# Econometrica

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Опубликовано на портале: 28-04-2005

*John Donald Roberts*,

*Paul R. Milgrom*Econometrica. 1990. Vol. 58. No. 6. P. 1255-1277.

We study a class of non-cooperative games that includes many standard oligopoly games,macro
economic coordination games, network and production externality games, and others.
Forthese games, the sets of rationalizable strategies, pure Nash equilibrium strategies,
and correlated equilibrium strategies are non-empty and have identical upper and
lower bounds. Also,a large class of dynamic learning processes - including both best-response
dynamics and Bayesian learning - lead eventually to behavior that lies between these
same bounds. General comparative static and welfare theorems are provided. We study
the class of (non-cooperative) supermodular games introduced by Topkis (1979) and
further studied by Vives (1988). These are games in which each player's strategy
set is partially ordered, the marginal returns to increasing one's strategy rise
with increases in the competitors' strategies and, if a player's strategies are multidimensional,
the marginal returns to any one component of the player's strategy increase with
increases in the other components. The simplest examples of such games arise in oligopoly
theory. These include the Cournot duopoly game with a wide range of demand specifications
and arbitrary continuous cost functions, Bertrand oligopoly games, R&D racing games,
and oligopoly games with endogenous choice of production technologies. Additional
applications drawn from industrial organization include the Hendricks-Kovenock (1988)
drilling game (in which oil firms decide whether to drill exploratory wells when
the information obtained is a public good), network externality games, and certain
oligopoly games that arise in connection with the Milgrom-Roberts (1988) model of
manufacturing. Other examples include games used to model

**Rationalizable Strategic Behavior**[статья]

Опубликовано на портале: 30-01-2007

*B. Douglas Bernheim*Econometrica. 2006. Vol. 52. No. 4. P. 1007-1028.

This paper examines the nature of rational choice in strategic games. Although there
are many reasons why an agent might select a Nash equilibrium strategy in a particular
game, rationality alone does not require him to do so. A natural extension of widely
accepted axioms for rational choice under uncertainty to strategic environments generates
an alternative class of strategies, labelled "rationalizable." It is argued that
no rationalizable strategy can be discarded on the basis of rationality alone, and
that all rationally justifiable strategies are members of the rationalizable set.
The properties of rationalizable strategies are studied, and refinements are considered.

Опубликовано на портале: 22-01-2007

*Ehud Lehrer*,

*Ehud Kalai*Econometrica. 1993. Vol. 61. No. 5. P. 1019-1045.

Subjective utility maximizers, in an infinitely repeated game, will learn to predict
opponents' future strategies and will converge to play according to a Nash equilibrium
of the repeated game. Players' initial uncertainty is placed directly on opponents'
strategies and the above result is obtained under the assumption that the individual
beliefs are compatible with the chosen strategies. An immediate corollary is that,
when playing a Harsanyi-Nash equilibrium of a repeated game of incomplete information
about opponents' payoff matrices, players will eventually play a Nash equilibrium
of the real game, as if they had complete information.

**Sequential Equilibria**[статья]

Опубликовано на портале: 31-01-2007

*David M. Kreps*,

*Robert B. Wilson*Econometrica. 1982. Vol. 50. No. 4. P. 863-94.

We propose a new criterion for equilibria of extensive games, in the spirit of Selten's
perfectness criteria. This criterion requires that players' strategies be sequentially
rational: Every decision must be part of an optimal strategy for the remainder of
the game. This entails specification of players' beliefs concerning how the game
has evolved for each information set, including information sets off the equilibrium
path. The properties of sequential equilibria are developed; in particular, we study
the topological structure of the set of sequential equilibria. The connections with
Selten's trembling-hand perfect equilibria are given.

Опубликовано на портале: 30-01-2007

*Susan Carleton Athey*Econometrica. 2001. Vol. 69. No. 4. P. 861-89.

This paper analyzes a class of games of incomplete information where each agent has
private information about her own type, and the types are drawn from an atomless
joint probability distribution. The main result establishes existence of pure strategy
Nash equilibria (PSNE) under a condition we call the single crossing condition (SCC),
roughly described as follows: whenever each opponent uses a nondecreasing strategy
(in the sense that higher types choose higher actions), a player's best response
strategy is also nondecreasing. When the SCC holds, a PSNE exists in every finite-action
game. Further, for games with continuous payoffs and a continuum of actions, there
exists a sequence of PSNE to finite-action games that converges to a PSNE of the
continuum-action game. These convergence and existence results also extend to some
classes of games with discontinuous payoffs, such as first-price auctions, where
bidders may be heterogeneous and reserve prices are permitted. Finally, the paper
characterizes the SCC based on properties of utility functions and probability distributions
over types. Applications include first-price, multi-unit, and all-pay auctions; pricing
games with incomplete information about costs; and noisy signaling games.

**Strategic Information Transmission**[статья]

Опубликовано на портале: 24-01-2007

*Vincent P. Crawford*,

*Joel Sobel*Econometrica. 1982. Vol. 50. No. 6. P. 1431-1451.

This paper develops a model of strategic communication, in which a better-informed
Sender (S) sends a possibly noisy signal to a Reciever (R), who then takes an action
that determines the welfare of both. We characterize the set of Bayesian Nash equilibria
under standart assumptions, and show that equilibrium signaling always takes a strikingly
simple form, in which S partitions the support of the (scalar) variable that represents
his private information and introduces noise into his signal by reporting, in effect,
only which element of the partition his observation actually lies in. We show under
further assumptions that before S observes his private information, the equilibrium
whose partition has the greatest number of elements is Pareto-superior to all other
equilibria, and that if agents coordinate on this equilibrium, R`s equilibrium expected
utility rises when agents` preferences become more similar. Since R bases his choice
of action on rational expectations, this establishes a sense in which equilibrium
signaling is more informative when agents` preferences are more similar.

**The Bargaining Problem**[статья]

Опубликовано на портале: 08-07-2005

*John Forbes Nash*Econometrica. 1950. Vol. 18. No. 2. P. 155-162.

A new treatment is presented of a classical economic problem, one which occurs in
many forms, as bargaining, bilateral monopoly, etc. It may also be regarded as a
nonzero-sum two-person game. In this treatment a few general assumptions are made
concerning the behavior of a single individual and of a group of two individuals
in certain economic environments. From these, the solution (in the sense of this
paper) of classical problem may be obtained. In the terms of game theory, values
are found for the game.
См. также: Two-person cooperative games, автор - Джо Нэш.

**The Evolution of Walrasian Behavior**[статья]

Опубликовано на портале: 24-01-2007

*Fernando Vega-Redondo*Econometrica. 1997. Vol. 65. No. 2. P. 375-384.

This article describes an evolutionary approach to understanding Walrasian behavior.
It avoids any considerations related to the absence of monopoly power or related
notion of a large enough population. Walrasian behavior may evolve within any quantity-setting
oligopoly producing a homogenous good, provided that the law of demand is satisfied.
Evolutionary models may produce interesting behavior that does not correspond to
a Nash equilibrium.

Опубликовано на портале: 31-01-2007

*Dilip Abreu*,

*Prajit K. Dutta*,

*Lones Smith*Econometrica. 1994. Vol. 62. No. 4. P. 939-948.

The paper discusses perfect "folk theorems" for infinitely repeated games with complete
information. Folk theorems assert that any feasible and individually rational payoff
vector of the stage game is a subgame perfect equilibrium payoff in the associated
infinitely repeated game with little or no discounting. It is obvious that feasibility
and individual rationality are necessary conditions for a payoff vector to be an
equilibrium payoff. The content of the folk theorems is that these conditions are
also sufficient. Perhaps the first folk theorem type result is due to Friedman (1971)
who showed that any feasible payoff which Pareto dominates a Nash equilibrium payoff
of the stage game will be an equilibrium payoff in the associated repeated game with
sufficiently patient players.

**Two-person cooperative games**[статья]

Опубликовано на портале: 13-03-2003

*John Forbes Nash*Econometrica. 1953. Vol. 21. No. 1. P. 128-140.

In this paper, the autor extends his previous treatment of «The Bargaining Problem»
to a wider class of situations in which threats can play a role/ A new approach
is introduced involving the elaboration of the threat concept.

Опубликовано на портале: 22-01-2007

*Dilip Abreu*,

*Hitoshi Matsushima*Econometrica. 1992. Vol. 60. No. 5. P. 993-1008.

The authors investigate the implementation of social choice functions that map to
lotteries over alternatives. They require virtual implementation in iteratively undominated
strategies. Under very weak domain restrictions, they show that if there are three
or more players, any social choice function may be so implemented. The literature
on implementation in Nash equilibrium and its refinements is compromised by its reliance
on game forms with unnatural features (for example, "integer games") or "modulo"
constructions with mixed strategies arbitrarily excluded. In contrast, the authors'
results employ finite (consequently "well-behaved") mechanisms and allow for mixed
strategies.