2 Take any \u03c3 NE \u03b3 and suppose that there is someG a \u03b3 such that \u03b3 G 6 NE G if

# 2 take any σ ne γ and suppose that there is someg a

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2
Take anyσNE(γ), and suppose that there is someGa(γ) such thatγ|G6∈NE(G) (ifnot, thenσis a SPNE and we are done). Now pick any ˜σNE(γ) and create a strategyprofileσ0such thatσ0|h=(˜σif h=Gσ|hif h6 3. Consider the following 3 player game. Player 1 announces a numbera1[0,1]. After this,player 2 observesa1and then announces a numbera2[0,1]. Ifa1+a21, then thegame ends, and the players’ payoffs are given byu1(a1, a2) =a1,u2(a1, a2) =a2,u3(a1, a2) = 1-a1-a2.Ifa1+a2>1, however, player 3 is allowed to choose either player 1 or player 2 as acoalition partner. If player 3 chooses player 1, then the payoffs areu1(a1, a2) =a1,u2(a1, a2) = 0,u3(a1, a2) = 1-a1.If player 3 chooses player 2 as a partner, the payoffs areu1(a1, a2) = 0,u2(a1, a2) =a2,u3(a1, a2) = 1-a2.Describe the set of subgame perfect Nash equilibria to this game.

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• Spring '11