Advice on Statistical Analysis for Circulation Research by Hideo Kusuoka and Julien I.E. Hoffman

By Hideo Kusuoka and Julien I.E. Hoffman

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The independence assumptions built into normal form games are met. g. Luce and Raiffa, pp. 52-53). g. Selten, 1965, 1975; Krepps and Wilson, 1982). In Section 4 I shall say something about how causal decision theory can illuminate some of the problems with the normal form. For the present, I want to show how causal decision theory can illuminate games (such as our Death in Damascus example) where the normal form is appropriate. The basic idea built into the normal form is that rational players with complete knowledge of the game structure and of the utilities of all the players ought to be able to choose their strategies (a strategy is a complete plan for how you would play out the game under any of the contingencies it would allow for) ahead of time - without any appeal to a posteriori knowledge of the strategy any other player actually ends up playing out.

B =? C, A E '0, and B, C E V(x,A v B) = PAvH(A) V(x,A) + PAvB(B) V(x,B) whenever x,A, x,B E X and A & B = <1>; furthermore. the PA'S are unique and V is unique up to positive linear transformation. IV. Clause (iii) of this theorem provides measures PH even when Py{B) O. Note that whenever A =? Band Py{B) > 0, PH(A) = Py{A I B). Clause (iv) of the theorem is the general form of our theory's rule (FI) relating the utilities of a proposition or mixture under different conditional hypotheses. = Proof: Fishburn's system does not include the restrictions that appear in A2 (a), the closure condition on mixing, which restrict mixing to incompatible propositions.

H SUM; y'i' U(x,Bi) all y' E Y Since P is a best response prior. we have 6. FCv,P(vIX)dP = 1 so that 7. / ev Ix)' ,vi dp for any appropriate integral over Yx. By 4 and 7 we have, 8. Bi))dp so that y* is a mixture of best responses to x. ') This theorem shows that the answer to McClennen provided by best response reasoning together with ratifiability formulated in callsal decision theory also provides a general argument recommending that solutions to two-person non-cooperative games be restricted to Nash equilibria.

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