Decision Making Beyond Sometimes It Is . . . Cheating May Hurt . . - PowerPoint PPT Presentation

Group Decision . . . Case of Three or More . . . Nash’s Solution as a . . . Decision Making Beyond Sometimes It Is . . . Cheating May Hurt . . . Arrow’s “Impossibility For Territorial . . . How to Find Individual . . . We Must Take . . . Theorem”, with the Analysis Paradox of Love of Effects of Collusion and Title Page Mutual Attraction ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Department of Computer Science Page 1 of 31 University of Texas at El Paso Go Back El Paso, TX 79968 vladik@utep.edu Full Screen (based on a joint work Close with H. T. Nguyen and O. Kosheleva) Quit

Group Decision . . . 1. Group Decision Making and Arrow’s Impossibility Case of Three or More . . . Theorem Nash’s Solution as a . . . Sometimes It Is . . . • In 1951, Kenneth J. Arrow published his famous result Cheating May Hurt . . . about group decision making. For Territorial . . . • This result that became one of the main reasons for his How to Find Individual . . . 1972 Nobel Prize. We Must Take . . . • The problem: Paradox of Love Title Page – A group of n participants P 1 , . . . , P n needs to select ◭◭ ◮◮ between one of m alternatives A 1 , . . . , A m . – To find individual preferences, we ask each partic- ◭ ◮ ipant P i to rank the alternatives A j : Page 2 of 31 A j 1 ≻ i A j 2 ≻ i . . . ≻ i A j n . Go Back Full Screen – Based on these n rankings, we must form a single group ranking (equivalence ∼ is allowed). Close Quit

Group Decision . . . 2. Case of Two Alternatives Is Easy Case of Three or More . . . Nash’s Solution as a . . . • Simplest case: Sometimes It Is . . . – we have only two alternatives A 1 and A 2 , Cheating May Hurt . . . – each participant either prefers A 1 or prefers A 2 . For Territorial . . . How to Find Individual . . . • Solution: it is reasonable, for a group: We Must Take . . . – to prefer A 1 if the majority prefers A 1 , Paradox of Love – to prefer A 2 if the majority prefers A 2 , and Title Page – to claim A 1 and A 2 to be of equal quality for the ◭◭ ◮◮ group (denoted A 1 ∼ A 2 ) if there is a tie. ◭ ◮ Page 3 of 31 Go Back Full Screen Close Quit

Group Decision . . . 3. Case of Three or More Alternatives Is Not Easy Case of Three or More . . . Nash’s Solution as a . . . • Arrow’s result: no group decision rule can satisfy the Sometimes It Is . . . following natural conditions. Cheating May Hurt . . . • Pareto condition: if all participants prefer A j to A k , For Territorial . . . then the group should also prefer A j to A k . How to Find Individual . . . • Independence from Irrelevant Alternatives: the group We Must Take . . . ranking of A j vs. A k should not depend on other A i s. Paradox of Love Title Page • Arrow’s theorem: every group decision rule which sat- ◭◭ ◮◮ isfies these two condition is a dictatorship rule: ◭ ◮ – the group accepts the preferences of one of the par- ticipants as the group decision and Page 4 of 31 – ignores the preferences of all other participants. Go Back • This violates symmetry : that the group decision rules Full Screen should not depend on the order of the participants. Close Quit

Group Decision . . . 4. Beyond Arrow’s Impossibility Theorem Case of Three or More . . . Nash’s Solution as a . . . • Usual claim: Arrow’s Impossibility Theorem proves Sometimes It Is . . . that reasonable group decision making is impossible. Cheating May Hurt . . . • Our claim: Arrow’s result is only valid if we have bi- For Territorial . . . nary (“yes”-“no”) individual preferences. How to Find Individual . . . • Fact: this information does not fully describe a per- We Must Take . . . sons’ preferences. Paradox of Love Title Page • Example: the preference A 1 ≻ A 2 ≻ A 3 : ◭◭ ◮◮ – it may indicate that a person strongly prefers A 1 ◭ ◮ to A 2 , and strongly prefers A 2 to A 3 , and Page 5 of 31 – it may also indicate that this person strongly prefers A 1 to A 2 , and at the same time, A 2 ≈ A 3 . Go Back • How can this distinction be described: researchers in Full Screen decision making use the notion of utility . Close Quit

Group Decision . . . 5. Why Utility Case of Three or More . . . Nash’s Solution as a . . . • Idea of value: a person’s rational decisions are based on Sometimes It Is . . . the relative values to the person of different outcomes. Cheating May Hurt . . . • Monetary value is often used: in financial applications, For Territorial . . . the value is usually measured in monetary units (e.g., $). How to Find Individual . . . • Problem with monetary value: the same monetary amount We Must Take . . . may have different values for different people: Paradox of Love Title Page – a single dollar is likely to have more value to a poor ◭◭ ◮◮ person – than to a rich one. ◭ ◮ Page 6 of 31 • Thus, a new scale is needed: in view of this difference, in decision theory, researchers use a special utility scale. Go Back Full Screen Close Quit

Group Decision . . . 6. What Is Utility: a Reminder Case of Three or More . . . Nash’s Solution as a . . . • Main idea behind utility: a common approach is based Sometimes It Is . . . on preferences of a decision maker among lotteries . Cheating May Hurt . . . • Specifics: For Territorial . . . – take a very undesirable outcome A − and a very How to Find Individual . . . desirable outcome A + ; We Must Take . . . – consider the lottery A ( p ) in which we get A + with Paradox of Love given probability p and A − with probability 1 − p ; Title Page – a utility u ( B ) of an outcome B is defined as the ◭◭ ◮◮ probability p s.t. B is of the same quality as A ( p ): ◭ ◮ B ∼ A ( p ) = A ( u ( B )) . Page 7 of 31 • Assumptions behind this definition: Go Back – clearly, the larger p , the more preferable A ( p ): Full Screen p < p ′ ⇒ A ( p ) < A ( p ′ ); Close – the comparison amongst lotteries is a total order. Quit

Group Decision . . . 7. Different Utility Scales Case of Three or More . . . Nash’s Solution as a . . . • Fact: the numerical value u ( B ) of the utility depends on the choice of A − and A + . Sometimes It Is . . . Cheating May Hurt . . . • Natural question: relate u ( B ) with the values u ′ ( B ) For Territorial . . . corr. to another choice of A − and A + . How to Find Individual . . . • Answer: the utilities u ( B ) and u ′ ( B ) corresponding to We Must Take . . . different choices are related by a linear transformation: Paradox of Love Title Page u ′ ( B ) = a · u ( B ) + b for some a > 0 and b. ◭◭ ◮◮ • Conclusion: by using appropriate values a and b , we ◭ ◮ can re-scale utilities to make them more convenient. Page 8 of 31 • Example: in financial applications, we can make the Go Back scale closer to the monetary scale. Full Screen Close Quit

Group Decision . . . 8. Problem Case of Three or More . . . Nash’s Solution as a . . . • Situation: we have n incompatible events E 1 , . . . , E n Sometimes It Is . . . occurring with known probabilities p 1 , . . . , p n . Cheating May Hurt . . . • If E i occurs, we get the outcome B i . For Territorial . . . • Examples of events: How to Find Individual . . . We Must Take . . . – coins can fall heads or tails; Paradox of Love – dice can show 1 to 6. Title Page • We know: the utility u i = u ( B i ) of each outcome B i . ◭◭ ◮◮ • Find: the utility of the corresponding lottery. ◭ ◮ Page 9 of 31 Go Back Full Screen Close Quit

Group Decision . . . 9. Solution: Expected Utility Case of Three or More . . . Nash’s Solution as a . . . • Main idea: u ( B i ) = u i means that B i is equiv. to get- ting A + w/prob. u i and A − w/prob. 1 − u i . Sometimes It Is . . . Cheating May Hurt . . . • Conclusion: the lottery “ B i if E i ” is equivalent to the For Territorial . . . following two-step lottery: How to Find Individual . . . – first, we select E i with probability p i , and We Must Take . . . – then, for each i , we select A + with probability u i Paradox of Love and A − with the probability 1 − u i . Title Page ◭◭ ◮◮ • In this two-step lottery, the probability of getting A + is equal to ◭ ◮ p 1 · u 1 + . . . + p n · u n . Page 10 of 31 • Result: the utility of the lottery “if E i then B i ” is Go Back n n Full Screen � � p i · u i = p ( E i ) · u ( B i ) . u = Close i =1 i =1 Quit

Group Decision . . . 10. Nash’s Bargaining Solution Case of Three or More . . . Nash’s Solution as a . . . • How to describe preferences: for each participant P i , Sometimes It Is . . . def we can determine the utility u ij = u i ( A j ) of all A j . Cheating May Hurt . . . • Question: how to transform these utilities into a rea- For Territorial . . . sonable group decision rule? How to Find Individual . . . • Solution: was provided by another future Nobelist John We Must Take . . . Nash. Paradox of Love Title Page • Nash’s assumptions: ◭◭ ◮◮ – symmetry, ◭ ◮ – independence from irrelevant alternatives, and Page 11 of 31 – scale invariance – under replacing function u i ( A ) Go Back with an equivalent function a · u i ( A ), Full Screen Close Quit