Chapter 4 Mixed aTwo kind of strategies: `pure Mixed Strategies `mixed aTwo kinds of equilibrium and `pure strategy Mixed Strategy `mixed strategy aTwo games with mixed strategy equilibria: Equilibrium `Matching Pennies `Market Niche 1 2

Matching Pennies: The payoff matrix Matching Pennies: No equilibrium in (All payoffs in cents) pure strategies

All Best Responses are underlined. Player 2 Player 2 Player 1 Heads Tails Player 1 Heads Tails

Heads Heads +1, -1 -1, +1 +1, -1 -1, +1

Tails -1, +1 +1, -1 Tails -1, +1 +1, -1

3 4

Computing Mixed Strategy Matching Pennies: Equilibria in 2×2 Games What about mixed strategies? aSolution criterion: each pure strategy in a probability y 1- y

mixed strategy equilibrium pays the same 1 2 ht at equilibrium probability aEach pure strategy not in a mixed strategy x H +1, -1 -1, +1 equilibrium pays less aDetailed calculations for Matching Pennies 1- x T -1, +1 +1, -1 and Market Niche aAn appealing condition on equilibria: x, y between 0 and 1 payoff dominance 5 That is, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 6 Need to calculate player 1’s expected Need to calculate player 2’s expected utility from player 2’s mixed strategy utility from player 1’s mixed strategy

probability y 1- y 2 1 1 2 ht htEU1: probability x H +1, -1 -1, +1 H +1, -1 -1, +1 2y - 1

1- x T -1, +1 +1, -1 T -1, +1 +1, -1 1 - 2y

EU2: 1 - 2x 2x - 1 EU1(H) = y × 1 + (1- y) × -1 = 2y - 1 EU2(h) = x × -1 + (1- x) × 1 = 1 - 2x

EU1(T) = y × -1 + (1- y) × 1 = 1 - 2y 7 EU2(t) = x × 1 + (1- x) × -1 = 2x - 1 8

In equilibrium, Player 1 is willing to randomize Similarly, Player 2 is willing to randomize only when he is indifferent between H and T only when she is indifferent between h and t

Player 1’s Conditions: EU1(H) = y × 1 + (1- y) × -1 = 2y - 1 EU1(H) = EU1(T) EU1(T) = y × -1 + (1- y) × 1 = 1 - 2y Player 2’s Conditions: In equilibrium: EU1(H) = EU1(T) EU2(h) = x × -1 + (1- x) × 1 = 1 - 2x ∴ 2y - 1 = 1 - 2y EU2(t) = x × 1 + (1- x) × -1 = 2x - 1 ⇒ 4y = 2 In equilibrium: EU2(h) = EU2(t) ⇒ y = ½ ∴ 1 - 2x = 2x - 1 ⇒ 1 - y = 1 - ½ = ½ ⇒ x = ½ and 1 - x = 1 - ½ = ½ ∴ y = 1 - y = ½ 9 ∴ x = 1 - x = ½ 10

Matching Pennies: Mixed strategies are not intuitive: Equilibrium in mixed strategies You randomize to make me indifferent.

probability ½ ½ Row randomizes to make Column 1 2 htEU : indifferent. probability 1

½ H +1, -1 -1, +1 0 Column randomizes to make Row || indifferent. ½ T -1, +1 +1, -1 0 Then each is playing a to the other. EU2: 0 = 0 Each is playing a best response to the other! 11 12 Market Niche: Two pure strategy Market Niche: The payoff matrix equilibria

Mutual best responses form an equilibrium. Firm 2 Firm 2 Firm 1 Enter Stay Out Firm 1 Enter Stay Out

Enter Enter -50, -50 100, 0 -50, -50 100, 0

Stay Out 0, 100 0, 0 Stay Out 0, 100 0, 0

13 14

Market Niche: Need to calculate firm 1’s expected What about mixed strategies? utility from firm 2’s mixed strategy

probability y 1- y probability y 1- y

1 2 1 2 es esEU1: probability

x E -50, -50 100, 0 E -50, -50 100, 0 100 - 150y

S 1- x S 0, 100 0, 0 0, 100 0, 0 0

x, y between 0 and 1 EU1(E) = y × -50 + (1- y) × 100 = 100 - 150y EU (S) = y × 0 + (1- y) × 0 = 0 That is, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 15 1 16

Need to calculate firm 2’s expected In equilibrium, Firm 1 is willing to randomize utility from firm 1’s mixed strategy only when it is indifferent between E and S

1 2 EU1(E) = y × -50 + (1- y) × 100 = 100 - 150y es EU (S) = y × 0 + (1- y) × 0 = 0 probability 1

x E -50, -50 100, 0 In equilibrium: EU1(E) = EU1(S) ∴ 100 - 150y = 0 1- x ⇒ 150y = 100 S 0, 100 0, 0 ⇒ y = 2/3

EU2: 100-150x 0 ⇒ 1 - y = 1 - 2/3 = 1/3 EU (e) = x × -50 + (1- x) × 100 = 100 - 150x 2 ∴ y = 2/3 and 1 - y = 1/3 EU2(s) = x × 0 + (1- x) × 0 = 0 17 18 Similarly, Firm 2 is willing to randomize only Market Niche: when it is indifferent between h and t Equilibrium in mixed strategies

Firm 1’s Conditions: probability 2/3 1/3 EU1(E) = EU1(S) 1 2 esEU : Firm 2’s Conditions: probability 1

EU2(e) = x × -50 + (1- x) × 100 = 100 - 150x E -50, -50 100, 0 0 EU2(s) = x × 0 + (1- x) × 0 = 0 2/3 || In equilibrium: EU2(e) = EU2(s) 1/3 S 0, 100 0, 0 0 ∴ 100 - 150x = 0 ⇒ 150x = 100 EU2: 0 = 0 Each firm is playing a best response to the other! ∴ x = 2/3 and 1 - x = 1/3 19 20

Mixed Strategies and bluffing: Liar’s Poker Liar’s Poker: extensive form aMixed strategies as a way to be Call 1, -1 unpredictable Says ace 0.5, -0.5 aBluffing and mixed strategies Fold Ace 2 Call -1, 1 1/2 aLiar’s poker, a game where bluffing pays 1 Says ace Fold 0.5, -0.5 0 1/2 King 1 Says king 0, 0

21 22

Liar’s Poker: No pure strategy Liar’s Poker: normal form equilibrium

1 2 1 2 Call Fold Call Fold

Say A when K 0, 0 0.5, -0.5 Say A when K 0, 0 0.5, -0.5

Say K when K 0.5, -0.5 0.25, -0.25 Say K when K 0.5, -0.5 0.25, -0.25

23 24 Liar’s Poker: Each player calculates his expected What about mixed strategies? utility from other’s mixed strategy

probability y 1- y probability y 1- y

1 2 1 2 cf cfEU1: probability probability

x A when K 0, 0 0.5, -0.5 x A when K 0, 0 0.5, -0.5 0.5 - 0.5y

K when K 1- x K when K 0.5, -0.5 0.25, -0.25 1- x 0.5, -0.5 0.25, -0.25 0.25 + 0.25y

x, y between 0 and 1 EU2: 0.5x - 0.5 -0.25x - 0.25

That is, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 25 26

In equilibrium, player 1 is willing to randomize Similarly, Player 2 is willing to randomize only when he is indifferent between A and K only when she is indifferent between c and f

Player 1’s Conditions: EU1(A) = y × 0 + (1- y) × 0.5 = 0.5 - 0.5y EU1(A) = EU1(K) EU1(K) = y × 0.5 + (1- y) × 0.25 = 0.25 + 0.25y Player 2’s Conditions: In equilibrium: EU1(A) = EU1(K) EU2(c) = x × 0 + (1- x) × -0.5 = 0.5x - 0.5 ∴ 0.5 - 0.5y = 0.25 + 0.25y EU2(f) = x × -0.5 + (1- x) × -0.25 = -0.25x - 0.25 ⇒ 0.75y = 0.25 In equilibrium: EU2(c) = EU2(f) ⇒ y = 1/3 ∴ 0.5x - 0.5 = -0.25x - 0.25 ⇒ 1 - y = 1 - 1/3 = 2/3 ⇒ 0.75x = 0.25 ∴ y = 1/3 and 1 - y = 2/3 27 ∴ x = 1/3 and 1 - x = 2/3 28

Liar’s Poker: Mixed Strategy Equilibria of Coordination Games and Coordination Equilibrium in mixed strategies Problems

probability 1/3 2/3 aGames with mixed strategy equilibria 1 2 which cannot be detected by the arrow cfEU : probability 1 diagram 1/3 A 0, 0 0.5, -0.5 1/3 aThe mixed strategy equilibrium of Video || System Coordination is not efficient

2/3 K 0.5, -0.5 0.25, -0.25 1/3

EU2: -1/3 = -1/3 Each player is playing a best response to the other! 29 30 Asymmetric Mixed Strategy Equilibria aMixed strategy Nash equilibria tend to aMaking a game asymmetric often makes have low efficiency its mixed strategy equilibrium asymmetric aCorrelated equilibria aAsymmetric Market Niche is an example `public signal ` in game that follows

31 32

Asymmetrical Market Niche: Asymmetrical Market Niche: The payoff matrix Two pure strategy equilibria

Firm 2 Firm 2 Firm 1 Enter Stay Out Firm 1 Enter Stay Out

Enter Enter -50, -50 150, 0 -50, -50 150, 0

Stay Out 0, 100 0, 0 Stay Out 0, 100 0, 0

33 34

Asymmetrical Market Niche: Need to calculate each firm’s expected What about mixed strategies? utility from the firm’s mixed strategy

probability y 1- y probability y 1- y

1 2 1 2 es esEU1: probability probability x x E -50, -50 150, 0 E -50, -50 150, 0 150 - 200y

1- x S 1- x S 0, 100 0, 0 0, 100 0, 0 0

x, y between 0 and 1 EU2: 100 - 150x 0

That is, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 35 36 In equilibrium, Firm 1 is willing to randomize Similarly, Firm 2 is willing to randomize only only when it is indifferent between E and S when it is indifferent between h and t

EU1(E) = y × -50 + (1- y) × 150 = 150 - 200y Firm 1’s Conditions: EU1(S) = y × 0 + (1- y) × 0 = 0 EU1(E) = EU1(S)

In equilibrium: EU1(E) = EU1(S) Firm 2’s Conditions: EU (e) = x × -50 + (1- x) × 100 = 100 - 150x ∴ 150 - 200y = 0 2 EU2(s) = x × 0 + (1- x) × 0 = 0 ⇒ 200y = 150 In equilibrium: EU (e) = EU (s) ⇒ y = 3/4 2 2 ⇒ 1 - y = 1- 3/4 = 1/4 ∴ 100 - 150x = 0 ⇒ 150x = 100 ∴ y = 3/4 and 1 - y = 1/4

37 ∴ x = 2/3 and 1 - x = 1/3 38

Asymmetrical Market Niche: Asymmetrical Market Niche: Equilibrium in mixed strategies Equilibrium in mixed strategies

probability 3/4 1/4 Although the two pure strategy 1 2 esEU : probability 1 equilibria (E,s) and (S,e) did not change in Asymmetrical Market 2/3 E -50, -50 150, 0 0 Niche, the mixed strategies || equilibrium did change. 1/3 S 0, 100 0, 0 0

EU2: 0 = 0 Each firm is playing a best response to the other! 39 40

Chicken: Chicken The payoff matrix

aTwo drivers race toward a cliff player 2 drive straight aStrategy choice: player 1 ahead swerve `swerve drive straight `straight ahead -10, -10 1, -1 ahead aMore general version of the game: `back down `do not back down swerve -1, 1 0, 0 aSolution as in Market Niche Game

41 42 Chicken: Chicken: strategy for player 1 strategy for player 2

player 2 drive straight player 2 drive straight player 1 ahead swerve player 1 ahead swerve

drive straight drive straight -10, -10 1, -1 -10, -10 1, -1 ahead ahead

swerve -1, 1 0, 0 swerve -1, 1 0, 0

43 44

Chicken: Chicken: two pure strategy Nash equilibria The payoff matrix

player 2 player 2 probability drive straight y 1- y player 1 ahead swerve player 1 straight swerve probability drive straight straight -10, -10 1, -1 -10, -10 1, -1 x ahead

1- x swerve -1, 1 0, 0 swerve -1, 1 0, 0 x, y between 0 and 1 That is, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 45 46

Chicken: In equilibrium, player 1 is willing to randomize only The payoff matrix when she is indifferent between “swerve” and “straight”

player 2 probability y 1- y EU1(straight) = y × (-10) + (1-y) × 1 = 1 – 11y EU player 1 straight swerve 1 EU1(swerve) = y × (-1) + (1- y) × 0 = -y probability In equilibrium: EU1(swerve) = EU1(straight) straight -10, -10 1, -1 1 – 11y x ∴ 1 – 11y = - y ⇒ 1 = 10y 1- x swerve -1, 1 0, 0 -y ⇒ y = 1/10 ⇒ 1 - y = 1 - 1/10 = 9/3 ∴ y = 1/10 and 1 - y = 9/10 EU2 1 – 11x -x 47 48 Similarly, player 2 is willing to randomize only when Chicken: he is indifferent between “swerve” and “straight” The payoff matrix

Player 1’s Conditions: player 2 probability 1/10 9/10 EU1(swerve) = EU1(straight) EU player 1 straight swerve 1 Player 2’s Conditions: probability -0.1 EU2(straight) = x × (-10) + (1-x) × 1 = 1 – 11x 1/10 straight -10, -10 1, -1 EU2(swerve) = x × (-1) + (1- x) × 0 = -x

In equilibrium: EU2(swerve) = EU2(straight) 9/10 swerve -1, 1 0, 0 -0.1 ∴ 1 – 11x = - x ⇒ x = 1/10

∴ x = 1/10 and 1 - x = 9/10 EU2 -0.1 -0.1 49 50

Everyday Low Pricing: Everyday Low Prices The payoff matrix

Retailer 2 Normal Sale aSales are mixed strategies price price Retailer 1 aSears’ marketing campaign to do away np sp with sales, called Everyday Low Prices aTwo types of buyers: NP 7500, 7500 7500, 8500 `informed `uninformed aA mixed strategy equilibrium tells how SP 8500, 7500 5500, 5500 often to run sales

51 52

Everyday Low Pricing: Everyday Low pricing: Two pure strategy equilibria What about mixed strategies?

Retailer 2 probability y 1- y Retailer 1 np sp 1 2 np sp probability

NP 7500, 7500 7500, 8500 x NP 7500, 7500 7500, 8500

1- x SP 8500, 7500 5500, 5500 SP 8500, 7500 5500, 5500 x, y between 0 and 1

53 That is, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 54 Each retailer calculates its expected In equilibrium, Retailer 1 is willing to randomize utility from other’s mixed strategy only when it is indifferent between NP and SP

probability y 1- y EU1(NP) = y × 7500 + (1- y) × 7500 = 7500 EU (SP) = y × 8500 + (1- y) × 5500 = 3000y + 5500 1 2 EU 1 np sp 1 probability In equilibrium: EU1(NP) = EU1(SP) x NP 7500, 7500 7500, 8500 7500 ∴ 7500 = 3000y + 5500 ⇒ 3000y = 2000 1- x SP 8500, 7500 5500, 5500 3000y + 5500 ⇒ y = 2/3 ⇒ 1 - y = 1 - 2/3 = 1/3 EU : 7500 3000x + 5500 2 ∴ y = 2/3 and 1 - y = 1/3 55 56

Similarly, Retailer 2 is willing to randomize Everyday Low Pricing: only when it is indifferent between c and f Equilibrium in mixed strategies

Retailer 1’s Conditions: probability

EU1(SP) = EU1(NP) 2/3 1/3 1 2 np sp EU : Retailer 2’s Conditions: probability 1

EU2(np) = x × 7500 + (1- x) × 7500 = 7500 NP 7500, 7500 7500, 8500 7500 EU2(sp) = x × 8500 + (1- x) × 5500 = 3000x + 5500 2/3 || In equilibrium: EU (np) = EU (sp) 2 2 1/3 SP 8500, 7500 5500, 5500 7500 ∴ 7500 = 3000x + 5500 ⇒ 3000x = 200 EU2: 7500 = 7500 Each player is playing a best response to the other! ∴ x = 2/3 and 1 - x = 1/3 57 58

Mixed strategies are not intuitive: Appendix: Bluffing in 1-card You randomize to make me indifferent. Stud Poker aA version of poker with 3 kinds of cards R Row randomizes to make Column (ace, king, and queen), 1-card hands, and E indifferent. players who see their cards aFor some ratios of the ante to the bet, 1- M Column randomizes to make Row I card stud poker has a unique equilibrium N indifferent. which is in mixed strategies D Then each is playing a best aEquilibrium play in poker usually calls for E response to the other. some bluffing R aThe solution of poker has all players 59 breaking even 60 One-card Stud Poker One-card Stud Poker. Payoff matrix, player 1 Payoff matrix, player 1, a=$1, b=$1

Player 2 Player 2 Player 1 I: Bet AKQ II: Bet AK III: Bet AQ IV: Bet A Player 1 I: Bet AKQ II: Bet AK III: Bet AQ IV: Bet A (a-2b)/9, (4a-2b)/9, I: Bet AKQ 0, 0 3a/9, -3a/9 I: Bet AKQ 0, 0 -1/9, 1/9 3/9, -3/9 2/9, -2/9 (2b-a)/9 (2b-4a)/9

(2b-a)/9, (2a-b)/9, II: Bet AK 0, 0 (a+b)/9, II: Bet AK 1/9, -1/9 0, 0 2/9, -2/9 1/9, -1/9 (a-2b)/9 -(a+b)/9 (b-2a)/9

-(a+b)/9, (2a-b)/9, III: Bet AQ -3a/9, 3a/9 0, 0 III: Bet AQ -3/9, 3/9 -2/9, 2/9 0, 0 1/9, -1/9 (a+b)/9 (b-2a)/9

(2b-4a)/9, (b-2a)/9, (b-2a)/9, IV: Bet A IV: Bet A -2/9, 2/9 -1/9, 1/9 -1/9, 1/9 (4a-2b)/9 (2a-b)/9 (2a-b)/9 0, 0 0, 0 61 62

One-card Stud Poker. Payoff matrix, player 1, a=$1 b=$2

Player 2 Player 1 I: Bet AKQ II: Bet AK III: Bet AQ IV: Bet A

I: Bet AKQ 0, 0 -3/9, 3/9 3/9, -3/9 0, 0

II: Bet AK 3/9, -3/9 0, 0 3/9, -3/9 0, 0

III: Bet AQ -3/9, 3/9 -3/9, 3/9 0, 0 0, 0

IV: Bet A 0, 0 0, 0 0, 0 0, 0 63