Rejoinder to the discussion of ‘Adversarial risk analysis: Borel games’

Abstract

By a natural segue, this brings us back to Polson's discussion of the computational methods. For the game with simultaneous play, we find that the main computational question is whether there is a fixed point (F, G) to which the algorithm converges. In another paper, regarding simultaneous routing games, Wang and Banks (2011) develop sufficient conditions to ensure convergence, and the calculations are much like those that Polson suggests. For sequential play, such as the Borel game treated in this paper, the calculations are lengthy and generally tedious, but the underlying principle is straightforward. As stated in the paper, symbolic computation would enable solutions for games with multiple players, multiple rounds of betting, and continuous bets-each of these possibilities is difficult for classical game theory, and the combination seems nearly intractable (cf. Karlin and Restrepo, 1957; Ferguson and Ferguson, 2007).In closing we entirely agree with both discussants: the minimax principle is too pessimistic, and a poor description of human behavior. The ARA approach is rooted in expected utility maximization, and thus enjoys the advantages noted in Kadane's historical survey. Whether the ARA approach is a good reflection of human decision making is arguable-clearly it is not something that anyone actually does, but we feel it may represent the kind of calculation that humans aspire to do. Chess players certainly engage in this kind of 'if I do this then she'll do that and I'll do this ...' thinking, and if the game had a random component (e.g. fairy chess) then it would closely match the logic in this paper. And the ARA method also allows the decision-maker to directly factor in beliefs about the opponent's style, in the way that Polson believes that on-line poker players bluff more aggressively than old-school green eyeshade players would do.