1,290 research outputs found
The projection dynamic, the replicator dynamic and the geometry of population games
Every population game defines a vector field on the set of strategy distributions X. The
projection dynamic maps each population game to a new vector field: namely, the one closest
to the payoff vector field among those that never point outward from X. We investigate the
geometric underpinnings of the projection dynamic, describe its basic game-theoretic properties,
and establish a number of close connections between the projection dynamic and the replicator
dynamic
Riemannian game dynamics
We study a class of evolutionary game dynamics defined by balancing a gain
determined by the game's payoffs against a cost of motion that captures the
difficulty with which the population moves between states. Costs of motion are
represented by a Riemannian metric, i.e., a state-dependent inner product on
the set of population states. The replicator dynamics and the (Euclidean)
projection dynamics are the archetypal examples of the class we study. Like
these representative dynamics, all Riemannian game dynamics satisfy certain
basic desiderata, including positive correlation and global convergence in
potential games. Moreover, when the underlying Riemannian metric satisfies a
Hessian integrability condition, the resulting dynamics preserve many further
properties of the replicator and projection dynamics. We examine the close
connections between Hessian game dynamics and reinforcement learning in normal
form games, extending and elucidating a well-known link between the replicator
dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the
derivation of the dynamic
Escort Evolutionary Game Theory
A family of replicator-like dynamics, called the escort replicator equation,
is constructed using information-geometric concepts and generalized information
entropies and diverenges from statistical thermodynamics. Lyapunov functions
and escort generalizations of basic concepts and constructions in evolutionary
game theory are given, such as an escorted Fisher's Fundamental theorem and
generalizations of the Shahshahani geometry.Comment: Minor typo correctio
Attractive evolutionary equilibria
We present attractiveness, a refinement criterion for evolutionary equilibria. Equilibria surviving this criterion are robust to small perturbations of the underlying payoff system or the dynamics at hand. Furthermore, certain attractive equilibria are equivalent to others for certain evolutionary dynamics. For instance, each attractive evolutionarily stable strategy is an attractive evolutionarily stable equilibrium for certain barycentric ray-projection dynamics, and vice versa
Non-centralized Control for Flow-based Distribution Networks: A Game-theoretical Insight
This paper solves a data-driven control problem for a flow-based distribution network with two objectives: a resource allocation and a fair distribution of costs. These objectives represent both cooperation and competition directions. It is proposed a solution that combines either a centralized or distributed cooperative game approach using the Shapley value to determine
a proper partitioning of the system and a fair communication cost distribution. On the other hand, a decentralized noncooperative game approach computing the Nash equilibrium is used to achieve the control objective of the resource allocation under a non-complete information topology. Furthermore, an invariant-set property is presented and the closed-loop system stability is analyzed for the non cooperative game approach. Another contribution regarding the cooperative game approach is an alternative way to compute the Shapley value for the proposed specific characteristic function. Unlike the classical
cooperative-games approach, which has a limited application due to the combinatorial explosion issues, the alternative method allows calculating the Shapley value in polynomial time and hence can be applied to large-scale problems.Generalitat de Catalunya FI 2014Ministerio de Ciencia y Educación DPI2016-76493-C3-3-RMinisterio de Ciencia y Educación DPI2008-05818Proyecto europeo FP7-ICT DYMASO
The Kullback-Leibler Divergence as a Lyapunov Function for Incentive Based Game Dynamics
It has been shown that the Kullback-Leibler divergence is a Lyapunov function
for the replicator equations at evolutionary stable states, or ESS. In this
paper we extend the result to a more general class of game dynamics. As a
result, sufficient conditions can be given for the asymptotic stability of rest
points for the entire class of incentive dynamics. The previous known results
will be can be shown as corollaries to the main theorem
Information Geometry and Evolutionary Game Theory
The Shahshahani geometry of evolutionary game theory is realized as the
information geometry of the simplex, deriving from the Fisher information
metric of the manifold of categorical probability distributions. Some essential
concepts in evolutionary game theory are realized information-theoretically.
Results are extended to the Lotka-Volterra equation and to multiple population
systems.Comment: Added reference
On evolutionary ray-projection dynamics
We introduce the ray-projection dynamics in evolutionary game theory by employing a ray projection of the relative fitness (vector) function, i.e., a projection unto the unit simplex along a ray through the origin. Ray-projection dynamics are weakly compatible in the terminology of Friedman (Econometrica 59:637–666, 1991), each of their interior fixed points is an equilibrium and each interior equilibrium is one of its fixed points. Furthermore, every interior evolutionarily stable strategy is an asymptotically stable fixed point, and every strict equilibrium is an evolutionarily stable state and an evolutionarily stable equilibrium. We also employ the ray-projection on a set of functions related to the relative fitness function and show that several well-known evolutionary dynamics can be obtained in this manner
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