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[ Pobierz caÅ‚ość w formacie PDF ] 27 1¡° R s ¡° v 1¡® achieve pro& ts ¦(v) = ¦(°)+ ds, while players with valuations in [®; ¯] make ¯ 1¡° zero pro& ts. Ex ante pro& ts of valuations [¯; 1] must exceed v2=2; that is: Z ³ ´ v 1¡° ° ¢ v + s1¡® ¡ ° ds ¡ K > v2=2: ¯ 1¡° Note that the derivative of the left-hand side with respect to v is v1¡®, which is larger than the corresponding derivative of the right-hand side which is v. Hence, the in- equality will hold if it holds for v = ¯. Consider players with valuations in [®; ¯]. Ex ante pro& ts from deviating are ° ¢ v ¡ K. Marginal pro& ts, °, once again exceed marginal pro& ts v in the original P.B.E., since v 5 ¯ are indi¤erent between deviating and not deviating, players with lower valuations will prefer not to deviate. This is equivalent to requiring that ® ¢ ° ¡ K = ®2=2. Hence, provided that there exist °, ® such that 8 ® ¢ ° ¡ K = ®2=2; ¯¡® : ¢ ° = K: 1¡® the equilibrium is not a P.S.E. Suppose now that it is not worthwhile for player 2 to cover after a deviation, regardless of valuation. It follows that the original P.B.E. is not a P.S.E. if 8 ® ¡ K = ®2=2; e®¡1¡® : 1¡® ®¡1 (e ¡® = lim°!1 ¯¡® ¢ °). This system guarantees that it is indeed optimal for player 1¡® 1¡® 2 not to cover, regardless of valuation; that if player has a valuation in the interval (®; 1] he will strictly prefer the expected payo¤ from deviating to the original expected payo¤, and that all players with valuations in the interval [0; ®) will strictly prefer the expected payo¤ of the original P.B.E. to their expected payo¤ from deviating. Although 28 it is not di¢cult to show which case obtains a function of K, this is not even necessary. It is enough to note that for ® = 0, expected pro& ts from deviating are smaller than expected pro& ts from not deviating, whereas in both cases, since expected pro& ts from deviating are larger than ®2 ¡ K, they are also larger than ®2=2, provided that ® is close enough to 1. This result is based on the fact that if K is strictly less than 1=2. ° being a continuous function of K, there does then necessarily exist, for any K an ® 2 (0; 1) satisfying one of the two systems. We & nally need to verify that there does not exist a P.S.E. providing assured de- ¹ terrence outside the interval [K; 1=2]. It is then easy to show that an equilibrium with assured deterrence, as speci& ed in the text, is a P.S.E.. Recall that in an equilibrium with assured deterrence, there exists an ® 2 (0; 1) such that all player 1s with valua- tions strictly smaller than ® make a zero opening bid, while all players with valuations strictly larger than ® invest K. In this equilibrium player 2 never covers, regardless of valuation . In simultaneous bidding between players with valuation v 2 [0; ®] and players with w 2 [0; 1] ;after a zero opening bid , expected pro& ts of player 1 with valuation v = ® are equivalent to ®2= (1 + ®); since a player with valuation ® will be indi¤erent between this expected pro& t and the expected pro& t following a bid of K, which is ® ¡ K, it follows that ® = K=(1 ¡ K). Consider a deviation by player 2 in which he covers. More precisely, suppose that players with w 2 (°; 1] will cover while players with valuations lower than ° 2 (0; 1) will not . Obviously, if players with valuations arbitrarily close to ° from above have expected pro& ts from the devia- tion that are arbitrarily small, players with valuations strictly above ° obtain strictly positive expected pro& ts from deviating while players with valuations strictly below achieve strictly negative pro& ts. These two situations compare with the zero pro& ts that player 2 achieves in the original P.B.E., regardless of valuation. In the subgame following the deviation by player 2: Player 1 s with valuations v 2 (®; 1] play against player 2s with w 2 (°; 1] . In this subgame, player 2 with valuation °+ (a valuation 29 arbitrarily close to ° from above) can achieve pro& t K; only if 1¡® 1¡° ° ¡ ® ¢ ° = K. 1 ¡ ® Hence, the equilibrium with assured deterrence is a P.S.E. if and only if such a ° 2 (0; 1) cannot be found. Since the left-hand side is increasing in °, it is both necessary and su¢cient that 1¡® 1¡° ° ¡ ® e¡(1¡®) ¡ ® ® lim ¢ ° = 5 : °!1 1 ¡ ® 1 ¡ ® 1 + ® 1¡K The latter inequality can be rewritten as 1 + ¢ ln2K = 0; which precisely states 1¡2K ¹ that K = K. Finally, when K > 1=2, there is no equilibrium with assured deterrence, since when K meets this condition there is no value of ® in the open unit interval which satis& es ® = K= (1 ¡ K) Finally, equilibria with covering, as speci& ed in the ¹ text for K the equilibrium path . 30 References [1] Amann, E., and W. Leininger, Asymmetric all-pay auctions with incomplete in- formation: the two-player case, Games Econ. Behav. 14 (1996), pp. 1-18. [2] Bellman, R., and D. Blackwell, Some two-person games involving blu¢ng, Proc. Natl. Acad. Sci. 35 (1949), pp. 600-605. [3] Binmore, K., Fun and Games: A text on Game Theory , D.C. Heath and Co., Lexington, MA, 1992. [4] Bishop, D.T., C. Cannings, A Generalised War of Attrition, J. Theoret. Biol. 70 (1978), pp. 85-124. [5] Borel, E., [6] Dixit, A., Strategic Behavior in Contests, American Economic Review, 77 (5), (1987), pp. 891-8. 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Shapley, A Simple Three-Person Poker Game, in Dimand M.A, Dimand R., eds. The foundations of game theory. Volume 2. Elgar Reference Collection. Cheltenham, U.K. and Lyme, N.H.: Elgar; 1997, pp. 13-24. Previously published: [1950]. [16] Newman, D.J., A model for real poker, Operations Research 7 (1959), pp. 557-560. [17] Riechert, S.E., Games Spider Play: Behavioural Variability in Territorial Disputes, Behav. Ecol. Sociobiol. 3, (1978), pp. 135-162. [18] Sakai, S., A model for real poker with an upper bound of assets, J. Optim. Theory Appl. 50 (1986), pp. 149-163. [19] Rosen, S., Prizes and Incentives, American Economic Review, 76 (4), (1986), pp. 716-727. [20] Shepperd, J.A. and R.E. Socherman, On the Manipulative Behavior of Low Machi- avellians; Feigning Incompetence to Sandbag an Opponent, Journal of Person- ality and Social Psychology, 72 (1997), pp. 1448-1459. [21] Von Neumann, J. and O. Morgenstern, Theory of Games and Economic Behav- ior , Princeton Univ. Press, Princeton, 1944. 32 [ Pobierz caÅ‚ość w formacie PDF ] Darmowy hosting zapewnia PRV.PL |