Evolutionarily stable strategy in asymmetric games: Dynamical and information-theoretical perspectives

TL;DR

The paper addresses how ESS can be defined and analyzed in asymmetric and bimatrix games, where multiple roles and possibly different strategy sets complicate the classical symmetric narrative. It studies three main ESS definitions (and-, sum-, or-definitions), establishes equivalences and hierarchies among them, and connects these notions to dynamical stability via replicator dynamics. A core contribution is the information-theoretic framing of ESS using KL-divergence (and a Fisher-information–Shahshahani perspective), yielding Lyapunov functions that prove local stability for many ESS scenarios, including pure and certain mixed equilibria, and clarifying when mixed ESS can be dynamically meaningful. The work further extends ESS concepts to multiplayer hypermatrix games, deriving conditions for pure and mixed ESS, and shows that information-theoretic and gradient-structure viewpoints illuminate stability in symmetric multipartite interactions (partnership games) but face obstructions in fully asymmetric cases. Collectively, the results unify game-theoretic stability, dynamical systems, and information theory in the study of ESS under asymmetry, while outlining open questions for broader multiplayer contexts and proof completeness.

Abstract

Evolutionarily stable strategy (ESS) is the defining concept of evolutionary game theory. It has a fairly unanimously accepted definition for the case of symmetric games which are played in a homogeneous population where all individuals are in same role. However, in asymmetric games, which are played in a population with multiple subpopulations (each of which has individuals in one particular role), situation is not as clear. Various generalizations of ESS defined for such cases differ in how they correspond to fixed points of replicator equation which models evolutionary dynamics of frequencies of strategies in the population. Moreover, some of the definitions may even be equivalent, and hence, redundant in the scheme of things. Along with reporting some new results, this paper is partly indented as a contextual mini-review of some of the most important definitions of ESS in asymmetric games. We present the definitions coherently and scrutinize them closely while establishing equivalences -- some of them hitherto unreported -- between them wherever possible. Since it is desirable that a definition of ESS should correspond to asymptotically stable fixed points of replicator dynamics, we bring forward the connections between various definitions and their dynamical stabilities. Furthermore, we find the use of principle of relative entropy to gain information-theoretic insights into the concept of ESS in asymmetric games, thereby establishing a three-fold connection between game theory, dynamical system theory, and information theory in this context. We discuss our conclusions also in the backdrop of asymmetric hypermatrix games where more than two individuals interact simultaneously in the course of getting payoffs.

Figures (5)

• Figure 1: Schematic summary of logical relations among ESS definitions in bimatrix and asymmetric games as discussed in this paper. Here single-headed arrows indicate implication, i.e., if the arrow points from A to B, then it means that A implies B. Similarly, the double-headed arrows indicate equivalence between definitions, i.e., if there is a double-headed arrow between A and B, then it means A is equivalent to B.
• Figure 2: Phase portraits supplementing Examples 1--5 of the main text which should be referred for further details. The red circular marker indicates the ESS (fixed point) under focus. The blue lines are trajectories under replicator equation whereas the green dashed lines are trajectories under adjusted replicator equation.
• Figure 3: Venn diagrams showing how the set of ESSes ($\mathbb{E}$) is related to the set of strict NEs ($\mathbb{S}$), the set of weak pure NEs ($\mathbb{W}$), and the set of mixed NEs ($\mathbb{M}$): Subfigure (a) corresponds to the unprimed and-definition of ESS. Subfigure (b) shows the Venn diagram for the primed and-definition, the sum-definition, and the face-value or-definition. Finally, subfigure (c) depicts the Venn diagram for the simultaneous-deviation or-definition of ESS.
• Figure 4: Figure lists the 3-player game payoff hypermatrices: The four hypermatrices represent the payoffs ${\sf A}$, ${\sf B}$, ${\sf C}$ and ${\sf D}$. The number written at the corners of each of the four hypermatrices are payoff elements. In the first hypermatrix, we have illustrated the convention with which we have assigned the indices of the $2\times2\times2$ dimensional hypermatrices.
• Figure 5: Numerical demonstration that showing inequality (\ref{['r_definition_mult']}) is not always satisfied, yet Conjecture 1 may be valid. We use payoff hypermatrices from Fig. \ref{['fig:payoff_cube']} and multiply the parameter $\mu$ with matrices ${\sf A}$ and ${\sf D}$ to control the strength of intra-specific interactions. The figure illustrates two distinct regions: In the first region (green colour), inequality (\ref{['r_definition_mult']}) is satisfied for all $(x,y)$ and in second region (red colour), it doesn't. However, all the interior fixed points under consideration are checked to be locally asymptotically stable and 2ESS.