`Game Theory - W4415 Solutions to the Midterm Examination - Fall 2000 Levent Koçkesen1. (a) Let ®1 (T ) = p and ®1 (B) = q: The mixed strategy where p = q = 1=2 gives an expected payo¤ of 3=2 whereas the action M gives a payo¤ of 1 irrespective of what player 2's action is. (b) One needs 3q &gt; 1, 3p &gt; 1 and p + q · 1: Therefore, the mixed strategies that strictly dominate action M are such that q &gt; 1=3; p &gt; 1=3; and p + q · 1: (c) Since the action M is strictly dominated it is not used with positive probability in any Nash equilibrium. So let ®1 (T ) = p and ®2 (L) = q: Best response correspondences are given by 8 &lt; f1g ; if q &lt; 1=2 [0; 1] ; if q = 1=2 ; B1 (q) = : f0g ; if q &gt; 1=2 8 &lt; f1g ; if p &lt; 1=2 [0; 1] ; if p = 1=2 : B2 (p) = : f0g ; if p &gt; 1=2 Plotting shows that the pure Nash equilibria are (B; L) and (T; R) whereas the mixed strategy equilibrium is p = q = 1=2: ² N = f1; 2g ² Ai = R+ ² ui (t1 ; t2 ) = 2. (a)8 &lt; :¡ti ; 1 vi ¡ ti ; 2 vi ¡ tj ;if ti &lt; tj if ti = tj ; if ti &gt; tjwhere j 6= i: (b) Suppose t1 = t2 = t: In this case each player gets 1 vi ¡ t: By conceding later than the other player 2 it can get vi ¡ t &gt; 1 vi ¡ t: So, there is no Nash equilibrium where t1 = t2 : 2 (c) By part (b) either t1 &gt; t2 &gt; 0 or t2 &gt; t1 &gt; 0: In either case one of the players receives a negative payo¤ and can pro...tably deviate by conceding immediately which gives it a payo¤ of zero. i. Direct argument: From parts (b) and (c) we know that either t1 = 0 and t2 &gt; 0 or t2 = 0 and t1 &gt; 0 in all Nash equilibria. Also observe that there is no Nash equilibrium if which 0 = ti &lt; tj &lt; vi as player i can increase its payo¤ by conceding slightly later than player j: Therefore, in all Nash equilibria we have either t1 = 0 and t2 ¸ v1 or t2 = 0 and t1 ¸ v2 :Let's con...rm that all such strategy pro...les are Nash equilibria. In the ...rst type of equilibrium player 1 receives a payo¤ of zero. If it deviates and concedes before player 2, it gets a negative payo¤. If it concedes at the same time with or after player 2 then it receives a nonpositive payo¤. So, it is best responding. Player 2's equilibrium payo¤ is v2 and the only other action that gives it a di¤erent payo¤ is to concede immediately which yields a payo¤ of 1 v2 : So, it 2 is best responding as well. 1ii. Best response correspondences: Bi (tj ) = 8 &lt; if tj &lt; vi ti &gt; tj ; f0g or ti &gt; tj ; if tj = vi ; : f0g if tj &gt; viwhere j 6= i: Plotting gives the Nash equilibria as found above. 3. (a) ² N = f1; 2g ² Ai = [0; 1] ² ui (x1 ; x2 ) =½xi (1 ¡ x1 ¡ x2 ) ; if x1 + x2 · 1 : 0; otherwise(b) If the solution occurs in the interior then the ...rst order conditions must hold: 1 ¡ 2x1 ¡ x2 = 0 or x1 = Similarly, in the interior, 1 ¡ x1 : 2 It is easily checked that the second order conditions hold. Therefore, the best response correspondences are given by ½ 0; if xj &gt; 1 : Bi (xj ) = 1¡xj ; if xj · 1 2 x2 = (c) Plotting the best response correspondences yields x1 = x2 = 1=3 as the unique Nash equilibrium. (d) The pro...le x1 = x2 = 1=4 does better than the Nash equilibrium pro...le. (e) Nash equilibrium can be found by solvingxi 2[0;1]1 ¡ x2 : 2max xi [1 ¡ (x1 + : : : + xn )]for each player i = 1; : : : ; n: The ...rst order conditions are Pn 1 ¡ j=1 xj xi = : 2 This implies that xi = x =1 n+1at Nash equilibrium for all i = 1; : : : ; n:2`

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