Efficient incentives with social preferences

Daske, Thomas; March, Christoph Article Efficient incentives with social preferences Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Daske, Thomas; March, Christoph (2024) : Efficient incentives with social preferences, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 3, pp. 975-999, https://doi.org/10.3982/TE5335 This Version is available at: https://hdl.handle.net/10419/320257 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 975–999 1555-7561/20240975 Efficient incentives with social preferences Thomas Daske Department of Economics and Policy, Technical University of Munich Christoph March Department of Economics, University of Bamberg We explore mechanism design with outcome-based social preferences. Agents’ social preferences and private payoffs are all subject to asymmetric information. We assume quasi-linear utility and independent types. We show how the asymmetry of information about agents’ social preferences can be operationalized to satisfy agents’ participation constraints. Our main result is a possibility result for groups of at least three agents: Any such group can resolve any given allocation problem with an ex post budget-balanced mechanism that is Bayesian incentivecompatible, interim individually rational, and ex post Pareto-efficient. Keywords. Mechanism design, social preferences, Bayesian implementation, participation constraints, participation stimulation, contests, money pump. JEL classification. C72, C78, D62, D82. 1. Introduction How can allocation problems be resolved in an efficient and mutually acceptable way? The literature on mechanism design has postulated four desirable properties of incentive mechanisms: incentive compatibility, ex post Pareto efficiency, ex post budget balance, and interim individual rationality. Bayesian implementation is suitable to achieve the first three of these properties (see, e.g., Arrow (1979), d’Aspremont and GérardVaret (1979)). Often, however, Bayesian mechanisms violate agents’ participation constraints.1 Thomas Daske: [email protected] Christoph March: [email protected] An earlier version circulated under the title “Efficient incentives in social networks: Gamification and the Coase theorem” and is available at http://hdl.handle.net/10419/222527. For their helpful comments and critical remarks, we thank Claude d’Aspremont, Jacques Crémer, Benny Moldovanu, Marco Sahm, Klaus Schmidt, Johannes Schneider, Roland Strausz, and Robert von Weizsäcker as well as participants at the European Winter Meeting of the Econometric Society in Milan, the World Congress of the Game Theory Society in Maastricht, the Annual Meeting of the Association for Public Economic Theory in Strasbourg, the Annual Congress of the International Institute of Public Finance in Glasgow, the European Meeting of the Econometric Society in Manchester, the Annual Congress of the German Economic Association in Leipzig, the virtual Econometric Society World Congress, and the Annual Congress of the Society for the Advancement of Economic Theory in Canberra. We are particularly grateful to several anonymous referees for their patience and very helpful suggestions. 1For settings with independent private signals see, e.g., Myerson and Satterthwaite (1983), Mailath and Postlewaite (1990), Williams (1999), and Segal and Whinston (2016). ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5335 976 Daske and March Theoretical Economics 19 (2024) Bayesian mechanisms that reconcile all four properties exist if agents’ private signals (or types) are sufficiently correlated. Crémer and McLean (1985,1988) show that the designer can exploit this correlation to validate the agents’ reports, extract all information rents, and ensure participation en passant.2 Mezzetti (2004) shows that the logic of Crémer and McLean (1985,1988) can be extended to the case of independent private signals if the designer is permitted to implement a two-stage mechanism. The allocation problem can be resolved with unanimous participation by sequentially administering a social alternative and transfers, with agents first reporting their preference types and then their satisfaction with the chosen alternative before finally receiving (or paying) transfers. The present study enriches the set of possibility results. We assume that types are independent (in contrast to Crémer and McLean (1985,1988)) and that there is only one round of reporting (in contrast to Mezzetti (2004)). Specifically, we consider agents with outcome-based social preferences that are privately known (next to privately known preferences for consumption). That is, agents care about the overall distributive effects of a mechanism, and their distributive preferences are private information. We show how this kind of information asymmetry can be operationalized to satisfy agents’ participation constraints. Our main result, Theorem 1, states that any group of at least three agents can resolve any given allocation problem with an ex post budget-balanced mechanism that is Bayesian incentive-compatible, interim individually rational, and ex post Paretoefficient. It builds on the following insights. In quasi-linear environments, a mechanism can be designed such that the incentives to reveal payoff types and social types are separated. While the allocation problem can be resolved through payoff-type conditional budget-balanced transfers, participation can be stimulated through additional budgetbalanced transfers that condition on agents’ social types. The latter is possible for more than two agents when leveraging the differences in agents’ other-regarding concerns. Technically, we exploit that each agent’s utility is a linear combination of all agents’ private payoffs, which are weighted according to that agent’s other-regarding concerns. This linearity enables us to render the agents’ social types strategically inoperative in the payoff-type conditional mechanism, so we can use them in a separate, social-type conditional mechanism to cross-subsidize the former. In this manner, our solution bundles two strategically independent mechanisms. (Our bundling of two mechanisms resembles Mezzetti (2004). We detail the differences between his and our study in Section 6.4.) Until recently, the literature on efficient design has either neglected social preferences altogether or assumed them to be common knowledge.3An exception is Bierbrauer and Netzer (2016), who study mechanism design when agents have privately known intention-based social preferences. They show that this sort of social preferences allows for efficient, individually rational design if and only if all agents are (commonly known to be) conditionally pro-social. Our study differs from theirs in the kind of social preferences under consideration as well as in the conditions for and the driving forces 2Likewise, McAfee and Reny (1992), McLean and Postlewaite (2004), Kosenok and Severinov (2008). 3See, e.g., Desiraju and Sappington (2007), Kucuksenel (2012), and Tang and Sandholm (2012). Theoretical Economics 19 (2024) Efficient incentives 977 behind the possibility result. First, we consider unconditional outcome-based rather than intention-based social preferences, and next to altruism and selfishness, we allow for anti-social preferences such as spite.4Second, the revelation principle holds in our setup, but not in Bierbrauer and Netzer’s (2016), as their agents’ preferences depend on the set of actions (i.e., messages) available in the mechanism. Indeed, the independence of agents’ preferences from the mechanism distinguishes our paper from various others on mechanism design with intention-based social preferences (e.g., Antler (2015), Kozlovskaya and Nicoló (2019)). Finally, the possibility result of Bierbrauer and Netzer (2016) exploits the mechanism dependence of preferences by introducing additional messages that are not chosen in equilibrium, but manipulate the kindness of truth-telling; this construction only works in the absence of selfish types. In contrast, our result exploits the asymmetry of information about agents’ social preferences. Notably, our study relates to the literature on money pumps (or dutch books). This literature has a long tradition in individual-choice theory. It shows how nonrational individual decision-making can be exploited to pull agents into transactions they stand to lose from (see, e.g., Border and Segal (1994)andRubinstein and Spiegler (2008); for asurvey,seeYaari (1998)). In the multi-agent version, a group of agents is subject to a money pump if an outside party is “able to extract money from the agents without putting any money at risk” (Nau (1992, p. 380)). Our study shows that a group of at least three agents with privately known social preferences can be offered an ex post budgetbalanced (nonzero) transfer scheme that all of them accept ex interim. This implies that a transfer scheme can be constructed that extracts money from the group via participation fees and is still unanimously accepted, and, thus, becomes a money pump. While the literature has focused on nonrational expectations (see, e.g., Eliaz and Spiegler (2007, 2009), Chen, Micali, and Pass (2015), Werner (2022)), we show that multi-agent money pumps can be grounded in nonstandard rational preferences. As in Antler (2023)for nonrational expectations, we require sufficiently many agents, at least three in our case. The following example provides a basic intuition for how asymmetric information about agents’ social preferences can be exploited to generate a money pump. Consider two agents, each of whom is either selfish (caring only about her private payoff) or altruistic (weighting the other’s payoff half as much as her own). Types are independent and equally likely. If both report selfish (altruistic), each is taxed (rewarded) 1 dollar; if they report opposite types, the altruist must pay the selfish 2 dollars. Clearly, reporting selfish always yields a higher private payoff, incentivizing truth-telling for selfish agents. Reporting altruist always yields a considerably larger payoff to the opponent than reporting selfish, incentivizing truth-telling for altruists. As unanimous participation yields each type an interim-expected utility gain (as compared to a status quo of zero transfers), agents are willing to pay for playing this game. Thus, a third agent can offer to finance the game by balancing the budget (i.e., to tax or reward according to the rules) in return for a uniform participation fee. As transfers are zero ex ante, a suf4The behavioral relevance of unconditional outcome-based social preferences has been well established. For evidence on altruism and selfishness, see Andreoni and Miller (2002), Charness and Rabin (2002), and Bruhin, Fehr, and Schunk (2019). For evidence on spite, see Saijo and Nakamura (1995), Fehr, Hoff, and Kshetramade (2008), and Prediger, Vollan, and Herrmann (2014). 978 Daske and March Theoretical Economics 19 (2024) ficiently small fee guarantees that all three agents are wanting ex interim to participate in the extended game. Conforming this scenario to the quoted money-pump definition of “extracting money from the agents without putting any money at risk” resembles a government selling a casino license: An outside party may enter the scene and offer our “third agent” the platform on which she can let others play our game in return for half of the participation fees. In this example, when looking at the actual players (selfish or altruistic), money is redistributed ex interim to those agents who care least about others. On the other hand, a pro-social agent interim-expects to impose a positive monetary externality on her opponent, and this externality overcompensates her emotionally for interim-expected monetary losses. These distributive effects are a general feature of the various money pumps we develop in this paper, although the notions of caring least and pro-sociality will bear more intricate meanings. (We present this example in more detail in Section 5.1.) The paper proceeds as follows. Section 2outlines the model framework. Section 3 states and interprets our main result. Section 4details the proof. Section 5illustrates the intuition behind our participation-stimulating transfers. Finally, Section 6reflects upon the assumptions that are critical to our result, distinguishes our mechanism from Mezzetti’s (2004), and illustrates how participation stimulation can be implemented in practice. The Appendix provides additional proofs. 2. The model 2.1 The allocation problem There is a group I={1, ,n}of n≥2 agents and there is a finite set Kof social alternatives. From alternative k∈Kand a transfer ti∈R,agentigains a private payoff i(k,ti|θi)=πi(k|θi)+ti,withπi:K×i→R.Agenti’s payoff type θibelongs to a finite set i,with|i|≥2. The collection of agents’ payoff types is denoted by θ=(θi,θ−i)∈ =ii,whereθ−i=(θj)j=i. Agents exhibit social preferences in the form of altruism or spite: From the collection of private payoffs (j)j∈I,agentiderives ex post utility uik,(tj)j∈I,θ−i|θi,δi= j∈I δij j(k,tj|θj), where the value δij that iassigns to j’s payoff, j=i, belongs to a closed (proper) interval ij =[δmin ij ,δmax ij ]⊂(−1/(n−1);1 ), while δii =1foralli. We refer to δij as i’s degree of altruism toward j, to the collection δi=(δij )j=i∈i=j=iij as i’s social type, and to the pair (θi,δi)as i’s type. The information structure is as follows. Each agent is privately informed about her payoff type and social type. Hence, there is a type distribution on ×(where =ii) with strictly positive variance of payoff types and social types. Type realizations are independent across agents. An agent’s payoff type and social type realize independently according to strictly positive densities, but the various degrees of altruism determining this agent’s social type may correlate. We assume that agents will observe each other’s payoffs ex post. (We make the implicit assumption of continuous social-type distributions to keep the exposition simple, but all results are equally valid if a social-type set contains mass points; see Section 5.1.) Theoretical Economics 19 (2024) Efficient incentives 979 A few remarks are appropriate. First, the interval (−1/(n−1);1 )is the maximum range of altruism, or spite, for which agents care about overall material efficiency while still being selfish to the extent that every one of them prefers a dollar to be her own rather than having that same dollar distributed among the others. Second, despite the asymmetry of information, it can still be common knowledge who is a friend and who is a foe. For instance, if δmax k ,δmax k <0<δ min ij ,δmin ji , then, in comparison, iand jare friends, whereas kand are foes. Likewise, it can be common knowledge that ilikes jmore than k,whichisthecaseifδmax ik <δ min ij . Finally, while we assume that the variance of every δij is strictly positive, it is allowed to be arbitrarily small. Reciprocal social preferences can thus be captured by letting ij =ji and δmin ij ≈δmax ij . The agents’ problem is to choose a social alternative kand transfers (ti)i∈Isuch that the resulting allocation, i.e., the collection of private payoffs, is ex post Pareto efficient. We require that agents must do so without having access to an outside source of money, such that transfers must be weakly budget-balanced: i∈Iti≤0. 2.2 Revelation mechanisms A direct revelation mechanism involves the agents in a strategic game of incomplete information in which they are asked to report their types truthfully. Types are reported simultaneously. Based on their reports, a social alternative is chosen and transfers are made. As the revelation principle applies to the present setup (Myerson (1979)), there is no loss of generality in considering only direct mechanisms. Formally, a direct mechanism is given by a pair k,Twith allocation function k:×→Kand transfer scheme T=(ti)i∈I:×→Rn. Notice that transfers may take arbitrary negative values. Denote by Ui(ˆ θi,ˆ δi|θi,δi)agent i’s interim-expected utility from reporting (ˆ θi,ˆ δi) ifhertruetypeis(θi,δi)while all the other agents report their types truthfully: Ui(ˆ θi,ˆ δi|θi,δi)=j∈Iδij [¯πij (ˆ θi,ˆ δi)+¯ tij (ˆ θi,ˆ δi)],where ¯πij (θi,δi)=Eθ−i,δ−i[πj(k(θ,δ)| θj)] and ¯ tij (θi,δi)=Eθ−i,δ−i[tj(θ,δ)]. For convenience, Ui(θi,δi)=Ui(θi,δi|θi,δi).Then the mechanism k,Tis Bayesian incentive-compatible if, for all i∈Iand all (θi,δi)∈ i×i,wehaveUi(θi,δi)=max(ˆ θi,ˆ δi)∈i×iUi(ˆ θi,ˆ δi|θi,δi).5 2.3 Efficiency and participation The following lemma links material efficiency (the maximum surplus of private payoffs) to Pareto efficiency. It allows us to focus on allocation functions that are ex post materially efficient, k(θ,δ)=k(θ)∈argmaxk∈Ki∈Iπi(k|θi), and transfers (ti)i∈Ithat are (strictly or ex post) budget-balanced, i∈Iti=0. Lemma 1. A mechanism is ex post Pareto-efficient only if transfers are ex post budgetbalanced. If |δij |<1/(2n−3)for all iand all j= i, then an ex post materially efficient 5Bayesian implementation has been criticized for assuming that the distribution of agents’ types is common knowledge. Bergemann and Morris (2005)haveproposedex post implementation for environments with interdependent utilities, requiring that truthful revelation of types constitutes a Nash equilibrium. However, Jehiel, Meyer-ter Vehn, Moldovanu, and Zame (2006) show that ex post implementation is “generically” not feasible in the presence of informational externalities, a finding extended by Zik (2021)toour present context. 980 Daske and March Theoretical Economics 19 (2024) allocation function is also ex post Pareto-efficient; moreover, no ex post budget-balanced transfer scheme ex post Pareto-dominates another. The intuition behind Lemma 1is this: If agents switch from a social alternative that is materially efficient to one that is not or switch from one budget-balanced transfer scheme to another, then at least one agent must incur a material loss. Now consider the agent whose material loss is largest; if this agent iis sufficiently selfish, |δij |<1/(2n−3) for all j=i, then she would also incur a loss utility-wise. In contrast, the Pareto frontier can be indefinite for combinations of social types satisfying |δij |≥1/(2n−3),inwhich case a subgroup of agents might be willing to transfer arbitrary amounts of money to their joint favorite agent.6 Finally, k,Tis interim individually rational if it gains all agents’ approval at the interim stage (i.e., unanimous approval constitutes a Bayes–Nash equilibrium at the stage where agents’ types are private information). Following Segal and Whinston (2016), we represent reservation utilities by the interim-expected utilities that agents’ derive from a Bayesian mechanism k◦,T◦,withk◦:×→Kspecifying “property rights” and T◦=(t◦ i)i∈I:×→Rnspecifying “liability rules.” 3. A possibility result We establish our main result with the help of two concepts: preference-separating mechanisms and participation-stimulating transfers. Definition 1 (Preference Separation and Participation Stimulation). A preferenceseparating mechanism k,Tconsists of the ex post materially efficient allocation function k:→K,withk(θ)∈arg maxk∈Ki∈Iπi(k|θi), and an ex post budgetbalanced transfer scheme T=(t i)i∈I:×→Rndefined by t i(ˆ θ,ˆ δ)= j=iEθ−iπjk(ˆ θi,θ−i)|θj−Eθ−jπik(ˆ θj,θ−j)|θi   the terms of trade +s i(ˆ δ)  participationstimulating transfers , where participation-stimulating (PS) transfers s=(s i)i∈I:→Rnare defined by jointly satisfying the following conditions: (i) PS transfers sare strategy-proof. For all i∈I,allδ∈,andall ˆ δi∈i,  j∈I δij s j(δ)≥ j∈I δij s j(ˆ δi,δ−i). (ii) PS transfers sare ex post budget-balanced. For all δ∈,  j∈I s j(δ)=0. 6An example is the group of three agents with δ13 =δ23 >1/3, δ12 =δ21 =−1/3, and δ31 =δ32 =0, in which agents 1 and 2 are willing to jointly transfer arbitrary individual amounts t>0toagent3. Theoretical Economics 19 (2024) Efficient incentives 981 (iii) From unanimous participation in s, each agent derives a strictly positive interim-expected utility gain: For all i∈Iand all δi∈i,  j∈I δij Eδ−is j(δ)>0. Theorem 1(Efficient Implementation With at Least Three Agents). If n≥3,then there exists a preference-separating mechanism k,Tthat is Bayesian incentivecompatible, interim individually rational, ex post budget-balanced, and ex post materially efficient. If |δij |<1/(2n−3)for all iand all j=i,thenk,Tis necessarily ex post Pareto-efficient. Before we prove Theorem 1, we shall discuss the inner logic of our mechanism. Notice first that, despite the decoupling of incentives to reveal payoff types and social types, our mechanism asks agents to report these types simultaneously. Consider the terms of trade, which operate on agents’ payoff types. As we will see, the terms of trade are social-preference robust in that they leave agents’ social preferences strategically irrelevant. This is achieved by applying the mutual-concessions principle of the dyadical expected-externality mechanism (Arrow (1979); d’Aspremont and GérardVaret (1979)) to each and every single dyad. For the materially efficient social alternative k(ˆ θ), the transfer of agent ito every other jequals j’s expectation of i’s material payoff when jreports payoff type ˆ θj;thatis,itransfers Eθ−j[πi(k(ˆ θj,θ−j)|θi)] to jand receives Eθ−i[πj(k(ˆ θi,θ−i)|θj)] from j. For two other-regarding agents, Bierbrauer and Netzer (2016) show that the expected-externality mechanism is social-preference robust. Agents are incentivized to behave as if they are selfish. If −ireports her payoff type truthfully, then Eθ−i[π−i(k(ˆ θi,θ−i)| θ−i)+t −i(ˆ θi,θ−i)] =Eθ[πi(k(θ)|θi)];thereby,i’s degree of altruism is rendered strategically irrelevant. Bierbrauer and Netzer (2016, p. 570) also show that, in their framework, the conventional n-agents expected-externality mechanism (see Mas-Colell, Whinston, and Green (1995, pp. 886)) is social-preference robust only under an additional symmetry condition. In our framework, social-preference robustness can be established for groups of arbitrary size without any symmetry requirements. Consequently, the terms of trade preserve agents’ privately known social preferences as a strategic degree of freedom, which is utilized by participation-stimulating transfers. Those are independent of the actual allocation problem and serve the purpose of stimulating agents’ participation in the terms of trade. While being ex post budget-balanced, PS transfers yield agents an interim-expected Pareto improvement upon the terms of trade. If this interim-expected Pareto improvement is amplified sufficiently through uniformly scaling up the PS transfers, then agents’ interim-expected utilities from unanimous participation will outweigh their reservation utilities. Notice that the scaling-up is only possible if we allow transfers to take arbitrary negative values. Finally, we note that our participation-stimulation approach cannot succeed in dyads. Proposition 1. Participation-stimulating transfers do not exist if n=2. 982 Daske and March Theoretical Economics 19 (2024) Proof. Suppose the opposite is true. Then Definition 1(iii) requires that 0 < Eδ−i[s i(δ)] +δiEδ−i[s −i(δ)] for both i∈{1, 2}and all δi∈i⊂(−1, 1), while s −i(δ)= −s i(δ)due to ex post budget balance. Hence, 0 <(1−δi)Eδ−i[s i(δ)], implying that 0<Eδ−i[s i(δ)] for all i,δi.Butthen0<Eδ[s i(δ)] for both i, contradicting ex post budget balance. The intuition behind Proposition 1is straightforward: Unanimous participation requires each social type to interim-expect a utility gain, but as budgets must be balanced ex post while each agent values her own material wellbeing more than the other’s, this requires each social type to interim-expect a material benefit. These interim expectations cannot be mutually consistent for all social types, regardless of the specification of transfers s; otherwise, both agents would benefit materially ex ante, contradicting budget balance. 4. Proof of Theorem 1 The proof of Theorem 1proceeds in a series of lemmas. Throughout, n≥3. Lemma 2. Preference-separating mechanisms are Bayesian incentive-compatible and ex post materially efficient. If |δij |<1/(2n−3)for all iand all j= i, they are also ex post Pareto-efficient. Proof.Incentive Compatibility. Suppose the agents other than ireveal their types truthfully. Then the transfers that iinterim-expects for herself and every other jread ¯ tii(ˆ θi,ˆ δi)= =i Eθ−iπk(ˆ θi,θ−i)|θ−(n−1)Eθπik(θ)|θi+Eδ−is i(ˆ δi,δ−i), ¯ tij (ˆ θi,ˆ δi)j=i = =j Eθ−i,θ−jπk(θ)|θ− =i,j Eθ−i,θ−πjk(θ)|θj −Eθ−iπjk(ˆ θi,θ−i)|θj+Eδ−is j(ˆ δi,δ−i) = ∈I Eθπk(θ)|θ−(n−1)Eθπjk(θ)|θj −Eθ−iπjk(ˆ θi,θ−i)|θj+Eδ−is j(ˆ δi,δ−i). Agent i’s interim-expected utility from reporting (ˆ θi,ˆ δi)thus satisfies Ui(ˆ θi,ˆ δi|θi,δi)= j∈I δij Eθ−iπjk(ˆ θi,θ−i)|θj+¯ tij (ˆ θi,ˆ δi) =Eθ−i ∈I πk(ˆ θi,θ−i)|θ+ j=i δij Eθ ∈I πk(θ)|θ −(n−1)Eθ j∈I δij πjk(θ)|θj+ j∈I δij Eδ−is j(ˆ δi,δ−i).(1) Theoretical Economics 19 (2024) Efficient incentives 989 Figure 2. The utility gain gi(δi)=Eδ−i[s i(δ)] +δiEδ−i[s −i(δ)] >0thatasocialtypeδi interim-expects under the transfer scheme sof (11), for two different type distributions, δi∈[δmin i,δmax i]=[−4/5, 4/5],Eδi[δi]=∓2/5, and Varδi[δi]=1/5, such that Eδ−i[s i(δ)] =9/25 −δ2 i,Eδ−i[s −i(δ)] =2δi±4/5, and gi(δi)=(δi±2/5)2+1/5. about Mthan about the others interim-expects to invoke a redistribution from the others to M(from Mto the others), which overcompensates her emotionally for interimexpected monetary losses. 6. Discussion 6.1 What if social types are common knowledge? Asymmetric information about agents’ social preferences is a key assumption in the above analysis. We can easily rule out that participation stimulation in the manner of Definition 1would work for commonly known social types: Under common knowledge, Definition 1(iii) would transform into the requirement that participation-stimulating transfers ex post Pareto-dominate the transfer scheme (si=0)i∈Iof ex post budgetbalanced zero transfers (i.e., j∈Iδij s j(δ)>0foralliand all δ), which is impossible due to Lemma 1. We shall also discuss what is feasible if social types are common knowledge. Plausibly, if agents are sufficiently altruistic (i.e., δij →1foralli,j= i), then individual rationality is satisfied for materially efficient allocation functions and budget-balanced transfers; see also Kucuksenel (2012). Seeking solutions that work for arbitrary social types, let us consider the following example, which we owe to an anonymous referee. 990 Daske and March Theoretical Economics 19 (2024) Example 1. Suppose there are three agents and it is commonly known that δ12 =δ23 = δ31 =1/10 <δ 13 =δ21 =δ32 =1/5. Now consider the following liability rule:Ifagent1 refuses to participate while the other agents agree, then agent 3 must pay x>0to agent 2; if 2 refuses while the others agree, then 1 must pay xto 3; and if 3 refuses while the others agree, then 2 must pay xto 1. Under this liability rule, assuming the respective other agents participate, an agent who refuses incurs a utility loss of x/10. Letting xbe sufficiently large, every mechanism becomes individually rational in Bayes–Nash equilibrium. ♦ In Example 1, an agent who refuses to participate is (emotionally) penalized by forcing the agent she likes more to subsidize the agent she likes less. Obviously, this strategy works for every group in which each agent iprefers some agent jiover some other agent i. Commonly known social preferences can thus be exploited to push, rather than pull, agents into participation. Example 1relates to the branch of literature that considers more general property rights and liability rules, allowing for redistribution even if some agents refuse to participate (Segal and Whinston (2016)) or allowing the designer to impose other threats against nonparticipation (Jehiel and Moldovanu (2006, p. 108)). A caveat to such participation-enforcement strategies is that they presume substantial bargaining power for the designer. Moreover, the concept has a flavor of redundancy: Would the corresponding property rights and liability rules not require agents’ approval in advance, potentially ruling out participation in the overall mechanism by backward induction? In contrast, our participation-stimulation approach works for any specification of property rights and liability rules. It thereby accounts for both the designer’s limited bargaining power and the agents’ free will. 6.2 Practical implementation An important question regarding possibility results concerns their practical relevance; i.e., whether they show how efficient design is attainable in practice or whether they serve to point out practical difficulties in the manner of a “reductio ad absurdum critique.” We shall therefore discuss the possibilities for and limitations to practically implementing our participation-stimulating transfers. We argue that, in an abstract way, participation stimulation can be seen in practice. Observe that our PS transfers only require agents to report a one-dimensional sufficient statistic for their social type. Thus, reporting social types translates into agents selecting one-dimensional strategies in a strategic game. It is this strategic game that renders participation attractive. In what follows, we illustrate how participation may be stimulated through various game forms. The idea is to exploit the agents’ social preferences by having them choose among different levels of a one-dimensional strategic variable and thereby impose positive or negative externalities on each other. For this purpose, define for each agent i=Maset of dedicated supporters Si⊆I\{i,M}and denote by S−i=I\({i,M}∪Si)the set of i’s dedicated opponents. For instance, if we let Si=I\{i,M}for all i= M,thenour formalism shall capture a public-good game among I\{M}, whereas Si=∅for each i= Theoretical Economics 19 (2024) Efficient incentives 991 Mshall capture a competition between the agents other than M.Forgivensets(Si)i=M, participation stimulation can be implemented as follows. Proposition 3. Participation stimulation can be implemented with an indirect mechanism under which agents i=Minvest xi≥0to receive net returns ˆ si((xj)j=M)=−xi+ ci+2μ√xi+2j∈Si√xj−2j∈S−i√xjfor appropriate constants μand (ci)i=M,while ˆ sM=−i=Mˆ si. InthegameofProposition3, agents’ investments may take the form of monetary investments, labor effort, or physical exertion. An agent’s investment imposes a positive (negative) payoff externality on those other agents for whom she is a dedicated supporter (opponent). If Si=I\{i,M}for each i= M, agents are involved in a situation of team-performance pay, effectively a game of private contributions to a public good for I\{M}. Conversely, letting Si=∅for each i= Myields a contest-like situation with relative-performance pay. Participation stimulation thus becomes a principal– agent scenario with Mtaking the role of the principal. Mixtures of relativeand teamperformance pay are feasible too. For instance, the partition I=I1∪I2∪{M}with Si=I\{i}for all i∈Iand ∈{1, 2}leads to a team competition between teams I1 and I2. In all those cases, each i=Mhas the dominant strategy to invest xi=(μ+δS i)2, where δS i= =i:i∈S (δi −δiM )− =i:i∈S− (δi −δiM ) δii −δiM , (12) while letting μ=maxj=M,δ∈|δS j|ensures that the mechanism is well defined. Agent i’s investment is strictly increasing in δS i, and it increases (decreases) in i’s relative pro-sociality toward those agents for whom iis a dedicated supporter (opponent). Hence, whether a dedicated supporter (opponent) turns out to be an actual supporter (opponent) depends on that agent’s social preferences. Moreover, i’s investment increases (decreases) in i’s preference for Mif there are more (less) agents for whom i is a dedicated opponent rather than supporter. The transfer that iinterim-expects for herself is maximal if δS i=0(see(14) in Appendix A.2). Money is thus redistributed ex interim to those agents who are (nearly) indifferent about any form of redistribution between three parties: those they are meant to support, those they are meant to oppose, and, finally, M. On the other hand, an agent who has strong concerns about the distributive effects for and between these three parties will obey (if δS i0) or disobey (if δS i0) her dedicated roles; this results in an interim-expected monetary loss, which is overcompensated for emotionally. We accompany Proposition 3with a real-world example. Think of a community organizing a fundraiser in support of their elementary school (e.g., to fund a new basketball court). The hard-core allocation problem underlying this event is obviously one of public-good provision, and the mechanism to resolve it, if only second best, is actually quite simple, realistically speaking: “Once you’re in, you have to give” as a matter 992 Daske and March Theoretical Economics 19 (2024) of social norm. Events of this sort are often complemented with some soft-core incentive device, like awarding the best-dressed guest. The major purpose of such an addon contest is not to make guests dress well, but rather to suppress “free-riding at the doorstep” by compensating participants for their monetary losses (the lost returns from free-riding) with the social utility they derive from engaging in the contest. Awarding the best-dressed guest provides participants with a platform to live out their propensities to compete, and it is this attraction that helps pull them over the doorstep.9 6.3 Model limitations From the other angle, though, we must scrutinize the assumptions that render participation stimulation possible. As is standard in mechanism-design theory, we assume that transfers may take arbitrary negative values. This presumes that agents are endowed to pay these transfers. As interpersonal transfers play an important role in our study beyond standard theory, it is worthwhile to discuss the impact of budget constraints. As is evident from (5), an agent’s payment (i.e., negative transfer) increases with that agent’s altruism toward that special agent Mwho is designated to balance the budget, and as δiM →1(allelsefixed),agenti’s payment would exceed all limits. Hence, introducing budget constraints would conflict with allowing for arbitrary social-type sets. This raises the question of whether participation stimulation would still work for bounded transfers if we confined social-type sets appropriately. In fact, this is not the case, for any type-set confinement: As is obvious from our derivation of PS transfers in Section 5.2, and from Figure 2in particular, individual payments are likely largest for the extreme social types. On the other hand, narrowing the support tends to decrease the variance (which is the minimum value that the interim-expected utility gain from participation stimulation can take), so PS transfers must be amplified even further through the factor αin the proof of Theorem 1. Similar arguments hold for our various versions of PS transfers, so we must conclude that budget constraints limit the scope of participation stimulation (as we constructed it). A way to resolve this problem would be to meet budget constraints with constraints on agents’ reservation utilities. Our assumption that agents’ social preferences extend to each others’ transfers is critical to our main result, and it distinguishes ours from other papers on mechanism design with social preferences. It implies that agents care about the overall distributive effects of the mechanism, but it requires that agents learn all other agents’ full private payoffs ex post. As outlined by Sobel (2005, p. 400), the domain of social preferences is critical in models with interdependent preferences. Yet, the literature provides little guidance in this regard. Very recent experimental studies suggest that some subjects sometimes apply their social preferences narrowly, but they conclude that more work is needed to explore the extent and the drivers of narrow distributive concerns (see Ellis and Freedman (2024); Exley and Kessler (2024)). Our assumption that private payoffs are quasi-linear while utility is linear in private payoffs is crucial for both preference separation and participation stimulation. It implies that agents are risk-neutral with respect to transfers. We know from Section 6.1 9A similar point is frequently made in conceptual research on how to organize fundraisers; see, e.g., Webber (2004) and Peloza and Hassay (2007). Theoretical Economics 19 (2024) Efficient incentives 993 that participation stimulation relies on agents accepting a gamble over the composition of social types at play. Plausibly, then, risk-averse agents are less susceptible to participation stimulation. We contend that when relaxing these assumptions, participation stimulation, now generally understood as complementing a mechanism with an unrelated strategic game, may still prove helpful in attaining individually rational secondbest implementation. We leave this for future work. 6.4 Relation to Mezzetti (2004) Our bundling of two mechanisms resembles the approach of Mezzetti (2004; henceforth Mezzetti). The key differences between his study and ours are the following. In our model, agents’ social preferences, and thus the allocational and informational externalities associated with them, extend to all agents’ transfers. Mezzetti’s agents can be other-regarding with respect to social alternatives, but must disregard other agents’ transfers; that is, they do not account for the overall distributive effects of a mechanism. As we will see, Mezzetti’s mechanism is thus not incentive-compatible in our model. While we consider a specific framework of one-dimensional allocational and informational externalities, Mezzetti considers a more general framework in which these externalities can be multi-dimensional. Jehiel and Moldovanu (2001)hadshownthat, with multi-dimensional externalities, there exists no mechanism that is both incentivecompatible and efficient, but they restricted attention to one round of reporting mechanisms. Mezzetti shows that the conclusion changes when considering a two-stage mechanism: In the first round of reporting, each agent signals her preference type regarding a set of social alternatives; based on these reports, the designer ultimately chooses an alternative that maximizes aggregate utility. In the second round of reporting, each agent signals the payoff she realizes under this alternative, and interpersonal transfers are determined based on these reports. Specifically, the second-stage transfer scheme utilizes the principle of the VCG (Vickrey (1961); Clarke (1971); Groves (1973)) mechanism: Each agent is transferred the sum of all other agents’ reported outcomedecision payoffs; since this transfer is independent of one’s own report, each agent has the weakly dominant strategy to report her outcome-decision payoff truthfully. By backward induction, this mechanism makes each agent a residual claimant of the full surplus and thereby incentivizes truth-telling in the first reporting stage. Having sketched Mezzetti’s mechanism, we can rule out that it would be incentivecompatible in our model. It is appropriate to consider two versions of his mechanism. The first is a one-to-one adaption to our framework. In the first stage, agents report both their payoff types and social types; based on these reports, the designer chooses the alternative kthat maximizes aggregate utility (which is a weighted sum of all agents’ private payoffs under k). In the second stage, each agent reports the utility she derives under k; based on these reports, she receives a transfer that equals the sum of all the other agents’ reported utility levels. This mechanism is clearly not incentivecompatible in the second reporting stage: An agent’s reported utility level affects every other’s transfer, which she values according to her social type; she is indifferent only if the sum of her degrees of altruism toward the others equals zero and would otherwise underor overstate her outcome-decision utility level. The second version shall 994 Daske and March Theoretical Economics 19 (2024) account for our focusing on social alternatives that condition on payoff types. Whereas we observed that our terms of trade implement the materially efficient alternative, it is natural to ask whether a version of Mezzetti’s mechanism that merely operates on payoff types would achieve the same. However, here, too, the second-stage transfer scheme is not incentive-compatible: Transferring to each agent the sum of the others’ reported outcome-decision payoffs gives almost all social types the incentive to underor overstate these payoffs. Finally, Mezzetti’s and our mechanism differ in the way they attract participation and allow the designer to extract the resulting surplus. (These issues are not discussed in Mezzetti (2004), but are considered in Mezzetti (2003,2007).) When applied to settings in which the surplus from the mechanism is strictly positive for any realization of types, Mezzetti’s mechanism can be rendered individually rational through appropriate lump-sum transfers (see Mezzetti (2003, Proposition 3)). Deploying side bets that leverage the correlation in agents’ second-stage payoff reports (similar to those in Crémer and McLean (1985,1988)), the designer may extract nearly the full surplus (see Mezzetti (2007, Theorem 4)). In our model, in contrast, participation can be attracted whenever social-type distributions have strictly positive variance while transfers may take arbitrary negative values. By leveraging the differences in agents’ other-regarding concerns, the designer can generate a money pump and extract far more than the gains from trade. Appendix A.1 Proof of Lemma 1 Having required weak budget balance, Pareto efficiency implies strict budget balance. Suppose i∈Iti=−for some >0. Then a Pareto improvement can be achieved through transfers (ti+/n)i∈I, since j∈Iδij >0 by assumption. In the following discussion, let |δij |<1/(2n−3)for all iand all j=i. Suppose that, for any fixed transfers (ti)i∈I, there exists a social alternative k◦(θ)that Pareto-dominates the alternative k(θ)∈arg maxk∈Ki∈Iπi(k|θi)while i∈Iπi(k◦|θi)<i∈Iπi(k|θi). Then there must exist agents iwho make strict material losses when switching from k to k◦;thatis,πi(k◦|θi)−πi(k|θi)=−i<0. Let ibe one of the agents for whom this material loss is largest. Agent iis not worse off utility-wise under k◦than under kif and only if she is emotionally compensated through the distributive effects on the others: j=iδij[πj(k◦|θj)−πj(k|θj)] ≥i. We show that this is impossible. First suppose δij≤0forallj=i.Theniobtains the maximum emotional compensation feasible if each j= ialso realizes the maximum material loss of −iwhen switching from kto k◦;thatis,ifπj(k◦|θj)−πj(k|θj)=−i<0. But even then, j=iδij[πj(k◦|θj)−πj(k|θj)] =j=iδij(−i)< i, since 0 ≥δij>−1/(2n−3)≥ −1/(n−1). Now suppose maxj=iδij>0andletj∈arg maxj=iδijbe the favorite agent of i.Theniobtains the maximum emotional compensation feasible if jrealizes a maximum material gain when switching from kto k◦under the constraint that j∈Iπj(k◦|θj)<j∈Iπj(k|θj).Thisisthecaseifeachj=i,jalso realizes the maximum material loss of −iwhile aggregate losses, amounting to (n−1)i, serve as Theoretical Economics 19 (2024) Efficient incentives 995 a subsidy to agent j;thatis,ifπj(k◦|θj)−πj(k|θj)=−i<0forallj= i,jwhile πj(k◦|θj)−πj(k|θj)=(n−1)i. But even then, j=iδij[πj(k◦|θj)−πj(k|θj)] = j=i,jδij(−i)+δij(n−1)i< i(n−2)/(2n−3)+i(n−1)/(2n−3)=i, since |δij|<1/(2n−3)for all j= i.Hence,agentiis worse off under k◦than under k, implying kis Pareto-efficient. It remains to show that, for any fixed social alternative k, no ex post budget-balanced transfer scheme ex post Pareto-dominates another if |δij |<1/(2n−3)for all iand all j= i. Suppose the opposite is true and that transfers (t◦ i)i∈Iex post Pareto-dominate transfers (t i)i∈I, while both are ex post budget-balanced. Then there is an agent iwho suffers the maximum monetary loss when switching from (t i)i∈Ito (t◦ i)i∈I.Fromhere, the proof proceeds exactly as above. A.2 Proof of Proposition 3 For any given sets (Si)i=M, we obtain participation-stimulating transfers by modifying the transfer scheme (2)–(5)as s M(δ)=− j=M s j(δ)(13) s j(δ)=−C+gjδS j−δS jg jδS j+ =j,M (−1)1S−j()·g δS for j=M(14) gjδS j=Var δjδS j+δS j−EδjδS j2(15) δS j= =j,M (−1)1S−(j)·(δj −δjM ) δjj −δjM (16) for some constant C>0, where 1A(x)is the indicator function (i.e., 1A(x)=1ifx∈A and 1A(x)=0ifx/∈A). To see this, we follow the proof of Lemma 4. Strategy Proofness.Unders, each agent j= Mreports a social type ˆ δj,whichis strategically equivalent to reporting some signal ˆ δS j∈R. Her ex post utility is given by  =M (δj −δjM )s (ˆ δ) =(δjj −δjM )gjˆ δS j−ˆ δS jg jˆ δS j+ =j,M (−1)1S−j()·g ˆ δS  + =j,M (δj −δjM )gˆ δS −ˆ δS g ˆ δS + =,j,M (−1)1S−()·g ˆ δS  + =j,M (−1)1S−(j)·(δj −δjM )g jˆ δS j−C =M (δj −δjM ). Hence, when substituting for δS j==j,M(−1)1S−(j)·(δj −δjM )/(δjj −δjM ),agentj maximizes gj(ˆ δS j)+(δS j−ˆ δS j)g j(ˆ δS j)overthechoiceof ˆ δS j.Asg j>0, each j=Mhas the 996 Daske and March Theoretical Economics 19 (2024) strictly dominant strategy to report ˆ δS j=δS j.AsagentMis not involved strategically, she has the weakly dominant strategy to report her true social type δM. Ex Post Budget Balance.Thisisimmediatefrom(13). Interim-Expected Pareto Improvement. When substituting for δS jand Eδ[g (δS )] = 0=Eδ[g(δS )−δS g (δS )], due to Lemma 3,j’s interim-expected utility from sis  =M (δj −δjM )Eδ−js (δ)=(δjj −δjM )gjδS j−C =M (δj −δjM ) =(δjj −δjM )gjδS j−C(δjj −δjM )−C =j,M (δj −δjM ) =(δjj −δjM )gjδS j−C1+δ j for δ i==i,M(δi −δiM )/(δii −δiM ). Recall that δjj =1>δ jM and gj(δS j)≥Var δj[δS j]> 0, and that δ j<n−2, since δjj −δjM >δ j −δjM for all = j,M. We thus obtain that each j= Mderives positive interim-expected utility from unanimous participation if we let C≤minj=MVarδj[δS j]/(n−1). Finally, due to Lemma 3again, also M’s interimexpected utility is positive if all agents participate: i∈IδMiEδ−M[s i(δ)] =j=M(δMj − 1)Eδ[s j(δ)] =Cj=M(1−δMj )>0. To implement s:→Rwith an indirect mechanism ˆ s:[0, ∞)n→R,weobserve that s j(δ)=2 ∈SjδS −EδδS −2 ∈S−jδS −EδδS +EδjδS j2−δS j2−C =2ˆ cj−μ+δS j2+2μμ+δS j+2 ∈Sjμ+δS −2 ∈S−jμ+δS  =cj−xj+2μxj+2 ∈Sj √x−2 ∈S−j √x =ˆ sj(x)=M when letting √x=μ+δS for μ=maxj=M,δ∈|δS j|while letting cj=2ˆ cjfor ˆ cj=μ·|S−j|−μ·|Sj|−1 2μ2+1 2EδjδS j2− ∈Sj EδδS + ∈S−j EδδS −1 2C. Since agent j= Mhas the strictly dominant strategy to report δS junder sand since dxj/dδS j>0, she also has the dominant strategy to invest xj=(μ+δS j)2under ˆ s. References Andreoni, James and John Miller (2002), “Giving according to GARP: An experimental test of the consistency of preferences for altruism.” Econometrica, 70, 737–753. [977] Antler, Yair (2015), “Two-sided matching with endogenous preferences.” American Economic Journal: Microeconomics, 7, 241–258. 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