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.
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Notes
- 1.
See (Rahman and Clerbout, 2015).
- 2.
Dependent moves are important for building subplays triggered by P challenging an O-implication: the moves depending on O counterattacking must be separated from the moves depending on O defending her implication.
- 3.
For a discussion of this problem see section 1.6 in (Rahman , Clerbout & Keiff, 2009).
- 4.
The bottom up procedure can be understood through two aspects:
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from the viewpoint of O: she is trying to win a play but with the minimal cost; so if she can win by changing a minimum amount of moves, she will choose that option. The last decision reveals this economy: if O changed the first decision , then all of the play would have to be replayed; whereas simply changing the last decision allows to keep all the rest. If this last decision was not the faulty decision bringing her to lose, then she goes to the decision before last, etc. up to the very first decision.
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From the viewpoint of the logician: the bottom up procedure allows us to build the core of the strategy : it reveals what is important in the strategies.
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- 5.
The idea is that O has just lost the previous play with the left or right decision she made; so she is now looking for what has gone wrong and tries to change her way of playing in order to win. Her last decision was the left- or right-decision for the conjunction ; so a way of seeing if she can win is to start the game again with exactly the same moves up to this decision, choose the other decision-option this time, and play with that. This is the whole idea of the core: allowing O to go through all of her possible decisions, and if in this rational way of proceeding that takes all of her opportunities into account, P still wins, then P has a winning strategy . The core is all of these relevant plays, bringing the strategy to its minimal aspect.
- 6.
Notice that the structure of this thesis is the following one, using schematic letters:\( \left(\left(\left(X\vee Y\right)\wedge \neg X\right)\supset Y\right)\wedge \left(Z\supset Z\right). \)
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Rahman, S., McConaughey, Z., Klev, A., Clerbout, N. (2018). Advanced Dialogues: Strategy Level. In: Immanent Reasoning or Equality in Action. Logic, Argumentation & Reasoning, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-91149-6_5
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