[ Pobierz całość 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
� �
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-
�
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
�
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
�
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.
[7] Gibson, B., and Sachau, D., Sandbagging as a Self-Presentational Strategy: Claim-
ing to Be Less Than You Are, Personality and Social Psychology Bulletin, 26 (1),
(2000), pp. 56-70.
[8] Grossman, S., and M. Perry, Perfect Sequential Equilibrium, J. Econ. Theory 39
(1986), pp. 97-119.
[9] Karlin, S., and R. Restrepo, Multistage poker models , Annals of mathematics
studies No. 39, Edited by M. Dresher, A. W. Tucker, and P. Wolfe, Princeton
Univ. Press, Princeton, 1957.
[10] Maynard Smith, J., The Theory of Games and the Evolution of Animal Con! icts,
J. Theoret. Biol. 47 (1974), pp. 209-221.
[11] Maynard Smith, J., Evolution and the Theory of Games , Cambridge University
Press, Cambridge, U.K., 1982
31
[12] McAfee, P. and K. Hendricks, Feints, Mimeo, University of Texas, 2001.
[13] McDonald, J., Strategy in Poker, Business and War , Norton, New York, NY,
1950.
[14] Myerson R., Optimal Auction Design, Math. Oper. Res. 6 (1981), pp. 58-73.
[15] Nash, J. and L. 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 całość w formacie PDF ]
zanotowane.pldoc.pisz.plpdf.pisz.plkarro31.pev.pl
 Darmowy hosting zapewnia PRV.PL [X] |
