Osborne 117.2 (a). (The question is scanned on two different pages. See the second part of the question on the page with Ex 118.1.)

we can state it more precisely as follows. d strategy profile a is a weighted aver- L(0) C() R(3) d strategy profiles of the type (a;, a_), T () .2 3,3 1, 1 is the probability a; (a;) assigned to (116.1) M (0) 0, . 2, . B (4) ., 4 5,1 0,7 Figure 117.1 A partially specified strategic game, illustrating a method of checking whether a mixed strategy profile is a mixed strategy Nash equilibrium. The dots indicate irrelevant payoffs. a, (a; ) E, (a;, a_ ), to check whether a mixed strategy profile is an equilibrium. For example, in Bos ure strategies) and E;(a;, a_;) is (as before) (Section 4.3.3) the strategy pair ((3, 3), (3, 3)) is a mixed strategy Nash equilibrium pure strategy that assigns probability 1 to because given player 2's strategy (3, 3), player 1's expected payoffs to B and S are xed strategy aj. (See the end of Section 17.2 both equal to , and given player 1's strategy (3, 3), player 2's expected payoffs to explanation of the _ notation.) B and S are both equal to 3. racterization of mixed strategy Nash equi- The next example is slightly more complicated. h equilibrium and denote by Ef player i's EXAMPLE 117.1 (Checking whether a mixed strategy profile is a mixed strategy ; = Uj(a*)). Because a" is an equilibrium, Nash equilibrium) I claim that for the game in Figure 117.1 (in which the dots all her strategies, including all her pure indicate irrelevant payoffs), the indicated pair of strategies, (,0, ) for player 1 E; is a weighted average of player i's ex- and (0, 3, 3) for player 2, is a mixed strategy Nash equilibrium. To verify this hich a; assigns positive probability. Thus claim, it suffices, by Proposition 116.2, to study each player's expected payoffs to strategies are all equal to Et. (If any were her three pure strategies. For player 1 these payoffs are ld be smaller.) We conclude that the ex- assigns positive probability is Er, and the T : 3 . 3+ 3 . 1 =3 t most E' . Conversely, if these conditions M : J . 0+ 3 .2 =3 a mixed strategy Nash equilibrium: the ted payoff to any other mixed strategy is B : 3 . 5 + 3 . 0 = 3. hted average of E; and numbers that are Player I's mixed strategy assigns positive probability to T and B and probability g result. zero to M, so the two conditions in Proposition 116.2 are satisfied for player 1. The expected payoff to each of player 2's pure strategies is ? (4 . 2 + 4 . 4 = ) . 3 + 4 . nixed strategy Nash equilibrium of finite 1 = 1. 1+ .7 = 2), so the two conditions in Proposition 116.2 are satisfied also egic game with UNM preferences in which for her. ed strategy Nash equilibrium if and only if, Note that the expected payoff to player 2's action L, which she uses with prob- ability zero, is the same as the expected payoff to her other two actions. This equal- ity is consistent with Proposition 116.2, the second part of which requires only that ion to which at assigns positive probability the expected payoffs to actions used with probability zero be no greater than the ex- pected payoffs to actions used with positive probability (not that they necessarily ion to which at assigns zero probability is be less). Note also that the fact that player 2's expected payoff to L is the same as which at assigns positive probability. her expected payoffs to C and R does not imply that the game has a mixed strategy is her expected payoff to any of her actions Nash equilibrium in which player 2 uses L with positive probability-it may, or it may not, depending on the unspecified payoffs. gives conditions for a mixed strategy EXERCISE 117.2 (Choosing numbers) Players 1 and 2 each choose a positive integer pected payoffs only to her pure strate- up to K. If the players choose the same number, then player 2 pays $1 to player 1; itely many actions, it allows us easily otherwise no payment is made. Each player's preferences are represented by her expected monetary payoff.118 Chapter 4. Mixed Strategy Equilibrium 4.3 Mixed strategy Nash a. Show that the game has a mixed strategy Nash equilibrium in which each player chooses each positive integer up to K with probability 1/K. assigns positive p equilibrium mixed b. (More difficult.) Show that the game has no other mixed strategy Nash equi- positive probability libria. (Deduce from the fact that player 1 assigns positive probability to Any equilibrium some action k that player 2 must do so; then look at the implied restriction than a strict equilil on player 1's equilibrium strategy.) centive to choose EXERCISE 118.1 (Silverman's game) Each of two players chooses a positive inte- There is no reason ger. If player i's integer is greater than player j's integer and less than three times same time there is this integer, then player j pays $1 to player i. If player i's integer is at least three good. Many pure times player j's integer, then player i pays $1 to player j. If the integers are equal, strict, but among m no payment is made. Each player's preferences are represented by her expected Given that in a r monetary payoff. Show that the game has no Nash equilibrium in pure strate- choose her equilibr gies and that the pair of mixed strategies in which each player chooses 1, 2, and 5 rium? From the ex each with probability ? is a mixed strategy Nash equilibrium. (In fact, this pair of we see that a player mixed strategies is the unique mixed strategy Nash equilibrium.) (You cannot ap- other player indiffer peal to Proposition 116.2 because the number of actions of each player is not finite. domize. In the mix However, you can use the argument for the "if" part of this result.) B with probability willing to choose e @ EXERCISE 118.2 (Voter participation) Consider the game of voter participation in is not that the play Exercise 34.2. Assume that 2