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Game Theory Mixed Strategy Study Material
Typology: Study notes
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Mixed Strategy, Pure Strategy Nash Equilibrium, Mixed Strategy Nash Equilibrium, Constant Sum Games
June 2016
reconsider ‘Matching Pennies’
Row
Column Head Tail Head 1 , − 1 − 1 , 1 Tail − 1 , 1 1 , − 1 Matching Pennies
constant sum game (game of pure conflict)
important for players not to play in a predictable way
→ no Nash equilibrium in pure strategies
→ players randomize over their actions, they play mixed strategies
suppose p = probability Column plays Heads → 1 − p = probability Column plays Tails
suppose q = probability Row plays Heads → 1 − q = probability Row plays Tails
to find best-responses, compute players expected payoffs Column Heads (prob p)
Tails (prob 1 − p)
Row’s exp payoff Row Heads(prob q) 1 , − 1 − 1 , 1 2 p − 1 Tails(prob 1 − q) − 1 , 1 1 , − 1 1 − 2 p Column’s exp. payoff 1 − 2 q −1 + 2q
if 2 p − 1 > 0 Row is better off playing Heads if 2 p − 1 < 0 Row is better off playing Tails
if 2 p − 1 = 0, Row is equally well off playing Heads, Tails, or flipping a coin
→ Row is willing to randomize only if Column plays Heads with p = 1/ 2 ; otherwise, Row plays either Heads or Tails
randomization requires equality of expected payoffs
Nash equilibrium: Row’s choice of q is best response to Column’s choice of p and vice versa
Note: pure strategies can be seen as special cases of mixed strategies where p ∈ { 0 , 1 } and q ∈ { 0 , 1 }
in constant sum games, keeping your opponent indifferent means that he/she cannot recognize and exploit systematic patters in your behavior keeping your opponent indifferent is equivalent to keeping yourself indifferent same principle applies to non-constant sum games (where players’ interests are not totally opposed to each other)
two teenagers drive toward each other at a high rate of speed, the driver that swerves first is deemed a chicken and loses face with the rest of the crowd
James
Dean Swerve Don’t Swerve 0 , 0 − 1 , 1 Don’t 1 , − 1 − 4 , − 4 Chicken (Swerve,Don’t) and (Don’t, Swerve) are NE in pure strategies, game also has a mixed strategy NE!
let p = prob Dean swerves, q = prob James swerves
in mixed strategy NE, p∗^ = q∗^ = 3/ 4 → 1 / 16 = prob of a crash
compute players expected payoffs
Dean Swerve (prob p)
Dont’ (prob 1 − p)
James’ exp payoff James Swerve(prob q) 0 , 0 − 1 , 1 p − 1 Don’t(prob 1 − q) 1 , − 1 − 4 , − 4 5 p − 4 Dean’s exp. payoff q^ −^1 5 q^ −^4
