1. 1 CRP 834: Decision Analysis Week Four Notes

2. 2 Review Game Theory –Classification and Description –Two Person Zero-Sum Game Game w/ Dominating Strategy Game w/ Saddle Points Game w/ Mixed Strategies

3. 3 Criterion for evaluating mixed strategies: the expected payoff ProbabilityII y1y1 y2y2 y3y3 ProbabilityPure Strategy 123 Ix1x1 10-22 1-x 1 (x 2 ) 254-3 Example 1 Games with Mixed Strategies

4. 4 Step 1: Write the payoff functions for Player I. (y 1, y 2, y 3 )Expected payoff (1, 0, 0) (0, 1, 0) (0, 0, 1) 0x 1 + 5(1- x 1 ) = 5 - 5 x 1 -2x 1 + 4(1- x 1 ) = 4 - 6 x 1 2x 1 - 3(1- x 1 ) = -3 + 5x 1 Games with Mixed Strategies

5. 5 Step 2: Graphical Analysis for player 1 Solve for x* and v? 4 - 6 x1 = -3 + 5x1 x* = 7/11 (x1*, x2*) =(7/11, 4/11) v =2/11 Games with Mixed Strategies

6. 6 Step 3:write the payoff function for player II (a)According to the definition of the upper value and the minimax theorem, the expected payoff resulting from this strategy (y1, y2, y3) = (y1*, y2*, y3*) y1*(5 - 5 x1) + y2* (4 - 6 x1) + y3* (-3 + 5x1) ≤ = v =2/11. (b) x1= x* = 7/11, then (20/11) y1 + (2/11)y2 +(2/11) y3=2/11. (*) Where y1 + y2 + y3 = 1. (3) we can rewrite (*) as: (18/11) y1 + (2/11) ( y1 + y2 + y3 ) =2/11. Then, y1= 0. (4) The payoff function is then: F(x)= y2* (4 - 6 x1) + y3* (-3 + 5x1), y1* =0 Games with Mixed Strategies

7. 7 Step 3 (cont’d) (5) If we rearrange F(x), F(x) = (4 y2* - 3y3* )+ x (-6 y2* + 5y3*) ≤ 2/11 AB F(x) = A + Bx ≤ 2/11 A= 2/11, B=0 Then: 4y2* - 3y3* = 2/11 (a) -6y2* + 5y3* = 0 (b) Solve for Eq. (a) and (b): y2*=5/11, y3*=6/11, y1*=0.

8. 8 Games with Mixed Strategies (Example 2) II 12 I1-36 28-2 362 I 123 II13-8-6 2 2-3

9. 9 Games with Mixed Strategies (Example 2-Graphical Analysis)

10. 10 Games with Mixed Strategies—Linear Programming II y 1, y 2, ……………………………., y n Ix1,x2,...xmx1,x2,...xm p 11, p 12, ……………………………, p 1n p 21, p 22, ……………………………, p 2n. P m1, p m2, ……………………………, p mn By definition, the strategy (x 1, x 2,…,x i,…,x m ) is optimal if: for all opposing strategies (y 1, y 2,…,y i,…,y m ). Where x i is the probability that player I uses pure strategy i, y j is the probability that player II uses pure strategy j, and is the payoff if player I uses pure strategy i and player II uses pure strategy j.

11. 11 Mixed Strategies – Linear Programming Since the inequality must hold for any pure strategy of player II: ……

12. 12 Mixed Strategies – Linear Programming For Player I For Player II: DUAL ……

13. 13 Example – Linear Programming II 123Min I111 22-22 333-3 Max332 (Q) How to write a linear program for this example?

14. 14 Games in an extensive form Problem Statement –Two players (I & II) and referee –Referee flips a coin, notes the outcome (H/T), and shows the result to player I. –Player I can pass or bet. If pass  pays $1 to player II If bets  Game continues –Player 2 can pass or call. If pass  pays $1 to player II If call  referee shows the coin »H : pay $2 to player I »T: receives $2 from player 1. (Q) Build the game tree, develop the payoff table, and determine the optimal mixed strategy.

15. 15 Game Tree Games in an Extensive Form H (0.5)T (0.5) I II BetPassBet Pass Pass Call +1 +2-2

16. 16 Expected Payoff Table Games in an Extensive Form Possible Decisions HT If pp PP f pb PB f bp BP f bb BB IIgpgp Pass gcgc Call Expected Payoff II gpgp gcgc If pp f pb 0-1.5 f bp 00.5 f bb 10 II gpgp gcgc If bp 00.5 f bb 10

17. 17 Mixed Strategy Games in an Extensive Form II gpgp gcgc If bp (x)00.5 F bb (1-x)10 Player 1: P1=1-x; P2=0.5x; x*=2/3 and v=1/3 Player II: EP = y1(1-x) + y2(0.5x) = y1+ x(-y1+0.5y2) ≤ 1/3 y1*= 1/3, y2*=2/3

18. 18 Two-Person Non-Zero Sum Games Prisoner’s Dilemma –Two prisoners held in separated jail cells No confess : Both set free (4) Both Confess: Moderate jail term (2) One confesses, the other not: Reward (5) & severe punishment (0) Prisoner 2 Don’t ConfessConfess Prisoner 1Don’t Confess(4,4)(0,5) Confess(5,0)(2,2)

19. 19 Equilibrium Points with Non-Zero Sum Games Strategies: –Player 1: (x 1, x 2,…,x i,…,x m ) = X (1xm) –Player 2: (y 1, y 2,…,y i,…,y n ) = y (n*1) Payoffs: –Player 1: P 1 ij  P 1 (1xm) –Player 2: P 2 ij  P 2 (nx1) Expected payoffs: –Player 1: XP 1 Y=∑∑x i y i P 1 ij –Player 2: XP 2 Y=∑∑ x i y i P 2 ij X*, Y* is an equilibrium solution if (Player 1) XP 1 Y* ≤ X*P 1 Y* for all x (Player 2) X*P 2 Y ≤ X*P 1 Y* for all y.