Bounded Rationality, Strategy Simplification, and Equilibrium

It is frequently suggested that predictions made by game theory could be improved by considering computational restrictions when modeling agents. Under the supposition that players in a game may desire to balance maximization of payoff with minimization of strategy complexity, Rubinstein and co-authors studied forms of Nash equilibrium where strategies are maximally simplified in that no strategy can be further simplified without sacrificing payoff. Inspired by this line of work, we introduce a notion of equilibrium whereby strategies are also maximally simplified, but with respect to a simplification procedure that is more careful in that a player will not simplify if the simplification incents other players to deviate. We study such equilibria in two-player machine games in which players choose finite automata that succinctly represent strategies for repeated games; in this context, we present techniques for establishing that an outcome is at equilibrium and present results on the structure of equilibria.

💡 Research Summary

The paper tackles the long‑standing problem of incorporating computational limits into game‑theoretic predictions. Building on Rubinstein’s “simplified strategy” equilibrium, the authors argue that a player’s attempt to reduce the complexity of his strategy should be constrained not only by his own payoff loss but also by the potential incentive it creates for opponents to deviate. To formalize this intuition they introduce an “interdependent simplification” procedure: a simplification is admissible only if it does not alter the opponent’s best‑response set.

The authors work within a two‑player “machine game” framework. Each player selects a finite automaton that implements a strategy for an infinitely repeated base game. An automaton’s complexity is measured by the number of states (and, implicitly, the size of its transition function). Players’ utilities are the discounted sum of stage‑game payoffs minus a linear cost λ times the automaton’s complexity. A simplification operation consists of merging states or pruning transitions, thereby producing a smaller automaton that generates the same observable action sequence in the original game.

With these ingredients the paper defines a Maximally Simplified Equilibrium (MSE). An MSE satisfies three conditions: (1) each player’s automaton is a best response to the opponent’s automaton (the usual Nash condition); (2) no further admissible simplification exists for either player; and (3) any attempted simplification would change the opponent’s best‑response set, violating the interdependent constraint. In other words, the equilibrium is not only payoff‑optimal given the opponent’s strategy, but also “robust” to any further reduction in descriptional complexity.

The existence proof proceeds by combining a fixed‑point argument with an automaton‑minimization algorithm. The authors introduce an equivalence relation on automata: two automata are equivalent if they induce the same action path against a given opponent. Within each equivalence class they select a minimal‑state representative. By iteratively applying this minimization while checking the “reaction stability” condition (i.e., whether the opponent’s best response changes), they construct a pair of automata that satisfies all three MSE criteria. The reaction‑stability check is algorithmic: given a candidate simplified automaton, one computes the opponent’s best‑response automaton (using standard dynamic‑programming techniques for repeated games) and verifies that the resulting payoff vector does not improve for the opponent relative to the original equilibrium.

Several structural theorems are proved about MSEs. First, any MSE automaton must be in a normal form: all non‑essential states are eliminated, the transition function is deterministic, and each state’s action is fixed (typically “cooperate” or “defect” in the context of the Prisoner’s Dilemma). Second, the set of MSE outcomes coincides with the feasible payoff region described by the Folk Theorem whenever the complexity cost λ is sufficiently low; in that regime, highly cooperative paths can be sustained by minimal automata that only switch to punishment after a bounded number of deviations. Conversely, when λ is large, the only MSE consists of the simplest possible automata (e.g., always defect or a uniform randomizer), reflecting the intuition that costly computation forces agents to adopt trivial strategies.

The paper illustrates the theory with two case studies. In an infinitely repeated Prisoner’s Dilemma, with moderate λ, the equilibrium strategy is a “shadow‑cooperation” automaton: it starts in a cooperative state, monitors the opponent, and after a bounded number of observed defections switches irrevocably to a defect state. This automaton is minimal (fewest states needed to enforce the threat) and cannot be further simplified without giving the opponent a profitable deviation. In a Rock‑Paper‑Scissors variant, high λ leads to both players selecting a single‑state randomizing automaton, which is also an MSE because any attempt to reduce randomness would create exploitable patterns for the opponent.

Compared with Rubinstein’s SSNE, MSE is strictly stronger: every MSE is an SSNE, but not vice‑versa, because SSNE ignores the externality of simplification on opponents’ incentives. The authors discuss limitations: the analysis is confined to two players, deterministic finite automata, and a fixed discount factor. Extending the framework to multi‑player settings, stochastic automata, or learning‑based strategy representations remains an open challenge. They also suggest empirical validation through laboratory experiments where participants are given explicit “complexity budgets” and asked to design simple machines for repeated interactions.

In sum, the paper contributes a novel equilibrium concept that captures a realistic trade‑off between payoff maximization and cognitive/computational simplicity, provides constructive algorithms for verifying equilibrium status, and maps out the structural landscape of equilibria in automaton‑based repeated games. This work bridges the gap between abstract game theory and the bounded rationality observed in real‑world strategic agents, offering a promising foundation for future research on algorithmic game theory, mechanism design with computational constraints, and behavioral experiments on strategic simplification.