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

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