1602 2 Person Total-Conflict Games: MIXED Strategies

1. 1602 2 Person Total-Conflict Games: MIXED Strategies

2. Pure Strategy = what we did last time. Mixed Strategy = what we are going to do today. Expected Value = (outcome) (probability) + (outcome) (probability) + (outcome) (probability) … + (outcome) (probability) Fair game = when the Expected value is zero Symmetrical game is when the “winnings” are the same Nonsymmetrical game is when the “winnings” are different but add up to the same amount. Zero sum game = if one person win the other loses.

3. Penny game If two players have pennies. It is determined ahead of time that Fred will win if both players present the same side of the coin and Fred win’s the Bre’s penny. If Fred and Bre show different sides of the coin then Bre wins Fred’s penny. Matrix looks like Bre H T Fred H 1 -1 T -1 1

4. Penny game Cont. What if the payoff was different? What if Fred got $5 for double heads & $1 for double tails? What if Bre got $3 (or -3 from Fred’s perspective) for a head and a tail. The “payoff” Matrix would look like this: Bre H T Fred H 5 -3 T -3 1

5. Penny game Cont. Here’s how we do a “mixed strategy.” Take expected value for Fred for Heads and Tails. Remember all probabilities must add up to 1. E(H) = (5)(p) + (-3)(1-p) E(T) = (-3)(p) + (1)(1-p) Bre H T Fred H 5 -3 T -3 1

6. Penny game Cont. Set the Expected Values equal to each other. (5)(p) + (-3)(1-p) = (-3)(p) + (1)(1-p) 5p – 3 + 3p = -3p + 1 – p 8p – 3 = -4p + 1 12p = 4 or p = 1/3 Bre H T Fred H 5 -3 T -3 1

7. Penny game Cont. So p = 1/3 So Fred should choose Heads 1/3 of the time and Tails 2/3 of the time (must add up to 1). Written as (1/3, 2/3) Bre H T Fred H 5 -3 T -3 1

8. Penny game Cont. So the value is E(H) = (5)(1/3) + (-3)(1-1/3) 5/3 + -3 + 1 Value = $-0.333 per play Bre H T Fred H 5 -3 T -3 1

9. Baseball Duel Assume that Nick can throw either a blazing fastball or a slow curve into the strike zone and so has two strategies: Fast (denoted by F) and curve (C). The pitcher and batter who attempts to guess, before each pitch is thrown, whether it be a fastball or a curve. Assume the following data is known by both players

10. Baseball Duel Cont. .300 if the batter guesses fast and the pitcher throws fast .200 if the batter guesses fast and the pitcher throws curve. .100 if the batter guesses curve and the pitcher throws fast .500 if the batter guesses curve and the pitcher throws curve Create the matrix Pitcher F C F .200 .300 Batter C .100 .500

11. Baseball Duel Cont. 1st see if there is a saddlepoint by doing pure strategy. Does the batter want big or small #’s? Big Does the pitcher want big or small #’s? Small Is this a saddlepoint? No Pitcher F C F .200 .200 .300 .200 Batter C .100 .500 .100 .300 .300 .500

12. Pitcher F C F .200 .300 Batter C .100 .500 Baseball Duel Cont. If no saddlepoint then, use mixed strategy. Find the Pitcher’s expected value for “fast” and “curve.” (it’s backwards) E(F) = .300(p) + .200(1-p) E(C) = .100(p) + .500(1-p) Set them equal .300(p) + .200(1-p) = .100(p) + .500(1-p) .300p + .200 – .200p = .100p + .500 – .500p .100p + .200 = -.400p + .500 .500p = .300 P = 3/5

13. Pitcher F C F .200 .300 Batter C .100 .500 Baseball Duel Cont. P(F) = 3/5 & P(C) = 2/5 Pitcher’s strategy is (3/5, 2/5) Find the value: (by plugging in the prob.) Value = E(F) = .300(3/5) + .200(1 – (3/5)) Value = .18 + .2 – 0.12 Value = 0.260

14. Pitcher F C F .200 .300 Batter C .100 .500 Baseball Duel Cont. Find the batter’s strategy. Opposite of what you think E(F) = .3q + .1(1-q) E(C) = .2q + .5(1-q) Set equal .3q + .1(1-q) = .2q + .5(1-q) .3q + .1 – .1q = .2q + .5 – .5q .2q + .1 = -.3q + .5 .5q = .4 Q = 4/5

15. Pitcher F C F .200 .300 Batter C .100 .500 Baseball Duel Cont. P(F) = 4/5 & P(C) = 1/5 Batter’s strategy is (4/5, 1/5) Value will be the same as before V = .260 So what’s the answer Pitcher Strategy (3/5,2/5); Batter Strategy (4/5,1/5); Value .260 HW P587: 6-10, 11a,b, 16 #9 is really hard to set up.