Why Utility What Is Utility: a . . . Different Utility Scales Expected Utility Nash’s Bargaining . . . Properties of Nash’s . . . Decision Making Beyond How We Can . . . Problem: Sometimes . . . Arrow’s “Impossibility In Case of Uncertainty, . . . Case Study: Territorial . . . Theorem”, with the Analysis Decision Making . . . For Territorial . . . How to Find Individual . . . of Effects of Collusion and Comments How to Find Individual . . . Mutual Attraction We Must Take . . . Paradox of Love Why Two and not Three? Hung T. Nguyen Emotional vs. . . . New Mexico State University Acknowledgments hunguyen@nmsu.edu Title Page Olga Kosheleva and Vladik Kreinovich ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ vladik@utep.edu Page 1 of 25 Go Back Full Screen
Why Utility What Is Utility: a . . . 1. Group Decision Making and Arrow’s Impossibility Different Utility Scales Expected Utility Theorem Nash’s Bargaining . . . Properties of Nash’s . . . • In 1951, Kenneth J. Arrow published his famous result about group decision How We Can . . . making. Problem: Sometimes . . . • This result that became one of the main reasons for his 1972 Nobel Prize. In Case of Uncertainty, . . . Case Study: Territorial . . . • The problem: Decision Making . . . – A group of n participants P 1 , . . . , P n needs to select between one of m For Territorial . . . alternatives A 1 , . . . , A m . How to Find Individual . . . Comments – To find individual preferences, we ask each participant P i to rank the alternatives A j from the most desirable to the least desirable: How to Find Individual . . . We Must Take . . . A j 1 ≻ i A j 2 ≻ i . . . ≻ i A j n . Paradox of Love Why Two and not Three? – Based on these n rankings, we must form a single group ranking (in the Emotional vs. . . . group ranking, equivalence ∼ is allowed). Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 25 Go Back Full Screen
Why Utility What Is Utility: a . . . 2. Case of Two Alternatives Is Easy Different Utility Scales Expected Utility • Simplest case: Nash’s Bargaining . . . Properties of Nash’s . . . – we have only two alternatives A 1 and A 2 , How We Can . . . – each participant either prefers A 1 or prefers A 2 . Problem: Sometimes . . . In Case of Uncertainty, . . . • Solution: it is reasonable, for a group: Case Study: Territorial . . . – to prefer A 1 if the majority prefers A 1 , Decision Making . . . For Territorial . . . – to prefer A 2 if the majority prefers A 2 , and How to Find Individual . . . – to claim A 1 and A 2 to be of equal quality for the group (denoted A 1 ∼ Comments A 2 ) if there is a tie. How to Find Individual . . . We Must Take . . . Paradox of Love Why Two and not Three? Emotional vs. . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 25 Go Back Full Screen
Why Utility What Is Utility: a . . . 3. Case of Three or More Alternatives Is Not Easy Different Utility Scales Expected Utility • Arrow’s result: no group decision rule can satisfy the following natural con- Nash’s Bargaining . . . ditions. Properties of Nash’s . . . How We Can . . . • Pareto condition: if all participants prefer A j to A k , then the group should Problem: Sometimes . . . also prefer A j to A k . In Case of Uncertainty, . . . • Independence from Irrelevant Alternatives: the group ranking between A j and Case Study: Territorial . . . A k should depend only on how participants rank A j and A k – and should Decision Making . . . not depend on how they rank other alternatives. For Territorial . . . How to Find Individual . . . • Arrow’s theorem: every group decision rule which satisfies these two condition Comments is a dictatorship rule: How to Find Individual . . . – the group accepts the preferences of one of the participants as the group We Must Take . . . decision and Paradox of Love – ignores the preferences of all other participants. Why Two and not Three? Emotional vs. . . . • This clearly violates another reasonable condition of symmetry : that the Acknowledgments group decision rules should not depend on the order in which we list the Title Page participants. ◭◭ ◮◮ ◭ ◮ Page 4 of 25 Go Back Full Screen
Why Utility What Is Utility: a . . . 4. Beyond Arrow’s Impossibility Theorem: Nash’s Bar- Different Utility Scales Expected Utility gaining Solution Nash’s Bargaining . . . Properties of Nash’s . . . • Usual claim: Arrow’s Impossibility Theorem is often cited as a proof that How We Can . . . reasonable group decision making is impossible – e.g., that a perfect voting Problem: Sometimes . . . procedure is impossible. In Case of Uncertainty, . . . • Our claim: Arrow’s result is only valid if we have binary (partial) information Case Study: Territorial . . . about individual preferences. Decision Making . . . For Territorial . . . • Conclusion: that the pessimistic interpretation of Arrow’s result is, well, too How to Find Individual . . . pessimistic :-) Comments • Implicit assumption behind Arrow’s result: the only information we have How to Find Individual . . . about individual preferences is the binary (“yes”-“no”) preferences between We Must Take . . . the alternatives. Paradox of Love • Fact: this information does not fully describe a persons’ preferences. Why Two and not Three? Emotional vs. . . . • Example: the preference A 1 ≻ A 2 ≻ A 3 : Acknowledgments – it may indicate that a person strongly prefers A 1 to A 2 , and strongly Title Page prefers A 2 to A 3 , and ◭◭ ◮◮ – it may also indicate that this person strongly prefers A 1 to A 2 , and at the same time, A 2 is almost of the same quality as A 3 . ◭ ◮ • How can this distinction be described: to describe this degree of preference, Page 5 of 25 researchers in decision making use the notion of utility . Go Back Full Screen
Why Utility What Is Utility: a . . . 5. Why Utility Different Utility Scales Expected Utility • Idea of value: a person’s rational decisions are based on the relative values Nash’s Bargaining . . . to the person of different outcomes. Properties of Nash’s . . . How We Can . . . • Monetary value is often used: in financial applications, the value is usually Problem: Sometimes . . . measured in monetary units such as dollars. In Case of Uncertainty, . . . • Problem with monetary value: the same monetary amount may have different Case Study: Territorial . . . values for different people: Decision Making . . . For Territorial . . . – a single dollar is likely to have more value to a poor person How to Find Individual . . . – than to a rich one. Comments How to Find Individual . . . • Thus, a new scale is needed: in view of this difference, in decision theory, to describe the relative values of different outcomes, researchers use a special We Must Take . . . utility scale instead of the more traditional monetary scales. Paradox of Love Why Two and not Three? Emotional vs. . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 25 Go Back Full Screen
Why Utility What Is Utility: a . . . 6. What Is Utility: a Reminder Different Utility Scales Expected Utility • Main idea behind utility: a common approach is based on preferences of a Nash’s Bargaining . . . decision maker among lotteries . Properties of Nash’s . . . How We Can . . . • Specifics: Problem: Sometimes . . . – take a very undesirable outcome A − and a very desirable outcome A + ; In Case of Uncertainty, . . . – consider the lottery A ( p ) in which we get A + with given probability p Case Study: Territorial . . . and A − with probability 1 − p ; Decision Making . . . For Territorial . . . – a utility u ( B ) of an outcome B is defined as the probability p for which How to Find Individual . . . B is of the same quality as A ( p ): B ∼ A ( p ) = A ( u ( B )) . Comments • Assumptions behind this definition: How to Find Individual . . . We Must Take . . . – clearly, the larger p , the more preferable A ( p ): Paradox of Love p < p ′ ⇒ A ( p ) < A ( p ′ ); Why Two and not Three? Emotional vs. . . . – the comparison amongst lotteries is a linear order – i.e., a person can Acknowledgments always select one of the two alternatives; Title Page – comparison is Archimedean – i.e. if for all ε > 0, an outcome B is better than A ( p − ε ) and worse than A ( p + ε ), then it is of the same quality as ◭◭ ◮◮ A ( p ): B ∼ A ( p ). ◭ ◮ Page 7 of 25 Go Back Full Screen
Recommend
More recommend
