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Game Theory - W4415 Solutions to the Midterm Examination - Fall 2000 Levent Koçkesen

1. (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 > 1, 3p > 1 and p + q · 1: Therefore, the mixed strategies that strictly dominate action M are such that q > 1=3; p > 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 < f1g ; if q < 1=2 [0; 1] ; if q = 1=2 ; B1 (q) = : f0g ; if q > 1=2 8 < f1g ; if p < 1=2 [0; 1] ; if p = 1=2 : B2 (p) = : f0g ; if p > 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 < :

¡ti ; 1 vi ¡ ti ; 2 vi ¡ tj ;

if ti < tj if ti = tj ; if ti > tj

where 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 > 1 vi ¡ t: So, there is no Nash equilibrium where t1 = t2 : 2 (c) By part (b) either t1 > t2 > 0 or t2 > t1 > 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 > 0 or t2 = 0 and t1 > 0 in all Nash equilibria. Also observe that there is no Nash equilibrium if which 0 = ti < tj < 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 < if tj < vi ti > tj ; f0g or ti > tj ; if tj = vi ; : f0g if tj > vi

where 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 ) =

½

xi (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 > 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 solving

xi 2[0;1]

1 ¡ x2 : 2

max 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+1

at Nash equilibrium for all i = 1; : : : ; n:

2

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