Coalitional Bargaining Equilibria

This paper takes up the foundational issue of existence of stationary subgame perfect equi- libria in a general class of coalitional bargaining games that includes many known bargaining models and models of coalition formation. General sufficient conditions for existence of equilib- ria are currently lacking in many interesting environments: bargaining models with non-concave stage utility functions, models with a Pareto optimal status quo alternative and heterogeneous discount factors, and models of coalition formation in public good economies with consumption lower bounds. This paper establishes existence of stationary equilibrium under compactness and continuity conditions, without the structure of convexity or comprehensiveness used in the extant literature. The proof requires a precise selection of voting equilibria following different proposals. The result is applied to obtain equilibria in models of bargaining over taxes, coalition formation in NTU environments, and collective dynamic programming problems.

Uncovered Sets

This paper covers the theory of the uncovered set used in the literatures on tournaments and spatial voting. I discern three main extant definitions, and I introduce two new concepts that bound exist- ing sets from above and below: the deep uncovered set and the shallow uncovered set. In a general topological setting, I provide relationships to other solutions and give results on existence and external stability for all of the covering concepts, and I establish continuity properties of the two new uncovered sets. Of note, I characterize each of the uncovered sets in terms of a decomposition into choices from externally stable sets; I define the minimal generalized covering solution, a nonempty refinement of the deep uncovered set that employs both of the new relations; and I define the acyclic Banks set, a nonempty generalization of the Banks set.

Noisy Stochastic Games

This paper establishes existence of a stationary Markov perfect equilibrium in general stochastic games with noise—a component of the state that is nonatomically distributed and not directly affected by the previous period’s state and actions. Noise may be simply a payoff-irrelevant public randomization device, delivering known results on existence of correlated equilibrium as a special case. More generally, noise can take the form of shocks that enter into players’ stage payoffs and the transition probability on states. The existence result is applied to a model of industry dynamics and to a model of dynamic electoral competition.

General Conditions for Existence of Maximal Elements via the Uncovered Set

This paper disentangles the topological assumptions of classical results (e.g., Walker (1977)) on existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions-with no other restrictions on preferences-there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is non-empty. Adding a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is indeed maximal. A corollary is a result that weakens the semi-convexity condition of Yannelis and Prabhakar (1983).

Noisy Stochastic Games

This paper establishes existence of a stationary Markov perfect equilibrium in general stochastic games with noise a component of the state that is nonatomically distributed and not directly affected by the previous periods state and actions. Noise may be simply a payoff irrelevant public randomization device, delivering known results on existence of correlated equilibrium as a special case. More generally, noise can take the form of shocks that enter into players stage payoffs and the transition probability on states. The existence result is applied to a model of industry dynamics and to a model of dynamic partisan electoral competition.

A General Bargaining Model of Legislative Policy-making

We present a general model of legislative bargaining in which the status quo is an arbitrary point in a multidimensional policy space. In contrast to other bargaining models, the status quo is not assumed to be bad for all legislators, and delay may be Pareto efficient. We prove existence of stationary equilibria. We show that if all legislators are risk averse or if even limited transfers are possible, then delay is only possible if the status quo lies in the core. Thus, we expect immediate agreement in multidimensional models, where the core is typically empty. In one dimension, delay is possible if and only if the status quo lies in the core of the voting rule, and then it is the only possible outcome. Our comparative statics analysis yield two noteworthy insights: moderate status quos imply moderate policy outcomes, and legislative patience implies policy moderation

A dynamic model of democratic elections in multidimensional policy spaces

We propose a general model of repeated elections. In each period, a challenger is chosen from the electorate to run against an incumbent politician in a majority-rule election, and the winner then selects a policy from a multidimensional policy space. Individual policy preferences are private information, whereas policy choices are publicly observable. We prove existence and continuity of equilibria in "simple" voting and policy strategies; we provide examples to show the variety of possible equilibrium patterns in multiple dimensions; we analyze the effects of patience and office-holding benefits on the persistence of policies over time; and we identify relationships between equilibrium policies and the core of the underlying voting game. As a byproduct of our analysis, we show how equilibrium incentives maylead elected representatives to make policy compromises, even when binding commitments are unavailable. We provide an informational story for incumbency advantage. Finally, we give an asymptotic version of the median voter theorem for the one-dimensional model as voters become-arbitrarily patient

A Newton Collocation Method for Solving Dynamic Bargaining Games

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.

Extremal Choice Equilibrium: Existence and Purification with Infinite-Dimensional Externalities

We prove existence and purification results for equilibria in which players choose extreme points of their feasible actions in a class of strategic environments exhibiting a product structure. We assume finite-dimensional action sets and allow for infinite-dimensional externalities. Applied to large games, we obtain existence of Nash equilibrium in pure strategies while allowing a continuum of groups and general dependence of payoffs on average actions across groups, without resorting to saturated measure spaces. Applied to games of incomplete information, we obtain a new purification result for Bayes-Nash equilibria that permits substantial correlation across types, without assuming conditional independence given the realization of a finite environmental state. We highlight our results in examples of industrial organization, auctions, and voting.

Voting Equilibria in Multi-candidate Elections

We consider a general plurality voting game with multiple candidates, where voter preferences over candidates are exogenously given. In particular, we allow for arbitrary voter indierences, as may arise in voting subgames of citizen-candidate or locational models of elections. We prove that the voting game admits pure strategy equilibria in undominated strategies. The proof is constructive: we exhibit an algorithm, the “best winning deviation” algorithm, that produces such an equilibrium in finite time. A byproduct of the algorithm is a simple story for how voters might learn to coordinate on such an equilibrium.