Mixed Strategies and Mixed Strategy Equilibrium

Size: px

Start display at page:

Download "Mixed Strategies and Mixed Strategy Equilibrium"

Stella Norton
7 years ago
Views:

1 Chapter 4 Mied trategies and Mied trateg quilibrium Mied trateg Two kind of strategies: pure mied Two kinds of equilibrium pure strateg mied strateg Two games with mied strateg equilibria: Matching Pennies Market Niche Matching Pennies: The paoff matri (All paoffs in cents) Matching Pennies: No equilibrium in pure strategies Plaer Plaer Heads Tails Plaer Plaer All Best Responses are underlined. Heads Tails Heads +, - -, + Heads +, - -, + Tails -, + +, - Tails -, + +, Computing Mied trateg quilibria in Games olution criterion: each pure strateg in a mied strateg equilibrium pas the same at equilibrium ach pure strateg not in a mied strateg equilibrium pas less Detailed calculations for Matching Pennies and Market Niche An appealing condition on equilibria: paoff dominance 5 Matching Pennies: What about mied strategies? - H T h +, - -, + - t -, + +, -, between and That is, and 6

Heads Tails Heads +, - -, + Heads +, - -, + Tails -, + +, - Tails -, + +, - 3 4 Computing Mied trateg quilibria in Games olution criterion: each pure strateg in a mied strateg equilibrium pas the

2 Need to calculate plaer s epected utilit from plaer s mied strateg Need to calculate plaer s epected utilit from plaer s mied strateg h - t U : h t H +, - -, + - H +, - -, + T -, + +, T -, + +, - U (H) = + (- ) - = - U (T) = - + (- ) = - 7 U : - - U (h) = - + (- ) = - U (t) = + (- ) - = - 8 In equilibrium, Plaer is willing to randomize onl when he is indifferent between H and T imilarl, Plaer is willing to randomize onl when she is indifferent between h and t U (H) = + (- ) - = - U (T) = - + (- ) = - Plaer s Conditions: U (H) = U (T) In equilibrium: U (H) = U (T) - = - Plaer s Conditions: U (h) = - + (- ) = - U (t) = + (- ) - = - 4 = = ½ - = - ½ = ½ = - = ½ 9 In equilibrium: U (h) = U (t) - = - = ½ and - = - ½ = ½ = - = ½ Matching Pennies: quilibrium in mied strategies Mied strategies are not intuitive: You randomize to make me indifferent. ½ h ½ t U : Row randomizes to make Column indifferent. ½ H +, - -, + Column randomizes to make Row indifferent. ½ T -, + +, - U : = Then each is plaing a best response to the other. ach is plaing a best response to the other!

Plaer is willing to randomize onl when she is indifferent between h and t U (H) = + (- ) - = - U (T) = - + (- ) = - Plaer s Conditions: U (H) = U (T) In equilibrium: U (H) = U (T) - = - Plaer s

3 Market Niche: The paoff matri Market Niche: Two pure strateg equilibria Firm Firm nter ta Out Mutual best responses form an equilibrium. Firm Firm nter ta Out nter nter ta Out, ta Out, 3 4 Market Niche: What about mied strategies? Need to calculate firm s epected utilit from firm s mied strateg - - e s e s U : - 5 -,,, between and U () = -5 + (- ) = - 5 That is, and 5 U () = + (- ) = 6 Need to calculate firm s epected utilit from firm s mied strateg In equilibrium, Firm is willing to randomize onl when it is indifferent between and e s U () = -5 + (- ) = - 5 U () = + (- ) = In equilibrium: U () = U () - 5 = -, 5 = = /3 U : -5 U (e) = -5 + (- ) = - 5 U (s) = + (- ) = 7 - = - /3 = /3 = /3 and - = /3 8

Need to calculate firm s epected utilit from firm s mied strateg - - e s e s U : - 5 -,,, between and U () = -5 + (- ) = - 5 That is, and 5 U () = + (- ) = 6 Need to calculate

4 imilarl, Firm is willing to randomize onl when it is indifferent between h and t Market Niche: quilibrium in mied strategies Firm s Conditions: U () = U () Firm s Conditions: U (e) = -5 + (- ) = - 5 U (s) = + (- ) = In equilibrium: U (e) = U (s) - 5 = /3 /3 e s /3 /3, U : 5 = = /3 and - = /3 9 U : = ach firm is plaing a best response to the other! Mied trategies and bluffing: Liar s Poker Liar s Poker: etensive form Mied strategies as a wa to be unpredictable Bluffing and mied strategies Liar s poker, a game where bluffing pas Ace / / King as ace as ace as king Call Fold Call Fold, -.5, -.5 -,.5, -.5 Liar s Poker: normal form Liar s Poker: No pure strateg equilibrium Call Fold Call Fold a A when K.5, -.5 a A when K.5, -.5 a K when K.5, -.5.5, -.5 a K when K.5, -.5.5,

Mied trategies and bluffing: Liar s Poker Liar s Poker: etensive form Mied strategies as a wa to be unpredictable Bluffing and mied strategies Liar s poker, a game where bluffing pas Ace / / King

5 Liar s Poker: What about mied strategies? ach plaer calculates his epected utilit from other s mied strateg - - c f c f U : A when K.5, -.5 A when K.5, K when K.5, -.5.5, K when K.5, -.5.5, , between and U : That is, and 5 6 In equilibrium, plaer is willing to randomize onl when he is indifferent between A and K imilarl, Plaer is willing to randomize onl when she is indifferent between c and f U (A) = + (- ).5 = U (K) =.5 + (- ).5 = Plaer s Conditions: U (A) = U (K) In equilibrium: U (A) = U (K) = Plaer s Conditions: U (c) = + (- ) -.5 = U (f) = (- ) -.5 = =.5 = /3 - = - /3 = /3 = /3 and - = /3 7 In equilibrium: U (c) = U (f) = =.5 = /3 and - = /3 8 Liar s Poker: quilibrium in mied strategies Mied trateg quilibria of Coordination Games and Coordination Problems /3 /3 c f /3 /3 A K.5, -.5.5, -.5.5, -.5 U : /3 /3 Games with mied strateg equilibria which cannot be detected b the arrow diagram The mied strateg equilibrium of Video stem Coordination is not efficient U : -/3 = -/3 ach plaer is plaing a best response to the other! 9 3

5 -.5 -.5 That is, and 5 6 In equilibrium, plaer is willing to randomize onl when he is indifferent between A and K imilarl, Plaer is willing to randomize onl when she is indifferent between c and f

6 Correlated quilibrium Mied strateg Nash equilibria tend to have low efficienc Correlated equilibria public signal Nash equilibrium in game that follows Asmmetric Mied trateg quilibria Making a game asmmetric often makes its mied strateg equilibrium asmmetric Asmmetric Market Niche is an eample 3 3 Asmmetrical Market Niche: The paoff matri Asmmetrical Market Niche: Two pure strateg equilibria Firm Firm nter ta Out Firm Firm nter ta Out nter 5 nter 5 ta Out, ta Out, Asmmetrical Market Niche: What about mied strategies? Need to calculate each firm s epected utilit from the firm s mied strateg - - e s e s U : , -,, between and U : - 5 That is, and 35 36

matri Asmmetrical Market Niche: Two pure strateg equilibria Firm Firm nter ta Out Firm Firm nter ta Out nter 5 nter 5 ta Out, ta Out, 33 34 Asmmetrical Market Niche:

7 In equilibrium, Firm is willing to randomize onl when it is indifferent between and imilarl, Firm is willing to randomize onl when it is indifferent between h and t U () = -5 + (- ) 5 = 5 - U () = + (- ) = Firm s Conditions: U () = U () In equilibrium: U () = U () 5 - = = 5 = 3/4 - = - 3/4 = /4 = 3/4 and - = /4 37 Firm s Conditions: U (e) = -5 + (- ) = - 5 U (s) = + (- ) = In equilibrium: U (e) = U (s) - 5 = 5 = = /3 and - = /3 38 Asmmetrical Market Niche: quilibrium in mied strategies Asmmetrical Market Niche: quilibrium in mied strategies 3/4 /4 e s /3 /3, 5 U : Although the two pure strateg equilibria (,s) and (,e) did not change in Asmmetrical Market Niche, the mied strategies equilibrium did change. U : = ach firm is plaing a best response to the other! 39 4 Chicken The paoff matri Two drivers race toward a cliff trateg choice: straight More general version of the game: back down do not back down olution as in Market Niche Game plaer plaer -, - -,, - 4 4

= U (s) - 5 = 5 = = /3 and - = /3 38 Asmmetrical Market Niche: quilibrium in mied strategies Asmmetrical Market Niche: quilibrium in mied strategies 3/4 /4 e s /3 /3, 5 U : Although the two pure

8 strateg for plaer strateg for plaer plaer plaer plaer plaer -, -, - -, -, - -, -, two pure strateg Nash equilibria The paoff matri plaer plaer -, -, - plaer plaer straight straight -, - -, - -, - -,, between and 45 That is, and 46 The paoff matri In equilibrium, plaer is willing to randomize onl when she is indifferent between and straight plaer plaer straight straight -, - -, - U U (straight) = (-) + (-) = U () = (-) + (- ) = - In equilibrium: U () = U (straight) = - - -, - = = / - = - / = 9/3 U - 47 = / and - = 9/ 48

equilibrium, plaer is willing to randomize onl when she is indifferent between and straight plaer plaer straight straight -, - -, -

9 imilarl, plaer is willing to randomize onl when he is indifferent between and straight The paoff matri Plaer s Conditions: U () = U (straight) Plaer s Conditions: U (straight) = (-) + (-) = U () = (-) + (- ) = - plaer / plaer straight / straight -, - 9/, - U -. In equilibrium: U () = U (straight) = - 9/ -, -. = / = / and - = 9/ 49 U verda Low Prices verda Low Pricing: The paoff matri ales are mied strategies ears marketing campaign to do awa with sales, called verda Low Prices Two tpes of buers: informed uninformed A mied strateg equilibrium tells how often to run sales Retailer Retailer NP P Normal price np 75, 75 85, 75 ale price sp 75, 85 55, verda Low Pricing: Two pure strateg equilibria verda Low pricing: What about mied strategies? Retailer Retailer np sp np - sp NP 75, 75 75, 85 NP 75, 75 75, 85 P 85, 75 55, 55 - P 85, 75 55, 55, between and 53 That is, and 54

10 ach retailer calculates its epected utilit from other s mied strateg In equilibrium, Retailer is willing to randomize onl when it is indifferent between NP and P - np sp U NP 75, 75 75, P 85, 75 55, U : U (NP) = 75 + (- ) 75 = 75 U (P) = 85 + (- ) 55 = In equilibrium: U (NP) = U (P) 75 = = = /3 - = - /3 = /3 = /3 and - = /3 56 imilarl, Retailer is willing to randomize onl when it is indifferent between c and f verda Low Pricing: quilibrium in mied strategies Retailer s Conditions: U (P) = U (NP) Retailer s Conditions: U (np) = 75 + (- ) 75 = 75 U (sp) = 85 + (- ) 55 = In equilibrium: U (np) = U (sp) 75 = /3 /3 np sp /3 NP 75, 75 75, 85 /3 P 85, 75 55, 55 U : = = /3 and - = /3 57 U : 75 = 75 ach plaer is plaing a best response to the other! 58 Mied strategies are not intuitive: You randomize to make me indifferent. R M I N D R Row randomizes to make Column indifferent. Column randomizes to make Row indifferent. Then each is plaing a best response to the other. 59 Appendi: Bluffing in -card tud Poker A version of poker with 3 kinds of cards (ace, king, and queen), -card hands, and plaers who see their cards For some ratios of the ante to the bet, - card stud poker has a unique equilibrium which is in mied strategies quilibrium pla in poker usuall calls for some bluffing The solution of poker has all plaers breaking even 6

willing to randomize onl when it is indifferent between c and f verda Low Pricing: quilibrium in mied strategies Retailer s Conditions: U (P) = U (NP) Retailer s Conditions: U (np) = 75 + (- ) 75 =

11 Plaer Plaer I: Bet AKQ One-card tud Poker Paoff matri, plaer I: Bet AKQ II: Bet AK III: Bet AQ IV: Bet A (a-b)/9, (b-a)/9 3a/9, -3a/9 (4a-b)/9, (b-4a)/9 One-card tud Poker. Paoff matri, plaer, a=$, b=$ Plaer Plaer I: Bet AKQ I: Bet AKQ II: Bet AK III: Bet AQ IV: Bet A -/9, /9 3/9, -3/9 /9, -/9 II: Bet AK (b-a)/9, (a-b)/9 (a+b)/9, -(a+b)/9 (a-b)/9, (b-a)/9 II: Bet AK /9, -/9 /9, -/9 /9, -/9 III: Bet AQ -3a/9, 3a/9 -(a+b)/9, (a+b)/9 (a-b)/9, (b-a)/9 III: Bet AQ -3/9, 3/9 -/9, /9 /9, -/9 IV: Bet A (b-4a)/9, (4a-b)/9 (b-a)/9, (a-b)/9 (b-a)/9, (a-b)/9 6 IV: Bet A -/9, /9 -/9, /9 -/9, /9 6 One-card tud Poker. Paoff matri, plaer, a=$ b=$ Plaer Plaer I: Bet AKQ I: Bet AKQ II: Bet AK III: Bet AQ IV: Bet A -3/9, 3/9 3/9, -3/9 II: Bet AK 3/9, -3/9 3/9, -3/9 III: Bet AQ -3/9, 3/9-3/9, 3/9 IV: Bet A 63

-/9 /9, -/9 /9, -/9 III: Bet AQ -3a/9, 3a/9 -(a+b)/9, (a+b)/9 (a-b)/9, (b-a)/9 III: Bet AQ -3/9, 3/9 -/9, /9 /9, -/9 IV: Bet A (b-4a)/9, (4a-b)/9 (b-a)/9, (a-b)/9 (b-a)/9, (a-b)/9 6 IV: Bet A -/9, /9

Poker with a Three Card Deck 1

Poker with a Three Card Deck 1 Poker with a Three Card Deck We start with a three card deck containing one Ace, one King and one Queen. Alice and Bob are each dealt one card at random. There is a pot of $ P (and we assume P 0). Alice