UTILITY THEORY

1. UTILITY THEORY QUANTIFING OUR PREFERENCES

2. PAYOFF TABLES States of Nature Alternatives • This example is Decision-Making under UNCERTAINTYbecause the probabilities of the states of nature are unknown. • MaxiMax, MaxiMin or MinMax Regret techniques can be applied.

3. PAYOFF TABLESMaxiMaxSolution States of Nature Alternatives • Find the maximum payoff for each alternative. • The alternative with the highest payoff is the solution. • The MaxiMaxsolution here is a three-way tie. (A, B, C)

4. PAYOFF TABLESMaxiMinSolution States of Nature Alternatives • Find the minimum payoff for each alternative. • Pick the maximum these (maximum of the minimums). • MaxiMinsolution is “D” because 4 is the largest of the minimum payoffs.

5. PAYOFF TABLESMin-Max Regret Solution States of Nature 0 0 5 0 0 5 5 4 3 0 Alternatives 7 4 3 0 2 2 0 4 0 2 2 1 1 2 4 1 2 1 For each alternative, take one cell at a time and ask yourself, “if I end up here, do I have any regrets about my decision?” Could you have done better? How much better? That amount is your regret. Add up the total regrets for each alternative and select the alternative with the least total regret. ( C & D)

6. PAYOFF TABLESSummary States of Nature Alternatives SUMMARY: Maximax Solution is A, B, or C. MaximinSolution is D Min-max Regret solution is C or D. Conclusion: Decision Making under Uncertainty Sucks.

7. Expected-Value Analysis (Baysian Decision Rule) Here, the probabilities of the states-of-nature are known, so it is decision making under risk. Alternatives Each payoff (cell value) is multiplied by its probability of occurring. Values are then add for each alternative.

8. Expected Value (Baysian Decision Rule) States of Nature and their probabilities of occurring Alternatives The highest expected value is for alternative "A."

9. Applying Expected Value Analysis to Insurance • When I bought a home in Colorado, I wanted to buy fire insurance. • Market value of my house was $200,000. • Fire Insurance costs $800 per year. • I discover that there was a .002% chance of a fire destroying my home. • Should I buy the insurance?

10. Expected Value Analysis(In this case, expected loss analysis) .002 x (-$200,000) + $0 = -$400 We prefer to buy the insurance even though doing so has a higherexpected loss. This is because of the peace of mind that we get from having insurance is not accounted for by expected value analysis. We are risk averse! We have a greater preference for avoiding losses than for equal gains.

11. Gambling Example Choice #2 No coin flip! I pay you $20 without flipping the coin. Choice #1 I flip a coin. Heads: I pay you $200. Tails: You pay me$100. Which choice do you prefer?

12. Expected Value Analysis of the coin flip option • Once again we are risk averse. • Given a choice between a risky gamble with a higher expected value, and a certain return with a lower value, we almost always avoid the risk in favor of the sure thingif the sure thing is a positive outcome.

13. Conclusions About Expected Value Analysis • It is decision-making under Risk • It is better than decision-making under uncertainty. • It does NOT account for our risk preferences or for other preferences about a situation. • Making decisions solely on expected value analysis is NOT good decision making. • It is only one factor to be considered when making a choice.

14. Utility-Theory Concept(Quantifying one’s preferences) • For a set of choices, ranked by our preferences, we can assign utility values that represent our preferences. • Our preferences, represented by utility values, can then be mathematically manipulated. • We call these utility values “Utils.”

15. ASSUMPTIONS OF UTILITY THEORY • An individual is assumed to behave consistently in accordance with his or her own tastes. • If A is preferred to B, & B is preferred to C, then A is preferred to C • This property is called "Transitivity" • The values obtained using utility theory pertain only to a single individual. • Different people have different preferences, and thus have different utility functions.

16. CERTAINTY EQUIVALENT • Certainty equivalent is that amount (payoff) that makes you indifferentbetween two choices. • Example: You flip a coin. • 1. If it’s heads, you win $200 • 2. If it’s tails, you win $10 • What certain (guaranteed) amount of money would make you indifferent about this gambling opportunity?

17. PREPARING A UTILITY FUNCTION Alternatives States of nature Payoffs are in millions. 100 100 We want to convert each payoff to a utility value that represents our preference for that payoff, relative to the other possible payoffs. We will assign utility values between 0 & 100. We assign the maximum payoff ($7 mil.) a utility value of 100 and the minimum payoff ($1 mil.) a utility value of 0. Utility 0 0 0 1 2 3 4 5 6 7 Payoffs

18. The Payoffs and their Utilities Payoff Utility 1--------------------0 2 3 4 5 6 7------------------100 100 100 Utility 0 0 0 1 2 3 4 5 6 7 Payoffs (millions)

19. Filling in the missing values using Certainty Equivalents • Receipt of what dollar amount would make you indifferent to the choice of 7 million or 1 million if you had a 50/50 chance of either? My indifference point (certainty equivalent) is $3 million.

20. Finding the utility value of 3 million • Substituting the utility value of 100 for $7 and 0 for $1 gives you an expected utility of 50 (.5 x $7) + (.5 x $1) = $4 Dollars Utils (.5 x 100) + (.5 x 0) = 50

21. The Payoffs and their Utilities Payoff Utility 1--------------------0 2 3-------------------50 4 5 6 7------------------100 100 Utility 0 0 1 2 3 4 5 6 7 Payoffs

22. Filling in the missing values using Certainty Equivalents • Receipt of what dollar amount would make you indifferent to the payoff of 3 million and 1 million if you had a 50/50 chance of either? Dollars Utils

23. The Payoffs and their Utilities Payoff Utility 1--------------------0 2-------------------25 3-------------------50 4 5 6 7------------------100 100 Utility 0 0 1 2 3 4 5 6 7 Payoffs

24. Filling in the missing values using Certainty Equivalents • Receipt of what dollar amount would make you indifferent to the payoff of 3 million and 7 million if you had a 50/50 chance of either? Dollars Utils

25. The Payoffs and their Utilities Payoff Utility 1--------------------0 2-------------------25 3-------------------50 4 5-------------------75 6 7------------------100 100 Utility 0 0 1 2 3 4 5 6 7 Payoffs

26. The Payoffs and their Utilities Payoff Utility 1--------------------0 2-------------------25 3-------------------50 4------------------- ? 5-------------------75 6------------------- ? 7------------------100 100 88 63 Utility 0 0 1 2 3 4 5 6 7 Payoffs

27. The Payoffs and their Utilities Payoff Utility 1--------------------0 2-------------------25 3-------------------50 4-------------------63 5-------------------75 6-------------------88 7------------------100 100 Utility 0 0 1 2 3 4 5 6 7 Payoffs

28. Substituting Utility Values for Payoffs in our table. Payoff Utility 1----0 2---25 3---50 4---63 5---75 6---88 7---100

29. UTILITY FUNCTIONS Risk Neutral Risk Averse Risk Seeking Utils Payoffs Payoffs Payoffs Generally, people are risk seeking (or less risk averse) with small amounts (payoffs), but become more risk averse with larger amounts. Thus, over a large range of payoffs, a typical utility function might start out risk seeking and become risk averse at higher payoffs.

30. Utility Theory Strengths • Accounts for our individual risk preferences in decision making • Quantifies the judgmental and intuitive aspects of a situation.

31. DISADVANTAGES • Each situation is different and requires re-computation of the utility function. • Time consuming • An individuals preference function for a given situation can vary depending on the individual’s state of mind, mood, health, etc. • Your utility function for the same situation may be different in the morning from what it is at night.

32. Game Theory A Quantitative Approach to Competitive Situations

33. PRISONER’S DILEMMA • Two women who committed a crime together are arrested and put in separate cells so they cannot communicate. They are charged with felonies and with being accessories to murder. The jail term for accessory to murder is 20 years, and for a felony it is 5 years. The district attorney has evidence to convict both women for the felony, but not for accessory to murder unless one or both women confess. • He speaks to each woman separately, hoping to get a confession from one or both. In order to get a conviction, he offers each woman the possibility of a reduced sentence if she testifies against the other. • If one woman confesses to the charge of accessory to murder, but the other woman does not confess, the one who confesses gets to go free, and the other woman gets 20 years. • If both women confess, they both get 10 years. • If neither confesses, they each get 5 years for the felony. • You are one of the women. What will you do?

34. Visual Analysis

35. MaxiMinAnalysis MaxiMin analysis is usually the preferred approach.

36. MaxiMax Analysis

37. The optimal result for both occurs ifthere is trust that neither will confess.

38. Research Results • In research on this dilemma, researchers have found two different results. • Pairs who have a continuing relationship in similar situations and have built up trust, tend to not confess. (They each get 5 years) • Pairs who experience a “single-trial” situation, tend to confess. (10 years each)

39. Theory of Games • In game theory, you have two or more contestants (people, groups, companies, countries) • Normally there is competition rather than cooperation. • With three or more, two can cooperate to gain advantage over others. • Game theory mathematically analyzes such situations.

40. Types of Games • Zero-Sum Games • There is a fixed amount of reward. What one person loses, the other gains • EG: Two companies competing in a saturated market. • Non-Zero-Sum Games • There is no fixed amount of reward. • EG: Two companies competing in a new market. Both can increase market share.

41. Types of Games • One-Person against Nature • Nature is the competitor • Two-Person Games • Only two contestants (prisoner’s dilemma) • N-Person Games • Any number beyond two contestants.

42. Two-Person, Zero-Sum Game • Two firms are competing for a particular market, where the market is fairly fixed (saturated). A1 and A2 are different pricing strategies for Firm A, andB1, and B2 are pricing strategies for Firm B. • The values in the matrix represent respective % of market share. For example, if Firm A uses pricing strategy A1 and Firm B uses strategy B2, Firm A will end up with 80% (.8) of the market and firm B will end up with 20% (.2)

43. MaxiMin Analysisw/Stable Outcome It is a “Stable Outcome” because each firm has an optimal (Dominate) strategy that will not be harmed, regardless of what the other firm does. This makes it a stable outcome.

44. Analysis Nuclear War between U.S.A. & China Russia USA Dominate strategy for both is a first strike.

45. Analysis If Both countries added a retaliatory capability, then the dominate strategy is to do nothing, regardless of whether there is trust or not. Russia USA

46. Terms and Concepts to Know • Zero-Sum and Non-Zero Sum Games • Maximin Analysis • Expected Value Analysis • Dominate Strategy • When regardless of what your opponent does, you benefit by keeping the same strategy. • Stable Outcome: • When each player has a dominant strategy.

47. GAME THEORY SUMMARY • Game Theory is a highly-quantitative technique for analyzing competitive situations involving two or more competitors. • It provides a conceptual framework for formulating and analyzing problems in simple competitive situations. • There is a considerable gap between what the theory can handle and the complexity of most competitive situations in practice. • The conceptual tools of game theory usually play only a supplementary role in dealing with practical situations. • Research is continuing, with some success, to extend the theory to more complex situations.