[Paper Review] Adversarial Scheduling in Evolutionary Game Dynamics

Gabriel Istrate, Madhav Marathe+1 more|ArXiv.org|Dec 5, 2008

Evolutionary Game Theory and Cooperation28 references3 citations

TL;DR

This paper investigates adversarial scheduling in evolutionary game dynamics, focusing on the Iterated Prisoner’s Dilemma with win-stay lose-shift strategies. It shows that while random scheduling leads to convergence to full cooperation, adversarial schedulers that control both players can prevent convergence on most graphs, but nonadaptive schedulers (choosing only one player) are no more powerful than random ones, even when adaptive, under fairness constraints.

ABSTRACT

Consider a system in which players at nodes of an underlying graph G repeatedly play Prisoner's Dilemma against their neighbors. The players adapt their strategies based on the past behavior of their opponents by applying the so-called win-stay lose-shift strategy. This dynamics has been studied in (Kittock 94), (Dyer et al. 2002), (Mossel and Roch, 2006). With random scheduling, starting from any initial configuration with high probability the system reaches the unique fixed point in which all players cooperate. This paper investigates the validity of this result under various classes of adversarial schedulers. Our results can be sumarized as follows: 1. An adversarial scheduler that can select both participants to the game can preclude the system from reaching the unique fixed point on most graph topologies. 2. A nonadaptive scheduler that is only allowed to choose one of the participants is no more powerful than a random scheduler. With this restriction even an adaptive scheduler is not significantly more powerful than the random scheduler, provided it is "reasonably fair". The results exemplify the adversarial scheduling approach we propose as a foundational basis for the generative approach to social science (Epstein 2007).

Motivation & Objective

• To assess the robustness of convergence to full cooperation in evolutionary game dynamics under adversarial scheduling, challenging the assumption of random or uniform scheduling.
• To identify which structural properties of schedulers (e.g., fairness, adaptiveness, endogeneity) are essential for maintaining convergence to the fixed point of full cooperation.
• To explore whether results derived under random scheduling remain valid under adversarial conditions, particularly in networked populations.
• To establish a foundation for adversarial analysis in agent-based social science, using game-theoretic models as a testbed.
• To demonstrate that fairness and endogeneity are critical for convergence, while adaptiveness and nonadaptiveness are less consequential under reasonable fairness conditions.

Proposed method

• The study models a population of agents on a graph, each playing the Iterated Prisoner’s Dilemma with neighbors using the win-stay lose-shift strategy.
• It introduces a formal classification of schedulers: edge, node, adaptive, nonadaptive, and fair, based on control over interaction selection.
• It analyzes convergence to the all-cooperate fixed point under various adversarial scheduling models, comparing with the known random-scheduling result.
• It uses probabilistic and combinatorial arguments, including the Coupon Collector Lemma, to establish fairness thresholds and convergence bounds.
• It applies techniques from self-stabilization theory, such as scheduler-luck games and threshold-based monotonicity, to prove convergence under restricted adversarial conditions.
• It constructs explicit sequences of moves that funnel the system toward the fixed point under node schedulers, demonstrating that endogeneity enables convergence even adversarially.

Experimental results

Research questions

• RQ1 Can an adversarial scheduler that selects both players in a game prevent the system from reaching the all-cooperate fixed point on most graph topologies?
• RQ2 Is a nonadaptive scheduler (selecting only one player per update) more powerful than a random scheduler in preventing convergence to full cooperation?
• RQ3 Does adaptiveness in scheduling significantly increase the power of an adversary to disrupt convergence, especially when fairness is maintained?
• RQ4 What structural properties of schedulers—such as fairness, endogeneity, or adaptiveness—are necessary or sufficient for convergence to the fixed point?
• RQ5 Can the convergence result under random scheduling be generalized to adversarial settings by reformulating the conditions using fairness and endogeneity?

Key findings

• An adversarial scheduler that controls both players in a game can prevent convergence to the all-cooperate fixed point on most graph topologies, demonstrating that the random-scheduling result is fragile under full adversarial control.
• A nonadaptive scheduler that selects only one player is no more powerful than a random scheduler, even when adaptive, provided the scheduler is reasonably fair (e.g., O(n log n)-fair with high probability).
• Fairness is a necessary condition for convergence under adversarial scheduling; without it, convergence cannot be guaranteed, even on simple topologies.
• Endogeneity—where agents can choose which neighbor to interact with—enables convergence even under adversarial node scheduling, as shown by explicit construction of funneling move sequences.
• Adaptiveness in scheduling does not significantly increase adversarial power unless fairness is compromised; under fairness, convergence holds generically across Erdős–Rényi random graphs.
• The results highlight that fairness and endogeneity are critical for robustness, while adaptiveness is inessential when fairness is maintained, suggesting that the original random-scheduling result can be generalized under minimal structural assumptions.