Traditional . . . Actual Choices Are . . . McFadden’s Formulas . . . Econometric Models Analysis of the Problem Deriving McFadden’s . . . of Probabilistic Choice: Discussion Our Main Idea Beyond McFadden’s Conclusions and . . . Formulas Home Page Title Page Olga Kosheleva 1 , Vladik Kreinovich 1 ◭◭ ◮◮ and Songsak Sriboonchitta 2 ◭ ◮ 1 University of Texas at El Paso El Paso, Texas 79968, USA Page 1 of 17 olgak@utep.edu, vladik@utep.edu 2 Faculty of Economics, Chiang Mai University Go Back Chiang Mai, Thailand, songsakecon@gmail.com Full Screen Close Quit
Traditional . . . 1. Traditional (Deterministic Choice) Approach Actual Choices Are . . . to Decision Making McFadden’s Formulas . . . Analysis of the Problem • In the traditional approach to decision making, we as- Deriving McFadden’s . . . sume that for every two alternatives a and b : Discussion • either the decision maker always prefers a , Our Main Idea • or the decision maker always prefers b , Conclusions and . . . • or, to the decision maker, a and b are equivalent. Home Page Title Page • Then, decision maker’s preferences can be described by utilities defined as follows. ◭◭ ◮◮ • We select two alternatives which are not present in the ◭ ◮ original choices: Page 2 of 17 • a very bad alternative a 0 , and Go Back • a very good alternative a 1 . Full Screen • Then, each actual alternative a is better than a 0 and Close worse than a 1 : a 0 < a < a 1 . Quit
Traditional . . . 2. Traditional Decision Making (cont-d) Actual Choices Are . . . McFadden’s Formulas . . . • To gauge the quality of the alternative a to the decision Analysis of the Problem maker, we can consider lotteries L ( p ) in which: Deriving McFadden’s . . . • we get a 1 with probability p and Discussion • we get a 0 with the remaining probability 1 − p . Our Main Idea Conclusions and . . . • For every p , we either have L ( p ) < a or a < L ( p ) or Home Page L ( p ) ∼ a . Title Page • When p = 1, L (1) = a 1 , thus a < L (1). ◭◭ ◮◮ • When p = 0, L (0) = a 0 , thus L (0) < a . ◭ ◮ • Clearly, the larger the probability p of the very good Page 3 of 17 outcome, the better the lottery; thus, if p < p ′ , then: Go Back • a < L ( p ) implies a < L ( p ′ ), and Full Screen • L ( p ′ ) < a implies L ( p ) < a . Close Quit
Traditional . . . 3. Traditional Decision Making (cont-d) Actual Choices Are . . . McFadden’s Formulas . . . • Therefore, we can conclude that Analysis of the Problem sup { p : L ( p ) < a } = inf { p : a < L ( p ) } . Deriving McFadden’s . . . Discussion def • u ( a ) = sup { p : L ( p ) < a } = inf { p : a < L ( p ) } has the Our Main Idea following properties: Conclusions and . . . • if p < u , then L ( p ) < a ; and Home Page • if p > a , then a < L ( p ). Title Page • In particular, for every small ε > 0, we have ◭◭ ◮◮ L ( u ( a ) − ε ) < a < L ( u ( a ) + ε ) . ◭ ◮ • So, a is “equivalent” to the lottery L ( p ) in which a 1 is Page 4 of 17 selected with the probability p = u ( a ): a ≡ L ( u ( a )) . Go Back • This probability u ( a ) is what is known as utility . Full Screen • Once we know all the utility values, we select the al- Close ternative with the largest utility. Quit
Traditional . . . 4. Traditional Decision Making (final) Actual Choices Are . . . McFadden’s Formulas . . . • Indeed, as we have mentioned, p < p ′ implies that Analysis of the Problem L ( p ) < L ( p ′ ), so when u ( a ) < u ( b ), we have Deriving McFadden’s . . . a ≡ L ( u ( a )) < L ( u ( b )) ≡ b and thus a < b. Discussion Our Main Idea • The above definition of utility depends on the choice Conclusions and . . . of two alternatives a 0 and a 1 . Home Page • If we select different a ′ 0 and a ′ 1 , then, as one can show, Title Page we get u ′ ( a ) = k · u ( a ) + ℓ for some k > 0 and ℓ . ◭◭ ◮◮ • Thus, utility is defined modulo linear transformation . ◭ ◮ Page 5 of 17 Go Back Full Screen Close Quit
Traditional . . . 5. Actual Choices Are Often Probabilistic Actual Choices Are . . . McFadden’s Formulas . . . • People sometimes make different choices when repeat- Analysis of the Problem edly presented with the same alternatives a and b . Deriving McFadden’s . . . • This is especially true when the compared alternatives Discussion a and b are close in value. Our Main Idea Conclusions and . . . • In such situations, we would like to predict the proba- Home Page bility P ( a, A ) of a from a set A . Title Page • We can still have a deterministic distinction: b > a if the person selects a more frequently than b : ◭◭ ◮◮ ◭ ◮ P ( a, { a, b } ) > 0 . 5 . Page 6 of 17 • Based on > , we can determine the utilities u ( a ). Go Back • It is reasonable to assume that P ( a, A ) depends only Full Screen on the utilities u ( a ), . . . , u ( b ). Close Quit
Traditional . . . 6. McFadden’s Formulas for Probabilistic Selec- Actual Choices Are . . . tion McFadden’s Formulas . . . Analysis of the Problem • The 2001 Nobelist D. McFadden proposed the follow- Deriving McFadden’s . . . ing formula for the desired probability P ( a, A ): Discussion exp( β · u ( a )) Our Main Idea P ( a, A ) = exp( β · u ( b )) . � Conclusions and . . . b ∈ A Home Page • In many practical situations, this formula indeed de- Title Page scribes people’s choices really well. ◭◭ ◮◮ • In some case, alternative formulas provide a better ex- ◭ ◮ planation of the empirical choices. Page 7 of 17 • In this talk, we use natural symmetries to come up with Go Back an appropriate generalization of McFadden’s formulas. Full Screen Close Quit
Traditional . . . 7. Analysis of the Problem Actual Choices Are . . . McFadden’s Formulas . . . • We may have many different alternatives a , b , . . . Analysis of the Problem • In some cases, we prefer a , in other cases, we prefer b . Deriving McFadden’s . . . Discussion • It is reasonable to require that: Our Main Idea • once we have decided on selecting either a or b , then Conclusions and . . . • the relative frequency of selecting a should be the Home Page same as when we simply select between a and b : Title Page P ( b, A ) = P ( a, { a, b } ) P ( a, A ) P ( a, { a, b } ) P ( b, { a, b } ) = 1 − P ( a, { a, b } ) . ◭◭ ◮◮ ◭ ◮ • Let us add a new alternative a n to our list, then: Page 8 of 17 P ( a, A ) P ( a, { a, a n } ) P ( a n , A ) = 1 − P ( a, { a, a n } ) , so P ( a, A ) = P ( a n , A ) · f ( a ) , Go Back Full Screen P ( a, { a, a n } ) def def where c = P ( a n , A ) and f ( a ) = 1 − P ( a, { a, a n } ) . Close Quit
Traditional . . . 8. Analysis of the Problem (cont-d) Actual Choices Are . . . McFadden’s Formulas . . . • c can be found from the condition that one of b ∈ A will Analysis of the Problem f ( a ) be selected: � P ( b, A ) = 1, so P ( a, A ) = f ( b ) . Deriving McFadden’s . . . � b ∈ A b ∈ A Discussion • We assumed that the probabilities depend only on the Our Main Idea utilities u ( a ). Conclusions and . . . Home Page • We thus conclude that f ( a ) must depend only on the Title Page utilities: f ( a ) = F ( u ( a )) for some F ( u ), and ◭◭ ◮◮ F ( u ( a )) P ( a, A ) = F ( u ( b )) . � ◭ ◮ b ∈ A Page 9 of 17 • Thus, all we need is to find an appropriate function Go Back F ( u ). Full Screen Close Quit
Traditional . . . 9. Properties of F ( u ) Actual Choices Are . . . McFadden’s Formulas . . . • The better the alternative a , i.e., the larger u ( a ), the Analysis of the Problem higher should be the probability P ( a, A ). Deriving McFadden’s . . . • Thus, F ( u ) is an increasing function of the utility u . Discussion • If we multiply all the values of F ( u ) by a constant, we Our Main Idea will get the exact same probabilities. Conclusions and . . . Home Page • Utilities are defined modulo a general linear transfor- Title Page mation. ◭◭ ◮◮ • In particular, it is possible to add a constant to all the utility values u ( a ) → u ′ ( a ) = u ( a ) + c. ◭ ◮ • Since this shift does not change the preferences, it is Page 10 of 17 reasonable to require that for u ′ ( a ), we get the same Go Back probabilities. Full Screen • Using new utility values u ′ ( a ) = u ( a ) + c means that Close we replace F ( u ( a )) with F ( u ′ ( a )) = F ( u ( a ) + c ). Quit
Traditional . . . 10. Deriving McFadden’s Formula Actual Choices Are . . . McFadden’s Formulas . . . • Using new utility values u ′ ( a ) = u ( a ) + c means that Analysis of the Problem we replace F ( u ( a )) with F ( u ′ ( a )) = F ( u ( a ) + c ). Deriving McFadden’s . . . • This is equivalent to using the original utility values Discussion def but with a new function F ′ ( u ) = F ( u + c ). Our Main Idea Conclusions and . . . • The functions F ( u ) and F ′ ( u ) describe the same prob- Home Page abilities if and only if F ′ ( u ) = C · F ( u ) for some C . Title Page • So, F ( u + c ) = C ( c ) · F ( u ) for some C ( c ). ◭◭ ◮◮ • It is known that every monotonic solution to this func- ◭ ◮ tion equation has the form F ( u ) = C 0 · exp( β · u ). Page 11 of 17 • This is exactly McFadden’s formula. Go Back Full Screen Close Quit
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