Decision Analysis

1. R. Keeney April 4, 2012 Decision Analysis

2. Game Theory • A decision maker wants to behave optimally but is faced with an opponent • Nature – offers uncertain outcomes • Competition – another optimizing decision maker • 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)

3. 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)

4. 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

5. Uncertainty and Maxi-min / Safety First Rule • I know that rain is possible, but no idea how likely it is to occur • Maxi-min decision making helps us formulate a plan in an optimal fashion • Maximize the minimums for each decision • If I take my umbrella, what is the worst I could do? • If I don’t take my umbrella, what is the worst I could do?

6. What’s the best worst case scenario? • 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

7. Maxi-min (Safety First) • A lot of decisions are made this way • Identify the worst that could happen, choose a course that has a “worst case scenario” that is least detrimental • Framework implies that people are risk averse • Focus on downside outcomes and try to avoid the worst of these • Assumes probabilistic knowledge of outcomes is not available or not able to be processed

8. Expected Value Criteria (Mixed strategy) • 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

9. Fifty percent chance of rain • Given the probability of rain, the EV for taking my umbrella is higher so that is the optimal decision

10. 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

11. Common Rule for EV: A breakeven probability of rain • 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.

12. 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

13. 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.

14. Dominant Decisions • Decision is whether or not to wear clothing • If it rains prefer to wear clothing • Get sick from rain and get arrested • If it doesn’t rain prefer to wear clothing • Don’t get sick but still arrested • Wearing clothing is a Dominant Decision • Nature’s play has no influence on the decision • Weather effects how much and what type of clothing just as it effects our decision on umbrella (where we saw a split decision)

15. Competitive Games: Bimatrix Each player has two actions and each player’s action has an impact on their own and the opponent’s payoff. Payoffs are listed in each intersecting cell for player 1 (P1) and player 2 (P2).

16. Prisoner’s Dilemma • Two criminals apprehended with enough evidence to prosecute for 1 year sentences • Suspected of also committing a murder • Outcomes range from going free to death penalty

17. 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

18. 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

19. They 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

20. Price Setting Competitors • Two companies set prices and earn profits • If C2 sets low price, C1 sets low price 2000>0 • If C2 sets high price, C1 sets low price 13000>10000 • Low prices are a dominant decision for C1

21. Price Setting Competitors • C2 faces the same payoffs • Also has low prices as a dominant decision • Both earn 2000 • If they collude (with a contract) they could both earn 10000 • Illegal contract in most cases

22. 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)