Game theory:(P.B.E) Consider the following two-players signaling game with a sender and a receiver. The sender has a type [0,1] picked according to an uniform distribution and this type is private information. First the sender chooses some message m [0,1]. The receiver observes the message m and chooses an a [0,1]. The receiver's payoff function is ur(,m,a) = (a )² and the sender's is us(,m,a) = (a ( + b))² for some fixed b known by both players. Consider the following semi-pooling equilibrium: Suppose all types in one interval choose the same message and this message is different from all other intervals. [0,x₁),...,[x_k,x_k+1),...,[x_n1,1], where 0 k n 1, x₀ = 0 and x_n = 1. Solve first for n = 2. Then, for n 2 and given b, how much larger than its precedent interval should an interval be for this equilibrium to exist? Besides, for n 2 and given b, for an equilibrium to exist, which upper bound should n have ?

Hint: Suppose that x_k x_k1 = l. To make the boundary x_k indifferent between the intervals [x_k1,x_k) and [x_k,x_k+1), how much above the optimal action for x_k should the receiver's action, associated with the latter interval, be? Compute x_k+1 x_k and compare it to x_k x_k1 = l.