Decision Theory: Basics

1. Decision Theory: Basics

2. Decision theory A decision involves a choice between two or more acts, each of which will produce one of (possibly) several outcomes. Divide decisions into three categories: • Decisions under certainty • Decisions under risk • Decisions under uncertainty (a.k.a. Decisions under ignorance)

3. Decisions Under Certainty The agent is certain what outcome each act will produce. They are the easiest decisions to make: Choose the act that will produce the outcome you most prefer!  

4. Example If you raise your right hand, I will give you $100. If you raise your left hand, I will give you $10. If you raise neither hand, I will not give you anything.

5. Example If you raise your right hand, I will give you $100. If you raise your left hand, I will give you $10. If you raise neither hand, I will not give you anything.

6. Decisions Under Risk These are decisions where : 1.) You lack certainty about the what outcomes the actions will have. 2.) But you can at least assign probabilities to what outcomes the actions will have.

7. Example You have two options. Option A: I’ll flip a coin. If it lands heads, I’ll give you $50. If it lands tails, I won’t give you anything. Option B: I’ll pick a number between 21-30 inclusive. If you guess the number, I’ll give you $70. If you don’t guess number, I won’t give you anything.

8. Which option do you choose? • Option A • Option B

9. Decisions Under Ignorance These are decisions where : 1.) You lack certainty about the what outcomes the actions will have. 2.) You can’t even assign probabilities to the outcomes the actions will have.

10. Example Suppose that just after graduating college you are offered three jobs. First, the Exe Company offers you a salary of $40,000. Exe is well established and secure. Your next offer comes from the Wye Company. Here, the salary is $60,000. But Wye is a new company and is less secure. You do not know how likely it is to last. It might go bankrupt and then you’ll be left without a job. The final offer comes from the Zee Company, which is as well established and secure as Exe. Here, the salary is $65,000. In all other respects, the jobs are similar. Which job do you take?

12. Taking the job at Zee dominates the other acts.

13. Now what?

14. Sonny and Cher

15. Sonny is “The Gambler” Chooses the best of the best. Maximax Strategy

16. Cher is “Cautious” Chooses the best of the worst. Maximin strategy

17. Who is right?

18. Maximax: The Gambler This strategy tells us to maximize the maximum. To apply maximax we first need to find the maximum outcome for each act. Simply read across a row and at the end of the row write down the maximum value; then choose the act with the maximum of the maximum values. According to maximax, you should choose A1.

19. Which action does maximax pick? • A1 • A2

20. Criticism of Maximax Too risky! Prohibits us from avoiding huge losses where there are only slight gains.

21. Which action does maximin pick? • A1 • A2

22. Maximin: The Cautious Player This strategy tells us to maximize the minimum. To find the minimum for each act, read across the row and write down the minimum value; then choose the act that has highest of lowest values According to maximin, you should choose A2.

23. Criticism of Maximin Too conservative. Prohibits us from taking advantage of huge gains where there are only slight losses.

24. Principle of Insufficient Reason When faced with a decision under ignorance, treat all outcomes as equiprobable. In effect, you chose in terms of expected utility. Exe: ($40k x .5) + ($40k x .5) = $40,000 Wye: ($60k x .5) + ($00k x .5) = $30,000

25. Criticism of the Principle of Insufficient Reason ARBITRARY! Gives no reason for preferring one probability distribution over another.

26. Decision Strategies Principle of Insufficient Reason: Assume every state is equally probable and choose the act that, on this assumption, maximizes expected utility. Dominance: If, no matter what happens, an act is never worse and sometimes better than every other, then choose it. Maximax: Choose the best of the best. Maximin: Choose the best of the worst. Minimax Regret: Choose the act that minimizes the maximum amount of regret.