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Zero-determinant strategies in repeated continuously-relaxed games

Masahiko Ueda, Ayaka Fujita

TL;DR

The paper extends zero-determinant (ZD) strategies to repeated games whose stage-action spaces are continuously relaxed via mixed extensions, showing that this continuous relaxation broadens the set of payoff-control capabilities than in the original finite-action setting. It presents a formal condition for the existence of two-point ZD strategies in the mixed-extension framework and demonstrates the broadened scope through Prisoner’s Dilemma, Public Goods, and a three-action symmetric game. A new construct, one-point ZD strategies, is introduced and shown to reproduce several equilibrium-payoff properties in classic games via fixed mixed strategies (e.g., Matching Pennies, Battle of the Sexes, Rock–Paper–Scissors). The results imply that mixing over actions not only facilitates tractable analysis but also expands the strategic toolbox for payoff control in repeated interactions, with connections to evolutionary dynamics and potential applications to broader continuous-action settings.

Abstract

Mixed extension has played an important role in game theory, especially in the proof of the existence of Nash equilibria in strategic form games. Mixed extension can be regarded as continuous relaxation of a strategic form game. Recently, in repeated games, a class of behavior strategies, called zero-determinant strategies, was introduced. Zero-determinant strategies control payoffs of players by unilaterally enforcing linear relations between payoffs. There are many attempts to extend zero-determinant strategies so as to apply them to broader situations. Here, we extend zero-determinant strategies to repeated games where action sets of players in stage game are continuously relaxed. We see that continuous relaxation broadens the range of possible zero-determinant strategies, compared to the original repeated games. Furthermore, we introduce a special type of zero-determinant strategies, called one-point zero-determinant strategies, which repeat only one continuously-relaxed action in all rounds. By investigating several examples, we show that some property of mixed-strategy Nash equilibria can be reinterpreted as a payoff-control property of one-point zero-determinant strategies.

Zero-determinant strategies in repeated continuously-relaxed games

TL;DR

The paper extends zero-determinant (ZD) strategies to repeated games whose stage-action spaces are continuously relaxed via mixed extensions, showing that this continuous relaxation broadens the set of payoff-control capabilities than in the original finite-action setting. It presents a formal condition for the existence of two-point ZD strategies in the mixed-extension framework and demonstrates the broadened scope through Prisoner’s Dilemma, Public Goods, and a three-action symmetric game. A new construct, one-point ZD strategies, is introduced and shown to reproduce several equilibrium-payoff properties in classic games via fixed mixed strategies (e.g., Matching Pennies, Battle of the Sexes, Rock–Paper–Scissors). The results imply that mixing over actions not only facilitates tractable analysis but also expands the strategic toolbox for payoff control in repeated interactions, with connections to evolutionary dynamics and potential applications to broader continuous-action settings.

Abstract

Mixed extension has played an important role in game theory, especially in the proof of the existence of Nash equilibria in strategic form games. Mixed extension can be regarded as continuous relaxation of a strategic form game. Recently, in repeated games, a class of behavior strategies, called zero-determinant strategies, was introduced. Zero-determinant strategies control payoffs of players by unilaterally enforcing linear relations between payoffs. There are many attempts to extend zero-determinant strategies so as to apply them to broader situations. Here, we extend zero-determinant strategies to repeated games where action sets of players in stage game are continuously relaxed. We see that continuous relaxation broadens the range of possible zero-determinant strategies, compared to the original repeated games. Furthermore, we introduce a special type of zero-determinant strategies, called one-point zero-determinant strategies, which repeat only one continuously-relaxed action in all rounds. By investigating several examples, we show that some property of mixed-strategy Nash equilibria can be reinterpreted as a payoff-control property of one-point zero-determinant strategies.
Paper Structure (12 sections, 6 theorems, 74 equations, 3 figures, 5 tables)

This paper contains 12 sections, 6 theorems, 74 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Preliminaries
  4. Existence of two-point ZD strategies
  5. Prisoner's dilemma game
  6. Public goods game
  7. A two-player three-action symmetric game
  8. One-point ZD strategies
  9. Matching pennies
  10. Battle of the sexes
  11. Rock-paper-scissors game
  12. Concluding remarks

Key Result

Proposition 1

If there exist two mixed strategies $\underline{p}_j, \overline{p}_j \in \Delta(A_j)$ and a constant $W>0$ such that then the memory-one strategy of player $j$, where $\Delta\left(A_j\right)^\prime:=\left\{ \underline{p}_j, \overline{p}_j \right\}$ is a restricted action set of player $j$, is a ZD strategy controlling $\tilde{B}$.

Figures (3)

  • Figure 1: A linear relation between $\mathcal{U}_1$ and $\mathcal{U}_2$ when player $1$ uses the two-point ZD strategy (\ref{['eq:transition_two-point']}) and player $2$ uses $10^3$ randomly-chosen two-point memory-one strategies. Parameters are set to $(R, S, T, P)=(3, 0, 5, 1)$, $r=2$, $W=3$, $\overline{p}_1(C)=2/3$, and $\underline{p}_1(C)=1/4$. Each $\mathcal{U}_j$ is calculated by time average over $10^5$ time steps.
  • Figure 2: A linear relation between $\mathcal{U}_1$ and $\mathcal{U}_2$ when player $1$ uses the two-point ZD strategy (\ref{['eq:transition_two-point']}) and player $2$ uses $10^4$ randomly-chosen two-point memory-one strategies. Parameter is set to $W=3$. Each $\mathcal{U}_j$ is calculated by time average over $10^5$ time steps.
  • Figure 3: Feasible regions of $\underline{p}_1$ and $\overline{p}_1$ in a probability simplex $\Delta(A_1)$. We define $b^{(a_2)}(a_1):= B(a_1, a_2)$ for simplicity. Any $p_1$ in the green region can be used as $\underline{p}_1$, and any $p_1$ in the red region can be used as $\overline{p}_1$.

Theorems & Definitions (9)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • Corollary 1