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Multi-type random game dynamics: limits at discontinuities and cyclic limits

Raghupati Vyas, Kousik Das, Veeraruna Kavitha, Souvik Roy

TL;DR

The paper develops a differential inclusion (DI) framework to analyze turn-by-turn, one-shot dynamics in multi-type populations with heterogeneous rational and behavioral agents. It shows that the stochastic population measures converge almost surely to internally chain transitive (ICT) sets of the driving DI, revealing both classical and non-classical limit behaviors, including cycles that arise from discontinuities. Singleton ICTs are linked to multi-type mean-field Nash equilibria (MT-MFE), while non-singleton ICTs yield cyclic outcomes predicted by a Region-Vertex (RV) graph analysis, with a numerical procedure to identify such ICTs. A detailed queuing-game example demonstrates cyclic ICTs under realistic mixtures of agent types and preferences, illustrating the practical impact of the theory for systems with priority-based services and crowd-effects. Overall, the DI-based approach unifies stochastic dynamics with discontinuous best-response regions to characterize long-run outcomes beyond classical ODE analyses.

Abstract

We consider (random) strategic interactions in a large population consisting of a variety of players. A rational player chooses actions that maximize certain utility functions, while a behavioral player chooses actions based on preferences such as avoid-the-crowd or follow-the-majority. We specifically study a turn-by-turn dynamic process in which players choose their actions sequentially and once; the utilities are realized either immediately or at the end of the game. In the literature, such dynamical systems are often analyzed using an appropriate approximating ordinary differential equation (ODE). However, the ODEs approximating the dynamics with pure actions are typically discontinuous. We adopt a differential inclusion (DI) based stochastic-approximation framework to derive the limiting analysis. The limits of the dynamics are characterized through the internally chain transitive (ICT) sets. We identify the presence of non-classical zeros as potential limits of the dynamics, a phenomenon not observed in classical settings involving continuous ODEs. These new limits arise precisely at the points of discontinuity of the dynamics. We further provide the conditions under which cyclic outcomes may occur at the limit. Finally, we study a queuing game with differential priority-based services and examine the impact of the proportions of avoid-the-crowd and two types of rational populations on the long-run outcomes of the strategic interactions. We identify potential point limits and establish the possibility of cyclic outcomes for certain parameter configurations.

Multi-type random game dynamics: limits at discontinuities and cyclic limits

TL;DR

The paper develops a differential inclusion (DI) framework to analyze turn-by-turn, one-shot dynamics in multi-type populations with heterogeneous rational and behavioral agents. It shows that the stochastic population measures converge almost surely to internally chain transitive (ICT) sets of the driving DI, revealing both classical and non-classical limit behaviors, including cycles that arise from discontinuities. Singleton ICTs are linked to multi-type mean-field Nash equilibria (MT-MFE), while non-singleton ICTs yield cyclic outcomes predicted by a Region-Vertex (RV) graph analysis, with a numerical procedure to identify such ICTs. A detailed queuing-game example demonstrates cyclic ICTs under realistic mixtures of agent types and preferences, illustrating the practical impact of the theory for systems with priority-based services and crowd-effects. Overall, the DI-based approach unifies stochastic dynamics with discontinuous best-response regions to characterize long-run outcomes beyond classical ODE analyses.

Abstract

We consider (random) strategic interactions in a large population consisting of a variety of players. A rational player chooses actions that maximize certain utility functions, while a behavioral player chooses actions based on preferences such as avoid-the-crowd or follow-the-majority. We specifically study a turn-by-turn dynamic process in which players choose their actions sequentially and once; the utilities are realized either immediately or at the end of the game. In the literature, such dynamical systems are often analyzed using an appropriate approximating ordinary differential equation (ODE). However, the ODEs approximating the dynamics with pure actions are typically discontinuous. We adopt a differential inclusion (DI) based stochastic-approximation framework to derive the limiting analysis. The limits of the dynamics are characterized through the internally chain transitive (ICT) sets. We identify the presence of non-classical zeros as potential limits of the dynamics, a phenomenon not observed in classical settings involving continuous ODEs. These new limits arise precisely at the points of discontinuity of the dynamics. We further provide the conditions under which cyclic outcomes may occur at the limit. Finally, we study a queuing game with differential priority-based services and examine the impact of the proportions of avoid-the-crowd and two types of rational populations on the long-run outcomes of the strategic interactions. We identify potential point limits and establish the possibility of cyclic outcomes for certain parameter configurations.
Paper Structure (33 sections, 14 theorems, 140 equations, 5 figures, 1 table)

This paper contains 33 sections, 14 theorems, 140 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. A multi-type population Game
  3. Some examples and applications
  4. Turn by turn dynamics among multi-type players
  5. Queuing game with a variety of agents and priorities-based services
  6. Differential inclusion (DI) based stochastic approximation
  7. Asymptotic analysis of \ref{['eqn_iterates_nu']} via Characterization of ICT Sets
  8. Games with two actions
  9. Generic games with finite actions
  10. Solutions of DI \ref{['eqn_general_DI']}
  11. Cyclic ICT sets
  12. Game with three actions: solution and ICT sets on ${\mathcal{H}}$
  13. Connection of singleton ICT sets with MT-MFE
  14. Methodology for deriving outcomes of the game-dynamics
  15. Analytical approach
  16. ...and 18 more sections

Key Result

Lemma 1

Consider a collection of measurable sets $\{ {\mathcal{C}}_{i\delta}\}_{i \le m}$ in Euclidean space $\mathbb{R}^k$, one for each $\delta \in (0, 1)$ and for some finite integer $m \ge 2.$ Consider a collection of 'centers' $\{\bm g^c_i\}_{i\le m}$ that satisfy the following for each $i$ and $\delta

Figures (5)

  • Figure 1: Representative picture for 2 types, 3 actions
  • Figure 2: Representative picture for 2 types and actions
  • Figure 3: Cycle in an instance
  • Figure 4: Queuing game \ref{['sub_sub_sec_queuing_application']} with two types of rational agents:$\{\mathcal{Q}_v\}$ regions (Left), Numerical cycle (Right)
  • Figure 5: Queuing game \ref{['sub_sub_sec_queuing_application']} with three types of agents:$\{\mathcal{Q}_v\}$ regions (Left), Numerical cycle (Right)

Theorems & Definitions (19)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Lemma 3
  • Lemma 4
  • Theorem 4
  • Lemma 5
  • ...and 9 more