348 research outputs found
Pure Nash Equilibria and Best-Response Dynamics in Random Games
In finite games mixed Nash equilibria always exist, but pure equilibria may
fail to exist. To assess the relevance of this nonexistence, we consider games
where the payoffs are drawn at random. In particular, we focus on games where a
large number of players can each choose one of two possible strategies, and the
payoffs are i.i.d. with the possibility of ties. We provide asymptotic results
about the random number of pure Nash equilibria, such as fast growth and a
central limit theorem, with bounds for the approximation error. Moreover, by
using a new link between percolation models and game theory, we describe in
detail the geometry of Nash equilibria and show that, when the probability of
ties is small, a best-response dynamics reaches a Nash equilibrium with a
probability that quickly approaches one as the number of players grows. We show
that a multitude of phase transitions depend only on a single parameter of the
model, that is, the probability of having ties.Comment: 29 pages, 7 figure
Multiple Stochastically Stable Equilibria in Coordination Games
In an (n,m)-coordination game, each of the n players has two alternative strategies. A strategy generates positive payoff only if there are at least m-1 others who choose the same, where m>n/2. The payoff is nondecreasing in the number of such others so that there are exactly two strict equilibria. Applying the adaptive play with mistakes (Young 1993) to (n,m)-coordination games, we point out potential complications inherent in many-person games. Focusing on games that admit simple analysis, we show that there is a nonempty open set of (n,m)-coordination games that possess multiple stochastically stable equilibria, which may be Pareto ranked, if and only if m>(n+3)/2, which in turn is equivalent to the condition that there is a strategy profile against which every player has alternative best responses.Equilibrium selection, stochastic stability, unanimity game, coordination game, collective decision making
Monotone methods for equilibrium selection under perfect foresight dynamics
This paper studies equilibrium selection in supermodular games
based on perfect foresight dynamics. A normal form game is played
repeatedly in a large society of rational agents. There are frictions:
opportunities to revise actions follow independent Poisson processes.
Each agent forms his belief about the future evolution of action distribution
in the society to take an action that maximizes his expected
discounted payo�. A perfect foresight path is de�ned to be a feasible
path of the action distribution along which every agent with a revision
opportunity takes a best response to this path itself. A Nash
equilibrium is said to be absorbing if there exists no perfect foresight
path escaping from a neighborhood of this equilibrium; a Nash equilibrium
is said to be globally accessible if for each initial distribution,
there exists a perfect foresight path converging to this equilibrium.
By exploiting the monotone structure of the dynamics, a unique Nash
equilibrium that is absorbing and globally accessible for any small degree
of friction is identi�ed for certain classes of supermodular games.
For games with monotone potentials, the selection of the monotone
potential maximizer is obtained. Complete characterizations of absorbing
equilibrium and globally accessible equilibrium are given for
binary supermodular games. An example demonstrates that unanimity
games may have multiple globally accessible equilibria for a small
friction
Entropic selection of Nash equilibrium
This study argues that Nash equilibria with less variations in players' best responses are more appealing. To that regard, a notion measuring such variations, the entropic selection of Nash equilibrium, is presented: For any given Nash equilibrium, we consider the cardinality of the support of a player's best response against others' strategies that are sufficiently close to the
behavior specified. These cardinalities across players are then aggregated with a real-valued function on whose form we impose no restrictions apart from the natural limitation to nondecreasingness in order to obtain equilibria with less variations. We prove that the entropic selection of Nash equilibrium is non-empty and admit desirable properties. Some well-known games, each of which display important insights about virtues / problems of various equilibrium
notions, are considered; and, in all of these games our notion displays none of the criticisms associated with these examples. These examples also show that our notion does not have any containment relations with other associated and well-known refinements, perfection, properness and persistence
Joint strategy fictitious play with inertia for potential games
We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these ldquolarge-scalerdquo games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in large-scale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as ldquoFictitious Playrdquo (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion
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