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The emergence of chaos in population game dynamics induced by comparisons

Jakub Bielawski, Łukasz Cholewa, Fryderyk Falniowski

TL;DR

This work shows that discrete-time population dynamics driven by revision protocols can exhibit Li–Yorke chaos in a simple $2\times2$ anti-coordination game with a unique mixed Nash equilibrium. By constructing explicit piecewise linear bimodal update maps and using both innovative and imitative revision mechanisms, the authors demonstrate that, for sufficiently large time steps $\delta$, the interior NE can be repelling and the system supports chaotic behavior with periodic orbits of any order. They develop a natural imitative protocol—a perturbed pairwise proportional imitation (PPI) with parameters $\eta$ and $\xi$—and prove Li–Yorke chaos, including observability via bifurcation analysis; they further extend these results with truncated imitation rates to broaden the parameter regime while preserving chaos. The results are linked via topological conjugacy between symmetric cases and rely on a key one-dimensional chaos criterion for bimodal maps, illustrating that chaos can be an intrinsic feature of discrete-time revision dynamics, not just a pathological edge case. These findings challenge the predictive reliability of equilibrium concepts in such learning dynamics and point to rich, machine-learning-like behavior in economic population models.

Abstract

Precise description of population game dynamics introduced by revision protocols - an economic model describing the agent's propensity to switch to a better-performing strategy - is of importance in economics and social sciences in general. In this setting innovation or imitation of others is the force which drives the evolution of the economic system. As the continuous-time game dynamics is relatively well understood, the same cannot be said about revision driven dynamics in the discrete time. We investigate the behavior of agents in a $2\times 2$ anti-coordination game with symmetric random matching and a unique mixed Nash equilibrium. In continuous time the Nash equilibrium is attracting and induces a global evolutionary stable state. We show that in the discrete time one can construct (either innovative or imitative) revision protocol and choose a level of the time step, under which the game dynamics is Li-Yorke chaotic, inducing complex and unpredictable behavior of the system, precluding stable predictions of equilibrium. Moreover, we reveal that this unpredictability is encoded into any imitative revision protocol. Furthermore, we show that for any such game there exists a perturbed pairwise proportional imitation protocol introducing chaotic behavior of the agents for sufficiently large time step.

The emergence of chaos in population game dynamics induced by comparisons

TL;DR

This work shows that discrete-time population dynamics driven by revision protocols can exhibit Li–Yorke chaos in a simple anti-coordination game with a unique mixed Nash equilibrium. By constructing explicit piecewise linear bimodal update maps and using both innovative and imitative revision mechanisms, the authors demonstrate that, for sufficiently large time steps , the interior NE can be repelling and the system supports chaotic behavior with periodic orbits of any order. They develop a natural imitative protocol—a perturbed pairwise proportional imitation (PPI) with parameters and —and prove Li–Yorke chaos, including observability via bifurcation analysis; they further extend these results with truncated imitation rates to broaden the parameter regime while preserving chaos. The results are linked via topological conjugacy between symmetric cases and rely on a key one-dimensional chaos criterion for bimodal maps, illustrating that chaos can be an intrinsic feature of discrete-time revision dynamics, not just a pathological edge case. These findings challenge the predictive reliability of equilibrium concepts in such learning dynamics and point to rich, machine-learning-like behavior in economic population models.

Abstract

Precise description of population game dynamics introduced by revision protocols - an economic model describing the agent's propensity to switch to a better-performing strategy - is of importance in economics and social sciences in general. In this setting innovation or imitation of others is the force which drives the evolution of the economic system. As the continuous-time game dynamics is relatively well understood, the same cannot be said about revision driven dynamics in the discrete time. We investigate the behavior of agents in a anti-coordination game with symmetric random matching and a unique mixed Nash equilibrium. In continuous time the Nash equilibrium is attracting and induces a global evolutionary stable state. We show that in the discrete time one can construct (either innovative or imitative) revision protocol and choose a level of the time step, under which the game dynamics is Li-Yorke chaotic, inducing complex and unpredictable behavior of the system, precluding stable predictions of equilibrium. Moreover, we reveal that this unpredictability is encoded into any imitative revision protocol. Furthermore, we show that for any such game there exists a perturbed pairwise proportional imitation protocol introducing chaotic behavior of the agents for sufficiently large time step.

Paper Structure

This paper contains 16 sections, 22 theorems, 131 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work.
  3. Settings
  4. Review on discrete-time dynamical systems
  5. Chaos for innovative dynamics
  6. Skeleton of the proof of Theorem \ref{['thm:anticoord-chaos']} for an innovative protocol
  7. A natural imitative protocol leading to chaos
  8. Skeleton of the proof of Theorem \ref{['thm:perturbedPPIchaos']}.
  9. Observability of chaos.
  10. Truncated dynamics
  11. Skeleton of the proof of Theorem \ref{['thm:truncatedPPIchaos']}
  12. Conclusions and future work
  13. Proof of Theorem \ref{['thm:anticoord-chaos']}
  14. Proof of Theorem \ref{['thm:anticoord-chaos']} for imitative revision protocols
  15. Proofs from Section \ref{['sec:thm2']}
  16. ...and 1 more sections

Key Result

Proposition 1

Let $f\colon[0,1]\to[0,1]$ be a continuous map. If there exist points $0\leq z_l<z_r\leq 1$ such that $f^2(z_l)>z_r$ and either hold or hold, then $f(x)<x<f^3(x)$ for some point $x\in(z_l,z_r)$.

Figures (8)

  • Figure 1: Diagrams illustrating conditions \ref{['lem:chaos_cond1']} and \ref{['lem:chaos_cond2']} of Proposition \ref{['lem:chaos_cond']} are in the first row. Diagrams illustrating conditions \ref{['lem:chaos_condA']} and \ref{['lem:chaos_condB']} are in the second row. Cobweb diagrams (columns 1. and 3.) and the first 50 iterations of a starting point (columns 2. and 4.). Here we test conditions of Proposition \ref{['lem:chaos_cond']} for $z_l=0.2$ and $z_r=0.6$ when they are critical points of piecewise linear maps: (a) None of the conditions is satisfied; (c) The condition \ref{['lem:chaos_condA']} holds; [ (b) and (d)] Conditions of Proposition \ref{['lem:chaos_cond']} are satisfied; we see a chaotic behavior of the trajectory of the starting point.
  • Figure 2: Switch rates $\rho_{AB}$ (on the left) and $\rho_{BA}$ (on the right) from Proposition \ref{['prop:inn_switch_rates']} for the game \ref{['eq:game']} with Nash equilibrium $p=0.2$, and parameters $\beta_2=2$, $\beta_3=-\frac{1}{3}$.
  • Figure 3: The graphs of maps $F$ induced by the switch rates $\rho_{AB}$ and $\rho_{BA}$ from Proposition \ref{['prop:inn_switch_rates']} for $p=0.2$ (on the left) and $\widetilde{F}$ from Proposition \ref{['prop:chaos_inn_p>']} for $\widetilde{p}=0.8$ (on the right) and for parameters $\beta_2=2$, $\beta_3=-\frac{1}{3}$.
  • Figure 4: Graphs of the elements of the dynamical system induced by an imitative revision protocol. [ (a) and (b)] Graphs of the switch rates for Nash equilibrium $p=0{.}4$. (c) Graph of the update map for Nash equilibrium $p=0{.}4$. (d) Graph of the update map for Nash equilibrium $\widetilde{p}=0{.}6$.
  • Figure 5: Bifurcation diagrams for the dynamical system of the map $F$ in \ref{['eq:PPI_map_pert']} for the maximal values of $\eta$ and $\xi$ and for $p=0.4$. The horizontal axis is the parameter $\delta \in (0,1]$. For each $\delta \in (0,1]$, 20000 iterations of the starting points were made and then next 100 iterations of the map $F$ were plotted. On the left diagram first the iterations of the right critical point of $F$ were plotted in blue, then the iterations of the left critical point of $F$ were plotted in red. On the right diagram the order of plotting is reversed. For small values of $\delta$ the fixed point $p$ attracts all trajectories of the dynamical system. As $\delta$ increases the point $p$ loses stability and we observe the period-doubling route to chaos.
  • ...and 3 more figures

Theorems & Definitions (46)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Remark 4
  • ...and 36 more