Question

Consider the Princess Bride signaling game picture below, where Nature chooses the type of Westley (strong or weak, with equal probability), Westley observes Nature's choice and decides whether to stay in bed (B. B') or get out of bed (O, O'), and then the Prince chooses whether to surrender (S) or fight (F) after observing Westley's action but not observing Nature's selecti At each terminal node, Westley's payoff is listed first, followed by the Prince's payoff. Assume that the cost to the weak type of getting out of bed, c, is positive 1,0 1,0 0.-2 Strong (1/2) 0,-2 1,0 Weak (1/2) 1-c,0 1 -r) 2,1 2-c,1 By choosing one the options below, explain whether there is a Perfect Bayesian Equilibrium (PBE) of this game in which the Prince surrenders if Westley stays in bed and fights if Westley gets out of bed (that is, the Prince uses the strategy SF) O There is no PBE in which the Prince uses the strategy SF', regardless of the cost parameter c O There is a PBE in which the Prince uses the strategy SF, regardless of the cost parameter c. ○ If the cost parameter c is larger than 1. then there is a PBE in which the Prince uses the strategy SF: otherwise, there is no such PBE. 0 If the cost parameter c is between 0 and 1, then there is a PBE in which the Prince uses the strategy SF'; otherwise, there is no such PBE.

Answer 1

Considering the Princess Bride signalling game, given that player 1 wesley knows his nature {strong, weak}

chooses to stay in bed(B) or stay out of bed(O) then player 2 the prince observing only the actions of player 1 not knowing the type of the player chooses to surrender(S) or fight(F).

Note that if player one is the weak type and chooses (O) then he has to incur some cost on his payoffs in either strategy of player two hence B^' dominates O^' regardless of the value of c.

Now the strategy of player two SF' can only be played for a separating equilibrium when strategy of player one is B for strong type and O for weak type when r=0,q=1 but comparing player two's payoff in this case the optimal strategy is to play FS^'. hence, it is not a perfect bayesian equilibrium.

Hence, the first option is true that there is no PBE in which Prince uses SF^',regardless of the cost parameter.