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**Additional resources for A Course in Game Theory. SOLUTIONS**

**Sample text**

Ak ) be a history, let h0 = ∅, and let hr = (a1 , . . , ar ) for 1 ≤ r ≤ k−1. Let R(i) be the set of history lengths of subhistories of h after which player i moves; that is, let R(i) = {r: hr ∈ Ii for some Ii ∈ Ii } and denote by Iir the information set of player i that contains hr when r ∈ R(i). Then Xi (h) = (Iir1 , ar1 +1 , . . , Iir , ar +1 ), where rj is the jth smallest member of R(i) and = |R(i)|. 1 (One-player games and principles of equivalence) 1 Inflation–deflation: The extensive game Γ is equivalent to the extensive game Γ if Γ differs from Γ only in that the player has an information set in Γ that is a union of information sets in Γ .

A∗n ) and hence g(a1 , a∗2 , . . , a∗n ) = g(a1 , a2 , a∗3 , . . , a∗n ). Continuing iteratively we deduce that g(a∗ ) = g(a ) and hence g(a ) = f ( ). 1 (Example of non-DSE implementable rule) Consider a preference profile in which for some outcome a we have x 1 a 1 a∗ for all x ∈ / {a, a∗ }, and for all i = 1 we have a i x for all x. Let 1 be a preference relation in which a 1 x 1 a∗ for all x ∈ / {a, a∗ }. 4 b. But f ( ) = a∗ and f ( −1 , i ) = a, so that 1 is not a dominant strategy for player 1 in G∗ , .

Now suppose that the negligent type offers 5 and the non-negligent type offers 3. Then both offers are accepted and the negligent type would be better off if she offered 3. The only sequential equilibria in which the two types of player 1 make the same offer are as follows. • If p1 (Not ) ≥ 25 then the following assessment is a sequential equilibrium. Both types of player 1 offer the compensation of 3 and player 2 accepts any offer. If the compensation of 3 is offered then player 2 believes that player 1 is not negligent with probability p1 (Not ); if the compensation 5 is offered then player 2 may hold any belief about player 1.