Party formation in single-issue politics [revised]

We study the implications of economies of party size in a model of party formation. We show that when the policy space is one-dimensional, candidates form at most two parties. This result does not depend on the magnitude of the economies of party size or sensitively on the nature of the individuals' preferences. It does depend on our assumptions that the policy space is one-dimensional and that uncertainty is absent; we study how modifications of these assumptions affect our conclusions.Political parties, party formation, economies of party size

Imitation Games and Computation

TAn imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.

PARTY FORMATION INCOLLECTIVE DECISION-MAKING

We study party formation in a general model of collective decisionmaking, modeling parties as agglomerations of policy positions championed by decision-makers. We show that if there are economies of party size and the policy chosen is not beaten by another policy in pairwise voting, then players agglomerate into exactly two parties. This result does not depend on the magnitude of the economies of party size or sensitively on the nature of the individuals' preferences. Our analysis encompasses a wide range of models, including decision-making in committees with costly participation and representative democracy in which the legislature is elected by citizens.

General equilibrium analysis in ordered topological vector spaces

The second welfare theorem and the core-equivalence theorem have been proved to be fundamental tools for obtaining equilibrium existence theorems, especially in an infinite dimensional setting. For well-behaved exchange economies that we call proper economies, this paper gives (minimal) conditions for supporting with prices Pareto optimal allocations and decentralizing Edgeworth equilibrium allocations as non-trivial equilibria. As we assume neither transitivity nor monotonicity on the preferences of consumers, most of the existing equilibrium existence results are a consequence of our results. A natural application is in Finance, where our conditions lead to new equilibrium existence results, and also explain why some financial economies fail to have equilibrium.Equilibrium; Valuation equilibrium; Pareto-optimum; Edgeworth equilibrium; Properness; ordered topological vector spaces; Riesz-Kantorovich formula; sup-convolution

On the Existence of Equilibrium in Bayesian Games Without Complementarities

This paper presents new results on the existence of pure-strategy Bayesian equilibria in specified functional forms. These results broaden the scope of methods developed by Reny (2011) well beyond monotone pure strategies. Applications include natural models of first-price and all-pay auctions not covered by previous existence results. To illustrate the scope of our results, we provide an analysis of three auctions: (i) a first-price auction of objects that are heterogeneous and imperfect substitutes; (ii) a first-price auction in which bidders’ payoffs have a very general interdependence structure; and (iii) an all-pay auction with non-monotone equilibrium

Production equilibria

This paper studies production economies in a commodity space that is an ordered locally convex space. We establish a general theorem on the existence of equilibrium without requiring that the commodity space or its dual be a vector lattice. Such commodity spaces arise in models of portfolio trading where the absence of some option usually means the absence of a vector lattice structure. The conditions on preferences and production sets are at least as general as those imposed in the literature dealing with vector lattice commodity spaces. The main assumption on the order structure is that the Riesz-Kantorovich functionals satisfy a uniform properness condition that can be formulated in terms of a duality property that is readily checked. This condition is satisfied in a vector lattice commodity space but there are many examples of other commodity spaces that satisfy the condition, which are not vector lattices, have no order unit, and do not have either the decomposition property or its approximate versions.Production economies; Equilibrium; Edgeworth equilibrium; Properness; Riesz-Kantorovich functional; Sup-convolution

Imitation Games and Computation

An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths

Linear and non-linear price decentralization

Compendious and thorough solutions to the existence of a linear price equilibrium problem, the second welfare theorem, and the limit theorem on the core are provided for exchange economies whose consomption sets are the positive cone of arbitrary ordered Fréchet-dispensing entirely with the assumption that the vector ordering of the commodity space is a lattice. The motivation comes from economic applications showing the need to bring within the scope of equilibrium theory vector orderings that are not lattices, which arise in the typical model of portfolio trading with missing options. The assumptions are on the primitives of the model. They are bounds on the marginals of non-linear prives and for omega-proper economies they are both sufficient and necessary.Linear and non-linear prices; equilibrium; welfare theorems

On the Existence of Equilibrium in Bayesian Games Without Complementarities

In a recent paper, Reny (2011) generalized the results of Athey (2001) and McAdams (2003) on the existence of monotone strategy equilibrium in Bayesian games. Though the generalization is subtle, Reny introduces far-reaching new techniques applying the fixed point theorem of Eilenberg and Montgomery (1946, Theorem 5). This is done by showing that with atomless type spaces the set of monotone functions is an absolute retract and when the values of the best response correspondence are non-empty sub-semilattices of monotone functions, they too are absolute retracts. In this paper, we provide an extensive generalization of Reny (2011), McAdams (2003), and Athey (2001). We study the problem of existence of Bayesian equilibrium in pure strategies for a given partially ordered compact subset of strategies. The ordering need not be a semilattice and these strategies need not be monotone. The main innovation is the interplay between the homotopy structures of the order complexes that are the subject of the celebrated work of Quillen (1978), and the hulling of partially ordered sets, an innovation that extends the properties of Reny’s semilattices to the non-lattice setting. We then describe some auctions that illustrate how this framework can be applied to generalize the existing results and extend the class of models for which we can establish existence of equilibrium. As with Reny (2011), our proof utilizes the fixed point theorem in Eilenberg and Montgomery (1946)