Acyclicity of Preferences, Nash Equilibria, and Subgame Perfect Equilibria: a Formal and Constructive Equivalence

In 1953, Kuhn showed that every sequential game has a Nash equilibrium by showing that a procedure, named backward induction'' in game theory, yields a Nash equilibrium. It actually yields Nash equilibria that define a proper subclass of Nash equilibria. In 1965, Selten named this proper subclass subgame perfect equilibria. In game theory, payoffs are rewards usually granted at the end of a game. Although traditional game theory mainly focuses on real-valued payoffs that are implicitly ordered by the usual total order over the reals, works of Simon or Blackwell already involved partially ordered payoffs. This paper generalises the notion of sequential game by replacing real-valued payoff functions with abstract atomic objects, called outcomes, and by replacing the usual total order over the reals with arbitrary binary relations over outcomes, called preferences. This introduces a general abstract formalism where Nash equilibrium, subgame perfect equilibrium, and backward induction’’ can still be defined. This paper proves that the following three propositions are equivalent: 1) Preferences over the outcomes are acyclic. 2) Every sequential game has a Nash equilibrium. 3) Every sequential game has a subgame perfect equilibrium. The result is fully computer-certified using Coq. Beside the additional guarantee of correctness, the activity of formalisation using Coq also helps clearly identify the useful definitions and the main articulations of the proof.

💡 Research Summary

The paper revisits the classic result that every finite sequential game possesses a Nash equilibrium, originally proved by Kuhn through the backward‑induction procedure, and later refined by Selten who identified the equilibria produced by this procedure as subgame‑perfect equilibria (SPE). While the traditional literature assumes real‑valued payoffs ordered by the usual total order on ℝ, the authors deliberately abandon this restriction. They replace concrete numeric payoffs with abstract atomic objects called outcomes and replace the total order with an arbitrary binary relation on outcomes, termed a preference. In this setting, a preference need not be total, transitive, or even antisymmetric; the only structural requirement the authors impose is acyclicity—the absence of cycles of the form a₁ ≺ a₂ ≺ … ≺ aₖ ≺ a₁.

The authors formalise sequential games as finite rooted trees. Internal nodes are labelled by the player who moves at that node, and leaves are labelled by outcomes. A strategy for a player selects, at each node she controls, one of the outgoing edges; a strategy profile is a tuple of strategies, one per player. Given a preference relation for each player, a player can compare any two outcomes that may arise under a given profile and decide whether she strictly prefers one, is indifferent, or cannot compare them.

The central technical contribution is a constructive equivalence among three statements:

1. Acyclic preferences – for every player, the binary relation on outcomes contains no directed cycles.
2. Universal Nash equilibrium existence – every finite sequential game (with arbitrary preferences) admits at least one Nash equilibrium.
3. Universal subgame‑perfect equilibrium existence – every finite sequential game admits at least one SPE.

The proof proceeds in three implications:

• (1) ⇒ (3): Assuming acyclicity, the authors show that backward induction can be carried out even when preferences are only partially ordered. Because there are no cycles, at each node there exists at least one maximal child with respect to the player’s preference; the player can safely choose such a child. By recursively applying this selection from the leaves up to the root, a strategy profile is constructed that is optimal in every subgame, i.e., an SPE. The construction is fully constructive: the algorithm returns an explicit profile rather than merely asserting existence.

• (3) ⇒ (2): Every SPE is, by definition, a Nash equilibrium of the whole game, because the equilibrium condition holds in the root subgame as a special case. Hence the existence of an SPE guarantees a Nash equilibrium.

• (2) ⇒ (1): The authors use a counter‑example argument. If a player’s preference contains a cycle a₁ ≺ a₂ ≺ … ≺ aₖ ≺ a₁, they build a tiny sequential game whose leaves are precisely these outcomes arranged so that each player’s decision point forces a traversal of the cycle. In such a game, any strategy profile can be unilaterally improved by moving to the next outcome in the cycle, preventing the existence of a Nash equilibrium. Therefore, universal Nash‑equilibrium existence forces all preferences to be acyclic.

All definitions (games, strategies, preferences, equilibrium conditions) and the entire proof are mechanised in the Coq proof assistant. The Coq development introduces inductive types for game trees, records for preference relations, and predicates for Nash and subgame‑perfect equilibria. Proof scripts combine Coq’s automation tactics with manual reasoning to verify each implication. The mechanisation not only guarantees logical correctness but also clarifies the minimal concepts required: the notion of a maximal element in an acyclic relation, the recursive construction of a strategy, and the equivalence between equilibrium notions.

The paper’s contributions are threefold. First, it identifies acyclicity as the weakest possible condition on arbitrary preferences that still ensures the classic existence results for sequential games. Second, it demonstrates that the backward‑induction algorithm remains sound under this minimal assumption, thereby extending the reach of the algorithm to settings with partially ordered or even incomparable outcomes (e.g., multi‑criteria decision problems, qualitative payoffs, or preferences expressed by logical formulas). Third, the complete Coq certification provides a high‑assurance artifact that can serve as a foundation for further formal work in game theory, such as mechanised reasoning about equilibrium refinements, algorithmic synthesis of strategies, or verification of protocols in distributed systems where agents have non‑numeric utilities.

In practical terms, the results open the door to modelling strategic interactions where agents evaluate outcomes by lexicographic, Pareto, or other non‑numeric criteria, while still being able to compute equilibria algorithmically. The constructive nature of the proof yields an explicit backward‑induction procedure that can be implemented in software tools for automated game analysis, even when the underlying utility structure is only a partial order. Moreover, the formalisation in Coq can be extended to infinite horizons, stochastic elements, or richer solution concepts, offering a robust platform for future research at the intersection of formal methods and economic/game‑theoretic modelling.