Chapter 9 Utility Theory

1. Chapter 9Utility Theory Becky D’Amico Nick Gismondi

2. Overview • Utility theory is the process of assigning numbers to the various outcomes in the matrix games • Specifically, the science behind assigning these numbers in a way that reflects an actor’s preferences

3. Games with a Saddle Point • Saddle point (just in case you forgot) is when the column maximum equals the row minimum. Both Rose and Colin will prefer this strategy. • Game theory has to make a recommendation that makes sense, aka will represent the actual preferences of the players.

4. Games with a Saddle Point If we were to ask Rose to rank these strategies, she may or may not be able to depending on factors of the game. She may have intransitive preferences, or she may find that 2 choices are too different to compare. If the game is zero sum, Colin’s ranking would be opposite to Rose’s. If Rose could rank the strategies, we could apply numbers to them to match her preference. An example of this would be:

5. Games with a Saddle Point Saddle point would be BB, because it is the largest in its column and smallest in its row The saddle point will stay the same as long as order preserving transformations are done.

6. Order Preserving Transformations Multiply each value by 2, matrix becomes: Minimax = maximin, saddle point remains in the same spot

7. Ordinal Scale • From the book: “A scale on which higher numbers represent more preferred outcomes when only the number matters, not their absolute value or relative magnitude.”

8. Ordinal Scale… that makes sense • When items are classified according to whether they have more or less of a characteristic. The main characteristic is that the categories have a logical or ordered relationship to each other. These types of scales permit the measurement of degrees of difference, but not the specific amount of difference. This scale is very common in marketing, satisfaction and attitudinal research. For example, • How would you rate the service of our wait-staff? • Excellent 0 Very good 0 Good 0 Fair 0 Poor 0 • Although we would know that respondent X ("very good") thought the service to be better than respondent Y ("good"), we have no idea how much better nor can we even be sure that both respondents have the same understanding of what constitutes "good service" and therefore, whether they really differ in their opinion about its quality.

9. Ordinal Utilities • Numbers determined from preferences using ordinal scale. The numbers assigned to these preferences help us to determine dominance, saddle points, and etc. Yay! This kind of makes sense! 

10. Games Without a Saddle Point  • A mixed strategy is optimal • Numbers must be assigned to these letters in a way so that the ratio of differences (d-c) : (a-b) is meaningful • These scales where the order and the ratio is meaningful are known as interval scales • Numbers reflecting preferences on an interval scale are called cardinal utilities. • For a mixed strategy solution to be meaningful, the numbers in the matrix MUST be cardinal utilities.

11. So… what does all of that mean? • Say you have a matrix with values U, V, W, and X • Rose gives you her preferences: U, X, W, then V • You need to determine Rose’s utilities on a cardinal (meaningful) scale, and these numbers need to reflect her preferences accurately. • How on earth do you do this? LOTTERIES. (Not as fun as it sounds)

12. How To Run a Lottery • Assign numbers arbitrarily to V and U (in this case, U gets a higher number than V) • V = 0, V =100 • Ask her: “Now, which would you prefer: X for certain, or a lottery which would give you V ½ of the time and U ½ of the time?” • Rose responds that she prefers X. Therefore, X must rank higher than the midpoint between V and U. In this case, X > 50.

13. How To Run a Lottery • Now ask: “Would you prefer X for certain, or the lottery that would give you V ¼ of the time, and U ¾ of the time?” • Rose responds that she prefers the lottery. Therefore, X must be ranked below ¾ of the way from V to U. (In this case, below 75) • By continuing this process, we can eventually find a lottery that Rose would be indifferent towards. In this case, it would be the choice between X and getting V 4/10 of the time and U 6/10 of the time. • This will give us values for each of the options, and these values will represent cardinal utilities, giving them meaning. YAY! 

14. Details on the Process • Consistency means that Rose’s choices among any given lottery can be predicted by the line we create. • If Colin plays a mixed strategy (which he should), Rose’s game play is essentially a choice between lotteries, making the information about her choices in lotteries valuable to information about the game. • It is harder to get consistent cardinal utilities than ordinal utilities. (Real people might not do things consistently or rationally) • If we cannot assign cardinal utilities to a player’s preferences, we cannot obtain a meaningful mixed strategy solution.

15. Details on the Process • Colin’s preferences need to be opposite of Rose’s in order for it to be a zero sum game. • The endpoints for a cardinal utility scale are arbitrary (they don’t really matter), and they could have been chosen differently and lead to the same prediction in lottery choices. • You can transform one scale to another by using a positive linear function. This will not change the information they convey. • F(x) = ax + b where a>0

16. Example of Positive Linear Function • In constant sum games, games can become zero sum when some constant, C, is subtracted Transformed by positive linear function ½ (x-17):

17. How to Determine If a Game is Zero Sum • Plot the payoffs for each outcome on a coordinate plane. If all of the points lie along a line that has a negative slope, the game is zero sum.