WORKING PAPER. A Ranking Mechanism for Coupled Binary Decision Problems

Transcription

1 WORKING PAPER A Ranking Mechanism for Coupled Binary Decision Problems Kilian M. Russ 1 and Justus Winkelmann 1 1 Bonn Graduate School of Economics January 18, 2016 Abstract We propose a simple mechanism for coupled binary decisions, which is sensitive to preference intensities. Agents communicate not only whether they are in favor of rejecting or accepting the alternative for each decision problem, but also report an ordinal ranking over the decision problems themselves. The mechanism then picks for each problem the alternative that maximizes the sum of agents expected utilities given their reports. We prove that this indirect mechanism is incentive compatible and admits a full equilibrium characterization for finitely many decision problems. Furthermore our mechanism improves upon Separate Majority Voting and achieves first best efficiency in the limit, as the number of decision problems tends to infinity. We would like to thank Benny Moldovanu and Andy Kleiner for helpful comments.

2 1 Introduction In practice, almost all collective decisions in society be it in referenda, parliaments or committees are made by means of Majority Voting. Voting is commonly regarded as stemming from two underlying paradigms. First, everyone s opinion must be taken into account when making a decision and second all individuals are to be treated equally. The latter generally rules out money for mediation of conflicting interests in particular in the presence of wealth inequality. From an economic perspective the desirability of voting is less clear-cut and remains controversial. Usual voting procedures fail to elicit preference intensities, which in turn are indispensable for achieving efficient outcomes. Nevertheless for a simple Yes/No-decision voting has the invaluable property of being incentive compatible, meaning that individuals do not lie about the private information about their preferences. For instance, if one simply asked individuals how much they cared about an issue, which has to be approved or dismissed, they would certainly like to exaggerate their feelings to sway the decision in their favor regardless of how much they actually cared about it. So is it at all possible to elicit preference intensities in an incentive compatible way? In this paper we give an affirmative answer to this question for a special, but practically relevant class of decision problems, coupled binary decisions. We consider a setting with a closed agenda of several independent decision problems each consisting of two alternatives which have to be accepted or rejected. Rather than deciding upon each problem separately our mechanism makes use of the linkage of problems by eliciting agents relative preference intensities between problems. In our indirect mechanism agents do not only report whether they are in favor of accepting or rejecting the alternative for each decision problem, but also report an ordinal ranking over the decision problems themselves. The mechanism then picks for each decision problem the alternative that maximizes the sum of agents expected utilities given their reports. Put differently, the reports are regarded as truthful and alternatives are accepted if they are expected to increase overall welfare and rejected else. We prove that this procedure indeed induces the intended behavior (Proposition 3). Since our procedure uses all reported information including the ranking over problems we will refer to it as the Ranking Mechanism. It is worth emphasizing that the decision of whether an alternative is implemented for any given decision problem depends on both the number of individuals in favor to do so and their relative intensity towards the issue. In other words our mechanism is sensitive to cardinal information of preferences. We believe that the Ranking Mechanism offers a simple and novel alternative to existing voting procedures. We argue that it is easy to understand and induces intuitive behavior. The proposed mechanism arguably constitutes a rather significant departure from Majority Voting, but increases efficiency substantially, while retaining important properties such as ex-ante fairness or anonymity between agents and neutrality between alternatives. Furthermore our mechanism achieves full efficiency in the limit (Proposition 5), i.e. as the number of decision problems tends to infinity. 1

3 Related Literature Accumulating preferences of individuals has been proven to be a non-trivial problem. Arrow s [1951] impossibility result states that under relatively mild assumptions and ordinal preferences over three or more alternatives, there is no social choice function that is Pareto-efficient, independent of irrelevant alternatives and non-dictatorial. This insight has made its way from collective choice to the literature of mechanism design in the form of the Gibbard [1973] - Satterthwaite [1975] Theorem which shows that with ordinal preferences over at least three alternatives a social choice function is truthfully implementable if and only if it is dictatorial. The subsequent literature has either focused on the robustness of these impossibility results or on ways to circumvent them, e.g. through the weaker equilibrium concept of Bayes-Nash implementation or allowing for randomization. The idea of coupling decision problems is not new to the literature. It has been shown that coupling multiple independent copies of the same binary decision problem holds the promise of improving efficiency. However, for the setup with finitely many decision problems Hortala-Vallve [2010] shows that under dominant strategy implementation full efficiency can never be achieved and that any mechanism must be ordinal, i.e. insensitive to cardinal information of preferences. We show (Proposition 2) that even if one relaxes the implementation concept to Bayes-Nash incentive compatibility first best efficiency remains unachievable. However, there exist mechanisms that are sensitive to cardinal intensities and Pareto improve upon ordinal mechanisms such as Separate Majority Voting. Most notable examples thereof are Qualitative Voting due to Hortala-Vallve [2012], Storable Votes by Casella [2005] and a Rationing Mechanism by Jackson and Sonnenschein [2007]. Our Ranking Mechanism combines all desirable features of the existing literature and moreover is the only mechanism that admits a full equilibrium characterization for a finite number of decision problems. Section 5 introduces and summarizes these three competing mechanisms and their properties. Furthermore we present a comprehensive comparison between the performance, tractability and applicability of all alternatives and our Ranking Mechanism. A similar setting has been studied by Barbera et al. [1991]. Again, voters have to choose a subset of a finite set of objects but voters preferences are assumed to be linear orderings over all possible subsets that can be adopted. Their main result characterizes the set of dominant strategy implementable mechanisms, which implies that similarly to the result of Hortala- Vallve [2010] only the most preferred subset of each voter can be elicited truthfully. Our work is also related to the idea of cumulative voting in which agents are called to elect a subset of candidates. Similar to Qualitative Voting by Hortala-Vallve [2012], agents are endowed with a fixed number of votes that can be distributed on a set of candidates. Even if the endowment of several votes permits each agent to express some degree of preference intensities, cumulative voting primarily aims to increase the chance for minorities to elect a representative through concentrating votes (see Gerber et al. [1998]). Another related concept is the idea of vote trading which allows to elicit preference intensities through establishing a vote exchange market or introducing money as a medium of exchange. In the literature it is controversial whether vote trading is welfare enhancing (see, for example, 2

4 Buchanan and Tullock [1962]; Coleman [1966]; Philipson and Snyder Jr [1996]) or welfare reducing (see, for example, Piketty [1994]; Riker and Brams [1973]). Generally, the idea of vote trading is problematic since agents, who want to buy/sell their vote are heavily reliant on finding a compatible trading partner. Thus, usual market inefficiencies like lacking thickness, congestion or missing safety or trust in the market institutions may arise. Introducing money may be rejected on ethical grounds due to its conflict with the equal treatment of all individuals doctrine in an environment with unequal distribution of material wealth. On the contrary, our Ranking Mechanism elegantly solves this problem by endogenizing the costs of balancing preferences. A high-ranked decision problem has a negative effect on the rank of the other decision problems. In particular, every agent can personally express his/her preferences without relying on other agents or institutions. Another strand of the literature does not exclude monetary transfers from social choice settings per se. Recent contributions have identified two novel justifications to do so, by either taking the payment to the external agency explicitly into account for the welfare assessment (Kleiner and Drexl [2014] or considering more robust implementation requirements (Bierbrauer and Hellwig [forthcoming]). The rest of the paper is organized as follows: Section 2 illustrates our mechanism by means of an example. The formal model is presented in section 3. Section 4 introduces our Ranking Mechanism and studies its properties. In section 5 we assess and compare the performances of our and several benchmark mechanisms. Section 6 concludes. 2 An Illustrative Example Suppose there are three agents 1, 2, 3}, who have to decide over three binary decisions A, B, C}. Assume that preferences are independent and additively separable across different issues. For each decision problem, every agent has cardinal valuations for the different outcomes. 1 normalize the valuation to 0 if an issue is rejected and the status quo is maintained. valuation u k i of agent i 1, 2, 3} for accepted issue k A, B, C} can be positive or negative and we assume that it is drawn independently from a standard normal distribution. This means that if an agent i has a positive valuation u k i > 0, he is in favor of implementing issue k and if he has a negative valuation u k i < 0 he prefers to maintain the status quo. The absolute value u k i represents his preference intensity towards the issue. In our Ranking Mechanism, agents report whether they are in favor of accepting or rejecting the change in any given decision problem. Additionally they communicate an ordinal ranking over the decision problems themselves. E.g. a typical report of agent i is of the form m i = (111, ABC), indicating that agent i prefers accepting the change in every decision problem 2 and 1 One possible real life example for which these assumptions seem reasonable as a first order approximation are U.S. ballot measures. The constitutional amendments and statues put forward usually cover a variety of unrelated topics, for instance the vote on: business protection laws, assisted death and medical use of marijuana (2012 Massachusetts general election state ballot measures). 2 By convention the first number corresponds to the first issue on the agenda, the second number to the second issue on the agenda and so on, regardless of the individual s priority ranking. We The 3

5 cares most about A, then about B and least about C. Note that there are 8 6 = 48 possible reports which are equally probable ex-ante due to the symmetry of the standard normal distribution. After reporting these messages, the Ranking Mechanism picks for each decision problem the alternative that maximizes the sum of the agents expected utilities given their reports. We will establish later that reporting sincerely constitutes a Bayes-Nash equilibrium. Suppose for now that agent 1 reports m 1 = (100, ABC), agent 2 reports m 2 = (100, ACB) and agent 3 reports m 3 = (111, CAB). There is consensus on approving issue A and every agent is expected to derive positive utility from approval. We therefore omit the corresponding calculation and turn to the more interesting issues. Agent 1 and agent 2 reject issue B and agent 3 approves it. The expected utilities of approving B of the agents are: E[u B 1 m 1 = (100, ABC)] = 0.732, E[u B 2 m 2 = (100, ACB)] = and E[u B 3 m 3 = (111, CAB)] = The sum of expected utilities is negative and the Ranking Mechanism consequentially rejects B. Lastly, we turn to issue C. The expected utilities of the different agents are: E[u C 1 m 1 = (100, ABC)] = 0.335, E[u C 2 m 2 = (100, ACB)] = and E[u C 3 m 3 = (111, CAB)] = The sum of the expected utilities is and hence the Ranking Mechanism approves issue C, even though a majority prefers a rejection. The reason is that agents 1 and 2, who are against the approval, report relatively weak preference intensities, whereas agent 3, who favors an approval, indicates a strong preference intensity. The above example illustrates the Ranking Mechanism s sensitivity to cardinal information. For both issues B and C there are two agents against and one agent in favor of implementation. It is the reported ranking - capturing preference intensities - which ultimately determines how much weight is given to each agent and who is decisive. Consequentially the outcome in the two decision problems may very well be different, with B being rejected and C being accepted. This in particular highlights the rather strong departure from Majority Voting. Obviously there are many more combinations of reports for which the outcome of our Ranking Mechanism differs from that of Majority Voting. This leads to higher ex-ante expected welfare per person of our Ranking Mechanism (0.633) compared to Majority Voting (0.598), but strictly lower welfare compared to first-best (0.691). 4

6 3 The Model 3.1 The Setting There are n individuals, who have to decide on d binary decision problems. Each decision problem k = 1,..., d consists of two alternatives x k 0, 1}, where we interpret 0 as maintaining the status quo and 1 as implementing a change. We identify the overall outcome with a vector x 0, 1} d where the kth component x k is the outcome of decision problem k 1,..., d}. We normalize the utility of maintaining the status quo to 0 for every decision problem and every agent. Each agent i = 1,..., n draws a von Neumann-Morgenstern utility vector u i = (u 1 i,..., ud i ) U d i = U i R d representing his cardinal utility if the change is implemented in each of the different decision problems. We assume that u k i is drawn from an atom-free, symmetric (around zero) distribution with bounded support U i R. Furthermore u k i is independent across agents and across problems. We denote its cumulative distribution function (cdf) by P and probability density function (pdf) by ρ. In particular, these assumptions imply that indifferences both between alternatives and between problems occur with probability zero. Furthermore, we assume that an agent s utility is additively separable between problems, which allows us to summarize agent i s utility given overall outcome x 0, 1} d by V i (x) = d k=1 uk i x k. 3.2 Impossibility Results The following section offers a brief theoretical justification for our interest in the topic and demonstrates the necessity to work with Bayes-Nash implementation as an equilibrium concept. Albeit these results it should be noted that we do not pursue a mechanism design approach in this work, but rather propose a simple mechanism with several appealing properties. We start with defining when a mechanism takes cardinal information into account: Definition 1. A linking mechanism is sensitive whenever there exist two preference profiles with identical signs (of u k i ) in all decision problems k = 1,..., d}, which yet result in a different outcome in at least one decision problem. The first proposition originates from Hortala-Vallve [2010]: Proposition 1 (Hortala-Vallve [2010]). There exists no mechanism for d linked decision problems that is strategy-proof, unanimous and sensitive in an unrestricted domain. This means that under dominant strategy implementation, if one imposes unanimity, there is no advantage in linking decision problems. Thus, the best strategy-proof, anonymous and unanimous mechanism in a symmetric and neutral environment is Separate Majority Voting. 3 In the light of the above proposition the requirement of strategy-proofness appears to be too demanding. We believe quite often it is reasonable to assume that agents know less about other agents. The strictly weaker Bayes-Nash equilibrium concept might potentially enable us to 3 See Schmitz and Tröger [2012] for a proof of the optimality of Majority Voting in the class of all strategy-proof and anonymous mechanisms in a setting with two alternatives. 5

7 implement a much wider class of social choice functions. However, Bayes-Nash implementation does not render the problem uninteresting. Incentive considerations still posit a non-negligible restriction, as full efficiency remains unachievable: Proposition 2. There exists no mechanism for d linked decision problems that implements the first best social choice function in Bayes-Nash equilibrium in an unrestricted domain. Proof. See Appendix. Together the above results raise the following question: Is it possible to find Bayes-Nash incentive compatible mechanisms that are sensitive to cardinal information of preferences and Pareto improve upon Separate Majority Voting? In the next section we propose a simple, indirect mechanism, which gives an affirmative answer to this question. 4 The Ranking Mechanism As mentioned in the introduction our Ranking Mechanism asks agents to report two pieces of information: First, whether or not they support the proposed change in every decision problem and second a strict priority ranking over the decision problems themselves. The first part is essentially analogous to the type of information that is elicited in standard Voting procedures. It is the second part that allows us to make use of the link between problems by capturing relative preference intensities. Formally, agent s message space T = (a, π) a 0, 1} d, π permutation over 1,..., d} } consists of two parts. In the first part of his report m i T, agent i states whether he is in favor of maintaining the status quo (0) or implementing the change (1) in every problem. We summarize this part of his report by a vector a 0, 1} d. In the second part of his report agent i reports an ordinal ranking π over decision problems indicating his relative intensities for the different decisions. Since the Ranking Mechanism is indirect, we denote a strategy of agent i by a function t i : U i T. We intend to induce the following very simple and intuitive behavior by the agents: Definition 2. A strategy t : U i T is sincere if agent i reports his favored alternative x k 0, 1} for every problem k and a ranking which sorts all problems by the absolute value of u k i from high to low. Of course, we need to verify that agents indeed find it optimal to behave sincerely. For now suppose agents play sincere strategies and denote the report profile of all agents by m = (m 1,..., m n ) (T... T ) = T n. An indirect mechanism with message space T is a function f : T n L 0, 1} d}. 4 We define our Ranking Mechanism in the following way: Definition 3. The Ranking Mechanism g : T n 0, 1} d regards agents reports as sincere and picks the alternative that maximizes the expected sum of utilities of all agents conditional on 4 Formally, we allow for randomization. However, the assumptions on the type space imply that our mechanism is deterministic with probability one. 6

8 their reports. Ties are broken randomly. Formally, we define g(m) arg max x 0,1} d } E[V i (x) m i ]. In words, the Ranking Mechanism chooses the outcome, which implements the highest expected overall utility conditional on the report profile. It is in fact the best decision rule based on the available information. The crucial question however is whether agents have incentives to report their type in the same manner in which our Ranking Mechanism interprets it. 4.1 Incentive Compatibility In this section we prove the most important property of the Ranking Mechanism, namely incentive compatibility. More concretely, we provide an exact equilibrium characterization for an arbitrary finite number of coupled decision problems. To the best of our knowledge it is the only known sensitive mechanism on coupled binary decision problems with an exact equilibrium characterization: Proposition 3. The proposed mechanism is incentive compatible, i.e. all agents playing strategy t constitutes a Bayes-Nash equilibrium of the induced game. Proof. See Appendix. It is worth noting that our result is actually stronger: playing the sincere strategy is a best response to every set of strategies of the other agents in which no agent discriminates alternatives by their names (the proof of the proposition contains a formal definition). Incentive compatibility is essential for any further analysis of our mechanism. In particular, welfare results for settings with finitely many decision problems require an exact equilibrium characterization. Furthermore, we think that the above equilibrium is rather intuitive and hence very robust in terms of equilibrium selection. 4.2 Welfare In this section we first compare the Ranking Mechanism to Separate Majority Voting in terms of efficiency. We show that ex-ante expected welfare under the Ranking Mechanism is never lower and quite generally strictly higher than under Separate Majority Voting. Second, we provide a powerful limiting result stating that as the number of coupled decision problems goes to infinity, the Ranking Mechanism achieves full (unconstrained) efficiency. Proposition 4. The Ranking Mechanism (weakly) ex-ante Pareto dominates Separate Majority Voting. 5 5 Proposition 4 holds strict if either one of the following two conditions is met: 1. There are d 2 decision problems and the number n 2 of agents is even; 2. There are d 2 decision problems, the number n 2 of agents is odd and either the distribution of preferences is dispersed enough, or there are sufficiently many decision problems, or n is large enough. The exact conditions can be found in the Appendix. 7

9 Proof. See Appendix. Noteworthy the Ranking Mechanism strictly Pareto dominates Separate Majority Voting unless the problem is either trivial or small (see footnote 5). Perhaps unsurprisingly, our mechanism outperforms Separate Majority Voting whenever the number of agents is even, which is in large part due to the latter s inefficient resolution of ties and the consequential weak performance in that case. However, the Ranking Mechanism obviously constitutes a much greater departure not just tie breaking from Separate Majority Voting and hence holds the potential to increase welfare substantially. In Appendix B we illustrate this point by assessing its performance in settings with different numbers of agents and decision problems. In the remainder of this section we present our limiting result, which is reminiscent of Jackson and Sonnenschein [2007]. 6 For any number of agents and any distribution of preferences satisfying our assumptions above the Ranking Mechanism achieves full efficiency in the limit as the number of decision problems becomes arbitrary large: Proposition 5. As the number of decision problems tends to infinity, the corresponding sequence of sincere Bayes-Nash equilibria of the Ranking Mechanism results in expected utilities that converge to first best utility levels. 7 Proof. See Appendix. Apart from being theoretically appealing the above result offers a strong rationale for linking decision problems. 5 Comparison to Existing Mechanisms As mentioned in the introduction, we are not the first to propose a mechanism for the class of coupled binary decision problems. In this section we first compare our Ranking Mechanism to Qualitative Voting by Hortala-Vallve [2012], the paper most closely related to ours. We then present two alternative mechanisms, Storable Votes by Casella [2005] and the Rationing Mechanism by Jackson and Sonnenschein [2007], both however slightly differing either in modeling assumptions or focus. 5.1 Qualitative Voting Hortala-Vallve [2012] Hortala-Vallve [2012] works in the identical setting but proposes an alternative voting mechanism, which works as follows: All agents are endowed with a fixed number of votes that can be distributed freely among a set of issues that need to be approved or dismissed. An alternative is accepted if the total number of votes cast in favor of accepting is larger than the number of votes cast in favor of rejecting. If there is a tie, usual Majority Rule is applied. Qualitative Voting is sensitive to cardinal information of preferences due to voters ability to shift votes on issues that they care relatively more about. But despite its alleged simplicity Hortala-Vallve is not able 6 For a detailed comparison to Jackson and Sonnenschein [2007] see section 5. 7 We call an equilibrium sincere, if all agents play sincere strategies. 8

10 to construct a Bayes-Nash equilibrium of the induced game and his results are consequentially rather limited. The most advocated feature of Qualitative Voting, its simplicity, does not outweigh the complications due to strategic behavior by the agents. Our Ranking Mechanism is at least as simple and intuitive as Qualitative Voting and does not require agents to calculate some complicated strategy on how to cast their votes. In fact, the Ranking Mechanism shifts all computations to the designer and agents face a rather simple optimization problem. 5.2 Storable Votes by Casella [2005] Casella [2005] presents a simple, dynamic voting mechanism for binary decisions taken by a committee over time. Committee members, who meet regularly over time, decide by means of majority rule between two alternatives. Casting votes is not mandatory and votes can be withheld (thereby stored ) and used in later decisions. Even though the decision rule is ordinal, the mechanism is sensitive to preference intensities due to agents ability to intertemporally shift votes to decisions of high personal priority. However, the setup considerably differs from ours in terms of the resolution of uncertainty about agents preferences. We drop the assumption on sequential resolution of uncertainty because it seems rather unrealistic that a committee meets regularly over time and agents are completely naive about their preferences in the next period. In particular, the example of a central bank seems inappropriate since preferences for lowering or raising interest rates are presumably not independent over time. Apart from this modeling assumption there are at least two additional caveats to Storable Votes. First, recall that the Ranking Mechanism weakly Pareto dominates but in particular never performs worse than Separate Majority Voting. In contrast there are different scenarios in which Storable Votes achieves a strictly lower ex-ante expected welfare than Separate Majority Voting. Second, Casella [2005] does not provide a full equilibrium characterization. She shows that equilibrium strategies are monotone cut-off strategies. This result does not suffice to fully characterize equilibria, which is essential for any theoretical predictions and certainly necessary to assess welfare rigorously. 5.3 Rationing Mechanism by Jackson and Sonnenschein [2007] Jackson and Sonnenschein [2007] prove a rather universal limiting result stating that as the number of coupled decision problems tends to infinity, full ex-ante efficiency can be reached by a simple procedure that requires agents to ration their reports according to the underlying distribution. Their mechanism is more general in the sense that it works for decision problems with an arbitrary number of alternatives. The Rationing Mechanism works as follows: Agents type space is finite. For any number of decision problems an agent is asked to report his type, but he has to ration his reports so as to match the underlying distribution of his preferences as best as possible, meaning e.g. if an agent is likely to have a high type with small probability, he is only allowed to report relatively few high types. The Mechanism then picks the alternative 9

11 that maximizes agents expected welfare given their reports. However, as Jackson and Sonnenschein are solely interested in the limiting behavior, their procedure and method of proof do not readily extend to the case of finitely many problems. In particular, the Rationing Mechanism does not admit an equilibrium characterization in the case of finitely many decision problems. Thus, for finitely many decision problems their mechanism makes no clear predictions, with respect to neither behavior nor welfare. Furthermore, Jackson and Sonnenschein [2007] rationing procedure necessitates a restriction to finite type spaces. This induces another problem for a setting with finitely many decision problems apart from an exact equilibrium characterization. The number of decision problem and the number of possible types are intrinsically intertwined through their mechanism, i.e. for the rationing to make sense there have to be at least as many problems as types. For a reasonable application of the mechanism even a very small type space would require a relatively large number of decision problems. Lastly, the Rationing Mechanism relies heavily on knowledge about the distribution. Not only the mechanism designer must be perfectly informed, but also the agents must know their underlying distribution to ration reports accordingly. In contrast, in the Ranking Mechanism agents do not need to have any knowledge of their own distribution. 6 Conclusion All in all, the Ranking Mechanism combines all desirable properties of the existing alternative mechanisms discussed above. It enhances welfare due to its sensitivity to preference intensities, it is efficient in the limit and it operates on quite general settings allowing for continuous distributions of valuations. Last but not least, all predictions are rigorously founded in a full equilibrium characterization, which has been the major drawback of the existing literature. Furthermore we argue that the prescribed equilibrium is simple and intuitive, and therefore likely to resemble individuals behavior in practice. Our Ranking Mechanism requires a strong assumption regarding the designer, namely the knowledge of the true underlying distribution of preferences. In terms of applicability to real world problems it might prove very fruitful to investigate the mechanism s robustness with respect to that particular assumption. From a theoretical perspective, the full second best problem remains unsolved. 10

12 A Appendix: Proofs Proof of Proposition 2. For convenience we introduce the following notation: U = U 1... U n and U i = U 1... U i 1 U i+1... U n. A direct mechanism is a function h : U L 0, 1} d}. We denote the implemented outcome in decision problem k 1,..., d} by h k 0, 1}. We start by modifying Proposition 1 from Hortala-Vallve [2010] to the case of Bayes-Nash implementation. This step is straightforward and gives us: Corollary 1 (Hortala-Vallve [2010]). A linking mechanism h is Bayes-Nash incentive compatible if and only if the agents expected indirect utilities are homogeneous of degree one (HD1) and convex on their own preferences. That is V (u i ) = d k=1 uk i E u i [h k (u i, u i )] fulfills V (λ u i ) = λ V (u i ) for λ R + and is convex in u i. It follows, that an incentive compatible social choice function satisfies E u i [h k (u i, u i )] = E u i [h k (λ u i, u i )] for every j = 1,..., d. In words, all proportional types of agent i = 1,..., n} are bunched in expectations. We now proceed to prove that there exists no Bayes-Nash incentive compatible mechanism that implements the first best social choice function. For sake of contradiction suppose that h implements the first best scf in Bayes-Nash equilibrium. From the result above we know that proportional types have to be bunched in expectations (taken w.r.t. the other agents types). Consider the following two cases: 1) Proportional types are in fact bunched not only in expectations but also type by type; 2) Proportional types are bunched in expectations, but there exist type profiles (of the other agents) for which they are not bunched. Case 1: It is enough to consider the case of two agents and one decision problem. Note that for one decision problem all possible types are proportional and hence h has to be constant, which is clearly not optimal. The same line of reasoning extends to settings with more agents and more decision problems. Case 2: There exists k 1,..., d}, u i U i with u k i 0, λ R + \ 1} and u i U i such that h k (u i, u i ) h k(λ u i, u i ). Consider the case uk i > 0 and h k(u i, u i ) > h k(λ u i, u i ). All other cases follow by an analogous argument. First, note that because f bunches in expectations, there exists u i U i such that h k (u i, u i ) < h k(λ u i, u i ). Second, for fixed u i U i optimality of the function h k (, u i ) necessitates that it only depends on the value of u k i and not on the other components of u i. Further h k (, u i ) has to be non-decreasing in u k i > 0. For λ > 1 this contradicts the first inequality, for λ < 1 the second. Proof of Proposition 3. We prove the following result, which implies proposition 3: Playing the sincere strategy t is a best response when all other agents play a label-free strategy. We adopt Jackson and Sonnenschein s [2007] definition of label-freeness: A strategy t i : U i T for agent i is label-free, if it depends only on the realization of i s preferences and not the labels of the problems. Formally, given a permutation Π : 1,..., d} 1,..., d} and any u i = (u 1 i,..., ud i ) U i, let u Π i be defined by (u Π i )k = u Π(k) i for each k 1,..., d}. Given our indirect mechanism and its message space T = m = (a, π) a 0, 1} d, π is permutation over 1,..., d} } we analogously define t Π i (u i) = (a Π, Π(π)). A strategy t i is label-free if for any 11

13 permutation Π : 1,..., d} 1,..., d}, t Π i (uπ i ) = t i(u i ) for all u i U i. In the following we fix an agent i, a decision problem k 1,..., d} and assume that all other agents play label-free strategies. Note that the conditional expected utility E[u k i m i] of agent i in decision problem k 1,..., d} given a message m i depends on both the reported preferred outcome a k and the ordinal ranking π. By construction of the mechanism and because all other agents play label-free strategies it is always weakly better for agent i to report his preferred decision sincerely. This information determines only the sign of the conditional expected utility of the alternative and does not provide any information about the expected intensity. In particular, for any reported ordinal ranking π it is weakly optimal to report a sincerely. To prove incentive compatibility we can therefore restrict attention to reports of agent i consisting of the sincere a and an ordinal ranking π. In the following we will show that it is indeed optimal for agent i to also report a sincere priority ranking π, i.e. one that sorts all problems by the absolute value of u k i from high to low. The absolute value of the expected utility for decision problem k depends on the reported ordinal ranking. In particular, since all decision problems are identically distributed, it is enough to know the rank π(k) of decision problem k, i.e. E[u k i m i] = E[u k i π(k)]. Agent i is only interested in scenarios, in which he is pivotal, meaning that he is decisive and changes the outcome. We denote the number of other agents messages in which agent i reports a k and rank π(k) and is pivotal in decision problem k by m i j i E[uk j m j] < 0 < j i E[uk j m j] < E[u k i (a k, π(k))]} if ak = 1 p(π(k)) = m i E[u k i (a k, π(k))] < j i E[uk j m j] < 0 < j i E[uk j j]} m if ak = 0. It is more convenient to work with the probability of agent i being pivotal in decision problem k when reporting rank π(k). We denote this probability by P (π(k)) = p(π(k)) m i T n 1 }. The simple but crucial observation from the above is that the lower the rank π(k) (i.e. the higher the priority), the higher the probability of being pivotal P (π(k)), because a lower rank increases the absolute value of the conditional expected utility E[u k i m i ] of decision problem k. Therefore from agent i s perspective reporting a linear ordering can be equivalently be interpreted as distributing probabilities of being pivotal. Note that given label-free strategies of all other agents agent i gets is preferred outcome in any decision problem with probability 1 2 anyway and is not pivotal. Under the assumption that agent i sincerely reports a, he is left with the following maximization problem: max π k:a k =1 (P (π(k)) ) uk i + k:a k =0 (1 P (π(k)) 1 2 ) uk i 12

14 = max π = max π = max π k:a k =1 (P (π(k)) ) uk i + d P (π(k)) u k i k=1 d P (π(k)) u k i k=1 k:a k =1 k:a k =0 ( 1 2 P (π(k))) ( uk i ) u k i 1 2 k:a k =0 The expression is maximized by assigning the highest P (π(k)) to the highest u k i, the second highest P (π(k)) to the second highest u k i and so far. Hence, it is optimal to induce higher probability of being pivotal on problems which have a higher absolute value. This coincides with the sincere strategy t and thereby completes the proof. Proof of Proposition 4. First, we show that the Ranking Mechanism never performs worse than Separate Majority Voting. This follows by a rather simple type by type argument, because the Ranking Mechanism elicits strictly more information in an incentive compatible way than Separate Majority Voting. In particular, the first part the vector a of agents reports coincides with the message that agents communicate under Separate Majority Voting. As mentioned in the main text, the decision rule of the Ranking Mechanism trivially maximizes the expected utility given all reported information. Because all information that is elicited under Separate Majority Voting is also used in the Ranking Mechanism, the latter never performs worse. In particular, on any type profile under which the implemented outcome between the two mechanisms differs, Majority Voting on average achieves strictly lower, negative utility. u k i Second, we provide a technical condition under which the Ranking Mechanism strictly increases ex-ante expected welfare upon Separate Majority Voting. Again, we provide sufficient conditions for the existence of at least one report, on which Separate Majority Voting differs from the Ranking Mechanism. The most likely circumstance for this occurrence is a weak minority of agents with strong preference intensities outvoting a weak majority of agents with low preference intensities. More technically, suppose n 2 agents rank decision problem k as high as possible (least important) and n 2 agents rank the decision problem as low as possible (most important). Let X (r:d) denote the r-th order statistic of d draws from the distribution ρ. The minority outvotes the majority if n n E[X 2 (1) ] E[X 2 (d) ] n x (1 P(x)) d 1 ρ(x) dx 2 n 2 x (P(x)) d 1 ρ(x) dx This inequality is satisfied, if the distribution is dispersed enough, if there are sufficiently many agents, or if the number d of decision problems is high enough. Note that the above condition in particular implies that the Ranking Mechanism always outperforms Separate Majority Voting if the number of agents is even and there are at least two decision problems. 13

15 Proof of Proposition 5. A solution that is optimal for the coupled decision problem is optimal on every single decision, since agents utility is additively separable between problems. To see this, rewrite the mechanism as g(m) arg max x 0,1} d arg max x 0,1} d arg max x 0,1} d } E[V i (x) m i ] k=1 d k=1 } d E[u k i x k m i ] } E[u k i x k m i ] Thus, the Ranking Mechanism implements the optimal decision for every single problem. This allows us to restrict our attention to a single decision problem. The Ranking Mechanism restricted to a single decision problem k with alternatives 0 and 1 can be denoted by ḡ(m) arg max x k 0,1} [ ] } E u k i x k m i. As mentioned earlier, a report m i of agent i captures both the sign of the valuation of the alternative u k i for each decision problem k and a rank π(k) representing the relative importance of that particular problem. When analyzing the expression E[u k i m i ] it is useful to define a function sgn : 0, 1}, +}, such that + if a k = 1 sgn(a k ) = if a k = 0, which decodes the report into a sign of the valuation of the alternative for a decision problem. Since we work with identical independent copies of the same decision problem, for each agent the absolute value of the conditional expected utility of a decision problem ranked at π(k) is a random variable, which can be represented by the π(k)-th order statistic. Taken together, 8 ] [ ] E [u k i m i = E u k i (a k, π(k)) = E [ ] sgn(a k ) X π(k):d. A priori both signs, +} and every rank π(k) are equally probable for u k i, i = 1,..., n}. Furthermore, the distribution of all valuations are independent across both, agents and alternatives. Using this, we rewrite the ex-ante expected welfare of the mechanism ḡ for a single decision problem k: E U [ u k i ḡ(t (u)) ] = m T ( [ ]) P [m] E u k i ḡ(m) m 8 Note that we use the order statistic to calculate the absolute valuation of the alternative, whereas u k i can be both, positive and negative. For the calculations either the random variable X π(k):d or in the density function that is used in the expectation value has to be multiplied by the factor 2. 14

16 limit: = ( [ ]) P [m] ḡ(m) E u k i m m T = ( [ ] }) P [m] max 0, E u k i m i m T = ( P [m] max 0, E [ ] }) sgn(a k ) X π(k):d m i m T = } d d 1 2 n d n max 0, E [s i X pi :d] s,+} n p 1 =1 = 1 2 n d n = 1 2 n p n=1 d s,+} n p 1 =1 1 1 s,+} n 0 0 d max 0, p n=1 max 0, } E [s i X pi :d] E [ s i X d pi :d] } dp 1... dp n. We now proceed to study the ex-ante expected welfare of the Ranking Mechanism in the lim E [ ] 1 uḡ(t d ) = lim d 2 n = 1 2 n = 1 2 n = 1 2 n s,+} n 1 s,+} n 0 1 s,+} n 0 1 s,+} n =E U [max 0, 0 u k i }], 1 0 max max max max 0, 0, 0, 0, E [ s i X d pi :d] } dp 1... dp n lim E [ ] } s i X d pi :d dp 1... dp n d [ ] } E s i lim X d p d i :d dp 1... dp n } s i P 1 (p i ) dp 1... dp n where P 1 is the inverse of the cdf P of the absolute value of agents valuations. The second equation follows by continuity of the max-operator. The bounded support of ρ allows us to use the Bounded Convergence Theorem, which gives us the third equation. In the fourth equation we use the following result of order statistics, ( which implies convergence ) in probability of the d p -th order statistics: X d p AN P 1 (p),. The last term corresponds to p(1 p) n(ρ(p 1 (p)) 2 ) the first-best expected welfare, which completes the proof. 15

17 B Appendix: Numerical Results B.1 Figure 1 and 2 illustrate the performance of the Ranking Mechanism for an increasing number of decision problems and an increasing number of agents, respectively. Agents preferences follow a triangular distribution on the interval [ 2, 2]. Performance is expressed as a fraction of first best welfare. All calculations were carried out using Mathematica. Figure 1: This graph plots the performance of the Ranking Mechanism (blue) as a fraction of the first best welfare (yellow) for n=2 agents and an increasing number of decision problems. The performance of Separate Majority Voting is constant at 0.7 (green). Figure 2: This graph plots the performance of the Ranking Mechanism (blue) and Separate Majority Voting (green) as a fraction of the first best welfare (yellow) for d=3 decision problems and an increasing number of agents. 16

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