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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
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32 [ Pobierz całość w formacie PDF ]