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Abstract

What institutional arrangements allow veto players to secure maximal welfare when all agree on both the need for and the 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-maximizing 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

Institutional design mechanism design veto bargaining veto players

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. toward 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 change. 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 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 the scholarship on veto players (Cox and McCubbins, 2001; Diermeier et al., n.d.; Gelbach and Malesky, 2010; Tsebelis, 2002) and veto bargaining (Bueno de Mesquita and Stephenson, 2007; Callander and Krehbiel, 2014; Dragu and Board, 2015; Fox and Stephenson, 2011; Fox and Van Weelden, 2010; Romer and Rosenthal, 1978). There is an extensive literature on the policy and welfare implications of various political institutions under which multiple players must agree to effect policy change (e.g. Cameron and McCarthy, 2004; Indridason, 2011; Matthews, 1989). 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), then assess the players’ equilibrium payoffs under each of these games to determine which institutional arrangement leads to a higher payoff for the 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 normative assessment of the welfare properties of various veto bargaining institutions. In other words, using mechanism design allows us to conduct a welfare 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; Dragu et al., 2014; Dragu and Laver, 2017; Gailmard, 2009; Hörner et al., 2015; among others).¹ In related work, Dragu et al. (2014) have shown that the unique mechanism that satisfies individual rationality, 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 $U_{i} (\cdot)$ for $i \in {A, B}$ . 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. $a, b \geq 0$ ,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 $Γ = {S_{A}, S_{B}; p (\cdot)}$ 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 the elements of the strategy $S_{i}$ for $\in {A, B}$ 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 $x \in R_{+}$ , 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 $d (x) \in {yes, no}$ for every proposal of player $A$ . The rule function that stipulates the outcome under this mechanism is $p (x, d (x)) = x$ if $d = y e s$ and $p (x, d) = 0$ if $d = n o$ . 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. $b \sim U [0, L_{B}]$ ).

Now suppose the players operate within a “communication agenda-setting” mechanism in which player $B$ first sends a message $m \in [0, L_{B}]$ about its preferred policy; next, player $A$ observes the message and makes a policy proposal $x \in R_{+}$ ; 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, $s_{A} = x (m) \in R$ , and the strategy of player $B$ is a message $m$ and a binary decision $d (x) \in {yes, no}$ . The rule function that stipulates the outcome under this mechanism is $p (x (m); (m, d (x))) = x$ if $d (x) = y e s$ and $p (x (m); (m, d (x))) = 0$ if $d (x) = n o$ . 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 payoff of the two veto 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 assess which of the two mechanisms is better from this welfare perspective. While this technique is valuable, it does not 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, $D = {S_{A}, S_{B}; p (\cdot)}$ , the (message) strategy spaces are precisely the type spaces, that is, $S_{i} = [0, L_{i}]$ , 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 $p \in R$ 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 that ensure that the veto players have incentives to 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 $U_{A} (p (a, b), a) \geq U_{A} (p (\tilde{a}, b), a)$ and $U_{B} (p (a, b), b) \geq U_{B} (p (a, \tilde{b}), b)$ , for all $a$ , $b$ , $\tilde{a}$ and $\tilde{b}$ .

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. The mechanisms $p (a, b) = 0$ and $p (a, b) = m a x {a, b}$ are both incentive-compatible. The second mechanism is incentive-compatible 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 $p (a, b) = \frac{a + b}{2}$ is not dominant-strategy incentive-compatible as the players have clear incentives to misreport their preferred policy so as to induce an outcome that 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 $U_{A} (p (a, b), a) \geq U_{A} (0, a)$ and $U_{B} (p (a, b), b) \geq U_{A} (0, b)$ 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 $p (a, b) = m a x {a, b}$ does not 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 $a_{0} \in [0, L_{A}]$ is the solution to the following maximization problem:

\max_{p (a, b)} \int_{0}^{L B} U_{A} (p (a_{0}, b), a_{0}) f_{B} (b) d b,

subject to the incentive compatibility constraints

U_{A} (p (a, b), a) \geq U_{A} (p (\tilde{a}, b), a), \forall a, \tilde{a}, b,

U_{B} (p (a, b), b) \geq U_{B} (p (a, \tilde{b}), b), \forall b, \tilde{b}, a,

and the veto constraints

U_{A} (p (a, b), a) \geq U_{A} (0, a), \forall a, b,

U_{B} (p (a, b), b) \geq U_{B} (0, b), \forall 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 online 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 payoff is $p (a, b) = m i n {a, b}$ for $i \in {A, B}$ .

The proposition suggests that player $i$ ’s optimal mechanism is $p (a, b) = m i n {a, b}$ for $i \in {A, B}$ . To illustrate the intuition of the proposition suppose (without loss of generality) that player $B$ ’s ideal policy is closer to the status quo, that is $m i n {a, b} = b$ . It is straightforward why the mechanism $p (a, b) = m i n {a, b}$ maximizes player $B$ ’s expected payoff: it always implements player $B$ ’s most preferred policy. However, it is less obvious why this mechanism also maximizes $A$ ’s expected payoff since player $A$ would clearly prefer a policy higher than $b$ and closer to $a$ . Intuitively, the reason is that a policy higher than $b$ will not always satisfy player $B$ ’s veto constraint; thus, a mechanism that implemented a policy higher than $b$ (when the veto constraint $B$ is satisfied) would lead (from $A$ ’s perspective) to a volatile outcome: either a very good outcome (i.e. player $A$ gets a policy closer to $a$ ) or a very bad outcome (i.e. player $A$ gets the status quo when $B$ ’s veto constraint is not met). On the other hand, the mechanism $p (a, b) = m i n {a, b}$ induces a smooth and continuous outcome $b$ because $B$ ’s veto constraint is always satisfied. Since player $A$ has weakly-concave preferences, player $A$ ’s expected payoff is higher under the mechanism with the smoother, less volatile outcome.

To illustrate this intuition more formally, let $b \in [0, a]$ . Compare player $A$ ’s expected payoff under the mechanism $p (a, b) = b$ for $b \in [0, a]$ (i.e. the mechanism that always implements player $B$ ’s ideal policy) with player $A$ ’s expected payoff under the mechanism $p (a, b) = a$ for $b \in [\frac{a}{2}, a]$ and $p (a, b) = 0$ for $b \in [0, \frac{a}{2}]$ (i.e. the mechanism that implements player $A$ ’s ideal policy when $B$ ’s veto constraint is met and the status quo policy otherwise). Notice that the outcome is $b$ for $b \in [0, a]$ in the former mechanism while the (expected) outcome is $0$ with probability $1 / 2$ and $a$ with probability $1 / 2$ in the latter mechanism (since $b$ is uniformly distributed on $[0, a]$ ). If we compare the distribution of outcomes from the two mechanisms, the distribution under the latter mechanism is a mean preserving spread of the distribution under the first mechanism.⁵ Since players have weakly-concave preferences, player $A$ is better off under the first mechanism than under the second.⁶

By this proposition, for all possible institutional arrangements (mechanisms) under which these two veto players could interact, the institution in which $p (a, b) = m i n {a, b}$ is the unique equilibrium outcome is the institution under which both players obtain their best expected equilibrium payoff.⁷ 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 is $A$ ’s optimal outcome $p (a, b)$ .

Figure 1.

$A^{'} s$ optimal mechanism as a function of $B^{'} s$ ideal policy (for a fixed $a$ ).

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 $p (a, b) = m i n {a, b}$ for all $a, b \geq 0$ , then that institution maximizes each player’s expected utility. In contrast, if the equilibrium outcome is not $p (a, b) = m i n {a, b}$ for all $a, b \geq 0$ , 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 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 an 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 $x_{i} \in [0, L_{i}]$ for $i \in {A, B}$ . The timing of the game is the following:

The players simultaneously make a policy demand $x_{i} \in [0, L_{i}]$ for $i \in {A, B}$ .

The policy outcome is $m i n {x_{A}, x_{B}}$ .

Given this game, each player has a (weakly) dominant strategy to demand its ideal policy,that is, $x_{A}^{*} = a$ and $x_{B}^{*} = b$ , which implies that the unique equilibrium of this game is $p (a, b) = m i n {a, b}$ for any $a$ and $b$ . To show this, we show that, for any choice $x_{B}$ , the unique optimal strategy of player $A$ is $x_{A}^{*} = a$ . If player $A$ were to choose some policy $x_{A}^{*} < a$ , then such a choice either makes no difference to 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 $x \in ℝ$ . The timing of the game is the following:

Player $A$ chooses a policy bound $l$ .

Player $B$ chooses a policy $x \in ℝ$ .

The outcome of the game is $x$ if $x \leq ℓ$ and $q = 0$ if $x > ℓ$ .

We show that the unique equilibrium outcome of this game is $m i n {a, b}$ . 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 $x = m i n {b, ℓ}$ 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 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 $U_{B} (\cdot)$ is single-peaked, player $B$ ’s optimal strategy is $b$ if $b \leq ℓ$ and $ℓ$ if $b > ℓ$ . Therefore, player $B$ ’s unique optimal strategy is $x^{*} = m i n {b, ℓ}$ .

Given that the unique optimal strategy for player $B$ is $x^{*} = m i n {b, ℓ}$ 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 $b \leq ℓ^{'}$ , player $A$ receives the same payoff; for all $b \in (ℓ^{'}, a]$ , player $A$ is worse off because the outcome is $b$ if it chooses $a$ and $ℓ^{'} < b \leq a$ 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 $b \in [a, ℓ^{'}]$ 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 $m i n {a, b}$ 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.

Footnotes

Acknowledgements

We thank Xiaochen Fan, Indridi Indridason, Michael Laver, and Mattias Polborn for helpful comments and suggestions. All errors are ours.

Declaration of conflicting interest

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Supplementary material

The supplementary files are available at .

Notes

Carnegie Corporation of New York Grant

The open access article processing charge (APC) for this article was waived due to a grant awarded to Research & Politics from Carnegie Corporation of New York under its ‘Bridging the Gap’ initiative. The statements made and views expressed are solely the responsibility of the author.

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Supplementary Material

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Veto players,policy change,and institutional design

Abstract

Keywords

The model

Analysis

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interest

Funding

Supplementary material

Notes

Carnegie Corporation of New York Grant

References

Supplementary Material