4,878 research outputs found
Is Having a Unique Equilibrium Robust?
We investigate whether having a unique equilibrium (or a given number of
equilibria) is robust to perturbation of the payoffs, both for Nash equilibrium
and correlated equilibrium. We show that the set of n-player finite games with
a unique correlated equilibrium is open, while this is not true of Nash
equilibrium for n>2. The crucial lemma is that a unique correlated equilibrium
is a quasi-strict Nash equilibrium. Related results are studied. For instance,
we show that generic two-person zero-sum games have a unique correlated
equilibrium and that, while the set of symmetric bimatrix games with a unique
symmetric Nash equilibrium is not open, the set of symmetric bimatrix games
with a unique and quasi-strict symmetric Nash equilibrium is
Reputation and commitment in two-person repeated games without discounting
Two-person repeated games with no discounting are considered where there is uncertainty about the type of the players. If there is a possibility that a player is an automaton committed to a particular pure or mixed stage-game action, then this provides a lower bound on the Nash equilibrium payoffs to a normal type of this player. The lower bound is the best available and is robust to the existence of other types. The results are extended to the case of two-sided uncertainty. This work extends Schmidt (1993) who analyzed the restricted class of conflicting interest games
Decision Problems for Nash Equilibria in Stochastic Games
We analyse the computational complexity of finding Nash equilibria in
stochastic multiplayer games with -regular objectives. While the
existence of an equilibrium whose payoff falls into a certain interval may be
undecidable, we single out several decidable restrictions of the problem.
First, restricting the search space to stationary, or pure stationary,
equilibria results in problems that are typically contained in PSPACE and NP,
respectively. Second, we show that the existence of an equilibrium with a
binary payoff (i.e. an equilibrium where each player either wins or loses with
probability 1) is decidable. We also establish that the existence of a Nash
equilibrium with a certain binary payoff entails the existence of an
equilibrium with the same payoff in pure, finite-state strategies.Comment: 22 pages, revised versio
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
One for all, all for one---von Neumann, Wald, Rawls, and Pareto
Applications of the maximin criterion extend beyond economics to statistics,
computer science, politics, and operations research. However, the maximin
criterion---be it von Neumann's, Wald's, or Rawls'---draws fierce criticism due
to its extremely pessimistic stance. I propose a novel concept, dubbed the
optimin criterion, which is based on (Pareto) optimizing the worst-case payoffs
of tacit agreements. The optimin criterion generalizes and unifies results in
various fields: It not only coincides with (i) Wald's statistical
decision-making criterion when Nature is antagonistic, (ii) the core in
cooperative games when the core is nonempty, though it exists even if the core
is empty, but it also generalizes (iii) Nash equilibrium in -person
constant-sum games, (iv) stable matchings in matching models, and (v)
competitive equilibrium in the Arrow-Debreu economy. Moreover, every Nash
equilibrium satisfies the optimin criterion in an auxiliary game
- …