Decision Analysis

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)

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)

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

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?

8. Maximin (safety first rule) • 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

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?

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

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

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

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.

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

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 • 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

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

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

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

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