1 CRP 834: Decision Analysis Week Five Notes

2 Review Game Theory –Game w/ Mixed Strategies Graphic Method Linear Programming –Games In an Extensive Form

3 Two-Person Non-Zero Sum Games Prisoner’s Dilemma –Two prisoners held in separated jail cells No confession : 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)

4 Equilibrium Points w/ 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 j P 1 ij –Player 2: XP 2 Y=∑∑ x i y j 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.

5 Cooperative Game A nonconstant sum game in which the players can discuss their strategies before play and make binding agreements on strategies they will employ (i.e., they can form coalitions) Basic Problem: how to divide the coalition payoff among the members. Analyzed through the characteristic function (payoff function) of all possible coalitions.

6 Characteristic Function (Payoff function) V(N)  Grand coalition payoff (N) V(i)  Individual payoff V(S)  Coalition S payoff (1<S<N) Example:( Game in a 0-1 normalized form) Case of 3 players with benefits  (convention)    (Q) How to distribute the benefits of grand payoff among players?

7 Distribution of Benefits Grand Coalition Condition Group Rationality Individual Rationality CORE: a set of imputations satisfying “coalition rationality”

8 Shapley Value An imputation based on the power of each of the players A unique imputation is always produced by the Shapley value. Power of Player: Shapley value of player i: Probability to join the coalition s s = number of players in s n= number of players in total population.

9 Shapley Value  n (s) is simply the probability that a player joins coalition s. (A) Probability for the size of coalition S out of N  Binomial probability = # of coalition of size s that can be formed out of n participants (B) Probability that a specific coalition will be chosen – (C) # of players that can join this coalition s = (n-s) (D) Probability that a specific player I will join a coalition 1/(n-s) Therefore, the probability that a player I joins coalition s is:

10 Shaply Value of Player 1 Example: Player 1 (n=3) SPower of Player Weights(  )Shapley Value (  i 1 )   (1) -  (Ø) = 0,  (1,2)-  (2) = 0.1,  (1,3)-  (2) = 0.2,  (1,2,3)-  (2,3) = 0.8 2/6 1/6 2/6 0 1/60 2/60 16/60 ∑  i =1∑  i 1 =19/60  1 =19/60,  2 =19/60,  3 =22/60 ∑  i =1. V(1,2,3)=1

11 Application: Method for apportioning costs among participants in regional systems (Giglio and Wrightington, 1972) (Q) How to allocate cost for wastewater treatment facilities among three communities considering pollution standards and geography of the site. River A B C ? ?

12 For various combinations: Water PollutionCost P A =10 P B =20 P C =50 C A =20 C B =32 C C =48 C AB =45 C BC =60 C AC =60 C ABC =78 Application: Method for apportioning costs among participants in regional systems (Giglio and Wrightington, 1972)

13 Methods for Apportioning costs (Giglio and Wrightington, 1972) Cost sharing based on the measure of pollution Cost sharing based on single plant costs with a rebate proportional to the measure of pollution Cost sharing based on the separable costs remaining benefit method Cost Sharing Based on Bargaining Including the Regional Authority as a Participant

14 Method 1:Cost sharing based on the measure of pollution Each user pays an amount directly proportional to the amount of pollution he produces All polluters are treated equally. A pays 9.75 (=78*(10/80)), B 19.50, and C Notice: C pays more than he would if he decided to treat his waste alone This anomaly would arise if one user has a favorable geographic location (i.e. near to the river receiving the effluent)

15 Method 2: Cost sharing based on single plant costs with a rebate proportional to the measure of pollution Assumption: the rebate is proportional to the measure of pollution. C A + C B +C C = 100 (C A =20, C B =32, C C =48) C ABC = 78 (rebate = 22) A pays (=20-22*(10/80)), B 26.5, and C (Total = 78) Note that B+C = > C BC =60

16 Method 3: Cost sharing based on the separable costs remaining benefit method Uses the cost of alterative to determine the minimum cost that each community should assume. C ABC –C AB =33  C added 33 C ABC -C BC =18  A added 18 C ABC -C AC =18  B added 18 ∑ =69  C ABC =78 (need to add 9) Then allocate 9 proportional to the measure of pollution. A  , B  20.25, C  (∑ =78)

17 Method 4: Cost Sharing Based on Bargaining Including the Regional Authority as a Participant What is needed to make the regional plan work when there is no natural stable (feasible) solution (  subsidy & tax) Design of a specific core using different weighting factors: (cf: 2x 1 +2X 2 +10x 3 )