Veto Players, Policy Change and Institutional Design Tiberiu Dragu and Hannah K. Simpson New York University December 2016
Abstract What institutional arrangements allow veto players to secure maximal welfare when they all agree on the need for and direction of policy change? To answer this question, we conduct a mechanism design analysis. We focus on a system with two veto players, each with incomplete information about the other s policy preferences. We show that the unique welfare-optimizing mechanism is the mechanism that implements the preferred policy of the player whose ideal policy is closer to the status quo. We provide examples of institutional structures under which the unique equilibrium outcome of this two-player incomplete information game is the policy outcome implemented by this mechanism, and argue that our result can be used as a normative benchmark to assess the optimality of veto player institutions. Keywords: veto players; veto bargaining; mechanism design; institutional design
Veto players are a common feature of democracies. Generally, veto player institutions are studied in the context of their role in maintaining policy stability: increasing the number of veto players in a political system is thought to weakly increase policy stability because any one veto player with opposing preferences can block policy change (Tsebelis 2002). There is thus an extensive scholarship on the optimal number of veto institutions under different political and economic conditions: for example, more veto players may impede government adaptability to changing economic circumstances when society is divided (Cox and McCubbins 2001), but may facilitate policy change if special interests are weak (Gelbach and Malesky 2010) or (in the case of unanimity vs. majority voting rules) when there is no external policy enforcement mechanism (Maggi and Morelli 2006). Our paper asks a different question: what institutional arrangements allow veto players to secure maximal attainment of their welfare under circumstances where all veto players agree on the need for and direction of policy change? Often, shocks to the state of the world, like terrorist attacks or natural disasters, can shift all veto players preferences in the same direction, e.g., towards increasing security spending or disaster relief. In such cases, all veto players would have a common interest in changing the status quo, but their preferences might diverge regarding which policy reform is desirable. Under these circumstances, the institutional arrangement that structures the players interactions will affect the policy outcome, with important implications for the players welfare. To answer this question, we conduct a mechanism design analysis. A mechanism, for our purposes, is an institution that governs the process by which veto players decide on policy changes. We focus on a system with two veto players, each with incomplete information about the other s policy preferences. Our main result is that the mechanism that yields the best (expected) payoff to each player in such a setting is the mechanism that implements the preferred policy of the player whose ideal policy is closer to the status quo. We provide examples of institutional structures (formalized as non-cooperative games) under which the unique equilibrium outcome of this two-player incomplete information game is the policy 1
outcome implemented by this mechanism. We also discuss the usefulness of our analysis as a normative benchmark to assess a variety of veto player institutions: those institutions that yield a result other than the players less-extreme preferred policy are inefficient from a welfare perspective. Our analysis contributes to scholarship on veto players (Tsebelis 2002; Cox and McCubbins 2001; Gelbach and Malesky 2010) and veto bargaining (Romer and Rosenthal 1978; Bueno de Mesquita and Stephenson 2007; Fox and Van Weelden 2010; Fox and Stephenson 2011; Callander and Krehbiel 2014; Dragu and Board 2015). There is an extensive literature on the policy and welfare implications of various political-institutional structures under which multiple players must agree to effect policy change (e.g., Matthews 1989; Cameron 2000; Cameron and McCarthy 2004; Crombez et al. 2006). The general method of assessing the welfare implications of these institutions has been the following: formalize different institutions (usually two or three) as non-cooperative games, derive their equilibrium outcome(s) and then assess the players equilibrium payoffs under each of these games to determine which of these institutional arrangements leads to a higher payoff for players (for a description of this method, see Diermeier and Krehbiel 2003). Mechanism design analysis is an important next step in this theoretical literature because it facilitates the assessment of the welfare properties of all veto bargaining institutions. In other words, using mechanism design allows us to conduct a comprehensive evaluation of all possible institutional arrangements that could structure how veto players interact. This implies that the results of our analysis can serve as a normative benchmark by which to assess the optimality of veto bargaining models, regardless of their specific institutional characteristics. Our research also contributes to a literature that applies mechanism design to the study of political institutions and settings (Banks 1990; Baron 2000; Gailmard 2009; Dragu, Fan and Kuklinski 2014; Hörner et al. 2015, among others). 1 In related work, Dragu, Fan and Kuklinski (2014) have shown that the unique mechanism that satisfies certain procedural 1 This research note is also related to a literature that investigates the properties of strategy-proof social choice functions on single peaked domains (e.g. Moulin 1981; Penn Patty, and Gailmard 2011). 2
properties (i.e. individual rationality constraints, strategy proofness, and Pareto efficiency) is the mechanism that implements the ideal policy of the player preferring the less aggressive change from the policy status-quo. This research note focuses on a different normative criterion (which mechanism maximizes each player s expected payoff) to show that the unique welfare-maximizing mechanism is also the mechanism that implements the preferred policy of the player whose ideal policy is the closest to the status quo. The Model There are two veto players A and B. The players have preferences over a one-dimensional policy space. An exogenous status-quo q is in place. Without loss of generality, we fix the status-quo at q = 0. Each player s preference is represented by a twice continuously differentiable, single-peaked and symmetric (about an ideal position a and b respectively) and (weakly) concave utility function U i ( ) for i {A, B}. The players policy preferences are private information. Let a and b, the ideal positions of A and B, be independently distributed according to uniform distributions on [0, L A ] and [0, L B ], respectively. As mentioned, we focus our analysis on the scenario in which players agree on the direction of policy change, i.e., a, b 0, 2 because only in this setting does the resulting outcome depend on the institutional arrangement within which the players interact. That is, when players disagree about the direction of policy change (i.e., a < 0 < b or b < 0 < a), the outcome will be the status-quo policy regardless of the institutional arrangement within which the players interact, since this is the only outcome that satisfies the veto condition. A mechanism can be understood as the institution that governs the process by which the two veto players make a collective policy choice. Formally, a mechanism Γ = {S A, S B ; p( )} specifies the set of strategies available to each player, and a rule p(s A, s B ) that stipulates the policy outcome implemented by the mechanism for a given strategy profile s = (s A, s B ). Notice that a mechanism could, in principle, be a complex dynamic procedure, in which case 2 The scenario in which a, b < 0 is similar. 3
the elements of the strategy S i for {A, B} would consist of contingent plans of actions and messages. Notice also that a mechanism Γ, together with the players utility functions and beliefs about their preferred policy, induces a game of incomplete information. To illustrate, consider the following two examples of mechanisms that could structure the interaction of two veto players. First, suppose the players operate within a no communication agenda-setting mechanism in which player A proposes a policy x R +, after which player B decides to accept or veto x. The policy outcome is x if player B accepts player A s policy proposal and q = 0 if player B vetoes A s proposal. Here, player A s strategy is a policy proposal s A = x, and player B s strategy is a binary decision d(x) {yes, no} for every proposal of player A. The rule function that stipulates the outcome under this mechanism is p(x, d(x)) = x if d = yes and p(x, d) = 0 if d = no. Notice that in this mechanism there is no communication between the players regarding their preferred policies, which implies that player A s belief about player B s preferred policy when player A chooses x is the same as her prior (i.e b U[0, L B ]). Now suppose the players operate within a communication agenda-setting mechanism in which player B first sends a message m [0, L B ] about its preferred policy; next, player A observes the message and makes a policy proposal x R + ; and finally, player B accepts or vetoes x. In this mechanism, the strategy of player A is a policy choice as a function of the message player B sends, s A = x(m) R, and the strategy of player B is a message m and a binary decision d(x) {yes, no}. The rule function that stipulates the outcome under this mechanism is p(x(m); (m, d(x))) = x if d(x) = yes and p(x(m); (m, d(x))) = 0 if d(x) = no. In a similar vein, we can construct other mechanisms by permitting different communication protocols and/or by extending the timing of the interaction; in fact, there are infinitely many ways to specify the mechanism under which the two veto players interact, simply by varying the timing of the interaction, the policy proposals, and the messages each player can send. Our goal is to determine the optimal mechanism, by which we mean the mechanism that, among all possible mechanisms, maximizes the expected (interim) payoff of the two veto 4
players. This task would be difficult, if not impossible, to achieve via the standard approach to comparative institutional analysis: modeling each institution as a non-cooperative game, solving for the equilibrium policy outcome generated under each, and then comparing the expected equilibrium payoff for each player under the different institutional arrangements. Going back to the previous examples, for instance, one would solve for the equilibrium outcome under the no communication and communication agenda-setting mechanisms, and then compare the players equilibrium expected payoffs to asses which of the two mechanisms is better from this welfare perspective. While this technique is valuable, it doesn t allows us to conduct a comprehensive evaluation of all possible institutional arrangements that could structure how veto players interact. Instead, we employ a mechanism design approach and exploit the revelation principle. The revelation principle states that, for any equilibrium of a game of incomplete information that is induced by some mechanism under which the players interact, there exists an incentive-compatible direct revelation mechanism that is payoff-equivalent with that equilibrium (Myerson 1979). This implies that it is sufficient to find the optimal mechanism among the set of incentive compatible direct revelation mechanisms in order to determine which is the optimal mechanism among all possible mechanisms that could structure the players interaction. In a direct revelation mechanism, D = {S A, S B ; p( )}, the (message) strategy spaces are precisely the type spaces, i.e S i = [0, L i ], and a policy outcome results as a function of the reported types. One way of thinking about this is that instead of considering all possible institutional arrangements under which the players could interact, we need only study a simple setting: the set of mechanisms in which the players actions are to report their types and an outcome, p(a, b), results as a function of the players (true) types. In other words, an incentive compatible direct mechanism p(a, b) specifies a policy outcome p R as a function of A s and B s true types. To identify the optimal mechanism, we need only find the optimal mechanism from the set of direct revelation mechanisms subject to incentive compatibility constraints which ensure that the veto players have incentives to 5
truthfully reveal their types. In this context, the incentive compatibility constraints are as follows: Incentive Compatibility: A mechanism p(a, b) is dominant-strategy incentive compatible if and only if U A (p(a, b), a) U A (p(ã, b), a) and U B (p(a, b), b) U B (p(a, b), b), for all a,b, ã and b. This incentive compatibility condition requires that truthful revelation is an equilibrium in (weakly) dominant strategies in the game of incomplete information induced by the direct mechanism p(a, b). 3 To illustrate what the incentive compatibility condition entails, consider the following dominant-strategy incentive-compatible mechanisms. p(a, b) = max{a, b} are both incentive-compatible. The mechanisms p(a, b) = 0 and The second mechanism is incentivecompatible because, for each player, the outcome is either its own ideal policy or some policy higher than its ideal policy; in the former case, a player has no incentive to deviate and, in the latter case, the only way to change the outcome is to announce and implement an even higher policy, which would make the player worse off. The first policy mechanism is trivially dominant-strategy incentive-compatible. The outcome is the status quo, q = 0, regardless of what the players are doing; therefore, the players do not have an incentive to misreport their preferences. Note that dominant-strategy incentive-compatible mechanisms can be complicated in the sense that, in some intervals, a player s ideal policy is implemented and in other intervals a constant policy is implemented. Consider also the following examples of a mechanism that violates incentive compatibility. The mechanism p(a, b) = a+b 2 is not dominant-strategy incentive-compatible, as the players have clear incentives to misreport their preferred policy so as to induce an outcome that 3 Notice that the incentive compatible direct mechanisms are required to operate on the basis of veto players ideal points alone. In principle, these mechanisms could take into account all aspects of players preferences, however, the existing literature has shown that allowing for the use of additional information (in settings in which agents have single-peaked preferences over a one-dimensional policy space) does not enlarge the set of dominant strategy incentive compatible mechanisms (Barbera and Jackson 1994); thus our restriction is without loss of generality. 6
is closer to their most preferred policy outcome. For example, let a = 1 and b = 11. The outcome under this mechanism is p(a, b) = 6; player B has incentives to misreport its preferred policy to b = 21 so to change the outcome to p(a, b ) = 11, player B s ideal policy. Since each player can veto changes to the status-quo policy, players utility from the policy outcome resulting under an incentive-compatible mechanism p(a, b) must be at least as high as their payoffs from the status-quo. This gives rise to the following veto requirement: Veto Constraints: A policy mechanism p(a, b) satisfies the veto constraints if and only if U A (p(a, b), a) U A (0, a) and U B (p(a, b), b) U A (0, b) for all a and b. To illustrate what the veto requirement entails, consider the following mechanisms. First, the mechanism p(a, b) = 0 satisfies the veto condition since the outcome under this policy mechanism is always the status-quo policy. On the other hand, the mechanism p(a, b) = max{a, b} doesn t satisfy the veto requirement. To see this, let a = 1 and b = 10, which implies that the outcome of this mechanism is p(a, b) = 10. However, player A is better off with the status-quo policy than with the policy p(a, b) = 10; therefore, the veto constraint is not met. Analysis Given the mechanism design problem formulated previously, the mechanism p(a, b) that maximizes player A s expected payoff for any type a 0 [0, L A ] is the solution to the following maximization problem: LB max U A (p(a 0, b), a 0 )f B (b)db, p(a,b) 0 subject to the incentive compatibility constraints U A (p(a, b), a) U A (p(ã, b), a), a, ã, b, 7
U B (p(a, b), b) U B (p(a, b), b), b, b, a, and the veto constraints U A (p(a, b), a) U A (0, a), a, b, U B (p(a, b), b) U B (0, b), a, b. The maximization problem for player B is defined in an analogous manner. 4 Finding the optimal incentive-compatible mechanism for player B) (and A) that satisfies the veto conditions is a somewhat technical problem. We relegate the details of the proof to the appendix and state the main result below: Proposition. The unique incentive-compatible policy mechanism that satisfies the veto constraints and maximizes player i s expected (interim) payoff is p(a, b) = min{a, b} for i {A, B}. The proposition suggests that player i s optimal mechanism is p(a, b) = min{a, b} for i {A, B}. This result implies that, for example, even if player A s preferred policy is a further departure from the status-quo than player B, the mechanism that maximizes player A s expected (interim) payoff is the one that implements the ideal policy of player B. To provide some intuition for this result, let b [0, a] and let us compare player A s expected payoff under the mechanism p(a, b) = b for b [0, a] with player A s expected payoff under the mechanism p(a, b) = a for b [ a, a] and p(a, b) = 0 for b [0, a ]. Notice that, given that 2 2 player A is uncertain about player B s ideal policy, the outcome is b for any b [0, a] in the former mechanism while the outcome is 0 with probability 1/2 and a with probability 1/2 in the latter mechanism. Now consider the expected payoff of player A under two distributions: one in which the outcome is b for b [0, a] and one for which the outcome is 0 if b [0, a ] and a if 2 4 Notice that we are looking for the mechanism that maximizes player i s expected (interim) payoff given that player i knows her type but is uncertain about the other player s ideal policy for i {A, B}. 8
b [ a, a], given that b U[0, a]. The two distributions have the same mean a, but the second 2 2 distribution has a bigger variance. Thus the first distribution second-order stochastically dominates the second, and, because the players have weakly concave preferences, player A is better off under the first distribution. As a result, player A s expected payoff is better under the mechanism p(a, b) = b for b [0, a] than under the mechanism p(a, b) = a for b [ a 2, a] and p(a, b) = 0 for b [0, a 2 ].5 The previous example compares two mechanisms to illustrate the intuition of the proposition. But the proposition is more encompassing: it states that, for all possible institutional arrangements (i.e., mechanisms) within which the two veto players could interact, the institution where p(a, b) = min{a, b} is the unique equilibrium outcome that provides both players their best expected equilibrium payoff. In other words, the institution under which p(a, b) = min{a, b} for all a, b 0 is the unique equilibrium outcome is the optimal institutional arrangement among all possible institutions that could structure the interaction between the two veto players. Figure 1 illustrates A s optimal policy outcome as a function of all possible locations of B s preferred policy. The variable on the horizontal axis is B s preferred policy b, while the variable on the vertical axis A s optimal outcome p(a, b). Notice that this proposition can be used as a normative benchmark to assess various veto player institutions. In principle, we can take any institutional setting under which two veto players interact, formalize that institution as a game, and then analyze its equilibrium outcomes. If the equilibrium of that game is p(a, b) = min{a, b} for all a, b 0, then that institution maximizes each player s expected utility. In contrast, if the equilibrium outcome is not p(a, b) = min{a, b} for all a, b 0, then the players do not secure maximal attainment of their payoffs under that institution. For example, the no communication agenda-setting institution previously discussed is not welfare-optimal since there are conditions under which the equilibrium outcome of that game is the status-quo policy, although both players would 5 The same rationale applies when comparing the mechanism p(a, b) = b for b [0, t] with the mechanism p(a, b) = t for b [ t 2, t] and p(a, b) = 0 for b [0, t 2 ] where t [0, a]. 9
p(a, b) a...................... 0 a b Figure 1: A s optimal mechanism as a function of the B s ideal policy (for a fixed a). prefer some policy change (i.e. there is a positive equilibrium probability that player B rejects player A s policy proposal p and therefore the equilibrium outcome is the status-quo policy for some a, b > 0). It is also of interest to establish whether the outcome induced by the optimal mechanism can be obtained as the unique equilibrium outcome under some well specified non-cooperative game. In the remainder of this note, we show this by setting out two simple games of incomplete information, one simultaneous and one sequential, each of which generates outcome induced by the optimal mechanism as the unique equilibrium. First, consider the following simultaneous game in which a player i s action space is a policy demand x i [0, L i ] for i {A, B}. The timing of the game is the following 10
The players simultaneously make a policy demand x i [0, L i ] for i {A, B}. The policy outcome is min{x A, x B }. Given this game, each player has a (weakly) dominant strategy to demand its ideal policy, i.e. x A = a and x B = b, which implies that the unique equilibrium of this game is p(a, b) = min{a, b} for any a and b. To show this, we show that, for any choice x B, the unique optimal strategy of player B is x A = a. If player A were to choose some policy x A < a, then such a choice either makes no difference for the outcome or can result in a worse outcome for player A (a choice of x A < a makes no difference to the outcome if x B < a and results in a worse outcome for player A if a < x B ). Similarly, if player A were to choose some policy x A > a, then such a choice either makes no difference for the outcome or can result in a worse outcome for player A (a choice of x A > a makes no difference to the outcome if x B < a and results in a worse outcome for player A if a < x B ). Therefore, player A has a dominant strategy to choose x A = a. By a similar reasoning, player B has a dominant strategy to choose x B = b. Second, consider the following sequential game in which player A chooses a policy bound l and then player B chooses a policy x R. The timing of the game is the following: Player A chooses a policy bound l. Player B chooses a policy x R. The outcome of the game is x if x l and q = 0 if x > l. We show that the unique equilibrium outcome of this game is min{a, b}. We prove this by backward induction. In the second stage, player B s beliefs about player A s type after each observed policy bound are irrelevant because they do not affect player B s payoff. That is, if player A s choice in the first stage is l, the unique optimal strategy for player B is x = min{b, l} for any strategy of player A, and any beliefs of player B. To see this, let l be some policy bound chosen by player A in the first stage. After player B observes player A s 11
choice, player B can either choose a policy x l and the resulting outcome is x; or player B can choose a policy x > l and the resulting outcome is q = 0. Since U B ( ) is single-peaked, player B s optimal strategy is b if b l and l if b > l. Therefore, player b s unique optimal strategy is x = min{g, l}. Given that the unique optimal strategy for player B is x = min{g, l} for any l, we next prove that, in the first stage, player A s optimal strategy is l = a. If player A were to choose some l < a, then for all b l, player A receives the same payoff; for all b (l, a], player A is worse off because the outcome is b if it chooses a and l < b a if it chooses l ; and for all b > a, player A is worse off because the outcome is a if it chooses a and l < a if it chooses l. And if player A were to choose some l > a, then for all b a, player A receives the same payoff; for all b > a, player A is worse off because the outcome is b if b [a, l ] or l if b > l. Thus player A s optimal decision is l = a. As a result, the unique equilibrium outcome of this game is min{a, b} for any a, b 0, as claimed. Conclusion In this paper, we conduct a mechanism design analysis to determine how, under circumstances where all veto players agree on the direction of policy change but may diverge on the optimal amount, veto player institutions can be designed to facilitate the implementation of policy that maximizes the players expected payoff. We focus on a system with two veto players, each with incomplete information about the other s policy preferences. We show that the welfare-optimizing mechanism is the mechanism that implements the preferred policy of the player whose ideal policy is closer to the status quo. We provide examples of institutional structures under which the unique equilibrium outcome of this two-player incomplete information game is the policy outcome implemented by this mechanism, and argue that our result can be used as a normative benchmark to assess the optimality of veto player institutions. 12
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Online Appendix Before proceeding to prove the main proposition, we use the following lemmas 1 4 that describe some properties of dominant strategy incentive compatible mechanisms in this setting, properties that are useful to prove our main result (for proofs of these lemma, see Dragu, Fan and Kuklinski (2014)). Lemma 1. For any b, any mechanism p(a, b) that is dominant strategy incentive compatible for player A is weakly increasing in a; if p(a, b) is strictly increasing in a on an open interval (a 1, a 2 ), then p(a, b) = a on (a 1, a 2 ). For any a, any mechanism p(a, b) that is dominant strategy incentive compatible for player B is weakly increasing in b; if p(a, b) is strictly increasing in b on an open interval (b 1, b 2 ), then p(a, b) = b on (b 1, b 2 ). Lemma 2. Let p(a, b) be a dominant strategy incentive compatible mechanism. Then for any b, if p(a, b) = a on (a 1, a 2 ), then p(a, b) is continuous at both a 1 and a 2. Similarly, for any a, if p(a, b) = b on (b 1, b 2 ), then p(a, b) is continuous at both b 1 and b 2. Lemma 3. Let p(a, b) be a dominant strategy incentive compatible mechanism. Then for any b, if â is a discontinuity point of p(a, b), then â lim a â p(a, b) > 0 and lim a â +p(a, b) â > 0. Similarly, for any a, if ˆb is a discontinuity point of p(a, b), then ˆb lim b ˆb p(a, b) > 0 and lim b ˆb+p(a, b) ˆb > 0. Lemma 4. Let p(a, b) be a dominant strategy incentive compatible mechanism. Then for any b, if for some â, p(â, b) = c â, then p(a, b) = c for all a (min{c, â}, max{c, â}). Similarly, for any a, if for some ˆb, p(a, ˆb) = c ˆb, then p(a, b) = c for all b (min{c, ˆb}, max{c, ˆb}). Proof of Proposition. First, it s easy to see that p(a, b) = min{a, b} is a dominant strategy incentive compatible mechanism. For example, if player A were to report some â > a, then player A s payoff is either unchanged (if b a) or lower if (b > a). A similar reasoning 16
applies to why player B does not have an incentive to misreport its type if the outcome is p(a, b) = min{a, b}. Next we show that p(a, b) = min{a, b} is the unique mechanism that maximizes player A s expected (interim) payoff (among all dominant strategy incentive compatible mechanisms that satisfy the veto constraints). That is, we show that for any a 0 [0, L A ], p(a, b) = min{a, b} solves the following maximization problem subject to the incentive compatibility constraints LB max U A (p(a 0, b), a 0 )f B (b)db, (1) p(a,b) 0 U A (p(a, b), a) U A (p(ã, b), a), a, ã, b, U B (p(a, b), b) U B (p(a, b), b), b, b, a, and the veto constraints U A (p(a, b), a) U A (0, a), a, b, U B (p(a, b), b) U G (0, b) a, b. To show that p(a, b) = min{a, b} is the solution of (1), it suffices to show that p(a 0, b) = min{a 0, b} solves the following maximization problem LB max U A (p(a 0, b), a 0 )f B (b)db, (2) p(a 0,b) 0 subject to incentive compatibility and veto constraints of B when A is at a 0 U B (p(a 0, b), b) U B (p(a 0, b), b), b, b U B (p(a 0, b), b) U B (0, b), b. 17
Note that the second problem maximizes the same objective function from a larger pool of mechanisms than the first problem. As a result, if p(a 0, b) = min{a 0, b} is the solution to the second problem, (2), then p(a, b) = min{a, b} for all a and b, which satisfies all the constraints of the first problem, is the solution to the first problem, (1). Therefore, denote the solution to (2) by p (a 0, b), it suffices to show that p (a 0, b) = min{a 0, b}. First, we show that p (a 0, b) is continuous in b. Suppose p (a 0, b) is not continuous in b, and let ˆb be a discontinuity point of p (a 0, b). By Lemma 3, ˆb lim b ˆb p (a 0, b) > 0 and lim b ˆb+p (a 0, b) ˆb > 0. Since U B ( ) is symmetric, we have ˆb lim b ˆb p (a 0, b) = lim b ˆb+p (a 0, b) ˆb. We prove this last statement by contradiction. Suppose that ˆb lim b ˆb p (a 0, b) lim b ˆb+p (a 0, b) ˆb, and, with loss of generality, suppose that ˆb lim b ˆb p (a 0, b) > lim b ˆb+p (a 0, b) ˆb(> 0). By Lemma 1 and Lemma 2, any discontinuity point must be one that connects two flat segments. As a result, there exists ɛ > 0 such that p (a 0, ˆb ɛ) = lim b ˆb p (a 0, b) and p (a 0, ˆb + ɛ) = lim b ˆb+p (a 0, b). Since (ˆb ɛ) p (a 0, ĉ ɛ) > p (a 0, ˆb + ɛ) (ˆb ɛ), player B type ˆb ɛ has an incentive to misreport its type as ˆb + ɛ and thus p (a 0, b) violates the incentive compatibility constraint for b = ˆb ɛ. Denote m lim b ˆb p (a 0, b) and n lim b ˆb+p (a 0, b), from the above we have ˆb m = n ˆb > 0, that is ˆb = m+n 2. By Lemma 4, p (a 0, b) = m for all b [m, m+n 2 ), and p (a 0, b) = n for all b ( m+n 2, n]. Now consider p (a 0, b) such that p (a 0, b) = b for b [m, n] and p (a 0, b) = p (a 0, b) for all other b. Since p (a 0, b) satisfies the incentive compatibility and veto constraints for all b, it is easy to check that p (a 0, b) also satisfies the incentive compatibility and veto constraints for all b. We now show that p (a 0, b) yields a higher expected payoff for player A than p (a 0, b). Since p (a 0, b) and p (a 0, b) differ only on [m, n], it is sufficient to show that n U m A(a 0, b)f B (b)db > m+n 2 U m A (a 0, m)f B (b)db + b m+n U A (a 0, b)f B (b)db. The outcome of the 2 left hand side is a random variable, denoted by p, which is uniformly distributed on [m, n]; 18
while the outcome of the right hand side is a random variable, denoted by y, such that y = m with probability 1 and y = n with probability 1 m+n. We have Ep = Ey = ; Varp = (n m)2 ; 2 2 2 12 and Vary = 1 m+n (m ) 2 + 1 m+n (n ) 2 = (n m)2. Since Ep = Ey and Varp < Vary, y 2 2 2 2 4 is a mean-preserving spread of p. Since U A (a 0, ) is (weakly) concave, player A prefers p to y, that is, the left hand side is greater than the right hand side, a contradiction to p (a 0, b) being the solution to (2). Therefore, the solution p (a 0, b) is continuous in b. The veto constraint at b = 0 requires that p (a 0, 0) = 0. This together with Lemma 1 and the continuity of p (a 0, b) imply that p (a 0, b) can only take the following form: for some t [0, L B ], p (a 0, b) = b for b [0, t] and p (a 0, b) = t for b > t. As a result, p (a 0, b) is equivalent to the solution to the following problem. LB t max t [0,L B ] t 0 U A (a 0, b)f B (b)db + LB t U A (a 0, t)f B (b)db (3) We claim the solution to (3) is t = a 0. To see this, denote Φ(t) t 0 U A(a 0, b)f B (b)db + U A (a 0, t)f B (b)db. For any t < a 0, Φ(a 0 ) = t U 0 A(a 0, b)f B (b)db+ a 0 U t A (a 0, b)f B (b)db+ L B a 0 U A (a 0, a 0 )f B (b)db, and Φ(t) = t U 0 A(a 0, b)f B (b)db+ a 0 t a0 t U A (a 0, t)f B (b)db+ L B a 0 U A (a 0, t)f B (b)db. Φ(a 0 ) Φ(t) = [U A (a 0, b) U A (a 0, t)]f B (b)db+ L B a 0 [U A (a 0, a 0 ) U A (a 0, t)]f B (b)db > 0, because U A (a 0, b) U A (a 0, t) > 0 for any b (t, a 0 ) due to single-peakedness of U A (a 0, ), and U A (a 0, a 0 ) U a (a 0, t) > 0 since a 0 is the peak of U A (a 0, ). For any t > a 0, Φ(a 0 ) = a 0 U 0 A (a 0, b)f B (b)db+ t a 0 U A (a 0, a 0 )f B (b)db+ L B U t A (a 0, a 0 )f B (b)db, and Φ(t) = a 0 0 U A (a 0, b)f B (b)db + t a 0 U A (a 0, b)f B (b)db + L B t U A (a 0, t)f B (b)db. Φ(a 0 ) Φ(t) = t a 0 [U A (a 0, a 0 ) U A (a 0, b)]f B (b)db + L B [U t A (a 0, a 0 ) U A (a 0, t)]f B (b)db > 0, because U A (a 0, a 0 ) U A (a 0, b) > 0 and U A (a 0, a 0 ) U A (a 0, t) > 0 since a 0 is the peak of U A (a 0, ). Therefore, Φ(a 0 ) > Φ(t) for any t a 0, and therefore t = a 0. That is, p (a 0, b) = b for 19
c [0, a 0 ] and p (a 0, b) = a 0 for b > a 0, which implies that p (a 0, b) = min{b, a 0 }. The proof for p(a, b) = min{a, b} being also the unique mechanism that maximizes the player B s expected payoff (among all dominant strategy incentive compatible mechanisms that satisfy the veto constraints) is analogous. 20