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 q<0. 5, Head (r=1) 1/2 For q0.5, Tail(r=0) >Foq=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 q<0.5, Head (r=1) ➢ For q>0.5, 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 r<0.5, Tail(q=0) 1/2 For r>0. 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 r<0.5, Tail (q=0) ➢ For r>0.5, 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 q<0.5, Head (r=1) ➢ For q>0.5, Tail (r=0) ➢ For q=0.5, indifferent (0r1) ◼ Player 2’s best response B2(r): ➢ For r<0.5, Tail (q=0) ➢ For r>0.5, 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