Assume that two countries are on the verge of war and are simultaneously deciding whether or not to attack. A country’s military resources are its type, and their relevance is summarized in a parameter which influences the likelihood that they would win a war. Suppose the type space is made up of two values: p' and p", where 0 < p' < p" < 1. A country is type p" with probability q and type p' with probability 1 - q. Consider a country of type p (which equals either p' or p"). If it chooses to attack and it attacks first, then it believes it’ll win the war with probability xp, where x takes a value such that p < xp < 1. If the two countries both attack, then the probability that a type p country wins is p. If a type p country does not attack and the other country does attack, then the probability of victory for the type p country is yp, where y takes a value such that 0 < yp < p. Finally, if neither country attacks, then there is no war. A country is then more likely to win the war the higher is its type and if it attacks before the other country. A country’s payoff when there is no war is 0, from winning a war is W, and from losing a war is L. Assume that W > L. a. Derive the conditions for it to be a symmetric BNE for a country to attack regardless of its type. b. Derive the conditions for it to be a symmetric BNE for a country to attack only if its type is p
Assume that two countries are on the verge of war and are simultaneously deciding whether or not to attack. A country’s military resources are its type, and their relevance is summarized in a parameter which influences the likelihood that they would win a war. Suppose the type space is made up of two values: p' and p", where 0 < p' < p" < 1. A country is type p" with probability q and type p' with probability 1 - q. Consider a country of type
p (which equals either p' or p"). If it chooses to attack and it attacks first,
then it believes it’ll win the war with probability xp, where x takes a value
such that p < xp < 1. If the two countries both attack, then the probability
that a type p country wins is p. If a type p country does not attack and the
other country does attack, then the probability of victory for the type p
country is yp, where y takes a value such that 0 < yp < p. Finally, if neither
country attacks, then there is no war. A country is then more likely to win
the war the higher is its type and if it attacks before the other country. A
country’s payoff when there is no war is 0, from winning a war is W, and
from losing a war is L. Assume that W > L.
a. Derive the conditions for it to be a symmetric BNE for a country to
attack regardless of its type.
b. Derive the conditions for it to be a symmetric BNE for a country to
attack only if its type is p".
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