[Paper Review] Complex behaviour in cyclic competition bimatrix games

Cezary Olszowiec|arXiv (Cornell University)|May 2, 2016

Game Theory and Applications22 references3 citations

TL;DR

This paper re-examines a cyclic competition bimatrix game (Rock-Scissors-Paper with perfect memory) using rigorous dynamical systems analysis, correcting erroneous claims in prior literature about chaotic switching and stability. It demonstrates that infinite switching near the heteroclinic network is impossible under general conditions, identifies four distinct parameter regions with different dynamical behaviors, and shows that stability of cycles depends critically on parameter values, not just sign of $\epsilon_X + \epsilon_Y$, with chaotic behavior potentially arising via superhomoclinic orbits.

ABSTRACT

We consider an example of cyclic competition bimatrix game which is a Rock-Scissors-Paper game with assumption about perfect memory of the playing agents. At first we investigate the dynamics in the neighbourhood of the Nash equilibrium as well as the dynamics on the boundary of codimension 1 - that is when one of the strategies is not played by any agent. For the analysis of the asymptotic behaviour close to the boundary of state space, we provide the description of a naturally appearing heteroclinic network. Due to the symmetry, the heteroclinic network induces a quotient network. This quotient network is investigated as well. In the literature this model was already studied, but unfortunately the results stated there are invalid, as we will show in this paper. It turns out that certain types of behaviour are never possible or appear in the system only for some parameter values, contradicting what was stated before in the literature. Moreover, the parameter space (-1,1)x(-1,1) is divided into four regions where we observe either irregular behaviour, or preference to follow an itinerary consisting of strategies for which one or the other agent does not lose, or they alternate in winning. These regions in parameter space are separated by two analytical curves and lines where the game is either symmetric (system is not $C^1$ linearisable at stationary saddle points) or is zero-sum and the system is Hamiltonian. On each of these curves we observe different bifurcation scenarios: e.g. transition from order to chaos, or from one kind of stability to another kind, or just loss of one dimension of the local stable manifold of the subcycle.

Motivation & Objective

• To correct significant errors in prior analyses of cyclic competition bimatrix games, particularly regarding chaotic switching and stability of cycles.
• To provide a rigorous dynamical systems analysis of the Rock-Scissors-Paper game with perfect memory, focusing on heteroclinic networks and transition maps.
• To resolve inconsistencies between numerical results in the literature and analytical behavior near the heteroclinic network.
• To clarify the dependence of dynamical behavior—especially attainability of finite itineraries and stability—on the parameters $\epsilon_X$ and $\epsilon_Y$.
• To identify conditions under which chaotic behavior may emerge, particularly through transverse intersections of stable and unstable manifolds of cycles.

Proposed method

• Constructs the full heteroclinic network arising from the symmetry of the game, including all transition maps between equilibria.
• Computes the linear part of transition maps in local coordinates, proving they are affine, not identity as assumed in prior work.
• Uses rescaling and composition of local maps with transition maps to derive approximations resembling prior (incorrect) expressions.
• Analyzes bifurcations along curves where the system is symmetric or zero-sum, identifying transitions between stability types and loss of manifold dimensions.
• Performs numerical investigations to compare with Sato et al.’s simulations and to explore the existence of heteroclinic connections and superhomoclinic orbits.
• Proposes a framework for computer-assisted proofs of chaos in open parameter regions, especially near the $\beta$-curve.

Experimental results

Research questions

• RQ1 Why is the claim of infinite switching near the heteroclinic network in Aguiar & Castro (2010) invalid, and what conditions prevent it?
• RQ2 How do the transition maps between equilibria in the heteroclinic network differ from the identity maps assumed in prior work?
• RQ3 What is the true stability behavior of cycles $C_0$, $C_1$, and $C_2$ as a function of $\epsilon_X$ and $\epsilon_Y$?
• RQ4 Under what parameter conditions can finite itineraries along heteroclinic orbits be attained near the network?
• RQ5 What mechanisms could lead to chaotic dynamics, and is there evidence for superhomoclinic orbits in the system?

Key findings

• Finite itineraries of the form $tie \to win/loss \to loss/win \to tie$ are not attainable for sufficiently small $h$, contradicting Aguiar & Castro’s claim of infinite switching.
• The cycle $C_0$ is essentially asymptotically stable when $\epsilon_X + \epsilon_Y < 0$, correcting Aguiar & Castro’s assertion of relative asymptotic stability.
• For some open regions in parameter space, cycles $C_1$ and $C_2$ are neither almost completely unstable nor essentially asymptotically stable, but attract open sets of initial conditions.
• The parameter space $(-1,1)^2$ is divided into four regions with distinct dynamical behaviors: irregular motion, preference for non-losing itineraries, or alternation in winning.
• Chaos may arise via transverse intersections of stable and unstable manifolds of cycles $C_0$ and $C_2$, potentially forming a 3-dimensional set in parameter space.
• Numerical evidence suggests the existence of forward and backward heteroclinic connections to $C_2$ and $C_0$ near the bifurcation curve between white and yellow regions, supporting potential chaotic switching.