Richard D. McKelvey

A multistage game representation of sophisticated voting for binary procedures

Abstract Farquharson [5] was instrumental in creating the concept of sophisticated voting, in which voters are assumed to successively eliminate dominated strategies. However, his “matrix reduction” procedure is so cumbersome that little use has been made of his insights. Here we define, for binary procedures, a multistage method of sophisticated voting which is easy to apply and intuitively understandable. Our approach allows us to prove that if a majority alternative exists, sophisticated voting leads to an outcome at least as good as and sometimes preferred to the outcome of sincere voting. Our approach yields different sophisticated strategies than Farquharsons method, but we conjecture that both approaches always yield the same outcome.

The Competitive Solution for N-Person Games Without Transferable Utility, With an Application to Committee Games

This essay defines and experimentally tests a new solution concept for n-person cooperative games-the Competitive Solution. The need for a new solution concept derives from the fact that cooperatie game theory focuses for the most part on the special case of games with transferable utilii v, even though, as we argue here, this assumption excludes the possibility of modelling most interesting political coalition processes. For the more general case, though, standard solution cotlcepts are inadequate either because they are undefined or they fail to exist, and even if they do exist, theyfocus on predicting payoffs rather than the coalitions that are likely to form. The ComPetitive Solution seeks to avoid these problems, but it is not unrelated to existent theory in thit we can establish some relationships (see Theorems 1 and 2) between its payoff predictions and those of the core, the V-solution and the bargaining set. Additionally, owing to its definitio~i and motivation, nontrivial coalition predictions are made in conjunction with its payoff pt edictions. The Comb ptitive Solutions definition is entirely general, but a special class of gamesmajority ru: Kspatial games-are usedfor illustrations and the experimental test reported here consists of .ight plays of a 5-person spatial game that does not possess a main-simple Vsolution or i bargaining set. Overall, the data conform closely to the Competitive Solutions predictions.

Structural Instability of the Core

Let σ be a q-rule, where any coalition of size q, from the society of size n, is decisive. Let w(n,q)= 2q-n+1 and let W be a smooth ‘policy space’ of dimension w. Let U(W)N be the space of all smooth profiles on W, endowed with the Whitney topology. It is shown that there exists an ‘instability dimension’ w∗(σ) with 2≦w∗(σ)≦w(n,q) such that: 1. (i) if w≧w∗(σ), and W has no boundary, then the core of σ is empty for a dense set of profiles in U(W)N (i.e., almost always), 2. (ii) if w≧w∗(σ)+1, and W has a boundary, then the core of σ is empty, almost always, 3. (iii) if w≧w∗(σ)+1 then the cycle set is dense in W, almost always, 4. (iv) if w≧w∗(σ)+2 then the cycle set is also path connected, almost always. The method of proof is first of all to show that if a point belongs to the core, then certain generalized symmetry conditions in terms of ‘pivotal’ coalitions of size 2q−n must be satisfied. Secondly, it is shown that these symmetry conditions can almost never be satisfied when either W has empty boundary and is of dimension w(n,q) or when W has non-empty boundary and is of dimension w(n,q)+1.

An experimental study of constant-sum centipede games

In this paper, we report the results of a series of experiments on a version of the centipede game in which the total payoff to the two players is constant. Standard backward induction arguments lead to a unique Nash equilibrium outcome prediction, which is the same as the prediction made by theories of “fair” or “focal” outcomes.We find that subjects frequently fail to select the unique Nash outcome prediction. While this behavior was also observed in McKelvey and Palfrey (1992) in the “growing pie” version of the game they studied, the Nash outcome was not “fair”, and there was the possibility of Pareto improvement by deviating from Nash play. Their findings could therefore be explained by small amounts of altruistic behavior. There are no Pareto improvements available in the constant-sum games we examine. Hence, explanations based on altruism cannot account for these new data.We examine and compare two classes of models to explain these data. The first class consists of non-equilibrium modifications of the standard “Always Take” model. The other class we investigate, the Quantal Response Equilibrium model, describes an equilibrium in which subjects make mistakes in implementing their best replies and assume other players do so as well. One specification of this model fits the experimental data best, among the models we test, and is able to account for all the main features we observe in the data.

Voting Equilibria in Multidimensional Choice Spaces

This paper examines necessary and sufficient conditions for the existence of voting equilibria over a multidimensional issue space, following the lead of the seminal work of Davis and Hinich Davis, O. A., M. J. Hinich. 1966. A mathematical model of policy formation in a democratic society. J. Bernard, ed. Mathematical Applications in Political Science II. Southern Methodist University Press, Dallas, 175--208. and Plott Plott, C. R. 1967. A notion of equilibrium and its possibility under majority rule. Amer. Econom. Rev.57 787--806.. We assume a finite number of voters who vote among alternatives located in an m-dimensional vector space and that the voters vote according to their preferences defined over the space. Here, we generalize and strengthen many of the existing theorems to deal with global rather than local equilibria and plurality as opposed to majority voting rules.

Optimal research for cournot oligopolists

We extend the classical Cournot model to take account of uncertainty in either the cost function or the demand function. By undertaking research, firms can acquire private (asymmetric) information to help resolve their uncertainty and make a more informed production decision. The model is a two stage game: in the first stage research levels are chosen, and in the second stage, conditional on private research outcomes, production decisions are made. We find that for a linear, continuous information structure there is a unique Nash equilibrium to the game. In the equilibrium there may be an inefficient amount of aggregate research and there may be incomplete pooling as well. The model specializes to the classical case when the cost of research is zero (and each firm gains essentially the same information by doing an infinite amount of research) or when the cost of research is so high no firm undertakes research.

Limiting distributions for continuous state Markov voting models

This paper proves the existence of a stationary distribution for a class of Markov voting models. We assume that alternatives to replace the current status quo arise probabilistically, with the probability distribution at time t+1 having support set equal to the set of alternatives that defeat, according to some voting rule, the current status quo at time t. When preferences are based on Euclidean distance, it is shown that for a wide class of voting rules, a limiting distribution exists. For the special case of majority rule, not only does a limiting distribution always exist, but we obtain bounds for the concentration of the limiting distribution around a centrally located set. The implications are that under Markov voting models, small deviations from the conditions for a core point will still leave the limiting distribution quite concentrated around a generalized median point. Even though the majority relation is totally cyclic in such situations, our results show that such chaos is not probabilistically significant.

A Decade of Experimental Research on Spatial Models of Elections and Committees

The Euclidean representation of political issues and alternative outcomes, and the associated representation of preferences as quasiconcave utility functions, is by now a staple of formal models of committees and elections. This theoretical development, moreover, is accompanied by a considerable body of experimental research. We can view that research in two ways: as a test of the basic propositions about equilibria in specific institutional settings, and as an attempt to gain insights into those aspects of political processes that are poorly understood or imperfectly modeled, such as the robustness of theoretical results with respect to procedural details and bargaining environments. This essay reviews that research so that we can gain some sense of its overall import.

Sequential elections with limited information

We develop an incomplete information model of a sequence of elections in a one-dimensional policy space, where voters have no contemporaneous information about candidate positions, and candidates have no information about voter preferences. The only source of information is contemporaneous endorsement data and historical data on the policy positions of previous winning candidates. We define a notion of “stationary rational expectations equilibrium”, and show that such an equilibrium results in outcomes which are equivalent to those that would occur under full information.

The Effects of Payoff Magnitude and Heterogeneity on Behavior in 2 x 2 Games with Unique Mixed Strategy Equilibria

The Logit version of Quantal Response Equilibrium predicts that equilibrium behavior in games will vary systematically with payoff magnitudes, if all other factors are held constant (including the Nash equilibria of the game). We explore this in the context of a set of asymmetric 2x2 games with unique totally mixed strategy equilibria. The data provide little support for the payoff magnitude predictions of the Logit Equilibrium model. We extend the theoretical Quantal Response Equilibrium model to allow for heterogeneity, and find that the data fit the heterogeneous version of the theory significantly better.