Pareto Dominance of Deferred Acceptance through Early Decision

An early decision market is governed by rules that allow each student to apply to (at most) one college and require the student to attend this college if admitted. This market is ubiquitous in college admissions in the United States. We model this market as an extensive-form game of perfect information and study a refinement of subgame perfect equilibrium (SPE) that induces undominated Nash equilibria in every subgame (SPUE). Our main result shows that this game can be used to define a decentralized matching mechanism that weakly Pareto dominates student-proposing deferred acceptance.

Pareto Dominance of Deferred Acceptance through Early Decision

An early decision market is governed by rules that allow each student to apply to (at most) one college and require the student to attend this college if admitted. This market is ubiquitous in college admissions in the United States. We model this market as an extensive-form game of perfect information and study a refinement of subgame perfect equilibrium (SPE) that induces undominated Nash equilibria in every subgame (SPUE). Our main result shows that this game can be used to define a decentralized matching mechanism that weakly Pareto dominates student-proposing deferred acceptance.

A notion of subgame perfect Nash equilibrium under knightian uncertainty

We define a subgame perfect Nash equilibrium under Knightian uncertainty for two players, by means of a recursive backward induction procedure. We prove an extension of the Zermelo-von Neumann-Kuhn Theorem for games of perfect information, i. e., that the recursive procedure generates a Nash equilibrium under uncertainty (Dow and Werlang(1994)) of the whole game. We apply the notion for two well known games: the chain store and the centipede. On the one hand, we show that subgame perfection under Knightian uncertainty explains the chain store paradox in a one shot version. On the other hand, we show that subgame perfection under uncertainty does not account for the leaving behavior observed in the centipede game. This is in contrast to Dow, Orioli and Werlang(1996) where we explain by means of Nash equilibria under uncertainty (but not subgame perfect) the experiments of McKelvey and Palfrey(1992). Finally, we show that there may be nontrivial subgame perfect equilibria under uncertainty in more complex extensive form games, as in the case of the finitely repeated prisoner's dilemma, which accounts for cooperation in early stages of the game.

A notion of subgame perfect nash equilibrium under knightian uncertainty

We define a subgame perfect Nash equilibrium under Knightian uncertainty for two players, by means of a recursive backward induction procedure. We prove an extension of the Zermelo-von Neumann-Kuhn Theorem for games of perfect information, i. e., that the recursive procedure generates a Nash equilibrium under uncertainty (Dow and Werlang(1994)) of the whole game. We apply the notion for two well known games: the chain store and the centipede. On the one hand, we show that subgame perfection under Knightian uncertainty explains the chain store paradox in a one shot version. On the other hand, we show that subgame perfection under uncertainty does not account for the leaving behavior observed in the centipede game. This is in contrast to Dow, Orioli and Werlang(1996) where we explain by means of Nash equilibria under uncertainty (but not subgame perfect) the experiments of McKelvey and Palfrey(1992). Finally, we show that there may be nontrivial subgame perfect equilibria under uncertainty in more complex extensive form games, as in the case of the finitely repeated prisoner's dilemma, which accounts for cooperation in early stages of the game .

Efficient Nash Equilibrium under Adverse Selection

This paper revisits the problem of adverse selection in the insurance market of Rothschild and Stiglitz [28]. We propose a simple extension of the game-theoretic structure in Hellwig [14] under which Nash-type strategic interaction between the informed customers and the uninformed firms results always in a particular separating equilibrium. The equilibrium allocation is unique and Pareto-efficient in the interim sense subject to incentive-compatibility and individual rationality. In fact, it is the unique neutral optimum in the sense of Myerson [22].

A numerical algorithm for finding solutions of a generalized Nash equilibrium problem

A family of nonempty closed convex sets is built by using the data of the Generalized Nash equilibrium problem (GNEP). The sets are selected iteratively such that the intersection of the selected sets contains solutions of the GNEP. The algorithm introduced by Iusem-Sosa (2003) is adapted to obtain solutions of the GNEP. Finally some numerical experiments are given to illustrate the numerical behavior of the algorithm.

Nash equilibrium strategies in discrete-time finite-horizon dynamic games with risk-and effort-averse players

The objective of this paper is to re-examine the risk-and effort attitude in the context of strategic dynamic interactions stated as a discrete-time finite-horizon Nash game. The analysis is based on the assumption that players are endogenously risk-and effort-averse. Each player is characterized by distinct risk-and effort-aversion types that are unknown to his opponent. The goal of the game is the optimal risk-and effort-sharing between the players. It generally depends on the individual strategies adopted and, implicitly, on the the players' types or characteristics.

Bayesian nash equilibrium in ''linear'' cournot models with private information about costs

Calculating explicit closed form solutions of Cournot models where firms have private information about their costs is, in general, very cumbersome. Most authors consider therefore linear demands and constant marginal costs. However, within this framework, the nonnegativity constraint on prices (and quantities) has been ignored or not properly dealt with and the correct calculation of all Bayesian Nash equilibria is more complicated than expected. Moreover, multiple symmetric and interior Bayesianf equilibria may exist for an open set of parameters. The reason for this is that linear demand is not really linear, since there is a kink at zero price: the general ''linear'' inverse demand function is P (Q) = max{a - bQ, 0} rather than P (Q) = a - bQ.

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence to Nash equilibria, and present a number of possibility and impossibility results. Basically, we show that if in addition to random moves some recall is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it su±ces to recall the last two periods of play.

Nash equilibrium under knightian uncertainty: breaking down backward induction (extensively revised version)

We define Nash equilibrium for two-person normal form games in the presence of uncertainty, in the sense of Knight(1921). We use the fonna1iution of uncertainty due to Schmeidler and Gilboa. We show tbat there exist Nash equilibria for any degree of uncertainty, as measured by the uncertainty aversion (Dow anel Wer1ang(l992a». We show by example tbat prudent behaviour (maxmin) can be obtained as an outcome even when it is not rationaliuble in the usual sense. Next, we break down backward industion in the twice repeated prisoner's dilemma. We link these results with those on cooperation in the finitely repeated prisoner's dilemma obtained by Kreps-Milgrom-Roberts-Wdson(1982), and withthe 1iterature on epistemological conditions underlying Nash equilibrium. The knowledge notion implicit in this mode1 of equilibrium does not display logical omniscience.

Nash equilibrium under knightian uncertainty: breaking-down backward induction

We present two alternative definitions of Nash equilibrium for two person games in the presence af uncertainty, in the sense of Knight. We use the formalization of uncertainty due to Schmeidler and Gilboa. We show that, with one of the definitions, prudent behaviour (maxmin) can be obtained as an outcome even when it is not rationalizable in the usual sense. Most striking is that with the Same definition we break down backward induction in the twice repeated prisoner's dilemma. We also link these results with the Kreps-Milgrom-Roberts-Wilson explanation of cooperation in the finitely repeated prisoner's dilemma.

A comment on "Rational learning lead to nash equilibrium" by professors Ehud Kalai and Ehud Lehrer

Kalai and Lebrer (93a, b) have recently show that for the case of infinitely repeated games, a coordination assumption on beliefs and optimal strategies ensures convergence to Nash equilibrium. In this paper, we show that for the case of repeated games with long (but finite) horizon, their condition does not imply approximate Nash equilibrium play. Recently Kalai and Lehrer (93a, b) proved that a coordination assumption on beliefs and optimal strategies, ensures that pIayers of an infinitely repeated game eventually pIay 'E-close" to an E-Nash equilibrium. Their coordination assumption requires that if players believes that certain set of outcomes have positive probability then it must be the case that this set of outcomes have, in fact, positive probability. This coordination assumption is called absolute continuity. For the case of finitely repeated games, the absolute continuity assumption is a quite innocuous assumption that just ensures that pIayers' can revise their priors by Bayes' Law. However, for the case of infinitely repeated games, the absolute continuity assumption is a stronger requirement because it also refers to events that can never be observed in finite time.

Existence of nash equilibrium in competitive nonlinear pricing games with adverse selection

We show that for a large class of competitive nonlinear pricing games with adverse selection, the property of better-reply security is naturally satisfied - thus, resolving via a result due to Reny (1999) the issue of existence of Nash equilibrium for a large class of competitive nonlinear pricing games.

Weak local Nash equilibrium

In this paper, we consider a concept of local Nash equilibrium for non-cooperative games - the so-called weak local Nash equilibrium. We prove its existence for a significantly more general class of sets of strategies than compact convex sets. The theorems on existence of the weak local equilibrium presented here are applications of Brouwer and Lefschetz fixed point theorems. © 2013 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University.