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**Extra info for A Course in Game Theory. SOLUTIONS**

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Now let p = −α/β ≥ 0. Since ( i∈N zi∗ − zk∗ + zk , i∈N fi (zi∗ ) − fk (zk∗ ) + fk (zk )) ∈ X for any zk ∈ R+ we have fk (zk∗ ) − pzk∗ ≥ fk (zk ) − pzk for all zk ∈ R+ , so that (p, (zi∗ )i∈N ) is a competitive equilibrium. Comment This is not an exercise in game theory. 3). Now, the concavity of f (1, k) implies that k(f (1, k) − f (1, k − 1)) ≤ 2(f (1, k) − f (1, k/2)) (since k (f (1, j) − f (1, j − 1)) f (1, k) − f (1, k/2) = j=k/2+1 k (f (1, k) − f (1, k − 1)) ≥ j=k/2+1 ≥ (k/2)[f (1, k) − k(1, k − 1)]).

5, which player 2 immediately accepts. 40 Chapter 12. 1 (PBE is SE in Spence’s model ) It is necessary to show only that the assessments are consistent. Consider the pooling equilibrium. Suppose that a type θ1L worker chooses e∗ with probability 1 − and distributes the remaining probability over other actions, while a type θ1H worker chooses e∗ with probability 1 − 2 and distributes the remaining probability 2 over other actions. The employer’s belief that these completely mixed strategies induce converges to the one in the equilibrium as → 0, so that the equilibrium assessment is indeed consistent.

Then player i’s expected payoff conditional on reaching Ii is the sum of his payoffs to histories that do not reach another of his information sets, say Ei , and O(β , µ|Ii )(h) · ui (h). Ii ∈F (Ii ) h∈Z(Ii ) This is equal, using the equality in the first part of the problem, to O(β , µ|Ii )(h) · Pr(β , µ|Ii )(Ii ) · ui (h), Ei + Ii ∈F (Ii ) h∈Z(Ii ) which is equal to Ei + Pr(β , µ|Ii )(Ii ) · E(β ,µ) [ui |Ii ], Ii ∈F (Ii ) where E(β ,µ) [ui |Ii ] is the expected payoff under (β , µ) conditional on reaching Ii , which by the induction assumption is at most Ei + Pr(β , µ|Ii )(Ii ) · E(β,µ) [ui |Ii ].