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Games with infinite past

Galit Ashkenazi-Golan, János Flesch, Eilon Solan

TL;DR

This work extends equilibrium analysis to dynamic games with an infinite past, where stages run on ${\bf Z}_0$ and backward induction may fail due to nonexistence of consistent runs. It introduces two equilibrium notions and proves their equivalence under a natural activity condition; it establishes existence results for two-player win-lose games with winning sets of Borel-rank at most 2 and continuity-based equilibria for multi-player non-zero-sum games, using Cantor’s intersection theorem instead of traditional backward inductive arguments. The segmentation of runs into segments provides a framework to study segment-restricted games that resemble standard finite-horizon games and clarifies determinacy and value at the segment level. Collectively, the paper contributes both structural insights into infinite-past dynamics and precise existence results under descriptive-set-theoretic conditions, with implications for long-run strategic reasoning and applications to reversed-time payoff models.

Abstract

We study multi-player games with perfect information and general payoff function, where the set of stages is the set of non-positive integers $\{\ldots,-2,-1,0\}$. We define two related equilibrium concepts: one considering only deviations at finitely many stages and another considering all deviations. We show that (i) The sets of equilibrium plays coincide for the two equilibrium concepts, provided that at least two players are active along each infinite play. (ii) In win-lose games, the game has an equilibrium if the winning sets have Borel-rank at most 2, and we provide a counter-example showing that this is no longer true for Borel-rank 3. (iii) In general non-zero-sum games, the game has an equilibrium if the payoff functions are continuous, for example, with reversed-time discounted payoffs. The challenge for all these results is that not all strategy profiles admit a consistent infinite play, hampering the use of backward induction arguments.

Games with infinite past

TL;DR

This work extends equilibrium analysis to dynamic games with an infinite past, where stages run on and backward induction may fail due to nonexistence of consistent runs. It introduces two equilibrium notions and proves their equivalence under a natural activity condition; it establishes existence results for two-player win-lose games with winning sets of Borel-rank at most 2 and continuity-based equilibria for multi-player non-zero-sum games, using Cantor’s intersection theorem instead of traditional backward inductive arguments. The segmentation of runs into segments provides a framework to study segment-restricted games that resemble standard finite-horizon games and clarifies determinacy and value at the segment level. Collectively, the paper contributes both structural insights into infinite-past dynamics and precise existence results under descriptive-set-theoretic conditions, with implications for long-run strategic reasoning and applications to reversed-time payoff models.

Abstract

We study multi-player games with perfect information and general payoff function, where the set of stages is the set of non-positive integers . We define two related equilibrium concepts: one considering only deviations at finitely many stages and another considering all deviations. We show that (i) The sets of equilibrium plays coincide for the two equilibrium concepts, provided that at least two players are active along each infinite play. (ii) In win-lose games, the game has an equilibrium if the winning sets have Borel-rank at most 2, and we provide a counter-example showing that this is no longer true for Borel-rank 3. (iii) In general non-zero-sum games, the game has an equilibrium if the payoff functions are continuous, for example, with reversed-time discounted payoffs. The challenge for all these results is that not all strategy profiles admit a consistent infinite play, hampering the use of backward induction arguments.

Paper Structure

This paper contains 19 sections, 14 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries - the Borel Hierarchy
  3. The model
  4. The game
  5. Strategies and consistent runs
  6. Segments
  7. Equilibrium
  8. Two equilibrium concepts
  9. Examples
  10. Relation between the two equilibrium concepts
  11. Backward induction and games with infinite past
  12. Two-player win-lose games
  13. Reversed positions and open winning sets
  14. An auxiliary result for open winning sets
  15. Existence of equilibrium for winning sets of Borel-rank at most 2
  16. ...and 4 more sections

Key Result

Lemma 11

Consider a strategy profile $s=(s_i)_{i\in I}$ and a segment $\Omega$.

Figures (1)

  • Figure 1: Segments of a game with two actions, $\ell$ (left) and $r$ (right). The left (resp., right) third of the figure corresponds to the segment that contains the run $(\ldots,\ell,\ell)$ (resp., $(\ldots,r,r)$). The bottom-left node corresponds to the run $(\ldots,\ell,\ell)$, the node above it to the position $(\ldots,\ell,\ell)$ in stage $0$, the bottom-right node to the run $(\ldots,r,r)$, etc. The dotted vertical lines indicate that segments have no common position, and between the two dotted lines there are a continuum of segments.

Theorems & Definitions (48)

  • Definition 1: Game with infinite past
  • Remark 2: On the set of actions
  • Remark 3: Players' information in games with infinite past
  • Definition 4: Strategy
  • Definition 5: Consistent run
  • Example 6: A strategy with no consistent run
  • Example 7: A strategy with two consistent runs
  • Remark 8: On the set $R(s)$ of all consistent runs
  • Definition 9: Segment and corresponding positions
  • Definition 10: Strategy permitting a segment
  • ...and 38 more