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Sequential Games

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath.CA — Classical Analysis and ODEs

Contributed by Rajesh Dachiraju ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

We formulate a time-series game in which each player chooses a hidden Markov model (HMM) as a mixed strategy. The HMM generates a finite, variable-length sequence of probability vectors, and the payoff is assigned to the completed game as the sum of its stage rewards. We prove a conditional Nash-equilibrium existence result for games with a common finite bound on output lengths. The proof applies a fixed-point argument to the best-response correspondence when each player's admissible HMM-density set is nonempty, convex, and compact in (L1)(L^1), and the completed-game payoff is bounded and measurable. A common BV bound together with (L1)(L^1)-closedness can ensure compactness; closure under suitable disjoint-state mixtures can ensure convexity. These properties do not follow from the HMM representation or BV regularity alone and must be verified for a specified model. Consistent estimation of HMM densities may support computation, but by itself does not establish equilibrium existence or replace the theorem's assumptions.

Provenance statement

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References

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Version history

  1. v1Submitted by Rajesh DachirajuInitial depositCurrentOct 15, 2026