  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.
