1. Decision Analysis April 11, 2011

2. Game Theory Frame Work Players ◦ Decision maker: optimizing agent ◦ Opponent  Nature: offers uncertain outcome  Competition: other optimizing agent Strategies/actions Outcomes

3. Payoff Matrix We focus on simple examples using ‘payoff matrix’ Decisions for one actor are the rows and for the other are the columns Intersecting cells are the payoffs Bimatrix (two payoffs in the cells) State 1State 2 Act 1Payoff 1,1Payoff 2,1 Act 2Payoff 1,2Payoff 2,2

4. Decision Theory Nature is the opponent One decision maker has to decide whether or not to carry an umbrella Decisions are compared for each column ◦ If it rains, Umbrella is best (5>0) ◦ If no rain, No Umbrella is best (4>1) RainNo Rain Umbrella51 No Umbrella04

5. Split Decision The play made by nature (rain, no rain) determines the decision maker’s optimal strategy ◦ Assume I have to make the decision in advance of knowing whether or not it will rain RainNo Rain Umbrella51 No Umbrella04

6. Uncertainty In know that rain is possible, but I have no idea how likely it is to occur. How does the decision maker choose? Two Methods ◦ Maximin: largest minimum payoff (caution) ◦ Maximax: largest maximum payoff (optimism)

7. Maximin (safety first rule) Maximize the minimums for each decision ◦ If I take my umbrella, what is the worst I can do? ◦ If I don’t take my umbrella, what is the worst I can do? RainNo Rain Umbrella51 No Umbrella04

8.  Comparing the two worst case scenarios  Payoff of 1 for taking umbrella  Payoff of 0 for not taking umbrella  An optimal choice under this framework is then to take the umbrella no matter what since 1 > 0  Framework implies that people are risk averse  Focus on downside outcomes and try to avoid the worst of these Maximin (safety first rule)

9. Maximax Maximize the maximums for each decision ◦ If I take my umbrella, what’s the best I can do? ◦ If I don’t take my umbrella, what’s the best I can do? RainNo Rain Umbrella51 No Umbrella04

10. Maximax  Comparing the two best case scenarios  Payoff of 5 for taking umbrella  Payoff of 4 for not taking umbrella  An optimal choice under this framework is then to take the umbrella no matter what since 5 > 4  Both methods assume probabilistic knowledge of outcomes is not available or not able to be processed

11. Expected Value Criteria  What if I know probabilities of events?  Wake up and check the weather forecast, tells me 50% chance of rain  Take a weighted average (i.e. the expected value) of outcomes for each decision and compare them Rain (p=0.5) No Rain (p=0.5) Umbrella51 No Umbrella04

12. Fifty Percent Chance of Rain Given the probability of rain, the EV for taking my umbrella is higher so that is the optimal decision Rain (p=0.5) No Rain (p=0.5) EV (Sum over row) Umbrella5*0.51*0.53.0 No Umbrella 0*0.54*0.52.0

13. 25 Percent Chance of Rain Given the lower probability of rain, the EV for taking my umbrella is lower so no umbrella is my optimal decision Rain (p=0.25) No Rain (p=0.75) EV (Sum over row) Umbrella5*0.251*0.752.0 No Umbrella 0*0.254*0.253.0

14. Common Rule for EV: a breakeven probability of rain  Probability (x) that event happened and probability (1-x) that something else happens  Setting the two values in the last column equal gives me their EV’s in terms of x. Solving for x gives me a breakeven probability. Rain (p=x) No Rain (p=1-x) EV (Sum over row) Umbrella5*x1*(1-x)5x+(1-x) No Umbrella 0*x4*(1-x)0x+4(1-x)

15. Common Rule for EV: a breakeven probability of rain  Umbrella: 4x + 1  No Umbrella:4 – 4x  Setting equal: 4x + 1 = 4 – 4x -> 8x – 3 =0  X = 0.375  If rain forecast is > 37.5%, take umbrella  If rain forecast is < 37.5%, do not take umbrella Rain (p=x) No Rain (p=1-x) EV (Sum over row) Umbrella5*x1*(1-x)5x+(1-x) No Umbrella 0*x4*(1-x)0x+4(1-x)

16. In Practice The tough work is not the decision analysis it is in determining the appropriate probabilities and payoffs ◦ Probabilities  Consulting and market information firms specialize in forecasting earnings, prices, returns on investments etc. ◦ Payoffs  Economics and accounting provide the framework here  Profits, revenue, gross margins, costs, etc.

17. Competitive Games: Bimatrix Player 1 Player 2 Action 1Action 2 Action 1P1, P2 Action 2P1, P2 Each player has two actions and each player’s action has an impact on their own and the opponent’s payoff. Both players decide at once Payoffs are listed in each intersecting cell for player 1 (P1) and player 2 (P2).

18. Prisoner’s Dilemma Two criminals arrested for both murder and illegal weapon possession Police have proof of weapon violation (each get 1 year) Police need each prisoner to confess to convict for murder (death penalty) If both keep quiet, each only get 1 year If either confesses, both could be sentenced to death

19. Prisoner’s Dilemma Prisoners are separated for questioning Outcomes range from going free to death penalty Prisoner 1 Prisoner 2 ConfessDon’t Confess ConfessP1 = Life jail P2 = Life jail P1 = Free P2 = Death Don’t ConfessP1 = Death P2 = Free P1 = 1 year jail P2 = 1 year jail

20. What will they do? Prisoner 1’s decision  If Prisoner 2 confesses then prisoner 1 optimally confesses since: Life jail > Death  If Prisoner 2 does not confess then prisoner 1 optimally confesses since: Free > 1 year in jail  Confession is a dominant decision for prisoner 1  Optimally confesses no matter what prisoner 2 does Prisoner 1 Prisoner 2 ConfessDon’t Confess ConfessP1 = Life jailP1 = Free Don’t ConfessP1 = DeathP1 = 1 year jail

21. What will they do? Prisoner 2’s decision Prisoner 2 faces the same payoffs as prisoner 1 Prisoner 2 has same dominant decision to confess ◦ Optimally confesses no matter what prisoner 1 does Prisoner 2 Prisoner 1 ConfessDon’t Confess ConfessP2 = Life jailP2 = Free Don’t ConfessP2 = DeathP2 = 1 year jail

22. Both confess, Both get life sentences  This is far from the best outcome overall for the prisoners  If neither confesses, they get only one year in jail  But, if either does not confess, the other can go free just by confessing while the other gets the death penalty  Incentive is to agree to not confess, then confess to go free Prisoner 1 Prisoner 2 ConfessDon’t Confess ConfessP1 = Life jail P2 = Life jail P1 = Free P2 = Death Don’t ConfessP1 = Death P2 = Free P1 = 1 year jail P2 = 1 year jail

23. Summary  Decision analysis is a more complex world for looking at optimal plans for decision makers  Uncertain events and optimal decisions by competitors limit outcomes in interesting ways  In particular, the best outcome for both decision makers may be unreachable because of your opponent’s decision and the incentive to deviate from a jointly optimal plan when individual incentives dominate  Broad application: Companies spend a lot of time analyzing competition ▪ Implicit collusion: Take turns running sales (Coke and Pepsi)

24. And for Agriculture… Objective: maximize gross product ◦ St.: resource availability and requirement Decision variables: Cropping patterns Size and equipment types Uncertainties: ◦ Weather conditions ◦ Market prices ◦ Crop and animal disease