Robust Political Equilibria Under Plurality and Runoff Rule

In Ruling Majorities and Reasoning Pluralities, Professor Saul Levmore explores the “division of labor” between the various thresholds of agreement required for collective action—supermajority, simple majority, or plurality rule. His particular emphasis is on the choice between the last two options. To improve our understanding of this choice in various settings, Professor Levmore considers the relationship between two well-known contributions to the study of group decisionmaking, namely, the Condorcet Jury Theorem and the Condorcet Criterion, which have not generally been treated together. This essay explores the relationship between these two insights in the context of judicial decisionmaking. Counterintuitively, the essay demonstrates that while the Condorcet Criterion continues to hold great promise as a tool of decision in en banc appellate courts, the Condorcet Jury Theorem is most appealing in judicial contexts that appear less collegial and thus less like a jury. In contexts in which jurists engage in deliberation and in which they anticipate that their outputs will have precedential effect, the Jury Theorem has little if any role to play. In contrast, the Theorem might prove more useful in evaluating low-level judicial outputs or decisions that otherwise are not expected to be treated as precedent. Thus, common law trial court decisions and civil law court decisions are more likely to be useful benchmarks in a Condorcet Jury Theorem analysis. It is because such courts do not anticipate that their decisions will have the force of precedent that they are more likely to be derived from governing legal texts, and thus more prone to characterizations of accuracy, than are judicial outputs that are driven by policy concerns and are generated with the understanding that they will be treated as precedent.

The computational study of strategic interaction situations has recently deserved a lot of attention in multi--agent systems. A number of results on strategic--form games and zero--sum extensive--form games are known in the literature, while general--sum extensive--form games are not studied in depth. We focus on the problem to decide whether or not a solution is a refinement of the Nash equilibrium (NE) for extensive--form games. Refinements are needed because the NE concept is not satisfactory for this game class. While verifying whether a solution is an NE is in P, verifying whether it is a NE refinement may be not (all the results known so far show NP--hardness). In this paper, we provide the first positive result, showing that verifying a sequential equilibrium with any number of agents and a quasi perfect equilibrium with two agents are in P. We show also that when the input is expressed in (non--perturbed) sequence form even the problem to verify a subgame perfect equilibrium is NP--complete and that sequence form, if applicable, must be rethought to verify (and therefore to compute) an extensive--form perfect equilibrium.

Local perfection

This paper uses exogenous variation in electoral rules to test the predictions of strategic voting models and the causal validity of Duverger’s Law. Exploiting a regression discontinuity design in the assignment of single-ballot and dual-ballot (runoff) plurality systems in Brazilian mayoral races, the results indicate that single-ballot plurality rule causes voters to desert third placed candidates and vote for the top two vote getters. The effects are stronger in close elections and cannot be explained by differences in the number of candidates, as well as their party affiliation and observable characteristics.

This paper suggests an evaluation of the plurality rule according to how much it can be manipulated by individuals. We analyze strategic voting with reference not only to the usual notion of Nash equilibrium but also to some weaknesses of this notion, depending on the assumptions about how the individual agents choose their strategies. We provide two types of results for each concept of equilibrium: First, we give necessary and sufficient conditions for a profile to be an equilibrium when the plurality rule is used; and second, we present formulas giving the exact frequencies of unstable situations in the three-alternative case.

Strategic customer behavior facing possible stockout: An experimental study

This article reviews basic insights about compliance and "hard" enforcement that can be derived from various non-cooperative equilibrium concepts, and evaluates the Marrakesh Accords in light of these insights. Five different notions of equilibrium are considered – the Nash equilibrium, the subgame perfect equilibrium, the renegotiation proof equilibrium, the coalition proof equilibrium, and the perfect Bayesian equilibrium. These various types of equilibrium have a number of implications for effective enforcement: (1) Consequences of non-compliance should be more than proportionate. (2) Punishment needs to take place on the Pareto frontier, rather than by reversion to some suboptimal state. (3) An effective enforcement system must be able to curb collective as well as individual incentives to cheat. (4) A fully transparent enforcement regime is not unconditionally a good thing. It is concluded that constructing an effective system for "hard" enforcement of the Kyoto Protocol is a formidable task that has only partially been accomplished by the Marrakesh Accords. In practice, however, the design of the compliance system for the climate regime had to balance a desire to minimize non-compliance against other important goals, including the need for due process.

Refining Perfect Bayesian Equilibrium by Bayesian Iterative Conjectures Algorithm

A discussion is presented regarding the single-stage election procedures Plurality, Negative Plurality, and Borda Rules; with an emphasis on their Condorcet Efficiency, which measures their propensity to elect the Condorcet Winner. A theoretical analysis shows how the Condorcet Efficiency of these voting rules change as the various measures of group mutual coherence change. Both Plurality and Negative Plurality Rules are found to perform very poorly over identifiable ranges of the defined measures of group mutual coherence. While Borda Rule is not always found to be the most Condorcet Efficient single-stage rule in all scenarios, it never performs poorly under any scenario. These theoretical results are then reinforced by empirically-based analysis to show very solid support for using Borda Rule, despite some concerns that have been expressed that Borda Rule requires voters to rank candidates, while Plurality and Negative Plurality Rules do not.

This paper introduces a bottleneck game with finite sets of commuters and departing time slots as an extension of congestion games of Konishi et al. (J Econ Theory 72:225–237, 1997a). After characterizing Nash equilibrium of the game, we provide sufficient conditions for which the equivalence between Nash and strong equilibria holds. Somewhat surprisingly, unlike in congestion games, a Nash equilibrium in pure strategies may often fail to exist, even when players are homogeneous. In contrast, when there is a continuum of atomless players, the existence of a Nash equilibrium and the equivalence between the set of Nash and strong equilibria hold as in congestion games (Konishi et al. 1997a).

In this paper strategic situations of voting and abstentions are analysed in a three-candidate voting model where voters are indifferent to various alternatives and candidates are interested in winning the election and seeing their most preferred alternative being realized. A specific decision problem is analysed, described by an Indifference Trap Game, with respect to, e.g., the corresponding Nash equilibria, perfect equilibria, and maximin solution. A second-best outcome is contained in the choice set of all three solutions while the Nash equilibrium concept is compatible with the Pareto efficient outcomes of the game.

We study Hotelling's two-stage model of spatial competition, in which two firms first simultaneously choose locations in the unit interval, then simultaneously choose prices. Under Hotelling's assumptions (uniform distribution of consumers, travel cost proportional to distance, inelastic demand of one unit by each consumer) the price-setting subgames possess equilibria in pure strategies for only a limited set of location pairs. Because of this problem (pointed out independently by Vickrey (1964) and d'Aspremont et al. (1979)), Hotelling's claim that there is an equilibrium of the two-stage game in which the firms locate close to each other is incorrect. A result of Dasgupta and Maskin (1986) guarantees that each price-setting subgame has an equilibrium in mixed strategies. We first study these mixed strategy equilibria. We are unable to provide a complete characterization of them, although we show that for a subset of location pairs all equilibria are of a certain type. We reduce the problem of finding an equilibrium of this type to that of solving three or fewer highly nonlinear equations. At each of a large number of location pairs we have computed approximate solutions to the system of equations. Next, we use our analytical results and computations to study the equilibrium location choices of the firms. There is a unique (up to symmetry) subgame perfect equilibrium in which the location choices of the firms are pure; in it, the firms locate 0.27 from the ends of the market. At this equilibrium, the support of the subgame equilibrium price strategy is the union of two short intervals. Most of the probability weight is in the upper interval, so that this strategy is reminiscent of occasional sales by the firms. We also find a subgame perfect equilibrium in which each firm uses a mixed strategy in locations. In fact, in the class of strategy pairs in which the firms use the same mixed strategy over locations, and this strategy is symmetric about 0.5, there is a single equilibrium. In this equilibrium most of the probability weight of the common strategy is between 0.2 and 0.4, and between 0.6 and 0.8. There is a wide range of pure Nash (as opposed to subgame perfect) equilibrium location pairs: the subgame strategies in which each firm threatens to charge a price of zero in response to a deviation support all but those location pairs in which the firms are very close.

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.

Following recent studies of iterative voting and its effects on plurality vote outcomes, we provide characterisations and complexity results for three models of iterative voting under the plurality rule. Our focus is on providing a better understanding regarding the set of equilibria attainable by iterative voting processes. We start with the basic model of plurality voting. We first establish some useful properties of equilibria, reachable by iterative voting, which enable us to show that deciding whether a given profile is an iteratively reachable equilibrium is NP-complete. We then proceed to combine iterative voting with the concept of truth bias, a model where voters prefer to be truthful when they cannot affect the outcome. We fully characterise the set of attainable truth-biased equilibria, and show that it is possible to determine all such equilibria in polynomial time. Finally, we also examine the model of lazy voters, in which a voter may choose to abstain from the election. We establish convergence of the iterative process, albeit not necessarily to a Nash equilibrium. As in the case with truth bias, we also provide a polynomial time algorithm to find all the attainable equilibria.

Ogawa (Int Tax Public Financ 20(3):474–484, 2013) discusses the analysis of Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010a) by taking into account capital ownership in the government’s objective functions. He establishes that the unique subgame perfect Nash equilibrium (SPNE) of the endogenous timing game corresponds to the simultaneous Nash equilibrium. This result contrasts with Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010a) who conclude to the existence of two Stackelberg outcomes at the SPNEs. Highlighting the role of plain complementarity or substitutability, we obtain conditions under which leadership still emerges at the equilibrium of the endogenous timing game when capital ownership is considered. Numerical simulations confirm this finding and display a case where plain properties are not monotone and a well-identified Stackelberg outcome is the SPNE.