36 research outputs found
Separable and Low-Rank Continuous Games
In this paper, we study nonzero-sum separable games, which are continuous
games whose payoffs take a sum-of-products form. Included in this subclass are
all finite games and polynomial games. We investigate the structure of
equilibria in separable games. We show that these games admit finitely
supported Nash equilibria. Motivated by the bounds on the supports of mixed
equilibria in two-player finite games in terms of the ranks of the payoff
matrices, we define the notion of the rank of an n-player continuous game and
use this to provide bounds on the cardinality of the support of equilibrium
strategies. We present a general characterization theorem that states that a
continuous game has finite rank if and only if it is separable. Using our rank
results, we present an efficient algorithm for computing approximate equilibria
of two-player separable games with fixed strategy spaces in time polynomial in
the rank of the game
Structure of Extreme Correlated Equilibria: a Zero-Sum Example and its Implications
We exhibit the rich structure of the set of correlated equilibria by
analyzing the simplest of polynomial games: the mixed extension of matching
pennies. We show that while the correlated equilibrium set is convex and
compact, the structure of its extreme points can be quite complicated. In
finite games the ratio of extreme correlated to extreme Nash equilibria can be
greater than exponential in the size of the strategy spaces. In polynomial
games there can exist extreme correlated equilibria which are not finitely
supported; we construct a large family of examples using techniques from
ergodic theory. We show that in general the set of correlated equilibrium
distributions of a polynomial game cannot be described by conditions on
finitely many moments (means, covariances, etc.), in marked contrast to the set
of Nash equilibria which is always expressible in terms of finitely many
moments
Nash embedding and equilibrium in pure quantum states
With respect to probabilistic mixtures of the strategies in non-cooperative
games, quantum game theory provides guarantee of fixed-point stability, the
so-called Nash equilibrium. This permits players to choose mixed quantum
strategies that prepare mixed quantum states optimally under constraints. In
this letter, we show that fixed-point stability of Nash equilibrium can also be
guaranteed for pure quantum strategies via an application of the Nash embedding
theorem, permitting players to prepare pure quantum states optimally under
constraints.Comment: 7 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1609.0836
Nash Equilibria in Multi-Agent Motor Interactions
Social interactions in classic cognitive games like the ultimatum game or the
prisoner's dilemma typically lead to Nash equilibria when multiple
competitive decision makers with perfect knowledge select optimal strategies.
However, in evolutionary game theory it has been shown that Nash equilibria can
also arise as attractors in dynamical systems that can describe, for example,
the population dynamics of microorganisms. Similar to such evolutionary
dynamics, we find that Nash equilibria arise naturally in motor interactions in
which players vie for control and try to minimize effort. When confronted with
sensorimotor interaction tasks that correspond to the classical
prisoner's dilemma and the rope-pulling game, two-player motor
interactions led predominantly to Nash solutions. In contrast, when a single
player took both roles, playing the sensorimotor game bimanually, cooperative
solutions were found. Our methodology opens up a new avenue for the study of
human motor interactions within a game theoretic framework, suggesting that the
coupling of motor systems can lead to game theoretic solutions