1 CRP 834: Decision Analysis Week Three Notes

2 Review Decision-Flow Diagram –A road map of all possible strategies and outcomes –At a decision fork, the decision maker exercises control of the next choice –At a chance fork, the decision maker relinquishes control to the state of nature Averaging Out-Folding Back –Starting from the termini of a decision tree, repeat the following processes backwards –At each chance juncture, the decision maker carries out an averaging-out process –At decision junction, the decision maker makes a decision

3 L3L3 -$9.0 R B (L 3, R) Continue Stop Same as (L 1, R) -$20 -$5  AA AA   $4 $100  Replace R B -$20 -$5  AA AA   $4 $100  No replace R B Same as (L 2, RR) Same as (L 2, RB) -$4.5 Averaging out-Folding Back– L3 Path

4 L3L3 -$9.0 R B (L 3, B) Stop Same as (L 1, B) Continue -$4.5 No replace R B Same as (L 2, BR) Same as (L 2, BB) Replace -$20 -$5  AA AA   $4 $100  R B -$20 -$5  AA AA   $4 $100  Averaging out-Folding Back– L3 Path – continued

5 Refuse to play L:L: L:L: L:L: L:L: Cost: $0 EMV: $28 Cost: $0 EMV: $0 Cost: $-8 EMV: $35.2 Cost: $-12 EMV: $42.4 Cost: $-9 EMV: $40.15

6 What is your decision: –Make experiment or not? –If make experiment, which option? –Having decided to take an experiment option, what actions you will take according to experiment results? –What is the benefit of information?

7 Game Theory Game theory: the study of decision-making situations in which conflict and cooperation play important roles Player: decision-maker –The players may be individuals, groups of individuals, firms, and nations. Payoff function: the objective function, –This function gives numeric payoffs to each player. Game: a collection of rules known to all players which determines what players may do and the outcome and payoff resulting from their choice. Move: a point in the game at which players must make a choice between alternatives Play: any particular set of moves and choices Strategy: a set of decision formulated in advance of play specifying choices to be made in every possible contingency. Field of application: parlor game, military battle, political campaign, advertising and marketing campaign by competing firms, etc.

8 Classification and Description Classification –Number of Players: –Number of strategies: –Nature of payoff functions: Zero-Sum game: Constant difference game: Non-zero sum game: –Nature of preplay negotiation: cooperative game noncooperative game. Description –Game in extensive form: (game tree) –Game in normal form (payoff table) II I 12 1(100, 50)(-10,50) 2(20,50)(20, -10)

9 Two-Person Zero-Sum Game Problem Statement: Two politicians are running against each other for the United States Senate. Campaign plans must now be made for the final 2 days before the election, and the both politicians want to spend these two days campaigning in two cities: BigTown and Megapolis. –Strategy 1: Spend 1 day in each city –Strategy 2: Spend both days in Bigtown –Strategy 3: Spend both days in Megapolis

10 Case 1: A dominating strategy II I1124I II 1212 I112I With this solution, player I will receive a payoff of 1 from player II, so that the value of the game is said to be 1

11 Case 2: Games with Saddle Points II 123 I Rule: Player 1: maximize the minimum payoffs (maximin) Player 2: minimize the maximum payoffs (minimax)

12 Case 2: Games with Saddle Points (cont’d) II 123Min I Max506 Maximin strategy by I Lower value of the game Minimax strategy by II Upper value of the game Saddle point. Lower value ( ) = Upper value ( ) = V

13 Case 3: Games with no saddle point II 123Min I Max542 Upper value of the game Lower value of the game

14 Games with Mixed Strategies Criterion for evaluating mixed strategies: the expected payoff ProbabilityII y1y1 y2y2 y3y3 ProbabilityPure Strategy 123 Ix1x x 1 (x 2 ) Example 1

15 Games with Mixed Strategies 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 ) = x 1 -2x 1 + 4(1- x 1 ) = x 1 2x 1 - 3(1- x 1 ) = x 1

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

17 Games with Mixed Strategies 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

18 Games with Mixed Strategies 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.

19 Games with Mixed Strategies (Example 2) II 12 I I 123 II