Kali P. Rath

A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players

SummaryIn this note we provide a direct and simple proof of the existence of pure strategy Nash equilibria in large finite action games when the payoffs depend on own action and the average response of others. The result is then extended to the case where the action set of each player is a compact subset of ℜn.

The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games

In 1951, Dvoretzky, Wald and Wolfowitz (henceforth DWW) showed that corresponding to any mixed strategy into a finite action space, there exists a pure-strategy with an identical integral with respect to a finite set of atomless measures. DWW used their theorem for purification: the elimination of randomness in statistical decision procedures and in zero-sum two-person games. In this short essay, we apply a consequence of their theorem to a finite-action setting of finite games with incomplete and private information, as well as to that of large games. In addition to simplified proofs and conceptual clarifications, the unification of results offered here re-emphasizes the close connection between statistical decision theory and the theory of games.

Two stage equilibrium and product choice with elastic demand

Abstract This paper examines a two stage model of product choice with elastic demand. Duopolists choose locations and then prices. Consumers’ demand is linear in price. Transportation cost is quadratic and is a lump sum. For each pair of locations, there is a price equilibrium in the second stage. The first stage equilibrium locations are unique, symmetric and depend upon the ratio of the reservation price and the transportation cost parameter. When this ratio exceeds a certain critical value, the locations are at the extreme end points of the market. As the ratio decreases, the locations gradually move towards the center.

The nonexistence of symmetric equilibria in anonymous games with compact action spaces

Abstract In an anonymous game the payoff of a player depends upon the players own action and the action distribution of all the players. If the game is atomless and the set of actions is finite, or countably infinite and compact, then there is a symmetric equilibrium distribution. Furthermore, every equilibrium distribution can be symmetrized. This note provides three examples to the effect that the conditions in these results cannot be relaxed. The first example is a game with atoms and finite actions which has no symmetric equilibrium distribution. In the second example, the game is atomless, the action space is uncountable and not every equilibrium distribution can be symmetrized. The third example shows that a symmetric equilibrium distribution may not exist in an atomless game with the interval [ − 1, 1]as the action space. A general construction then exhibits that given any uncountable compact metric space, there is an atomless game over that space which has no symmetric equilibrium distribution. A sufficient condition for an equilibrium distribution to be symmetrized is also given.

On a private information game without pure strategy equilibria1

Abstract We present an example of a two-person game of private information in which there is no equilibrium in pure strategies. Our example satisfies all the hypotheses of the existence theorems present in the literature on the subject of pure strategy equilibria, except for the fact that the action set of each player is given by the interval [−1,1]. As such, it illustrates the limitations that pertain to the purification of equilibria in a standard setting.

Large Games with a Bio-Social Typology

We present a comprehensive theory of large games in which players have names and determinate social-types and/or biological traits, and identify through four decisive examples, essentially based on a matching-pennies type game, pathologies arising from the use of a Lebesgue interval for playerʼs names. In a sufficiently general context of traits and actions, we address this dissonance by showing a saturated probability space as being a necessary and sufficient name-space for the existence and upper hemi-continuity of pure-strategy Nash equilibria in large games with traits. We illustrate the idealized results by corresponding asymptotic results for an increasing sequence of finite games.

Existence and upper hemicontinuity of equilibrium distributions of anonymous games with discontinuous payoffs

Abstract This paper deals with large anonymous games with compact metric action spaces and payoff functions which are upper semicontinuous but not necessarily lower semicontinuous. Under the supremum norm topology on the space of payoff functions and a tightness assumption on the game, the existence of Nash equilibrium distributions and the closed graph property of the equilibrium distribution correspondence are proved. If the space of payoff functions is given the hypotopology instead, the equilibrium distributions exists, but the equilibrium distribution correspondence may not have a closed graph.

Stationary and nonstationary strategies in Hotelling's model of spatial competition with repeated pricing decisions

Abstract. This paper examines Hotellings model of location with linear transportation cost. Existence of pure strategy subgame perfect equilibria in the infinitely repeated price game with fixed locations is proved. These subgame perfect equilibria have a stick and carrot structure. Given firm locations, there are discount factors sufficiently high that there is a subgame perfect equilibrium with a two-phase structure. Given the discount factors, there are stationary subgame perfect equilibria for a wide range of locations. However, for some pairs of location, no symmetric simple penal code exists, all subgame perfect profiles are nonstationary, and there is only one seller in the market in infinitely many periods.

Pure-Strategy Nash Equilibrium Points in Large Non-Anonymous Games

We present an example of a nonaiomic game without pure Nash equilibria. In the example, the set of players is modelled on the Lebesgue unit interval with an equicontinuous family of payoff functions, and an identical action set given by [–1,1]. This example is sharper than that recently presented by Rath-Sun-Yamashige in that the relationship between societal responses and individual payoffs is linear. We also present a theorem on the existence of pure strategy Nash equilibria in nonatomic games in which the set of players is modelled on a nonatomic Loeb measure space.

On non-linear extensions of the Perron-Frobenius theorem

Abstract The note provides bounds for eigenvalues of non-negative, homogeneous mappings, which need not be linear. It also gives a simple proof of the result, that repeated iterations of a primitive mapping take any semipositive vector to the positive eigenvector.