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We have to answer the following question, and I can make some progress on the first couple of parts but get stuck in finishing them off. The third part I'm not sure where to start!

*Consider the following Bayesian game. Nature chooses the type of player 1 from the set {1, 2, 3, 4} where each type has equal probability. Player 1, the sender, observes his type and may send a costless message from the set {m1, m2, m3, m4} that does not affect either player's payoffs. Player 2, the receiver, does not observe player 1's type, and must choose an action a from the set of real numbers. The sender's payoff is given by U( φ ; a) = 1.5a 􀀀- (a - φ􀀀 )^2: The receiver's payoff is given by V (φ ; a) = 􀀀-(a - φ )^2:

a) Show that there is always an equilibrium where the sender plays the same action after every message. Interpret this equilibrium.

b) Show that there cannot be an equilibrium with full separation of types.

c) Solve for an equilibrium with partial separation of types. (Hint: Look for separation between unequally sized subsets of the set of types).

d) Provide an argument why there cannot be any other equilibrium with partial separation apart from the one you and in part (c).*

In part (a) I started by finding the integral of the receiver's utility function, but I only know how to work the answer out from there when the interval is continuous between 0 and 1, and not when there are 4 set states.

In part (b) I made progress towards showing the contradiction if we assume the receiver plays according to the message he receives, but am not sure what his ideal play would in fact be to finish off showing that such an equilibrium can't exist.

And I'm not sure how to go about part (c) so please help!

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