897 research outputs found
Deterministic Equations for Stochastic Spatial Evolutionary Games
Spatial evolutionary games model individuals who are distributed in a spatial
domain and update their strategies upon playing a normal form game with their
neighbors. We derive integro-differential equations as deterministic
approximations of the microscopic updating stochastic processes. This
generalizes the known mean-field ordinary differential equations and provide a
powerful tool to investigate the spatial effects in populations evolution. The
deterministic equations allow to identify many interesting features of the
evolution of strategy profiles in a population, such as standing and traveling
waves, and pattern formation, especially in replicator-type evolutions
Decompositions of two player games: potential, zero-sum, and stable games
We introduce several methods of decomposition for two player normal form
games. Viewing the set of all games as a vector space, we exhibit explicit
orthonormal bases for the subspaces of potential games, zero-sum games, and
their orthogonal complements which we call anti-potential games and
anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential
game comes either from the Rock-Paper-Scissors type games (in the case of
symmetric games) or from the Matching Pennies type games (in the case of
asymmetric games). Using these decompositions, we prove old (and some new)
cycle criteria for potential and zero-sum games (as orthogonality relations
between subspaces). We illustrate the usefulness of our decomposition by (a)
analyzing the generalized Rock-Paper-Scissors game, (b) completely
characterizing the set of all null-stable games, (c) providing a large class of
strict stable games, (d) relating the game decomposition to the decomposition
of vector fields for the replicator equations, (e) constructing Lyapunov
functions for some replicator dynamics, and (f) constructing Zeeman games
-games with an interior asymptotically stable Nash equilibrium and a pure
strategy ESS
Inertial game dynamics and applications to constrained optimization
Aiming to provide a new class of game dynamics with good long-term
rationality properties, we derive a second-order inertial system that builds on
the widely studied "heavy ball with friction" optimization method. By
exploiting a well-known link between the replicator dynamics and the
Shahshahani geometry on the space of mixed strategies, the dynamics are stated
in a Riemannian geometric framework where trajectories are accelerated by the
players' unilateral payoff gradients and they slow down near Nash equilibria.
Surprisingly (and in stark contrast to another second-order variant of the
replicator dynamics), the inertial replicator dynamics are not well-posed; on
the other hand, it is possible to obtain a well-posed system by endowing the
mixed strategy space with a different Hessian-Riemannian (HR) metric structure,
and we characterize those HR geometries that do so. In the single-agent version
of the dynamics (corresponding to constrained optimization over simplex-like
objects), we show that regular maximum points of smooth functions attract all
nearby solution orbits with low initial speed. More generally, we establish an
inertial variant of the so-called "folk theorem" of evolutionary game theory
and we show that strict equilibria are attracting in asymmetric
(multi-population) games - provided of course that the dynamics are well-posed.
A similar asymptotic stability result is obtained for evolutionarily stable
strategies in symmetric (single- population) games.Comment: 30 pages, 4 figures; significantly revised paper structure and added
new material on Euclidean embeddings and evolutionarily stable strategie
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