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This paper specifies an extensive form as a 5-ary relation (that is, as a setof quintuples) which satisfies eight abstract axioms. Each quintuple isunderstood to list a player, a situation (a concept which generalizes aninformation set), a decision node, an action, and a successor node.Accordingly, the axioms are understood to specify abstract relationshipsbetween players, situations, nodes, and actions. Such an extensive form iscalled a "pentaform". A "pentaform game" is then defined to be a pentaformtogether with utility functions. The paper's main result is to construct anintuitive bijection between pentaform games and $\mathbf{Gm}$ games (Streufert2021, arXiv:2105.11398), which are centrally located in the literature, andwhich encompass all finite-horizon or infinite-horizon discrete games. In thissense, pentaform games equivalently formulate almost all extensive-form games.Secondary results concern disaggregating pentaforms by subsets, constructingpentaforms by unions, and initial applications to Selten subgames andperfect-recall (an extensive application to dynamic programming is in Streufert2023, arXiv:2302.03855).

Specifying a Game-Theoretic Extensive Form as an Abstract 5-ary Relation