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CRP 834: Decision Analysis

CRP 834: Decision Analysis. Week Three Notes. 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

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CRP 834: Decision Analysis

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  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. R $4 $4 q1 q1 Replace A1 A1 -$20 -$20 q2 q2 (L3, R) A2 A2 Continue q1 q1 -$5 -$5 -$4.5 B $100 $100 q2 q2 Stop No replace R R Same as (L2, RR) Same as (L1, R) L3 B Same as (L2, RB) -$9.0 B Averaging out-Folding Back– L3 Path

  4. $4 $4 q1 q1 A1 A1 -$20 -$20 q2 q2 A2 A2 q1 q1 -$5 -$5 $100 $100 q2 q2 R L3 -$9.0 No replace Same as (L1, B) R Same as (L2, BR) Same as (L2, BB) Stop B B R (L3, B) Continue Replace -$4.5 B Averaging out-Folding Back– L3 Path– continued

  5. Cost: $0 EMV: $0 Refuse to play Cost: $0 EMV: $28 L0: L1: Cost: $-8 EMV: $35.2 L2: Cost: $-12 EMV: $42.4 L3: 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)

  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 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 • Rule: Player 1: maximize the minimum payoffs (maximin) Player 2: minimize the maximum payoffs (minimax)

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

  13. Case 3: Games with no saddle point Lower value of the game Upper value of the game

  14. Games with Mixed Strategies Example 1 • Criterion for evaluating mixed strategies: the expected payoff

  15. Games with Mixed Strategies Step 1: Write the payoff functions for Player I.

  16. Games with Mixed Strategies 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

  17. Games with Mixed Strategies Step 3:write the payoff function for player II • 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 A B 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)

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