Advanced Dialogues: Strategy Level

Abstract

The strategy standpoint is but a generalisation of the procedure which is implemented at the play level; it is a systematic exposition of all the relevant variants of a game—the relevancy of the variants being determined from the viewpoint of one of the two players. For a more intuitive approach of strategies and a step-by-step introduction of strategies as branching tables, see Sect. 3.5, p. 59. Such trees with branching tables are a good didactic approach to strategies, for the rules in building the tree are the same as those for building the plays: we simply use an algorithm yielding all the relevant plays for a player, keeping the table presentation we use for plays. The link from plays to strategies is thus clearly apparent. This method however is rather cumbersome and becomes unmanageable as soon as we deal with games involving more than two choices, the generated trees taking too much space. We will here present strategies from another perspective, that of extensive forms of dialogical games (more precisely from their core; see below, Sect. 5.3) rather than the table presentation; the extensive form presentation has this advantage over the table presentation that strategies can be linked more straightforwardly to demonstrations, which will be useful in Chap. 9. This link is crucial to the logical framework of dialogues, for the dialogical notion of validity is secured through the notion of a winning strategy for the Proponent. Many metalogical results in the dialogical framework are obtained by leaving the level of rules and plays to move to the level of strategies; winning strategies for a player are one of these metalogical results.