higher than O ( F ).In timeline B she accepts C if and only if her decision utility from C given bymax F ∈C E u ∼ F [ V ( u )] + E u ∼ F [ N ( u | F )] + E u ∼ F [ N ( u | F )] (3)is higher than O ( F ). She then picks an F from C where the maximum is attained.In timelines C and D different selves of the agent with distinct perspectives decide onwhether C should be accepted and then on the lottery picked out of C . For the case thatthe agent has accepted C her decision utility from choosing out of C ismax F ∈C E u ∼ F [ V ( u )] + E u ∼ F [ N ( u | F )] . Being sophisticated, she then accepts C if and only if her decision utility E u ∼ F [ V ( u )] + E u ∼ F [ N ( u | F )] + E u ∼ F [ N ( u | F )], evaluated at the F she expects to choose out of C , ishigher than O ( F ). If there are multiple F ∈ C which are optimal we assume she breaksties deterministically and that her self at the moment of deciding whether to accept C anticipates the tie-breaking correctly. This is in line with Assumption 1 above.Formally, we assume the following condition about possible tie-breaking. Deterministic tie-breaking.
Whenever the agent is indifferent between accepting C and rejecting, she accepts it. Whenever the agent is indifferent between distinct elementsof C at the choice-out-of-menu stage she breaks ties deterministically.
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Our Assumptions together with deterministic tie-breaking ensure a full characterizationof behavior. It is easy to see that the behavior in timeline D is the same as in timelineC. The following Proposition registers this property as well as another simple one whichhas important consequences in mechanism design settings. O stands for outside option . She experiences only one news utility term given by N ( u | F ) whenever u is realized and takes ex-pectation of it by weighting with F . This is because F is not a surprise, i.e. it is expected by the agentat the beginning of time. Deterministic tie-breaking is in line with Assumption 2 above. We assume it here so that we canfocus on the behavioral features of the model rather than technicalities. All settings considered in this paper correspond to menus whose elements are parametrized througha compact one-dimensional interval so that it is easy to write down deterministic tie-breaking rules. roposition 1.
1) Expected news utility terms of the form E u ∼ F [ N ( u | F )] are non-positiveand vanish if and only if F puts unit mass on a single element u .2) The behavior of the agent is the same in timelines C and D. The proof of the equivalence of C and D is contained in the text above. It reliescrucially on the no-discounting assumption as well as on Assumption 2. Part 1) is animplication of loss aversion. Because of loss aversion percentile comparisons between therealized u and the percentiles of the ‘reference’ distribution F are weighted asymmetricallydepending on whether they correspond to a loss or gain; losses in the dimension j get aweight of µ j λ j whereas gains only of µ j in units of intrinsic utility. Averaging out acrossrealizations of u results in a negative expected news utility effect.Due to Proposition 1 we identify timelines C and D in the rest of the paper. This paper connects to different strands of the mechanism design literature as well as ofthe applied behavioral literature. The focus on optimal timeline choice for preferenceswhich are sensitive to the timing of announcements and additionally feature loss aversionseems new in the literature.[Ely et al. ’15] analyzes the optimal way to disclose information to an agent whosepreferences depend on the path of the belief change. In their model the agent has pref-erence for late resolution of uncertainty as she likes to experience suspense and surprise.In our model the agent exhibits preference for early resolution of uncertainty due to lossaversion. Moreover, the goal of the designer in [Ely et al. ’15] is to maximize welfare ofthe agent whereas we focus on profit maximization. Finally, their model features a fixedtimeline whereas we also study the optimal choice of the timeline in our model.There are by now several screening models where sophisticated agents exhibit loss-aversion. [Carbajal, Ely ’14] proposes a screening model of reference dependent and lossaverse consumers where the reference point is non-stochastic and depends linearly onthe private information of the consumer. [Hahn et al. ’18] consider price discriminationwith loss-averse consumers. Similar to the model proposed in Section 2 they assume thebuyers don’t know their intrinsic valuation at the moment they face the menu of bundlesthe monopolist offers. They assume that the reference point is the menu of bundlesthe monopolist offers and work with ex-post participation and incentive compatibilityconstraints. Loss aversion parameters are known by the monopolist in their model. Ourmodel assumes the monopolist has imperfect information about loss aversion parameters.Moreover, our model introduces and studies the issue of designing the optimal screeningtimeline which is missing from both papers mentioned.The applied behavioral literature offers several models where a designer, say a mo-nopolist or a firm in a competitive market, screens on behavioral features of the agents.[Eliaz, Spiegler ’06] and [Eliaz, Spiegler ’08] consider a designer who faces agents who mayhold potentially incorrect beliefs about their future utility and screens respectively on thelevel of sophistication or on the level of optimism of the agent. [Heidhues, K¨oszegi ’17]offer a related model and study welfare consequences of screening for the sophistication In the mechanism design settings below there is no uniform result as to which of timelines C,D isbetter once discounting is allowed. Intuitively, discounting applies in timeline C to both intrinsic utilityand news utility at the choice-from-menu stage whereas it applies to neither in timeline D. Nevertheless, since in our multi-agent model uncertainty per-sists even with deterministic mechanisms, we get a related result to theirs in our setting:it may be optimal for the designer to not insure the agent against future uncertainty.Finally, we stress that their paper focuses on PPE types of equilibria which can’t arise inthis paper.To the best of our knowledge, [Eisenhuth ’17] is the first paper considering a full-fledged mechanism design model of auctions with preferences which exhibit expectation-based loss aversion. He uses the equilibrium concept from [K¨oszegi, Rabin ’07] (CPE)and solves for the optimal symmetric auction with symmetric bidders. His environmentis most similar to timeline B in this paper. [Herweg et al. ’10] introduce the same CPEpreferences in the classical principal-agent model with moral hazard and show how theoptimal contract is much simpler than in the classical model. In a new paper, [Benkert’ 17]considers the optimal mechanism problem for the bilateral trade model where both buyerand seller behave according to CPE. He solves for the optimal mechanisms for a specialdistribution of types. Within the same timeline our paper additionally looks at othertopics from classical Bayesian mechanism design: auctions, public goods. Moreover, wealso address the issue of the optimal timing of the realization of the mechanism, whichdoesn’t occur in either of the above-mentioned works.This paper contributes to the emerging literature on strategic interaction of agentswhose utility depends on their beliefs about present and future consumption and moneytransfers. Relatedly, [Dato et al. ’17] characterize existence properties of strategic equi-librium based on the preferences of [K¨oszegi, Rabin ’06] (their PE and PPE concepts) and[K¨oszegi, Rabin ’07] (their CPE concept) in finite normal-form games and focus mostlyon existence and uniqueness as well as characterizing when equilibrium play is the sameunder classical and expectation-based loss averse preferences. The fact that ex-post effi-ciency in the public goods setting with timelines B,C,D may fail incentive compatibility(see subsection 3.2.1) is a reflection of the same non-existence phenomenon they identify In fact, the result in [Heidhues, K¨oszegi ’14] ceases to hold if the agent is given the chance to decideabout the purchase before seeing the distribution of prices in period 1. Ex-post participation constraintsare less appropriate in settings where the value of the product to the consumer is revealed only uponconsumption and the agent can commit to the mechanism before consumption. Similarly to our results in the online appendix [Eisenhuth ’17] establishes that optimal auctions forCPE preferences are all-pay with a reference price. He also considers a model of wide-bracketing, which isassumed away in our model due to the separability assumption across the two dimensions, consumptionand money.
10n their model.
This section offers a tractable model of screening with news utility agents whose privateinformation concerns their loss aversion parameters. Naturally, assuming that privateinformation of an agent includes multiple parameters of her behavioral preferences leadsto a multidimensional screening problem. Here we avoid the technical difficulties of mul-tidimensional mechanism design by assuming symmetry for loss aversion in the moneyand good dimension and that the designer knows all news utility parameters but one: theloss aversion parameter. We focus on a model of monopolistic screening where the buyersdon’t know the realization of their intrinsic utility at the moment of participation decisionand choice of contract. Real life settings approximated by this model would be buying tickets to a concertfrom an unknown band, buying a book from an unknown author, or vacationing in anunknown destination, etc.
We assume the designer is a monopolist producing non-negative quantities of a gooddenoted by q at a fixed marginal cost c > A buyer has intrinsic utility from acontract ( q, t ) of the form v ( q ) θ − t. Here θ is the intrinsic value of the good. v fulfills standard assumptions: v (0) = 0 , v ≥ , v (cid:48) > , v (cid:48)(cid:48) < x → v (cid:48) ( x ) = + ∞ . Moreover, for purely technical reasons weadditionally require a weak growth condition on v : there exists some p > v (cid:48) ( x ) ≥ Kv p − ( x ) for some K > Figure 2 below depicts the relevant timelines with news utility. We assume the timeline T ∈ { A, B, C } is either given through technological constraints or it is chosen by thedesigner. The informational assumptions of the screening model are as follows. Assumption (S)
Intrinsic utility θ is distributed according to a probability distribution F with bounded support over the non-negative numbers. F is common knowledge and itfulfills m := E θ ∼ F [ θ ] > µ g = µ m = 1 on news utility and have λ g = λ m = λ . The distribution of λ hasa continuously differentiable and strictly positive density g on [1 , ¯ λ ] (2 ≥ ¯ λ >
1) and itsc.d.f. G is common knowledge. It is possible to construct a more complicated model with signals which partly reveal the incompleteinformation about θ after or before the contract is signed. If the signals are public knowledge however,the more general case is easily reduced to a model similar to the one in this section. This can be relaxed to a weakly increasing marginal cost c and none of the results would qualitativelychange. We keep a constant marginal cost throughout for ease of exposition. This weak condition is crucial for our proof of existence of an optimal mechanism in timeline C. Itsays that the marginal utility v (cid:48) doesn’t fall too fast with x . As [Masatlioglu, Raymond ’16] establish, λ ≤ DCB
Participationdecision contractchoiceƟ revealed,consumption,paymentsParticipationdecision delay delaydelay
Participationdecision delay
Participationdecision PlaydecisionOutcomeannounced Ɵ revealed,consumption,paymentscontractchoice contractchoice Ɵ revealed,consumption,payments
Figure 2: Relevant timelines for the monopolistic screening model.Denote for future reference M := E θ,s ∼ F [( θ − s ) { θ ≥ s } ]. It is easy to see that M < m .The outside option which determines the pre-mechanism beliefs of each buyer typeconsists of a zero utility: zero amount of good and zero transfers to the designer areexpected in the absence of any mechanism.We focus in this section on the case of deterministic mechanisms, i.e. we assume forsimplicity in exposition that the monopolist doesn’t possess a randomization device. Wecomment in the end of this section on how the results change with randomized mecha-nisms.Mechanisms consist of menus C = { ( q ( λ ) , t ( λ )) } λ ∈ [1 , ¯ λ ] the monopolist offers. This isa set of contracts indexed by loss aversion specifying the quantity of the good q and theprice of that good at that quantity. In the framework of subsection 1.1 the buyers areagents facing a menu of lotteries over ∆([0 , ∞ ) × Θ × R ) with the property that themarginals over the quantity q and payment t are degenerate. The lotteries in the menuare indexed by λ ∈ [1 , ¯ λ ].In the remaining part of this section we characterize individual rationality and incen-tive compatibility for all timelines and finally look at the optimal timeline choice for thedesigner. It is without loss of generality for optimal mechanisms to consider onlypayment schedules t ≥ q (ˆ λ ) , t (ˆ λ )) then the news in our setting of timelines B,C (and therefore also D). v ( q (ˆ λ )) m − λt (ˆ λ ) . (4)She also experiences consumption utility v ( q (ˆ λ )) θ − t (ˆ λ ), which at the decision momentis in expected utility terms v ( q (ˆ λ )) m − t (ˆ λ ) . Gathering the terms together, we see that utility of a buyer of type λ from declaring ˆ λ is2 mv ( q (ˆ λ )) − (1 + λ ) t (ˆ λ ) . Denote Γ A ( λ ) = 2 m λ . (5)We call Γ A ( λ ) the A-virtual type of the buyer. The term in the denominator reflectsthe negative surprise effect in the money dimension. The numerator reflects the surpriseeffect in the good dimension. The
A-virtual type is a decreasing function of the agent’sloss aversion parameter.Incentive compatibility is characterized byarg max ˆ λ ∈ [1 , ¯ λ ] (cid:110) (1 + λ ) (cid:104) Γ A ( λ ) v ( q (ˆ λ )) − t (ˆ λ ) (cid:105)(cid:111) = λ. Timeline B:
Again it is without loss of generality for optimal mechanisms to consideronly payment schedules t ≥ λ has decided to choose bundle ( q (ˆ λ ) , t (ˆ λ )), she willfirst experience news utility from the comparison with the pre-mechanism expectationsjust as in timeline A.When the buyer learns her draw of the material valuation θ she experiences newsutility in the good dimension from comparing the outcome to her previous belief v ( q (ˆ λ )) s where s is distributed according to F : v ( q (ˆ λ )) (cid:90) ( θ − s ) { θ ≥ s } + λ ( θ − s ) { θ
13e call this the
B-virtual type of the agent. It is a decreasing function of her loss aversionparameter. The denominator reflects the surprise effect in the money dimension while thenumerator reflects both the surprise as-well-as the realization effect in the consumptiondimension.
Incentive Compatibility is characterized byarg max ˆ λ ∈ [1 , ¯ λ ] (cid:110) (1 + λ ) (cid:104) Γ B ( λ ) v ( q (ˆ λ )) − t (ˆ λ ) (cid:105)(cid:111) = λ. Timeline C.
At the moment of contract choice the agent experiences no surprise effectas the contract doesn’t constitute news anymore but she still takes into account therealization effect in expectation. Given this,
Incentive Compatibility is characterized byarg max ˆ λ ∈ [1 , ¯ λ ] (cid:110) [ m + (1 − λ ) M ] v ( q (ˆ λ )) − t (ˆ λ ) (cid:111) = λ. Denote Γ C ( λ ) = m + (1 − λ ) M the C-virtual type . It includes the expected future newsutility from the realization of the mechanism as well as the expected value of consumption.It is decreasing in the loss aversion parameter λ .Standard methods yield then the following characterization of incentive compatiblemechanisms. Here we call an allocation rule q : [1 , ¯ λ ] → R + is implementable if there existsan incentive compatible mechanism C with allocation rule q . Proposition 2.
1) An allocation rule q : [1 , ¯ λ ] → R + is implementable if and only if q ( · ) is non-increasing.2) (Mirrlees Representation) For any implementable q the corresponding payments t : [1 , ¯ λ ] → R are given up to a type-independent constant by the following Mirrlees repre-sentations. ( A ) t A ( s ) = Γ A ( s ) v ( q ( s )) − m (cid:90) ¯ λs v ( q ( t ))(1 + t ) dt, ( B ) t B ( s ) = Γ B ( s ) v ( q ( s )) − m + M ) (cid:90) ¯ λs v ( q ( t ))(1 + t ) dt, ( C ) t C ( s ) = Γ C ( s ) v ( q ( s )) + M (cid:90) s v ( q ( σ )) dσ. The type-independent constant not depicted in part 2) of the Proposition is a fixedpayment to the monopolist (or transfer to the agents) which is independent of the pri-vate information of the agents. In an optimal mechanism its value is determined by theindividual rationality requirement.
The following Proposition gives the individual rationality characterization for incentivecompatible mechanisms where the payment schedules t are non-negative. Proposition 3.
Fix a timeline T ∈ { A, B, C } . An incentive compatible contract C = { ( q ( λ ) , t T ( λ )) } λ ∈ [1 , ¯ λ ] is individually rational for timeline T if the following respective setsof inequalities are satisfied. This is without loss of generality for optimal mechanisms as we show in the appendix. For timelines T = A, B Γ T ( λ ) v ( q ( λ )) − t T ( λ ) ≥ , λ ∈ [1 , ¯ λ ] . • For timeline C Γ B ( λ ) v ( q ( λ )) − t C ( λ ) ≥ , λ ∈ [1 , ¯ λ ] . The expressions for timelines A, B are similar to ones from classical models except forthe fact that one has to use modified virtual types which take into account news utilityeffects.For timeline C , at the participation stage the agent anticipates that she’ll be truthfullater, if she accepts an incentive compatible menu C . In all incentive compatible mecha-nisms in timeline C the payment schedule t ( λ ) is weakly decreasing in λ . Given this, andthe fact that the monopolist can always offer the bundle (0 ,
0) to any type it follows thatthe optimal contract will never feature a net subsidy t ( λ ) < λ ∈ [1 , ¯ λ ]. It follows that in case of acceptance her utility if she is of type λ is given by V ( λ ) = [2 m + (1 − λ ) M ] v ( q ( λ )) − (1 + λ ) t ( λ ) . Namely, the agent experiences the following utility items: news utility from accepting themechanism, intrinsic expected utility from future consumption and finally expected futurenews utility from the realization effect.We show in the appendix that V ( λ ) can be rewritten as V ( λ ) = (1 + λ ) (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) v ( q ( λ )) − (1 + λ ) f − (1 + λ ) M (cid:90) λ v ( q ( s )) ds. where f is a type-independent payment. The individual rationality requirement can thenbe written as (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) v ( q ( λ )) − f − M (cid:90) λ v ( q ( s )) ds ≥ , λ ∈ [1 , ¯ λ ] . The fact that different selves decide on participation and bundle choice creates a‘wedge’ between individual rationality and incentive compatibility requirement. A mea-sure for this discrepancy is precisely the multiplicative factor appearing in the individualrationality constraint: Γ B ( λ ) − Γ C ( λ ) = (1 − λ ) m + ( λ − λ ) M λ . We consider the timelines one after the other. Finally, for the case that the monopolistcan pick the timeline we establish general results about the optimality of the timeline. This follows immediately from the Mirrlees representation coupled with the envelope theorem. If this were not true then it would hold t ( λ ) < λ > λ . Going over to t ( λ ) = 0 and q ( λ ) = 0 for all λ ≥ λ preserves incentive compatibility but increases profits. Establishing the existence of an optimal mechanism in timelines A and B uses classical methods ofpointwise maximization whereas the problem in timeline C can be rewritten into a calculus of variationsproblem with constraints for which we show existence of a solution and give a recipe in the online appendixon how to find it in many typical examples. imeline A. We establish in the appendix that the profit function for an incentivecompatible and individually rational menu of contracts for timeline A looks as follows.Π A = (cid:90) ¯ λ (cid:2) Ψ A ( s ) v ( q ( s )) − cq ( s ) (cid:3) G ( ds ) , (6)with Ψ A ( s ) = Γ A ( s ) − m G ( s )(1 + s ) g ( s ) . Ψ A ( s ) is the virtual valuation for timeline A. The virtual type Γ A ( s ) is corrected forthe informational rent of the agent represented here by 2 m G ( s )(1+ s ) g ( s ) .The monopolist problem is thus maximizing (6) under the constraint that q be non-increasing and that individual rationality is fulfilled.We give a specific example of the solution for timeline A. Example 1.
Assume G = unif orm ([1 , . We take F = unif orm ([0 , , v ( q ) = √ q . The virtualvaluation is calculated to be Ψ A ( s ) = 2(1 + s ) , s ∈ [1 , . The optimal allocation rule is q A ( λ ) = Ψ A ( λ ) c . In particular no type is excluded from the mechanism. One calculates that optimal profitfor timeline A is R A ( c ) = · · c . Timeline B.
The profit function for an incentive compatible and individually rationalmechanism for timeline B looks as follows.Π B = (cid:90) λ (cid:2) Ψ B ( s ) v ( q ( s )) − cq ( s ) (cid:3) G ( ds ) , (7)with Ψ B ( s ) = Γ B ( s ) − m + M ) G ( s )(1 + s ) g ( s ) . Ψ B is the virtual valuation for timeline B. Note that it is weakly lower than the virtualvaluation for timeline A. This is because as noted in Proposition 1 the realization effectis negative in expectation due to loss aversion. In comparison to timeline A, this resultsin a lower virtual type due to lower informational rents.The problem of the monopolist is maximizing (7) under the constraint that q is non-increasing and that individual rationality is fulfilled. Note that this is always positive and decreasing. There is no need to exclude types of high lossaversion from the mechanism or to use ironing techniques. imeline C. We show in the appendix that the objective function of the designer canbe written asΠ C = f · G (ˆ λ ) + (cid:90) ˆ λ (cid:40)(cid:34) Γ C ( λ ) + M G (ˆ λ ) − G ( λ ) g ( λ ) (cid:35) v ( q ( λ )) − cq ( λ ) (cid:41) dG ( λ )for a threshold type ˆ λ so that ( q ( λ ) , t ( λ )) = (0 ,
0) whenever λ ≥ ˆ λ (exclusion).Thus the problem of the designer for timeline C ismax ˆ λ ∈ [1 , ¯ λ ] ,f ∈ R ,q ( · ) (cid:90) ˆ λ (cid:40)(cid:34) Γ C ( λ ) + M G (ˆ λ ) − G ( λ ) g ( λ ) + f (cid:35) v ( q ( λ )) − cq ( λ ) (cid:41) dG ( λ )s.t.(1) q ( · ) is non-increasing,(2) (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) v ( q ( λ )) − M (cid:90) λ v ( q ( s )) ds ≥ f, λ ∈ [1 , ˆ λ ] . (8)Here Ψ C ( s, f ) = Γ C ( s ) + M G (ˆ λ ) − G ( s ) g ( s ) + f for s ≤ ˆ λ is the virtual valuation for timelineB. The virtual type Γ C is again corrected for the information rent and the (net) lump-sumsubsidy f to the participation self. Condition (2) in the above program is just a rewritingof the individual rationality constraint. f corresponds to a type-independent subsidywhich may need to be paid to ensure individual rationality. This subsidy accounts for theexternality which the self who chooses the bundle exerts on the participation self.We show that a solution for timeline C always exists. In the online appendix wealso show how to characterize it completely under some regularity requirements whichcorrespond to a no ironing condition in our setting. The second part of the followinggeneral result has a straightforward proof though. Proposition 4.
1) There always exists an optimal mechanism for timeline C.2) If F has support in the non-negative numbers the optimal fixed payment f in anoptimal mechanism for timeline C is negative, whenever profits are positive.
2) implies that in many cases the optimal payment schedule t ( · ) in timeline C isdiscontinuous in λ : high loss aversion types are excluded from the mechanism whereas alltypes who are served may receive a fixed, type-independent transfer from the monopolistto the agents, whenever the monopolist sells some positive amount. Due to the discrepancybetween the self choosing the contract and the self deciding whether to participate theformer self exerts an externality on the latter by disregarding the surprise effect. Thisexternality can be partially alleviated without adversely affecting incentive compatibilityin the second period by optimally transferring a fixed amount f < Optimal timeline.
Assuming that the monopolist can pick the timeline the followingTheorem is the main result of this subsection.
Theorem 1.
For the screening model timeline A is weakly better than timeline B whichis weakly better than timeline C. B in (2.3) is strictly lower than the virtual valuation fortimeline A given by Ψ A . Timeline A and B share the surprise effects at the participationdecision moment but the realization effect in the good dimension is absent from timelineB. This results in an increased willingness to pay for every type when compared to timelineB. Intuitively, timeline C may have a more favorable incentive compatibility situationoverall than timeline A since the payments from the buyers are not scaled down by 1 + λ .The latter happens in timeline A because of the negative surprise effect in the moneydimension. On the other hand, as Proposition 4 shows, whenever employing timeline Cthe monopolist has to subsidize participation with a lump-sum payment independent oftypes. As it turns out for our specification of the problem this subsidy is too costly evenfor small marginal costs c .The optimality of timeline A relies on two assumptions. First, our Assumption (S)corresponds to assuming a relatively ‘small’ λ m µ m in subsection 1.1. This makes for asmall negative surprise effect in the money dimension in timeline A. Second, we haveassumed that the monopolist can not offer stochastic payment schedules t ( λ ) ∈ ∆( R ).If the latter was possible then the incentive compatibility situation in timeline C is onone hand better than with deterministic contracts since the monopolist can use type-dependent lotteries as an additional screening device and on the other hand worse as nowfor every type ceteris paribus the perceived payments are higher. We conjecture thatrelaxing these two assumptions may result in timeline C optimality in some cases whereastimeline A still dominates timeline B. In this section we look at multi-agent mechanism design under the assumption that pri-vate information concerns the intrinsic type of other players. Behavioral parameters ofnews utility are common knowledge. For simplicity of exposition we focus on agentswhose intrinsic type spaces are identical. The results about incentive compatibility andindividual rationality can be generalized to asymmetric agents without difficulty. We consider a group of agents i = 1 , . . . , N who can potentially take part in a mech-anism. We assume throughout the designer has standard Expected Utility risk neutralpreferences, is interested in revenue maximization and that she has full commitment.An agent i derives intrinsic utility from consumption of a profile a of consumptiongoods coming from a set of allocations A as well as from a (net) monetary transfer to theprincipal which is denoted by t i . Formally here A is assumed to be a compact, connectedsubset with non-empty interior of R N . We assume that intrinsic utility for each agent i with type θ i is quasilinear and of the form (1). Types are one-dimensional and are given Details that timeline A still dominates B when relaxing the two assumptions are available uponrequest. See online appendix for applications to optimal mechanisms with asymmetric agents and timeline Ain two cases: optimal auctions with asymmetric agents and bilateral trade.
18y the interval Θ = [ θ, ¯ θ ] consisting of non-negative numbers. θ i is agent i ’s (intrinsic)type and we denote Θ = Θ × · · · × Θ n the product of the type spaces.If the agent doesn’t participate in the mechanism the value of her intrinsic utility in theallocation dimension is given by a number denoted v i ( ∅ ). We require throughout that v i ( ∅ )is either the maximal or the minimal value that v i can take, i.e. ( v i ( ∅ ) ∈ { sup v i , inf v i } ). Moreover, in the money dimension we assume the agent doesn’t expect any transfers inthe absence of the mechanism. For the model presented in this section we make the following informational assump-tion.
Assumption (A):
The private information of the agent i consists of θ i (her intrinsic type). The agents know their type at the outset of any interaction with the designer(interim stage). All agents and the designer have a common knowledge prior for thetype profile ( θ , . . . , θ N ). The types across agents are i.i.d. and the common marginaldistribution of θ i , denoted F , has a continuously differentiable, strictly positive density f : [ θ, ¯ θ ] → R + . ADCB
Participationdecision PlaydecisionOutcomeannouncedPlaydecisionParticipationdecision Outcomeannounced delay delaydelay
Participationdecision Playdecision Outcomeannounced delay
Participationdecision PlaydecisionOutcomeannounced
Figure 3: Timelines for the multiple agent model with uncertainty about intrinsic types.The timeline of the mechanism is either fixed due to technological constraints or achoice variable of the designer. In the latter case he declares at the beginning the timelinehe commits to. As established in Proposition 1 the relevant timelines for the analysis arein Figure 3.For any fixed timeline the designer can in principle consider arbitrarily complicatedmechanisms. We restrict the analysis without loss of generality to direct mechanisms. Fora given timeline, a direct mechanism asks the agents to report their private informationand assigns as a function of their reports an allocation a from A and (net) transfers t i tothe designer. The restriction to direct mechanisms is justified by the revelation principle . All of our applications both in the main paper as well as in the online appendix fulfill this assumption. The online appendix comments on the case of non-trivial outside options in the money dimension.Incentive compatibility and individual rationality characterizations are similar to the ones in this section.We focus here on the trivial case for ease of exposition. The revelation principle holds true for all models we consider in this section. We establish this factin the Online Appendix. A as well as payments from the agents to the designer together with atimeline T ∈ { A, B, C } . Formally, for any T ∈ { A, B, C } a direct mechanism is a map asfollows. M T = ( q, t , t , . . . , t n ) : Θ × · · · × Θ n →A × R n . (9)The uncertainty each agent faces in a given mechanism M T derives only from notknowing the other agents’ types. In terms of subsection 1.1 we are considering agentswho are offered menus M i of lotteries over ∆( A × Θ − i × R ) parametrized by θ i ∈ [ θ, ¯ θ ] andwho have to time their decisions according to the timeline T . Given the general form ofmechanisms allowed and the full support assumption on F it is without loss of generalityto assume that the mechanisms are not randomized, i.e. that the designer doesn’t have arandomization device at her disposal. In the following we take as given a direct mechanism M T as in (9).Fix an agent i ∈ { , . . . , N } . For any distribution G ∈ ∆( A× R ) giving the distributionof pairs ( a, t i ) induced from the play under the mechanism over A × R the marginal of G over A is denoted by G a and over t i is denoted by G t . We call the following term the news utility of agent i from changing beliefs from H to G when the type of the agent is θ i . N i ( G | H | θ i ) = µ gi (cid:90) ξ gi ( v i ( c G a ( p )) θ i − v i ( c H a ( p )) θ i ) dp + µ mi (cid:90) ξ mi ( c H t ( p ) − c G t ( p )) dp. (10)Here µ gi , µ mi , λ gi , λ mi are the agent-specific behavioral parameters for news utility as insubsection 1.1. For future reference we also define the aggregate news utility parametersΛ gi = µ gi ( λ gi −
1) and Λ mi = µ mi ( λ mi − − i decide to participate and reporttheir types truthfully to the designer, whereas agent i of type θ i decides to report ˆ θ i upona positive participation decision. Define V i (ˆ θ ) as the expected value of v i and T i (ˆ θ i ) theexpected value of t i under these reporting strategies from the perspective of agent i . Finally, define T + i (ˆ θ i ) = E θ − i [max { t i (ˆ θ i , θ − i ) , } ], the the expected transfer of type ˆ θ i fromagent i to the designer.The news utility from the realization effect if the agent decides to participate for thecase v i ( ∅ ) = inf a ∈A v i ( a ) is N i (ˆ θ i |∅| θ i ) = µ gi V i (ˆ θ i ) θ i − µ gi v ( ∅ ) θ i − µ mi T i (ˆ θ ) − Λ mi T + i (ˆ θ ) , whereas for the case v i ( ∅ ) = sup a ∈A v i ( a ) it is N i (ˆ θ i |∅| θ i ) = λ gi µ gi V i (ˆ θ i ) θ i − λ gi µ gi v ( ∅ ) θ i − µ mi T i (ˆ θ ) − Λ mi T + i (ˆ θ ) . In contrast to the first case of v i ( ∅ ) = inf a ∈A v i ( a ), in the second case of v i ( ∅ ) = sup a ∈A v i ( a )the utility difference between pre-mechanism belief and post-participation decision isweighted additionally by λ gi . This is because of loss aversion. She can use the type draws θ to induce desired distributions on payments. Formally, V i (ˆ θ i ) = E θ − i [ q (ˆ θ i , θ − i )] and T i (ˆ θ i ) = E θ − i [ t (ˆ θ i , θ − i )]. news utility from the realization of the outcome of the mechanismfor timelines B and C. If agent i of type θ i has reported ˆ θ i and the realized part of theoutcome of the mechanism relevant to agent i is (cid:16) q (ˆ θ i , ˆ θ − i ) , t i (ˆ θ i , ˆ θ − i ) (cid:17) , in addition to intrinsic utility she experiences news utility in the good dimension of µ gi (cid:90) θ − i : v i ( q (ˆ θ i , ˆ θ − i )) >v i ( q (ˆ θ i ,θ − i )) ( v i ( q (ˆ θ i , ˆ θ − i )) − v i ( q (ˆ θ i , θ − i ))) dF − i ( θ − i )+ µ gi λ gi (cid:90) θ − i : v i ( q (ˆ θ i , ˆ θ − i ))
For the timelines in Figure 3, the decision utilities at the type reportingstage have a quasilinear, product form. Namely, the utility of agent i of type θ i fromreporting type ˆ θ i is of the form V ti (ˆ θ i | θ i ) = W ti (ˆ θ i ) θ i − Υ ti ( ˆ θ i ) , t = A, B, C. (14)
Depending on the timeline t the functions W ti , Υ ti : [ θ, ¯ θ ] → R have the following form inthe case v i ( ∅ ) = inf v i Timeline A: W Ai ( θ i ) = (1 + µ gi ) V i ( θ i ) − µ gi v ( ∅ ) , Υ Ai ( θ i ) = (1 + µ mi ) T i ( θ i ) + Λ mi T + i ( θ i ) . • Timeline B: W Bi ( θ i ) = (1+ µ gi ) V i ( θ i ) − µ gi v ( ∅ ) − Λ gi Γ gi ( θ i ) , Υ Bi ( θ i ) = (1+ µ mi ) T i ( θ i )+Λ mi ( T + i ( θ i )+ ω i ( θ i )) . • Timeline C: W Ci ( θ i ) = V i ( θ i ) − Λ gi Γ gi ( θ i ) , Υ Ci ( θ i ) = T i ( θ i ) + Λ mi ω i ( θ i ) . For the case v i ( ∅ ) = sup v i the only change is in timelines t = A, B where the terms W ti change into • Timeline A: W Ai ( θ i ) = (1 + λ gi µ gi ) V i ( θ i ) − λ gi µ gi v ( ∅ ) , • Timeline B: W Bi ( θ i ) = (1 + λ gi µ gi ) V i ( θ i ) − λ gi µ gi v ( ∅ ) − Λ gi Γ gi ( θ i ) . Note that (14) is reminiscent of the classical utility assumption (1). We name theterms W ti perceived valuations . In the equilibrium of the mechanism they give the marginalexpected valuation of the allocation for the agent after taking into account news utilityeffects, be it the surprise effect (timelines A,B), the realization effect (timelines B,C) orboth (timeline B). When news utility is absent, all perceived valuations W ti are equal tothe expected intrinsic valuation V i .We name the terms Υ ti the perceived transfers . In case news utility is absent, theseterms become equal to the expected interim transfers T i for all timelines. In the equilib-rium of the mechanism they give the effect of money transfers on the decision utility inthe type reporting stage after taking into account news utility effects.With these definitions, classical results help give a full characterization of incentivecompatibility for all timelines. Proposition 6 (Incentive Compatibility) . A direct mechanism M is incentive compat-ible for the timeline T ∈ { A, B, C } if and only if the perceived valuations W Ti are non-decreasing.If this is the case, we have the following Mirrlees representation for the interim utilityat the reporting stage V Ti ( θ i ) = V Ti ( θ i | θ i ) V Ti ( θ i ) = V Ti ( θ i | θ i ) + (cid:90) θ i θ i W Ti ( s ) ds. It also holds Υ Ti ( θ i ) = W Ti ( θ i ) θ i − V Ti ( θ i ) − (cid:90) θ i θ i W Ti ( s ) ds. perceived valuations instead of expected intrinsic valuations.We now turn to the individual rationality requirement for incentive compatible mech-anisms. For all timelines each agent experiences the surprise effect of the mechanism and,given sophistication, takes into account her own future behavior when deciding to par-ticipate. In particular, when facing an incentive compatible mechanism she knows she isgoing to reveal her true type to the designer at the reporting stage. Under the assumptionthat the other agents play truthfully reporting the correct type induces a distribution overgood consumption and money transfers.We assume the outside option of the mechanism is degenerate: for agent i of type θ i the outside option is v i ( ∅ ) θ i . The mechanism is individually rational for an agent i when the equilibrium utility it offers in period one is at least as high as v i ( ∅ ) θ i for alltypes θ i ∈ [ θ, ¯ θ ]. This equilibrium utility incorporates not only the expected consumptionand transfers from the realization of the mechanism, but also news utility in the form ofthe surprise effect for all timelines and in the form of the expected realization effect fortimelines B,C. The following Proposition summarizes this in participation utility formulas. Proposition 7. (Individual Rationality) An incentive compatible mechanism M is indi-vidually rational for the respective timelines if the following is fulfilled. • Timeline A: W Ai ( θ i ) θ i − Υ Ai ( θ i ) ≥ v i ( ∅ ) θ i , for all θ i ∈ [ θ, ¯ θ ] . • Timeline B: W Bi ( θ i ) θ i − Υ Bi ( θ i ) ≥ v i ( ∅ ) θ i , for all θ i ∈ [ θ, ¯ θ ] . • Timeline C: W Bi ( θ i ) θ i − Υ Bi ( θ i ) ≥ v i ( ∅ ) θ i , for all θ i ∈ [ θ, ¯ θ ] . Individual rationality for timelines A and B is equivalent to the requirement thatthe equilibrium decision utility in the reporting stage exceeds the value of the outsideoption v i ( ∅ ) θ i . This is because in both of these timelines there is no delay between theparticipation decision and the reporting decision. In contrast, in timeline C the reportingdecision doesn’t take into account the bygone surprise effect . The participation self has totake both news utility effects into account. Therefore the participation decision utilitiesare the same as for timeline B, except that the terms V i , T i , T + i are now determined in thereporting stage. In this subsection we illustrate how loss aversion affects incentive compatibility resultsfrom classical Bayesian mechanism design. Namely, we show how ex-post efficiency in asymmetric public good provision setting may fail to be incentive compatible, even thoughit never does so in the absence of news utility effects. We also illustrate how this issuemay be overcome whenever there are enough players in the game.Assume a society of N ≥ F with support [ θ, θ ] with θ ≥ Assume the agents have quasilinear intrinsic utility anddenote by q the probability of provision of the public good: the intrinsic valuation has The last requirement means the public good would always be (weakly) desirable if it is implementedwithout transfers. v ( q ) = q . Let c ( N ) be the average costs so that providing the public good willcost to the social planner N c ( N ).Maximizing welfare under complete information and under the assumption that thesocial planner has the funds for provision gives the ex-post efficiency rule: q ( θ , . . . , θ n ) = (cid:40) µ g ) (cid:80) ni =1 θ i ≥ N c ( N )0 otherwise. (15)This is the first-best rule under news utility. Denote by ˜ c ( N ) = c ( N )1+ µ g , the normalizedper-person provision costs. If an agent is of type θ her interim probability of public goodprovision under (15) is Q ( θ ) = 1 − F ∗ ( N − ( N ˜ c ( N ) − θ ). In the classical setting without news utility and loss aversion ex-post efficiency isalways incentive compatible. This is not always the case in the presence of news utility. Proposition 8.
Assume (1 + µ g ) θ < c ( N ) < (1 + µ g )¯ θ .1) Ex-post efficiency is always incentive compatible for timeline A.If Λ g ≤ µ g then ex-post efficiency is incentive compatible for timeline B.If Λ g ≤ then ex-post efficiency is incentive compatible for timeline C.2) If Λ g > µ g , it can happen that q is not incentive compatible for timeline B, for ex-ample if F is concentrated in the vicinity of θ and N c ( N ) is big enough. This impossibilitybecomes more common with higher loss aversion in the consumption dimension.The same kind of result holds true for timeline C, if Λ g > .3) Let E [ F ] be the mathematical expectation of the distribution F .If θ < lim sup n →∞ ˜ c ( N ) < E [ F ] , then q is always incentive compatible whenever N is highenough with every timeline. Note that in timeline A the model is isomorphic to a classical quasi-linear model wherethe type spaces are [(1 + µ g ) θ, (1 + µ g )¯ θ ] and the transfers of each agent are scaled downby λ m µ m . The same argumenst as in the classical setting deliver all of 1).The result of part 2) of Proposition 8 for timelines B, C is an instance of equilibriumnon-existence. It is similar to the result in Theorem 1 of [Dato et al. ’17] who show lackof existence of equilibrium in the related CPE model. When N is very large the comparison between average intrinsic valuation E [ F ] andaverage costs scaled by 1 + µ g decides on provision. Due to the law of large numbers then,if the average costs of provision fulfill the inequality in part 3) of the Proposition, the goodis provided with high ex-ante probability so that the expected news utility terms fromthe realization effect weigh less on the decision of the agent. This effect helps preserveincentive compatibility even under relatively high loss aversion. Here with F ∗ ( N ) we denote the N -times convolution of a distribution F , i.e. the distribution of thesum of N i.i.d. draws of F . Ex-post efficiency is always incentive compatible with news utility if c ( N ) ≤ (1 + µ g ) θ or if c ( N ) ≥ (1+ µ g )¯ θ . We omit these uninteresting cases and focus on the interesting case (1+ µ g ) θ < c ( N ) < (1+ µ g )¯ θ in the following. CPE is defined in [K¨oszegi, Rabin ’07] and has become popular in the applied behavioral literature.[Masatlioglu, Raymond ’16] show that when CPE is interpreted as a static risk preference it leads tochoice over lotteries which may violate monotonicity with respect to first-order-stochastic-dominance(FOSD), whenever the loss aversion parameter λ m is high enough. Part 2) of the above Proposition is aresult in the same spirit. .3 Optimal Symmetric Auctions In this subsection we focus on symmetric unit auctions. We first define the environment.Let ∆ = { ( q , q , . . . , q N ) ∈ R N + : (cid:80) ni =1 q i ≤ } be the feasible allocations of a singleunit of a good to be auctioned off between N bidders.A direct mechanism M for the auction with a fixed timeline T ∈ { A, B, C } is amapping ( q , . . . , q N , t , . . . , t N ) : Θ → ∆ × R n giving as a function of reports for each agent the probability that she gets the good andthe payment to the auctioneer.We make the classical assumption v i ( q ) = q i , i.e. the intrinsic value of the good of anagent i is equal to the probability that the good ends up with agent i .To ensure that there are no incentive compatibility issues as in subsection 3.2.1 weadd a parametric restriction for the news utility parameters to the classical regularityassumption for the virtual valuation of the agents. Assumption (A2) • No Dominance of News Utility in the good dimension:Λ g = µ g ( λ g − ≤ . • Regularity: The virtual valuation of F given by the function γ ( t ) : [ θ, ¯ θ ] → R , γ ( t ) = t − − F ( t ) f ( t ) is strictly increasing.The restriction for the aggregate news utility parameter ensures that no bidder showspreference for a stochastically dominated allocation in the good dimension in timelines Aand B. Regularity corresponds to the classical assumption first introduced in [Myerson ’81].It is fulfilled for many natural examples and allows for simple characterizations of theoptimal auction. Proposition 9.
1) Under Assumption (A2) the optimal allocation rule for timelines
A, B is the same as the Myersonian rule, that is, given θ ∗ such that γ ( θ ∗ ) = 0 sell to any ofagents with the highest type θ m = max i ≤ n θ i as long as θ m ≥ θ ∗ .2) Optimal auctions in timeline B are all-pay. That is, it is optimal to fully insure thebidder against the uncertainty she is facing in the transfers.3) Optimal auctions in timeline C are usually not all-pay. That is, it may be optimalto not fully insure a positive measure of types against the uncertainty they are facing inthe money dimension. Moreover, the optimal threshold type is usually different from theone in timelines A and B. The online appendix contains further results: optimal unit auctions for asymmetric agents in timelineA as well as optimal symmetric auctions for timeline B. In both cases we show how news utility changesresults and intuitions from the classical setting. Recall the related discussion in [Masatlioglu, Raymond ’16]. This parameter restriction also appearsin other settings in the applied behavioral literature, such as [Herweg et al. ’10]. For timelines A and B one could use the methods from [Toikka ’11] whenever Λ g >
1. The modelhere is separable according to the terminology in [Toikka ’11] (see section 3 of his paper).
25n indirect optimal auction is an auction with a reservation price and which followsthe respective timeline. In the case of timeline B it is additionally all-pay: each bidderhas to pay her bid. Bidders who would never bid above the reservation price bid zero andpay zero.Part 1) in the case of timeline A follows closely the standard classical proof for symmet-ric unit auctions. The case of timeline B follows immediately from (the more general)Proposition 6 in the online appendix.Part 2) follows from the general all-pay result for timeline B which is proven in theonline appendix (see Theorem 1 there). A related result is known in the classical literatureon Expected Utility agents: optimal revenue maximizing auctions feature degeneratetransfers whenever an agent’s Bernoulli utility is separable in the consumption and moneydimension and the agent is risk averse w.r.t. money (see [Maskin, Riley ’84]).In timeline C the auctioneer has an additional variable he can use to give incentivesfor truth-telling in the reporting stage: the expected news utility terms in the moneydimension given by ω ( θ ). Its usage comes at a cost as ceteris paribus an ω ( θ ) > θ . Thus, when compared totimeline A, timeline C besides the advantage that any expected payments are not shadeddown as in timeline A because of the missing surprise effect in the money dimension,it has the additional advantage of having one more choice variable for the auctioneer.Timeline A has the advantage of lacking the negative expected news utility term from therealization effect, which implies that there is no need to subsidize individual rationalityas it is usually necessary in timeline C. We show numerically that in many parameterconstellations for timeline C the auctioneer decides to make use of ω ( θ ), i.e. leaves someof the types with risk in the money dimension so as to help incentive compatibility inthe reporting stage. The next example illustrates this optimal distortion in the moneydimension. Example 3.
Consider an auction with two symmetric agents who satisfy the follow-ing assumptions: λ g = 1 . , µ g = 1 , µ m λ m = 1 and distribution of intrinsic type F = unif orm ([1 , , (cid:51) θ → c m ω ( θ ) , where c m = λ m µ m λ m µ m Λ m is a normalizing constant depending only on λ m , µ m . Note thatthe distortion in the money dimension is decreasing and that there is no distortion at thetop. Another feature of optimal auctions in timeline C is that now the threshold type theauctioneer uses to decide whether to sell the unit at all is different from the classical See e.g. chapter 3 of [B¨orgers ’15]. The monotonicity in distortion is not a general feature as other numerical exercises show (availableupon request). For example, an inverse-U shaped distortion is possible if F = unif orm ([0 , .0 1.2 1.4 1.6 1.8 2.0 . . . . type op t i m a l f r i c Figure 4: Friction in the money dimension as a function of type for Example 4.Myerson one γ ( θ ∗ ). The auctioneer has to weigh different effects in timeline C: types haveto be subsidized for participating because of the wedge between incentive compatibilityand individual rationality and she has to decide which subset of types to fully insure in themoney dimension so as to maximize incentives of truthtelling after a positive participationdecision. As news utility parameters are varied but so as to keep Λ g ≤ F fulfilling regularity in both ahigher or lower threshold than the classical threshold θ ∗ (which satisfies γ ( θ ∗ ) = 0). In this subsection we assume the timeline is a choice variable of the auctioneer and considerits optimality.
Theorem 2.
1) Timeline A always dominates timeline B in terms of revenue maximiza-tion.2) There is no uniform ranking of timelines A and C in terms of revenue maximization.
When compared to timeline A, timeline B features the same news utility effects exceptfor an additional negative realization effect coming from the negative expected news utilityin the good dimension. This lowers ceteris paribus for timeline B the willingness to payof a bidder so that timeline A is always preferred for revenue maximization.The second part is proven by example.
Example 4.
We take the same data as Example 4 with the only difference that now wedon’t fix µ m λ m but instead vary it in the interval [1 , A, C .27 .0 1.2 1.4 1.6 1.8 2.0 . . . . . . friction in the money dimension p r o f i t Figure 5: Auction revenues for timelines A (blue) and C (red) for friction in the moneydimension in the range [1 , µ m λ m is low the payments of the agent in timeline A are not shaded as muchdue to the negative surprise effect in the money dimension. This, and the fact thattimeline C features a negative realization effect as well as a possible lump-sum subsidy forparticipation yield optimality of timeline A whenever µ m λ m is low. When µ m λ m is highthe advantages of timeline C come to bear: there is no shading of payments due to thenegative surprise effect in the money dimension and there is an additional (albeit costly)variable which can be used to give incentives in the reporting stage. While it is truethat individual rationality may need a lump-sum subsidy, the subsidy is on average smallbecause the lowest possible intrinsic type ( θ = 1) loses a lot in terms of intrinsic utility incase of non-participation, her outside option being zero overall utility. In fact, numericalresults available upon request show that the optimal timeline in this example is indeed Aif we change the distribution of intrinsic types in Example 4 to F = unif orm ([0 , In this paper we have considered agents whose preferences are sensitive to changes ofbeliefs and additionally feature loss aversion. Under the assumption that agents aresophisticated about future behavior we have characterized features of optimal mechanismdesign in two different models: one where a monopolist screens a single agent according totheir loss aversion parameter and one where multiple agents face uncertainty regarding theintrinsic types of the other agents. This work can be extended along different directions.For simplicity we have assumed that there is no discounting of time. This makestwo of the possible timelines equivalent for all purposes. Relaxing that assumption is afruitful didactic exercise as it would give a more complete picture for the characterizationof optimal timeline design.A major assumption to relax in our model is sophistication. Allowing for naive orpartially naive agents in our setting will eventually lead to changes regarding timelineoptimality as well as changes in the features of optimal mechanisms. To see how, considertimeline C and assume that the self who decides about participation assumes erroneouslythat the self who decides about play will stick to her optimal plans. This will implythat the participation-decision self will behave the same in timelines B and C. Knowing28his, she won’t ask for a lump-sum subsidy in order to participate as was the case underthe sophistication assumption.
Ceteris paribus this lowers the implementation costs fortimeline C for the designer. We conjecture that in the case of naive agents timeline Cbecomes optimal in many more cases than it does in the case of sophisticated agents.We haven’t considered the case where different agents may be in different timelinesor the case where the timeline is not common knowledge for all agents at the start of thegame. Moreover, we haven’t solved for the optimal timeline in the case of auctions withasymmetric agents. These non-trivial extensions are left for future research.Finally, a new strand of literature started by papers like [Rey, Salanie ’01] and [Maskin, Moore ’99]considers designers who don’t have full commitment. Relaxing the full commitment as-sumption in our setting is a very interesting topic left for future research.
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Appendices
A Auxiliary Results
A result on expected future news utility terms.
Consider first a one dimensionalmodel. Here we skip the indices j = g, m for simplicity.Whenever G is a degenerate distribution corresponding to getting r with probabil-ity one, we write for the news utility term comparing the degenerate distribution to H N ( r | H ). It holds N ( r | H ) = E z ∼ H [ ξ ( r − z )] , where ξ is a piecewise linear gain-loss valuation function as in (2). The expected newsutility from the realization of H is denoted by − ω ( H ) and it holds ω ( H ) = E r ∼ H [ N ( r | H )] . The following technical Lemma is easy to prove. Lemma 1.
It holds ω ( H ) = Λ (cid:90) (cid:90) { z>w } ( z − w ) dH ( z ) dH ( w ) .ω ( H ) is nonnegative and equal to zero if and only if H is a degenerate distribution.Moreover, whenever H is supported in the non-negative numbers it holds ω ( H ) ≤ Λ E [ H ] . Note that a similar result has been proven in the CPE setting in [Eisenhuth ’17]. See Lemma 1 there.Since the proof of our Lemma follows word-for-word his argument we skip it here.
31e have considered a model, where the uncertainty is one-dimensional. Given theseparability assumptions we make in Subsection 1.1 this result implies immediately theproof of part 1) in Proposition 1.
Image of news utility terms as a function of distributions.
The following isa characterization for the image of the pair ( ω ( H ) , E [ H ]) as a function of the one-dimensional distribution H . Lemma 2.
For every element ( x, y ) ∈ R + × R there exists a binary distribution H = L ( p, b, d ) = pδ b + (1 − p ) δ d with b ≥ d and p ∈ (0 , such that ( ω ( H ) , E [ H ]) = ( x, y ) . Proof. If x = 0 then just pick H = δ y . So let’s focus on the case x > b > d . Then for a binary lottery, the system of equations we have to solve is (cid:40) d + p ( b − d ) = y,p (1 − p )( b − d ) = x. But this is clearly solvable in p, b, d . Pick for example p = , which leads to b = y + 2 x, d = y − x .This Lemma shows that the designer in the multi-agent model can use loss aversionin the money dimension to give additional incentives for truth-telling once the agents arelocked-in after a positive participation decision. Indeed, for a fixed agent i and type θ i in the auction model the distributions H correspond to the distributions on R generatedas an image distribution of t i ( θ i , θ − i ) under the product measure F N − ( θ − i ). Note thatbecause of our assumptions this measure has full-support on [ θ, ¯ θ ] N − . It follows thatone can induce any Borel measure H which is absolutely continuous w.r.t. the Lebesguemeasure by appropriate choice of the (deterministic) payment functions t i . Note that theexpected news utility term ω here corresponds to M in the screening model which wasexogenously given there. In the multi-agent model ω becomes endogenous. An abstract Incentive Compatibility characterizationProposition 10.
A direct mechanism M as in (9) is incentive compatible if and only ifthe corresponding perceived valuations W i are non-decreasing.If this is the case, we have the following Mirrlees representation for the interim utilityin equilibrium V i ( θ i ) = V i ( θ i | θ i ) V i ( θ i ) = V i ( θ i | θ i ) + (cid:90) θ i θ i W i ( s ) ds. (16) It also holds Υ i ( θ i ) = W i ( θ i ) θ i − V i ( θ ) − (cid:90) θ i θ i W i ( s ) ds. (17)The proof is a trivial adaptation of the proof of the Mirrlees representation in theclassical quasilinear utility model. See [B¨orgers ’15] for the classical proof.32 Proofs for Section 2
We start with the analysis for timeline B as that for timeline A is very similar, yet simpler.Analysis for timeline C is more involved due to the wedge between individual rationalityand incentive compatibility.
B.0.1 Analysis for timeline B
Let U ( s ) = (1 + s )[Γ B ( s ) v ( q ( s )) − t ( s )] = (1 + s ) W ( s ) be the utility of buyer of type s inan incentive compatible mechanism. Then individual rationality is tantamount to U ( s ) ≥ W ( s ) ≥ , ¯ λ ≥ s ≥ . Just as in the classical setting (see chapter 2 of [B¨orgers ’15]) one can show that W isdifferentiable a.e. with W (cid:48) ( s ) = U (cid:48) ( s )1 + s − U ( s )(1 + s ) = m − m − M s v ( q ( s )) − t ( s )1 + s − − t ( s ) + Γ B ( s ) v ( q ( s ))1 + s = − m + M ) v ( q ( s ))(1 + s ) . Here we have used the envelope theorem for the optimization problem of the agent. Forany incentive compatible mechanism it follows with the Mirrlees representation that W ( s ) − W (¯ λ ) = (cid:90) s ¯ λ W (cid:48) ( t ) dt = 2( m + M ) (cid:90) ¯ λs v ( q ( t ))(1 + t ) dt. We can therefore write t ( s ) = Γ B ( s ) v ( q ( s )) − m + M ) (cid:90) ¯ λs v ( q ( t ))(1 + t ) dt − W (¯ λ ) , λ ≥ s ≥ . The fact that W is decreasing for any incentive compatible mechanism implies that indi-vidual rationality is fulfilled if and only if W (¯ λ ) ≥
0. An optimal mechanism will have W (¯ λ ) = 0.In all, the profit function isΠ = (cid:90) ¯ λ (cid:34) Γ B ( s ) v ( q ( s )) − m + M ) (cid:90) ¯ λs v ( q ( t ))(1 + t ) dt − cq ( s ) (cid:35) G ( ds ) . The problem of the monopolist consists of maximizing this expression w.r.t. q non-increasing. Doing the usual Fubini transformation for the double integral one getsΠ = (cid:90) ¯ λ (cid:2) Ψ B ( s ) v ( q ( s )) − cq ( s ) (cid:3) G ( ds ) , ‘w.r.t.’ means ‘with respect to’. B ( s ) = Γ B ( s ) − m + M ) G ( s )(1 + s ) g ( s ) . Ψ B ( s ) is the virtual valuation in this model. If it is non-increasing, then one can maximizethe integrand point-wise to get the optimal solution. In that case one can calculate theFOC µ [Ψ( s ) v (cid:48) ( q ( s )) − c ] = 0 , where µ is a Kuhn Tucker parameter, which is zero, as long as Ψ B ( s ) ≤ Regularity for timeline B m + 7 M m + M ) ≥ G ( λ ) g ( λ ) (cid:20)
21 + λ + g (cid:48) ( λ ) g ( λ ) (cid:21) , λ ∈ [1 , ¯ λ ] . (18)If Ψ is not non-increasing, then one can use ironing techniques from [Toikka ’11] tosolve for bunching. Conditions in his paper are fulfilled since the model here is separable according to his terminology (see section 3 of his paper). B.0.2 Analysis for timeline A
The calculations for timeline A are virtually the same as for timeline B, except that theterms involving M are missing because there is no realization effect in timeline A. Theformal analysis is word-for-word the same as in timeline B except that now we have toset M = 0 everywhere in the calculations.In particular, the regularity assumption for timeline A doesn’t depend on the featuresof the distribution of intrinsic values F . Regularity for timeline A ≥ G ( λ ) g ( λ ) (cid:20)
21 + λ + g (cid:48) ( λ ) g ( λ ) (cid:21) , λ ∈ [1 , ¯ λ ] . Finding the optimal mechanism proceeds the same way as for timeline B.
B.0.3 Analysis for timeline CProof of Proposition 4, Part 1.
Step 0.
Assume in this preliminary step that thegrowth condition on v implies that | x − y | ≤ const | v ( x ) p − v ( y ) p | , where const in the following will be a non-specified positive constant number which maychange from line to line and always so that the respective inequality holds. It follows thenthat | v − ( x ) − v − ( y ) | ≤ c | x p − y p | . This just corresponds to the derivative of ψ B being non-positive. L p ([1 , ¯ λ ]) (cid:51) f → (cid:90) ¯ λ v − ( f ( s )) ds (19)is continuous in the L p -norm. The simple argument for this uses the fact that the function[0 , ∞ ) (cid:51) x → x p is H¨older continuous with exponent p . Proof of | x − y | ≤ const | v ( x ) p − v ( y ) p | This follows simply by Taylor’s expansion oforder one applied to x → v ( x ) p and the growth condition on v . Step 1.
Since the virtual type Γ C ( λ ) = m + (1 − λ ) M is decreasing in λ , just as for thetimeline B, it followsIncentive compatibility is equivalent to q ( · ) non-decreasing . By Mirrlees representation of the optimal payments for this situation, payments t ( λ )which ensure incentive compatibility are given up to a constant by the incentive compat-ibility requirement of the decision-play self who has already decided to participate in themechanism. We denote by U ( λ ) the play-decision utility in equilibrium of an incentivecompatible mechanism (this is determined in the second period). It is given by U ( λ ) = Γ C ( λ ) v ( q ( λ )) − t ( λ ) . The envelope theorem from the maximization problem of the agent gives U (cid:48) ( λ ) = − M v ( q ( λ )) . We can use this to write U ( λ ) = Γ C ( λ ) v ( q ( λ )) − t ( λ ) = U (1) − M (cid:90) λ v ( q ( s )) ds. We solve for t ( λ ) to write for a constant f = − U (1) ∈ R . t ( λ ) = f + Γ C ( λ ) v ( q ( λ )) + M (cid:90) λ v ( q ( s )) ds. (20) Fact: t ( · ) is weakly decreasing in any incentive compatible mechanism.This can be easily established through taking (one-sided) derivatives. The calculationuses Assumption (S) extensively.The Fact has an important implication for the optimal mechanism whenever it exists: t ( λ ) ≥ always in any optimal mechanism for timeline C. Proof of this implication isthrough contradiction. Suppose, for the sake of contradiction that there is an incentivecompatible and individually rational mechanism which is profit-optimal for timeline Cand so that t ( λ ) < λ . The Fact implies that t ( λ ) < λ ≥ λ . Butthen the monopolist can switch to ( q ( λ ) , t ( λ )) = (0 ,
0) for all λ ≥ λ (i.e. exclude all typeswith λ ≥ λ ). This preserves incentive compatibility and also individual rationality asthe former is equivalent to q non-increasing (which remains intact) and the latter depends35nly on the own type. Thus there is a threshold type ˆ λ ∈ [1 , ¯ λ ] so that the monopolistsets ( q ( λ ) , t ( λ )) = (0 ,
0) whenever λ > ˆ λ . Step 2.
The proof of this step doesn’t need the growth condition on v . Assume in thisstep that the threshold type ˆ λ is already given. As one can see easily, the arguments ofthis step hold the same for every particular value ˆ λ ∈ (1 , ¯ λ ].Given the t ( λ ) ≥ λ accepts the mechanism the agent experiencesthe following utility (see also discussion in main text): V ( λ ) = [(2 m + (1 − λ ) M ] v ( q ( λ )) − (1 + λ ) t ( λ ) , or using the definitions from timeline B V ( λ ) = (1 + λ )Γ B ( λ ) v ( q ( λ )) − (1 + λ ) t ( λ ) . (21)Replacing (20) into (21) we get V ( λ ) = (1 + λ ) (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) v ( q ( λ )) − (1 + λ ) f − (1 + λ ) M (cid:90) λ v ( q ( s )) ds. It follows that the individual rationality requirement can be written as (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) v ( q ( λ )) − f − M (cid:90) λ v ( q ( s )) ds ≥ , λ ∈ [1 , ˆ λ ] . The objective function of the designer becomes (cid:90) ˆ λ t ( s ) − cq ( s ) dG ( s ) = f · G (ˆ λ ) + (cid:90) ˆ λ (cid:20) Γ C ( λ ) v ( q ( λ )) + M (cid:90) λ v ( q ( s )) ds (cid:21) − cq ( s ) dG ( λ ) . (22)After the usual application of Fubini’s Theorem this expression turns into f · G (ˆ λ ) + (cid:90) ˆ λ (cid:40)(cid:34) Γ C ( λ ) + M G (ˆ λ ) − G ( λ ) g ( λ ) (cid:35) v ( q ( λ )) − cq ( λ ) (cid:41) dG ( λ ) . Thus the problem of the designer for timeline C can be rewritten as followsmax f ∈ R ,s (cid:55)→ q ( s ) f · G (ˆ λ ) + (cid:90) ˆ λ (cid:40)(cid:34) Γ C ( λ ) + M G (ˆ λ ) − G ( λ ) g ( λ ) (cid:35) v ( q ( λ )) − cq ( λ ) (cid:41) dG ( λ )s.t.(1) q ( s ) is non-increasing,(2) (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) v ( q ( λ )) − f − M (cid:90) λ v ( q ( s )) ds ≥ , λ ∈ [1 , ˆ λ ] . (23) Thus, types outside [ λ , ¯ λ ] retain individual rationality under the changed mechanism and types in[ λ , ¯ λ ] are excluded from the mechanism and thus get their outside option of zero utility. f in (22) into the integral and replacing the constraint (2)we get an upper bound for the profit. Namely, the maximum of the following integral (cid:90) ˆ λ Γ B ( λ ) v ( q ( λ )) − cq ( λ ) dG ( λ ) , (24)with respect to all non-increasing q . This program is easily solvable through point-wisemaximization given our constraints. In in all we have an upper bound for the programin (8). The point-wise maximum of (24) has the following property:(!) ∞ > q ( λ ) > B ( λ ) > q ( λ ) = 0 otherwise. This means theproblem in (24) has a finite value.In particular, it is not optimal in (24) to set q so that lim sup λ → + q ( λ ) = + ∞ , i.e. q is not locally unbounded near 1. That would only be profitable if the optimizedvalue in (8) were infinity, which contradicts (!). This implies that v ( q ( λ )) , λ ∈ [1 , ˆ λ ]remains bounded for any potential solution of (8).Moreover, it is easy to see from (8) and (!) that(!!) f is bounded and finite in any potential optimum.Introduce now for a given nondecreasing q ( · ) u ( λ ) = f + M (cid:90) λ v ( q ( s )) ds. (25)This results in u being continuous, non-decreasing and concave, but with free endpoint u (1) = f . Moreover, for every u non-decreasing and concave there exists a non-increasing q ( · ) and a f ∈ R such that u can be expressed as in (25). We make use of this equivalenceconsiderably in what follows.Inserting (25) in (22) we get that each feasible allocation q ( · ) for the program (8)results in a feasible ‘allocation’ u ( · ) for the following maximization program.max s (cid:55)→ u ( s ) (cid:90) ˆ λ (cid:26) Γ C ( λ ) u (cid:48) ( λ ) M + u ( λ ) − cv − ( u (cid:48) ( λ )) (cid:27) dG ( λ )s.t.(1) u ( s ) is continuous, non-decreasing, concave(2) (cid:2) Γ B ( λ ) − Γ C ( λ ) (cid:3) u (cid:48) ( λ ) M − u ( λ ) ≥ , a.e. λ ∈ [1 , ˆ λ ] . (26)The other direction is also trivially true: whenever we have a solution u to (26) we geta solution to (23) by the relations q ( s ) = v − (cid:16) u (cid:48) ( s ) M (cid:17) , t ( s ) = u ( s ) + Γ C ( s ) u (cid:48) ( s ) M for s ≤ ˆ λ and q ( s ) = 0 , t ( s ) = 0 otherwise. Note that it is w.l.o.g. to ask for constraint (2) to holda.e. This is because G is absolutely continuous w.r.t. the Lebesgue measure on [1 , ¯ λ ]. f · G (ˆ λ ) = (cid:82) ˆ λ f dG ( λ ). Recall that g is bounded away from zero on [1 , ¯ λ ]. u (cid:48) (1) and u (1) are bounded for all feasiblecandidates of (26). In combination with constraint (1) in (26) (continuity and concavity)this implies that we can restrict the class of u in the maximization in (26) by requiringadditionally the following condition. (3) functions u are bounded in the Sobolev p-norm: u (cid:55)→ (cid:16)(cid:82) ˆ λ | u ( λ ) | p dλ + (cid:82) ˆ λ | u (cid:48) ( λ ) | p dλ (cid:17) p by a constant c >
0. Here recall that p >
1. The constant c is chosen to be independent ofˆ λ given its boundedness and the upper and lower bounds for the optimal profit of timelineC. Moreover they are also bounded in the essential supremum-norm by a uniform boundholding for all u under consideration.Denote by C (ˆ λ ) the class of functions satisfying (1) , (2) , (3) where (1) and (2) are onlyrequired to hold almost everywhere.Note then the easy-to-show but important facts: C (ˆ λ ) is convex and bounded in theSobolev-p norm. This follows from the linearity of the two constraints and of the Sobolevnorm, as well as the requirement (3). C (ˆ λ ) is closed in the Sobolev-p norm. To show this assume that we have a sequence u n , n ≥ , (2) , (3) and which converge in Sobolev-p norm to u . Thenthat (2) , (3) hold true for u is easy to check: convergence in L p implies convergence of asuitable subsequence a.e.; this implies that u is non-decreasing and concave [1 , ˆ λ ] whichmeans that it is also continuous in (1 , ˆ λ ); (1) , (2) are then automatically fulfilled; (3) isfulfilled due to the definition of convergence in the Sobolev-p norm.Since C (ˆ λ ) is closed and bounded it is sequentially compact in the weak topologyinduced by Sobolev p-norm. This is because the space of Sobolev functions equippedwith the Sobolev p-norm is reflexive and one can apply the Banach-Alaoglu Theoremwhich implies sequential compactness w.r.t. weak topology for all bounded, closed sets. The functional being maximized has the form J ( u ) = (cid:90) ˆ λ L ( λ, u ( λ ) , u (cid:48) ( λ )) dλ with L ( λ, u, w ) = g ( λ ) (cid:0) Γ C ( λ ) wM + u − cv − ( w ) (cid:1) for ( λ, u, w ) coming from a bounded setof R (determined by the above discussion) and otherwise flattens to zero continuouslyfor all ( λ, u, w ) outside this bounded set. L is then a continuous function and satisfies L ( λ, u, w ) ≤ a ( λ ) w + b ( λ ) + c | u | , with some bounded functions a, b >
0. Finally, w (cid:55)→ L ( λ, u, w ) is strictly concave since v − is strictly convex. By Theorem 3.23, pg. 96 in [Dacorogna ’08] it follows that J issequential upper semi-continuous w.r.t. the weak topology.Because of upper-semicontinuity of J and compactness of the set C (ˆ λ ) (all in thesame topology), we know that maximizing J over C (ˆ λ ) has a solution. The solution isa Sobolev function, i.e. it can be chosen to be continuous everywhere and differentiablealmost everywhere. u (cid:48) is bounded from below by monotonicity. If it were optimal to have lim sup s → u (cid:48) ( s ) = + ∞ wewould get a contradiction to (!) since u (cid:48) ( s ) = M v ( q ( s )). Note that there is a natural continuous and onto embedding ι (ˆ λ, ˆ λ (cid:48) ) : C (ˆ λ (cid:48) ) →C (ˆ λ ) whenever ˆ λ ≤ ˆ λ (cid:48) .It is given by restricting the definition domain of the functions. See e.g. section 3.15 of [Rudin ’91]. tep 2. We now show how to calculate the optimal ˆ λ ∈ [1 , ¯ λ ] . For this, it sufficesto show that the value function of the maximization problem (26) is continuous w.r.t.ˆ λ ∈ [1 , ¯ λ ].Note that [1 , ¯ λ ] (cid:51) ˆ λ →C (ˆ λ ) is a continuous, compact-valued correspondence in ˆ λ . Herewe are considering all C (ˆ λ ) embedded in the ‘largest’ space C (¯ λ ), all equipped with theSobolev-p norm. Upper hemicontinuity follows by checking directly its sequential char-acterization. Lower hemicontinuity follows just as easily because of the continuous andonto embedding ι (ˆ λ, ˆ λ (cid:48) ) : C (ˆ λ (cid:48) ) →C (ˆ λ ) introduced in footnote 52. Finally, note that theobjective functional in (26) is jointly continuous in (ˆ λ, u ) where [1 , ¯ λ ] × ( Sobolev − p )([1 , ¯ λ ])is equipped with the product topology of the euclidean one and the one of the Sobolev-pnorm. This follows because of the H¨older inequality combined with the continuity of themap in (19). Proof of Proposition 4, part 2).
This follows from the fact that Γ B ( λ ) − Γ C ( λ ) = − λ λ ( m − λM ). Γ B ( λ ) − Γ C ( λ ) is zero for λ = 1 and negative for all small λ >
1. It followsfrom the second constraint in (8) that f ≤ q (1) = 0 then it would follow that q ≡
0, i.e. the profit would be zero. Thus,whenever the profit is not zero it means that for some range near λ = 1 the monopolistoffers positive amounts of the good. It follows that f < Proof of Theorem 1.
That A is weakly better than B follows directly from the factthat the virtual valuation Ψ A in timeline A is pointwise higher than the virtual valuationΨ B in timeline B and that otherwise the set of incentive compatible and individuallyrational mechanisms is the same for both timelines.That B is weakly better than C follows from Proposition 4. Namely, we show thereduring the existence proof for timeline C that:- the profit in timeline C is always bounded by the profit in timeline B, leaving out theindividual rationality requirement in timeline B (proof of part 1 of Proposition 4). The optimal IC mechanism in timeline B is automatically IR.- Individual rationality in timeline C weakly costs something positive to the monopolistwhenever profits in C are strictly positive (part 2) of Proposition 4).
C Proofs for Section 3
Proof of Proposition 5.
Straightforward algebra calculations show the formulas for (6)hold for both the case when v i ( ∅ ) is the minimum and the case when it is the maximumof v i with the definitions made in the main text. We have just split payments t i ( θ i , · ) intothe range where it is strictly positive and the range where it is negative. The positiverange is multiplied by λ mi . A similar split is done for the good dimension, but now it isthe negative range which is multiplied by λ gi . One uses Lemma 1 and algebra. The samesteps can be made for the terms ω i . Proof of Proposition 6.
Here we use Proposition 10 in a first step which ensures theexistence of the perceived payments. 39he proof is finished if we show the existence of the payment schedules t i for fixed Υ i -sdelivered by the proof in the classical setting. Some t i -s that do the trick are the following:set first t i ( θ i , θ − i ) = t i ( θ i , θ (cid:48)− i ) , ∀ θ i , θ − i , θ (cid:48)− i . This implies T i ( θ ) = t i ( θ, θ − i ) , ∀ θ, θ − i and T + i ( θ ) = t i ( θ, θ − i ) { t i ( θ,θ − i ) ≥ } = T i ( θ ) { T i ( θ ) ≥ } . Then, ω i ( θ ) = 0. If Υ i ( θ ) = 0 thenset T i ( θ ) = 0, otherwise if Υ i ( θ ) > T i ( θ ) = Υ i ( θ )1+ λ mi µ mi , while if Υ i ( θ ) < T i ( θ ) = Υ i ( θ )1+ µ mi . Note that T i ( θ ) are increasing in Υ( θ i ), albeit with a discontinuity atzero. This finishes the proof, since the Mirrlees representation also follows by the samearguments as in the classical proof with quasilinear utilities. In general, if one wants amechanism with non-trivial ω i -s one uses Lemma 2. In timeline B, where the T + i termsmatter, we show in the online appendix that for optimal mechanisms it holds ω i ≡
0. Itfollows that optimal mechanisms in all of our applications below will have t i ≥
0, so that T + i = T i . Proof of Proposition 8.
In the following we suppress the timeline superscript wheneverthe argument is valid for all timelines or it is clear from the proof context to which timelinethe statements correspond.1)-2) The case of timeline A is clear from the discussion in text. We show the resultfor timeline B . The proof for timeline C is similar.First note that q being symmetric we can write Q i ( θ ) = Q ( θ ) and thus W i = W for each i . Moreover, q being monotone increasing in each θ i , the same follows for Q .Indeed, we have Q ( θ ) = 1 − F ∗ ( N − ( N ˜ c ( N ) − θ ). Taking W as a function of Q, i.e. W ( Q ) = (1 + µ g ) Q − Λ g Q (1 − Q ) we have d W dQ = 1 + µ g − Λ g + 2Λ g Q. This derivative is always nonnegative for Λ g ≤ µ g , which establishes 1).For Λ g > µ g , substituting the formula for Q into the derivative above, we see that W is nondecreasing if and only if12 ( 1 + µ g + Λ g Λ g ) ≥ F ∗ ( N − ( N ˜ c ( N ) − θ ) for all θ ∈ [ θ, θ ] . (27)The left hand side of (27) is smaller than 1, if the premise of 2) is fulfilled. Also thecondition is more likely to be violated as ˜ c ( N ) approaches θ and as F becomes more andmore concentrated in lower θ -s. Note also that the upper bound on the left of (27) issmaller the higher Λ g is. This establishes 2).3) Note first the bound F ∗ ( N − ( N ˜ c ( N ) − θ ) ≤ F ∗ ( N − ( N ˜ c ( N ) − θ ), which is uniformin θ . We can write F ∗ ( N − ( N ˜ c ( N ) − θ ) = P ( (cid:80) N − i =1 θ i N − ≤ N ˜ c ( N ) − θN − Strong Law of Large Numbers we know that (cid:80) N − i =1 θ i N − converges almost surely to E [ F ]. Meanwhile, N ˜ c ( N ) N − has possible limit points allstrictly smaller than E [ F ] due to assumption. Thus we have that as N → ∞ the left handside of (27) converges to zero uniformly in θ . For timelines A and C this is clear from Proposition 5. See for example chapter 2 of [Durrett ’10]. roof of Theorem 2, 1). We know the rule for the optimal direct mechanism reads ‘give the good to the bidder i with the highest θ i , as long as θ i > θ ∗ ’ . Individual rationalityis realized in both timelines by setting the utility of the lowest types in equilibrium equalto zero. Recalling that in both cases the revenue in the optimal mechanism is given by N (cid:88) i =1 (cid:90) ¯ θθ W A/Bi ( θ i ) γ ( θ i ) dF ( θ i ) , one sees that the fact W Ai ( θ i ) ≥ W Bi ( θ i ) , together with the fact that in both cases theoptimal allocation rules prescribe W A/Bi ( θ i ) = 0 , whenever θ i < θ ∗ , the result follows. Analysis for timeline C in the case of symmetric auctions.
We assume through-out i.i.d. intrinsic types and that Λ g ≤ s has the following form. Q ( s ) = F ( s ) n − { s ≥ ˆ θ } . Given this, we can then build the perceived valuation W ( s ) = Q ( s )(1 − Λ g (1 − Q ( s ))) , as well as define the auxiliary functions h ( s ) = W ( s ) s − (cid:90) sθ W ( t ) dt. Define furthermore g ( s ) = W ( s ) s + µ g Q ( s ) s. Note that g and h depend on the threshold ˆ θ . They are both zero below ˆ θ . Moreover,both functions are weakly increasing due to IC and the assumption that Λ g ≤
1. Denotein the following by c the decision utility of the agent with the lowest type θ . Claim.
An incentive compatible and individually rational mechanism in the timelineC is a tuple (ˆ θ, ω, c ) s.t.( IC ) c + Λ m ω ( s ) + T ( s ) = h ( s ) , s ≥ ˆ θ ( IR ) c ≥ s m, ˆ θ ( s ) − λ m µ m λ m µ m Λ m ω ( s ) , s ≥ ˆ θ where s m, ˆ θ ( s ) = h ( s ) − λ m µ m g ( s ). Proof of Claim.
These follow directly by using Propositions 6 and 7. (IC) is immediatewhereas the individual rationality requirement can be written first as41 ( θ ) ≥ (1 + λ m µ m ) T ( θ ) + Λ m ω ( θ ) , s ≥ ˆ θ (28)which is then easily manipulated into (IR).Note that (IC) together with Assumption (A2) implies that Λ m ω ( s )+ T ( s ) is increasingin s . Thus, types with s < ˆ θ don’t get served at all and pay nothing whereas types s ≥ ˆ θ may receive a (net) subsidy of c . s m, ˆ θ is a piece-wise smooth function with at most one discontinuity of uniformlybounded size across all incentive compatible mechanisms. Due to symmetry and the above we can rewrite the maximization problem of thedesigner as max c ∈ R , ( T,ω ):Θ i → R × R + (cid:90) ¯ θθ T ( θ ) dF ( θ ) , s.t. ( IC ) and ( IR ) ∀ θ. The objective function of the problem for timeline C is (cid:90) ¯ θ ˆ θ h ( t ) dF ( t ) − (cid:90) ¯ θθ ( c + Λ m ω ( s )) dF ( s ) . We can split the maximization problem in two parts. Once a threshold ˆ θ has beenchosen, the rest of the mechanism is found by solvingmin c,ω ( · ) ≥ (cid:90) ¯ θ ˆ θ ( c + Λ m ω ( s )) dF ( s ) , under the constraint( IR ) c ≥ s m, ˆ θ ( s ) − λ m µ m λ m µ m Λ m ω ( s ) , s ≥ ˆ θ. Given any c , one can see that the optimal ω has to satisfyΛ m ω ( s ) = 1 + λ m µ m λ m µ m max { s m, ˆ θ ( s ) − c, } , s ≥ ˆ θ Note that payments are degenerate for all types (all-pay) if and only if c ≥ max s s m, ˆ θ ( s ).Otherwise, the optimal mechanism is not all-pay and some of the types will not be fullyinsured in the money dimension.Overall, given a threshold type ˆ θ the rest of the mechanism is determined by solvingmin c ∈ R λ m µ m λ m µ m (cid:90) ¯ θ ˆ θ max { s m, ˆ θ ( s ) −
11 + λ m µ m c, λ m µ m λ m µ m c } dF ( s ) − F (ˆ θ ) max {− λ m µ m c, c } Note that the presence of the last term in the minimization problem will usually makefor a non-smooth solution as one varies the parameter λ m µ m . Nevertheless, the valuefunction of this minimization problem is a continuous function of ˆ θ . Denote the valuefunction of this problem by H (ˆ θ ). The optimal threshold then solves It is increasing in the money friction λ m µ m . Its difference to h disappears uniformly as λ m µ m →∞ . The conditions for Berge’s maximum theorem are given because c can be taken to be bounded withoutloss of generality, whenever λ m µ m comes from a bounded interval. ˆ θ ∈ [ θ, ¯ θ ] (cid:90) ¯ θ ˆ θ h ( t ) dF ( t ) − H (ˆ θ ) . The maximand is a continuous function being maximized over a compact interval. There-fore there always exists a solution so that the discussion above implies that there alwaysexists an optimal auction for timeline C.
Optimal Timelines
Given that timeline B is never optimal the comparison is betweentimelines A and C.Recall that the revenue from one auction participant in timeline A is1 + µ g λ m µ m (cid:90) ¯ θθ (cid:18) Q ( s ) s − (cid:90) s Q ( t ) dt (cid:19) dF ( t ) = 1 + µ g λ m µ m (cid:90) θ ∗ Q ( s ) γ ( s ) ds Here, θ ∗ satisfies γ ( θ ∗∗