1. Consider an agent who takes an action after receiving some in- formation.   A decision  problem  is a tuple  (S, Π, C, u, p), where S is the state space, Π is the partition, C is the set of actions, u  : S x C  → R is the utility function and p is the prior over

S . Show that if partition Π 1  is finer than partition Π2 then decision problem A = (S, Π1 , C, u, p) is more valuable than decision problem B = (S, Π2 , C, u, p).

2. In the strategic bargaining model of Rubinstein, show that if the utility of player i, ui , is differentiable and axioms A0-A6 hold, we have that δu (x i ) < u (v i (xi , 1)), where vi (xi , 1) > 0 is the present value at period 0 of getting xi  at period 1. Moreover, show that if ui  is concave then u (x i ) < u (v i (xi , 1)).

3. Consider a state space  S  =  {s1 , s2 , s3 , s4 , s5 }.   There are two agents, i = 1, 2, with a common prior p = (0.3, 0.1, 0.2, 0.3, 0.1). Agent 1 has information partition Π 1  = {{s1 , s2 }, {s3 }, {s4 , s5 }} and 2 has information partition Π2 = {{s3 , s2 }, {s4 }, {s1 , s5 }}.

(a) Given common prior p and information structures Π 1 , Π2 , derive types t1 (s), t2 (s), for each s e S .

(b) Given the types t1  and t2 that you have derived in the previous question, find probability distribution p1  e ∆S that is a prior for agent 1 and p2   e ∆S that is a prior for agent 2, where p1 , p2     p.   Is there a common prior, different from p, that assigns positive probability to all states?

(c) Suppose that 1’s prior is p1   = (0.2, 0.1, 0.3, 0.2, 0.2) and 2’s prior is p2 = (0.3, 0.1, 0.2, 0.2, 0.2). Find an ex ante bet and an interim bet.

4. Show that in a bargaining game of alternating offers one can find, for every (x1 , x2 ) e X = {(x1 , x2 ) e R  : x1 + x2  < 1}, a Nash

equilibrium where player one gets x1  in period 0.  Explain why the Nash equilibrium you construct may not be subgame perfect.

5. Consider a state space S and two agents, 1 and 2, with information partitions Π 1 and Π2 , respectively. Describe what are types. Define what is interim betting among the two agents.  Show that there is a common prior if and only if there is no interim betting.