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Extending Games beyond the Finite Horizon

Kiri Sakahara, Takashi Sato

TL;DR

Extending Games beyond the Finite Horizon introduces Alternative Set Theory (AST) to model infinite-like cognitive perception in long-horizon games. It develops two perspectives on histories, Perspective View and Bird's Eye View, and analyzes payoff aggregation under three criteria Discounted/Simple Sum, Overtaking, and Limit of Means to derive new subgame-perfect equilibria. The framework resolves classic paradoxes such as Selten's chain store and Rosenthal's centipede by allowing huge indistinguishable payoffs to influence strategic choice, with dynamic consistency and huge transitivity playing key roles. Overall, the work broadens game theory's reach into ultra-long horizons by tying mathematical foundations to cognitive models of time and perception.

Abstract

This paper argues that the finite horizon paradox, where game theory contradicts intuition, stems from the limitations of standard number systems in modelling the cognitive perception of infinity. To address this issue, we propose a new framework based on Alternative Set Theory (AST). This framework represents different cognitive perspectives on a long history of events using distinct topologies. These topologies define an indiscernibility equivalence that formally treats huge, indistinguishable quantities as equivalent. This offers criterion-dependent resolutions to long-standing paradoxes, such as Selten's chain store paradox and Rosenthal's centipede game. Our framework reveals new intuitive subgame perfect equilibria, the characteristics of which depend on the chosen temporal perspective and payoff evaluation. Ultimately, by grounding its mathematical foundation in different modes of human cognition, our work expands the explanatory power of game theory for long-horizon scenarios.

Extending Games beyond the Finite Horizon

TL;DR

Extending Games beyond the Finite Horizon introduces Alternative Set Theory (AST) to model infinite-like cognitive perception in long-horizon games. It develops two perspectives on histories, Perspective View and Bird's Eye View, and analyzes payoff aggregation under three criteria Discounted/Simple Sum, Overtaking, and Limit of Means to derive new subgame-perfect equilibria. The framework resolves classic paradoxes such as Selten's chain store and Rosenthal's centipede by allowing huge indistinguishable payoffs to influence strategic choice, with dynamic consistency and huge transitivity playing key roles. Overall, the work broadens game theory's reach into ultra-long horizons by tying mathematical foundations to cognitive models of time and perception.

Abstract

This paper argues that the finite horizon paradox, where game theory contradicts intuition, stems from the limitations of standard number systems in modelling the cognitive perception of infinity. To address this issue, we propose a new framework based on Alternative Set Theory (AST). This framework represents different cognitive perspectives on a long history of events using distinct topologies. These topologies define an indiscernibility equivalence that formally treats huge, indistinguishable quantities as equivalent. This offers criterion-dependent resolutions to long-standing paradoxes, such as Selten's chain store paradox and Rosenthal's centipede game. Our framework reveals new intuitive subgame perfect equilibria, the characteristics of which depend on the chosen temporal perspective and payoff evaluation. Ultimately, by grounding its mathematical foundation in different modes of human cognition, our work expands the explanatory power of game theory for long-horizon scenarios.

Paper Structure

This paper contains 13 sections, 14 theorems, 67 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A System of Numbers
  3. Extensive Games with Perfect Information
  4. Generalised Repeated Games
  5. Strategic Games
  6. Perspectives on Whole Histories
  7. Perspective View
  8. Bird's Eye View
  9. Payoff Functions of $\mathbf{\Gamma}^\tau$
  10. Discounted/Simple Sum
  11. Overtaking
  12. Limit of Means
  13. Concluding Remarks

Key Result

Proposition 8

If a preference relation $\succsim_i^\tau$ on $\mathbf{Z}^\tau$ is compact, it cannot satisfy strict separability, which replaces $\succsim_i$ with strict preferences $\succ_i$.

Figures (11)

  • Figure 1: A tree of the constituent game of a chain store game.
  • Figure 2: A tree of the 2-repeated chain store game.
  • Figure 3: Hasse diagrams of the preference relations of (a) the chain store and (b) the local store.
  • Figure 4: A tree of the constituent game of a centipede game.
  • Figure 5: A tree of the 2-repeated centipede game.
  • ...and 6 more figures

Theorems & Definitions (62)

  • Definition 1: Definition 89.1 of OR
  • Definition 2: Definition 92.1 of OR
  • Definition 3: Definition 93.1 of OR
  • Definition 4: Definition 97.1 of OR
  • Definition 5: Definition 97.2 of OR
  • Definition 6
  • Definition 7
  • Proposition 8
  • proof
  • Definition 9
  • ...and 52 more