6. Two U.S. senators are considering entering the race for the Democratic nomination for U.S. president. Each candidate has a privately known personal cost to entering the race. Assume that the probability of having a low entry cost, /, is p and the probability of hav- ing a high entry cost, /y, is 1 - p. Thus, the type space has just two values. A candidate's payoff depends on whether he enters the race and whether the other senator enters as well. Let va be a candidate's payoff when he enters and the other senator does as well (so that there are two candidates), v, be a candidate's payoff when he enters and the other senator does not (so that there is one candidate), and 0 be the payoff when he does not enter. Assume that VI > V2>0, V2 -f. >0 > via V1 - (# > 0. a. Derive the conditions whereby it is a symmetric BNE for a candidate to enter only when she has a low personal cost from doing so. ANSWER: It is a symmetric Bayes-Nash equilibrium for a senator to enter only when she has low personal cost if and only if Low type: p(vz - () + (1 -p)( -[)=0=p2 + (1 -p) = High type: 0 =p(v2 - [n) + (1 -p)(vi -/m)= /m =pv2 + (1 -p)vi. These two conditions can be combined to yield fu =pv2 + (1 - p) = fu This condition holds when p is close to 1. b. Derive the conditions whereby it is a symmetric BNE for a candidate to enter for sure when she has a low personal cost and to enter with some probability strictly between O and 1 when she has a high personal cost