LECTURE 2 MIXED STRATEGY GAME Notes modified from Yongqin Wang@ Fudan University
LECTURE 2 MIXED STRATEGY GAME Notes modified from Yongqin Wang @ Fudan University 1
Matching pennies Player 2 Head Tail Head 1 1 1 Player 1 1 1 1 Head is Player 1's best response to Player 2's strategy tail Tail is player 2's best response to Player 1s strategy Tail Tail is Player 1,'s best response to Player 2's strategy Head Head is Player 2's best response to Player 1's strategy Head Hence, NO Nash equilibrium
Matching pennies -1 , 1 1 , -1 1 , -1 -1 , 1 2 ◼ Head is Player 1’s best response to Player 2’s strategy Tail ◼ Tail is Player 2’s best response to Player 1’s strategy Tail ◼ Tail is Player 1’s best response to Player 2’s strategy Head ◼ Head is Player 2’s best response to Player 1’s strategy Head ➢ Hence, NO Nash equilibrium Player 1 Player 2 Tail Head Tail Head
Solving matching pennies Player 2 Head Tail Head 1 1 1 Player 1 Tail 1 1 1 1-x Randomize your strategies Player 1 chooses Head and Tail with probabilities r and r, respectively Player 2 chooses Head and Tail with probabilities g and 1-q respectively ■ Mixed Strategy Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities
Solving matching pennies Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 3 ◼ Randomize your strategies ➢ Player 1 chooses Head and Tail with probabilities r and 1-r, respectively. ➢ Player 2 chooses Head and Tail with probabilities q and 1-q, respectively. ◼ Mixed Strategy: ➢ Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities. q 1-q r 1-r
M Mixed strategy A mixed strategy of a player is a probability distribution over player's(pure) strategies A mixed strategy for Chris is a probability distribution(p, 1-p) where p is the probability of playing Opera, and 1-p is that probability of playing Prize Fight If p=1 then Chris actually plays Opera. If p=0 then Chris actually plays Prize Fight Battle of sexes Pat Opera Prize Fight Opera(p) 2,1 0 0 Chris Prize Fight(1-p) 0 0 1 2
Mixed strategy Battle of sexes Pat Opera Prize Fight Chris Opera (p) 2 , 1 0 , 0 Prize Fight (1-p) 0 , 0 1 , 2 4 ◼ A mixed strategy of a player is a probability distribution over player’s (pure) strategies. ➢ A mixed strategy for Chris is a probability distribution (p, 1-p), where p is the probability of playing Opera, and 1-p is that probability of playing Prize Fight. ➢ If p=1 then Chris actually plays Opera. If p=0 then Chris actually plays Prize Fight
Solving matching pennies Player 2 Expected Head Tail payoffs Head 1 1 1-2q Player 1 Tail 11q 1 1,11-x2g-1 1-g Player 1's expected payoffs If player 1 chooses Head, -g+(1-q)=1-2g If player 1 chooses Tail, g-(1-9)=2g-1
Solving matching pennies Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 5 ◼ Player 1’s expected payoffs ➢ If Player 1 chooses Head, -q+(1-q)=1-2q ➢ If Player 1 chooses Tail, q-(1-q)=2q-1 q 1-q 1-2q 2q-1 Expected payoffs r 1-r
Solving matching pennies Player 2 Expected Head Tail payoffs Head 1 1 1-2 Player 1 Tail 1 1 1 11-x2q-1 g Player 1s best response B1(q): For qFoq=0.5, indifferent(0≤r≤1) 1/2
Solving matching pennies Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 6 ◼ Player 1’s best response B1(q): ➢ For q0.5, Tail (r=0) ➢ For q=0.5, indifferent (0r1) 1 q r 1 1/2 1/2 q 1-q 1-2q 2q-1 Expected payoffs r 1-r
Solving matching pennies Player 2 Expected Head Tail payoUTs Head 1 1 1 1 1-2q Player 1 Tail 1 1 1,1」1-x2q-1 Expected payoff 2x-1 1-2y Player 2's expected payoffs If Player 2 chooses Head, r-(l-r)=2r-1 If Player 2 chooses Tail, -r+(1-r)=1-2r
Solving matching pennies Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 7 ◼ Player 2’s expected payoffs ➢ If Player 2 chooses Head, r-(1-r)=2r-1 ➢ If Player 2 chooses Tail, -r+(1-r)=1-2r 1-2q 2q-1 Expected payoffs r 1-r Expected q 1-q payoffs 2r-1 1-2r
Solving matching pennies Player 2 Expected Head Tail payoffs Head 1 1-2 1 g Player 1 Tail 1 1 1,11 2q-1 Expected 1-q payor 2x-1 1-2y a Player 2's best response 2 x): For r0. 5, Head (q=1) >For=0.5, indifferent(0≤q≤1) 1/2
Solving matching pennies Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 8 ◼ Player 2’s best response B2(r): ➢ For r0.5, Head (q=1) ➢ For r=0.5, indifferent (0q1) q 1-q 1-2q 2q-1 Expected payoffs r 1-r Expected payoffs 2r-1 1-2r 1 q r 1 1/2 1/2
Solving matching pennies Player 2 Head Player 1's best response B1(q): Playe Head 1 1 1 For g<0.5, Head(r1)1 1 1 1 1-x For q0.5, Tail(r0) Forq=0.5, indifferent(0≤r≤1) 1-q Player 2's best response Mixed strategy B2(x) Nash equilibrium For r<0.5, Tail(go) For r0.5, Head (q=1) Forr=0.5, indifferent(0≤q≤1) v Check 1/2 0.5∈B1(0.5) q=0.5∈B2(0.5) 1/2
Solving matching pennies Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 9 ◼ Player 1’s best response B1(q): ➢ For q0.5, Tail (r=0) ➢ For q=0.5, indifferent (0r1) ◼ Player 2’s best response B2(r): ➢ For r0.5, Head (q=1) ➢ For r=0.5, indifferent (0q1) ✓ Check r = 0.5 B1(0.5) q = 0.5 B2(0.5) 1 q r 1 1/2 1/2 r 1-r q 1-q Mixed strategy Nash equilibrium
Mixed strategy: example ■ Matching pennies Player 1 has two pure strategies: H and T (Oh)=0.5, 01 T=0.5)is a Mixed strategy That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively (O1H)=0.3,010=0.7)is another Mixed strategy That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively 10
Mixed strategy: example ◼ Matching pennies ◼ Player 1 has two pure strategies: H and T ( 1 (H)=0.5, 1 (T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively. ( 1 (H)=0.3, 1 (T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively. 10
