ABSTRACT
Proof It is sufficient to show that there exists an interval [a, b] ~ [0, 1] such that for all x E [a, b], uAx) ;::: p - CA and UB(1 - x) ;::: 1 - P - CB' Choose an E such that ° < E < min{cA' CB} and define a = max{O,p - E} and b = min{p + E, I}. Consider anyx' E [a, b]. By weak concavity, uAx') ;::: x' for all x' E [0, 1]. Further, x' > p - CA, since x' ;::: a ;::: p - E > p - CA, by definitions. Thus UA(X') > P - CA' A similar argument shows that uB(1 - x') > 1 - P - CB for all x' E [a, b]. 0
state B, CIJ, from a cumulative distribution H(z) on the nonnegative real numbers with a
strictly positive density function h(z) and a nondecreasing hazard rate h(z)1 (1 - H(z».7o State B observes CB but A does not. State A moves first, choosing a demand x E [0, 1]. B observes the demand and chooses whether to fight or not. As discussed in the text, payoffs are (p - CA, 1 - P - CB) if B fights and (x, 1 - x) if B does not fight.
