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Equalizer zero-determinant strategy in discounted repeated Stackelberg asymmetric game

Zhaoyang Cheng, Guanpu Chen, Yiguang Hong

TL;DR

This work addresses the computational burden of obtaining Strong Stackelberg Equilibrium (SSE) strategies in discounted repeated Stackelberg asymmetric games by introducing equalizer zero-determinant (ZD) strategies that unilaterally constrain opponents' utilities. The authors establish existence conditions for equalizer ZD strategies in the one-to-one case and derive robust bounds on performance loss relative to SSE; they extend the results to multi-player scenarios, providing bounds on the sum of opponents' utilities and a generalized SSE gap bound. Theoretical results are complemented by simulations in UAV and moving target defense (MTD) contexts, showing that equalizer ZD strategies can maintain near-SSE leader utilities while dramatically reducing computation. Overall, the approach offers a practical, scalable alternative for leadership in dynamic, asymmetric, and resource-constrained adversarial settings.

Abstract

This paper focuses on the performance of equalizer zero-determinant (ZD) strategies in discounted repeated Stackerberg asymmetric games. In the leader-follower adversarial scenario, the strong Stackelberg equilibrium (SSE) deriving from the opponents' best response (BR), is technically the optimal strategy for the leader. However, computing an SSE strategy may be difficult since it needs to solve a mixed-integer program and has exponential complexity in the number of states. To this end, we propose to adopt an equalizer ZD strategy, which can unilaterally restrict the opponent's expected utility. We first study the existence of an equalizer ZD strategy with one-to-one situations, and analyze an upper bound of its performance with the baseline SSE strategy. Then we turn to multi-player models, where there exists one player adopting an equalizer ZD strategy. We give bounds of the sum of opponents' utilities, and compare it with the SSE strategy. Finally, we give simulations on unmanned aerial vehicles (UAVs) and the moving target defense (MTD) to verify the effectiveness of our approach.

Equalizer zero-determinant strategy in discounted repeated Stackelberg asymmetric game

TL;DR

This work addresses the computational burden of obtaining Strong Stackelberg Equilibrium (SSE) strategies in discounted repeated Stackelberg asymmetric games by introducing equalizer zero-determinant (ZD) strategies that unilaterally constrain opponents' utilities. The authors establish existence conditions for equalizer ZD strategies in the one-to-one case and derive robust bounds on performance loss relative to SSE; they extend the results to multi-player scenarios, providing bounds on the sum of opponents' utilities and a generalized SSE gap bound. Theoretical results are complemented by simulations in UAV and moving target defense (MTD) contexts, showing that equalizer ZD strategies can maintain near-SSE leader utilities while dramatically reducing computation. Overall, the approach offers a practical, scalable alternative for leadership in dynamic, asymmetric, and resource-constrained adversarial settings.

Abstract

This paper focuses on the performance of equalizer zero-determinant (ZD) strategies in discounted repeated Stackerberg asymmetric games. In the leader-follower adversarial scenario, the strong Stackelberg equilibrium (SSE) deriving from the opponents' best response (BR), is technically the optimal strategy for the leader. However, computing an SSE strategy may be difficult since it needs to solve a mixed-integer program and has exponential complexity in the number of states. To this end, we propose to adopt an equalizer ZD strategy, which can unilaterally restrict the opponent's expected utility. We first study the existence of an equalizer ZD strategy with one-to-one situations, and analyze an upper bound of its performance with the baseline SSE strategy. Then we turn to multi-player models, where there exists one player adopting an equalizer ZD strategy. We give bounds of the sum of opponents' utilities, and compare it with the SSE strategy. Finally, we give simulations on unmanned aerial vehicles (UAVs) and the moving target defense (MTD) to verify the effectiveness of our approach.
Paper Structure (11 sections, 5 theorems, 44 equations, 4 figures, 1 table)

This paper contains 11 sections, 5 theorems, 44 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Repeated leader-follower game
  4. Strong Stackelberg equilibrium strategy
  5. Zero-determinant strategy
  6. One-to-one situation
  7. Multi-player situation
  8. Examples
  9. Utility performance in UAVs
  10. Action Interaction in MTD
  11. Conclusion

Key Result

lemma 1

An equalizer ZD strategy exists if $\Lambda(\delta)$ is nonempty.

Figures (4)

  • Figure 1: Performance of the equalizer ZD strategy compared with the SSE strategy in the one-to-one situation.
  • Figure 2: Performance of the equalizer ZD strategy compared with the SSE strategy in the three-player situation.
  • Figure 3: Target interaction between player 1 (the defender) and player 2 (the attacker). Blue solid lines show the target that player 1 protects, while red dotted lines show the target that player 2 attacks in each stage.
  • Figure 4: Target interaction among player 1 (a defender), player 2 (an attacker), and player 3 (an attacker). Blue solid lines show the target that player 1 protects, while red dotted lines and green dotted lines show the target that player 2 and player 3 attack, respectively.

Theorems & Definitions (7)

  • definition 1
  • definition 2
  • lemma 1
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4