Majority rule when voters like to win

I analyze symmetric majority rule voting equilibria when voters wish to elect the better candidate and to vote for the winner. When voters care only about the winning candidate (the standard formulation) a unique responsive equilibrium exists. The addition of a desire to win creates multiple equilibria, some with unusual properties. In most of these equilibria information is not aggregated effectively, and I uncover the novel possibility of negative information aggregation in which information aggregated in equilibrium is used to select the worse rather than the better candidate.I then characterize the efficiency of optimal equilibria as the population becomes large and show that a discontinuity arises in the information aggregation capabilities of the majority rule voting mechanism: in elections without a dominant front-running candidate the better candidate is almost surely elected, whereas in races with a front-runner information cannot be effectively aggregated in equilibrium.     

Would letting people vote for multiple candidates yield policy moderation?

We investigate whether letting people vote for multiple candidates would yield policy moderation. We do so in a setting that takes three key features of elections into account, namely, strategic voting, endogenous candidacy and policy motivation on the part of the candidates. We consider two classes of voting rules. One class consists of the voting rules where each voter casts several equally-weighed votes for the different candidates. The other class consists of the voting rules where each voter rank-orders the candidates. We identify conditions under which these voting rules yield policy moderation. We also show that these voting rules may yield policy extremism instead of policy moderation if one (or several) of the conditions is not satisfied! Finally, we find that amongst these voting rules the extent of policy moderation is maximal under the Borda Count if we consider only equilibria where all candidates are serious contenders. However, this result does not carry over to spoiler equilibria, where Approval Voting can yield more moderate policy outcomes than the Borda Count.     

Learning in Elections and Voter Turnout

Voter turnout in game theoretic models of voting has typically been difficult to predict because of the problem of multiple Nash equilibria ( Palfrey and Rosenthal 1983, 1985 ). Many of these equilibria require an extreme precision of beliefs among voters that is unlikely to be reached in real elections. At the same time, mechanisms like pre‐election polls exist to shape the beliefs of voters about expected turnout. We combine these two features in a model of voter learning in elections and characterize the asymptotically stable equilibria of both complete and incomplete information games in a simple symmetric setting with two candidates. We also show how the model can be used to qualitatively explain several phenomena observed in reality: increases in costs of voting affect turnout adversely but there may be persistence of turnout levels between elections even though costs and other parameters change. Increase in uncertainty increases turnout while increases in the size of the electorate decrease it, in line with intuition.     

Majority rule when voters like to win

I analyze symmetric majority rule voting equilibria when voters wish to elect the better candidate and to vote for the winner. When voters care only about the winning candidate (the standard formulation) a unique responsive equilibrium exists. The addition of a desire to win creates multiple equilibria, some with unusual properties. In most of these equilibria information is not aggregated effectively, and I uncover the novel possibility of negative information aggregation in which information aggregated in equilibrium is used to select the worse rather than the better candidate.I then characterize the efficiency of optimal equilibria as the population becomes large and show that a discontinuity arises in the information aggregation capabilities of the majority rule voting mechanism: in elections without a dominant front-running candidate the better candidate is almost surely elected, whereas in races with a front-runner information cannot be effectively aggregated in equilibrium.     

Approval voting with endogenous candidates

We present a formal model of political competition under approval voting which allows for endogenous candidate entry. Our analysis yields a number of novel insights. First, we develop a notion of sincere voting behavior under approval voting, called relative sincerity. We then show that the relatively sincere voting behavior is consistent with the strategic calculus of voting. Second, we show that in a one-dimensional model with distance preferences, equilibria in relatively sincere strategies and without spoiler candidates always generate outcomes close to the median voter. Moreover, approval voting satisfies Duverger's Law in the sense that there are at most two winning positions! Finally, we extend our analysis to arbitrary policy spaces. In the general setting, approval voting is shown to be susceptible to the same kinds of problems as the plurality rule, such as the possibility of non-majoritarian outcomes, failure to elect the Condorcet winner and existence of spoiler candidates.     

Voting on public goods in multi-jurisdictional systems

This paper studies an economy with multiple jurisdictions, each producing a (pure) public good. Decisions are taken by means of a centralized voting procedure, to which individual agents participate and whose rules are defined with respect to the jurisdictional structure. In particular, a winning alternative must gain the support of a given number of jurisdictions. A well-known example is qualified majority rules. We establish condition on voting rules for the set of voting equilibria to coincide with the set of ratio equilibria of the economy, under alternative assumptions on the possibility of transfers across jurisdictions. We employ the obtained results to characterize the equilibria of two-stage voting situations, in which jurisdictional delegates vote instructed by internal committees.     

Kantian Equilibrium

Consider a game whose strategies are "contributions". A strategy profile is a Kantian equilibrium if  no  player would like  all  players to alter their contributions by the  same multiplicative factor.  Kantian equilibria are Pareto efficient. We characterize the allocation rules on several domains of environments that can be implemented as Kantian equilibria. The concept unifies the  proportional solution  on production economies and the  linear cost-share equilibrium  on public-good economies. We study Kantian equilibrium in the prisoner's dilemma, in a voting problem, and in a political economy where redistribution is the issue. The Kantian dictum engenders considerable but not unqualified cooperation.     

Multi-unit auctions with private information: an indivisible unit continuous price model

We construct a model of multi-unit auctions in which I bidders bid for two indivisible units of a common value good. Using a first-order approach, we find that there are equilibria in which bidders bid the same price for both units in the discriminatory auction, but not in the uniform auction. When there are only two bidders, under certain conditions, there are linear equilibria for both the discriminatory and the uniform auction formats. In all equilibria, bidders equalize the expected marginal benefit of bidding to the marginal costs of bidding. We show that comparison of the seller??s expected revenue across auction formats depends only on the ratio of the precision of private information to the precision of public information.     

Convergence Results for Unanimous Voting

This paper derives a necessary condition for unanimous voting to converge to the perfect information outcome when voters are only imperfectly informed about the alternatives. Under some continuity assumptions, the condition is also sufficient for the existence of a sequence of equilibria that exhibits convergence. The requirement is equivalent to that found by Milgrom [1979, Econometrica47, 679-688] for information aggregation in single-prize auctions. An example illustrates that convergence may be reasonably fast for small committees. However, if voters have common preferences, unanimity is not the optimal voting rule. Unanimity rule makes sense only as a way to ensure minority views are respected. Journal of Economic Literature Classification Numbers: D72, D82, D44.     

Electoral competition with privately-informed candidates

This paper formulates and analyzes a general model of elections in which candidates receive private signals about voters' preferences prior to committing to political platforms. We fully characterize the unique pure-strategy equilibrium: After receiving her signal, each candidate locates at the median of the distribution of the median voter's location, conditional on the other candidate receiving the same signal. Sufficient conditions for the existence of pure strategy equilibrium are provided. Though the electoral game exhibits discontinuous payoffs for the candidates, we prove that mixed strategy equilibria exist generally, that equilibrium expected payoffs are continuous in the parameters of the model, and that mixed strategy equilibria are upper hemicontinuous. This allows us to study the robustness of the median voter theorem to private information: Pure strategy equilibria may fail to exist in models “close” to the Downsian model, but mixed strategy equilibria must, and they will be “close” to the Downsian equilibrium.     

Asymmetric Equilibria in a Model with Costly Voting

Most of the voting models limit their analysis to the investigation of symmetric equilibria where “similar” voters make “similar” voting decisions. In this paper we examine the validity of this restriction in a model with costly plurality voting. We first show that in any pure strategy equilibrium every two individuals who have the same preferences and participate in elections, would vote for the same candidate. However, this result does not hold for mixed strategies equilibria.     

Dynamic stability and reform of political institutions

This paper examines endogenous institutional change in a class of dynamic political games. The political aggregation rules used at date t+1 are instrumental choices under rules at date t. Effectively, rules are “players” who can strategically delegate future policy-making authority to different rules. A political rule is stable if it selects itself. A reform occurs when an alternative rule is selected. The stability of a political rule is shown to depend on whether its choices are dynamically consistent. For instance, simple majority rules can be shown to be dynamically consistent in many common environments where wealth-weighted voting rules are not. The result extends to political rules that incorporate private activities such as extra-legal protests, threats, or private investment. The approach is one way of understanding various explanations of institutional change proposed in the literature. A parametric model of public goods provision gives an illustration.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Voting with abstention

Most voting models in the literature neglect abstention, but is such a simplification justified? I investigate this question in a model with outside pressure on voters. For sequential voting (e.g., roll call votes), with and without an abstention option, there is a unique subgame perfect equilibrium, which implies that true majorities always succeed. Abstention can be an equilibrium strategy for some voters, in particular under complex decision rules (e.g., weighted voting, double majorities). Simultaneous voting often has a unique pure strategy equilibrium but also a plethora of mixed and pure/mixed strategy equilibria. Therefore, only with equilibrium selection, can we evaluate the consequences of neglecting abstention. For equal weight voting, equilibria selected by the procedure of Harsanyi and Selten change completely with an abstention option, even if abstention itself is not or rarely used. With small enough outside pressure, however, the selected equilibrium honors true majorities in both cases.     

The swing voter's curse with adversarial preferences

We analyze voting behavior in a large electorate in which voters have adversarial state-contingent preferences with incomplete information about the state of the world. We show that one type of voter can suffer from the swing voter's curse à la Feddersen and Pesendorfer [The swing voter's curse, Amer. Econ. Rev. 86 (1996) 408-424], and go on to characterize the symmetric Nash equilibria of this model under different parameter values. We prove that unlike settings with nonadversarial preferences, there are equilibria in which in one state of the world, a minority-preferred candidate almost surely wins the election and thus the election may fail to correctly aggregate information. Indeed, we show that the fraction of the electorate dissatisfied with the result can be as large as .     

Pitfalls in Restructuring the Electricity Industry

Abstract. This paper models some reductions in output that may follow the opening of electricity markets to competition. Specifically, we show that vertical separation of electricity generation, transmission and distribution could reduce welfare compared to the previous system of vertically integrated monopoly, if grid owners can act as monopolistic retailers or, alternatively, set access prices that maximize merchandizing surplus. Moreover, we show that a rule of non‐discriminatory pricing would not remedy any of these problems. Hence, to secure the efficiency gains of deregulation, the re‐regulation of the remaining monopolistic core – the electric grid – cannot rely on simple rules like cost‐based non‐discriminatory pricing.     

Multi-stage voting, sequential elimination and Condorcet consistency

A class of voting procedures based on repeated ballots and elimination of one candidate in each round is shown to always induce an outcome in the top cycle and is thus Condorcet consistent, when voters behave strategically. This is an important class as it covers multi-stage, sequential elimination extensions of all standard one-shot voting rules (with the exception of negative voting), the same one-shot rules that would fail Condorcet consistency. The necessity of repeated ballots and sequential elimination are demonstrated by further showing that Condorcet consistency would fail in all standard voting rules that violate one or both of these conditions.     

Simple Markovian equilibria in dynamic spatial legislative bargaining

The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period's status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player's opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in generic games with impatient players and in symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.     

Expectations and optimal monetary policy: A stability problem revisited

A landmark result in the optimal monetary policy design literature is that fundamental-based interest rate rules invariably lead to rational expectations equilibria (REE) that are not stable under adaptive learning. In this paper, we make a novel information assumption that private agents cannot observe aggregate fundamental shocks, and use simple linear forecasting rules for learning. We find that with fundamental-based rules, there exist limited information equilibria that are stable under learning. Moreover, there are multiple equilibria. Learning can be used as a selection tool to identify a unique equilibrium.     

On persistent disequilibrium

Consider a standard model of a Walrasian economy where the time derivative of price change is a sign preserving function of excess demand. The only steady states of such a system are the Walrasian equilibria. For if the system is not at a Walrasian equilibrium then there must be excess demands or supplies in some market and thus those prices must be changing. Walras' law implies that all prices cannot change in the same direction, and thus relative prices must change.However, it seems that in real world economies there have been states of persistent disequilibrium. How can this be? How can there be stationary states of an economy that are not Walrasian equilibria?The answer presented in this paper goes something like this: the demands actually presented to the market, i.e., the demands that affect price movements, are not the Walrasian demands. Rather, these demands, the effective demands, are a function of both price and quantity signals. There is no reason why there cannot be equilibria of the effective demand system that are not equilibria of the Walrasian demand system.In this paper, I present a simple abstract model of effective demand systems and given a condition for this system to have non-Walrasian equilibria. I also present a simple example of this phenomena. Finally I discuss the real world implications of this model. Related concepts of effective demand are discussed in [A1H], [2], [3], [L5], and [L6].