Outline Decision Making: . The Notion of Utility Subjective Probabilities Quantum Social Science Example of Seemingly . Relation to Quantum . From Individual to . (A review of a recent book by Alternative (Non- . E. Haven and A. Khrennikov) Why Humans Use . Home Page Presented by Vladik Kreinovich Title Page Department of Computer Science JJ II University of Texas at El Paso El Paso, TX 79968, USA J I [email protected] Page1 of 29 http://www.cs.utep.edu/vladik Go Back Full Screen Close Quit Outline Decision Making: . 1. Outline The Notion of Utility • Decision theory { how to describe preferences if people Subjective Probabilities behave rationally. Example of Seemingly . Relation to Quantum . • In practice, people sometimes deviate from rational be- havior. From Individual to . Alternative (Non- . • In some cases, quantum-type models are helpful in ex- Why Humans Use . plaining actual human behavior. Home Page • The authors use this to explain fluctuations of the stock Title Page market prices. JJ II • They also speculate on why humans use quantum-stype J I reasoning. Page2 of 29 Go Back Full Screen Close Quit Outline Decision Making: . 2. Decision Making: General Need and Traditional Approach The Notion of Utility Subjective Probabilities • To make a decision, we must: Example of Seemingly . Relation to Quantum . { find out the user's preference, and From Individual to . { help the user select an alternative which is the best Alternative (Non- . { according to these preferences. Why Humans Use . • Traditional approach is based on an assumption that Home Page 0 00 for each two alternatives A and A , a user can tell: Title Page { whether the first alternative is better for him/her; JJ II we will denote this by A00 < A0; J I { or the second alternative is better; we will denote this by A0 < A00; Page3 of 29 { or the two given alternatives are of equal value to Go Back 0 00 the user; we will denote this by A = A . Full Screen Close Quit Outline Decision Making: . 3. The Notion of Utility The Notion of Utility • Under the above assumption, we can form a natural Subjective Probabilities numerical scale for describing preferences. Example of Seemingly . Relation to Quantum . • Let us select a very bad alternative A0 and a very good From Individual to . alternative A1. Alternative (Non- . • Then, most other alternatives are better than A0 but Why Humans Use . worse than A1. Home Page • For every prob. p 2 [0; 1], we can form a lottery L(p) Title Page in which we get A1 w/prob. p and A0 w/prob. 1 − p. JJ II • When p = 0, this lottery simply coincides with the J I alternative A0: L(0) = A0. Page4 of 29 • The larger the probability p of the positive outcome Go Back increases, the better the result: Full Screen p0 < p00 implies L(p0) < L(p00): Close Quit Outline Decision Making: . 4. The Notion of Utility (cont-d) The Notion of Utility • Finally, for p = 1, the lottery coincides with the alter- Subjective Probabilities native A1: L(1) = A1. Example of Seemingly . Relation to Quantum . • Thus, we have a continuous scale of alternatives L(p) From Individual to . that monotonically goes from L(0) = A0 to L(1) = A1. Alternative (Non- . • Due to monotonicity, when p increases, we first have Why Humans Use . L(p) < A, then we have L(p) > A. Home Page • The threshold value is called the utility of the alterna- Title Page tive A: JJ II u(A) def= supfp : L(p) < Ag = inffp : L(p) > Ag: J I • Then, for every " > 0, we have Page5 of 29 L(u(A) − ") < A < L(u(A) + "): Go Back • We will describe such (almost) equivalence by ≡, i.e., Full Screen we will write that A ≡ L(u(A)). Close Quit Outline Decision Making: . 5. Fast Iterative Process for Determining u(A) The Notion of Utility • Initially: we know the values u = 0 and u = 1 such Subjective Probabilities that A ≡ L(u(A)) for some u(A) 2 [u; u]. Example of Seemingly . Relation to Quantum . • What we do: we compute the midpoint umid of the From Individual to . interval [u; u] and compare A with L(umid). Alternative (Non- . • Possibilities: A ≤ L(u ) and L(u ) ≤ A. mid mid Why Humans Use . • Case 1: if A ≤ L(umid), then u(A) ≤ umid, so Home Page Title Page u 2 [u; umid]: JJ II • Case 2: if L(umid) ≤ A, then umid ≤ u(A), so J I u 2 [umid; u]: Page6 of 29 • After each iteration, we decrease the width of the in- terval [u; u] by half. Go Back • After k iterations, we get an interval of width 2−k which Full Screen −k contains u(A) { i.e., we get u(A) w/accuracy 2 . Close Quit Outline Decision Making: . 6. How to Make a Decision Based on Utility Val- ues The Notion of Utility Subjective Probabilities • Suppose that we have found the utilities u(A0), u(A00), Example of Seemingly . 0 00 . , of the alternatives A , A ,... Relation to Quantum . • Which of these alternatives should we choose? From Individual to . Alternative (Non- . • By definition of utility, we have: Why Humans Use . • A ≡ L(u(A)) for every alternative A, and Home Page • L(p0) < L(p00) if and only if p0 < p00. Title Page • We can thus conclude that A0 is preferable to A00 if and JJ II 0 00 only if u(A ) > u(A ). J I • In other words, we should always select an alternative Page7 of 29 with the largest possible value of utility. Go Back Full Screen Close Quit Outline Decision Making: . 7. How to Estimate Utility of an Action The Notion of Utility • For each action, we usually know possible outcomes Subjective Probabilities S1;:::;Sn. Example of Seemingly . Relation to Quantum . • We can often estimate the prob. p1; : : : ; pn of these out- comes. From Individual to . Alternative (Non- . • By definition of utility, each situation S is equiv. to a i Why Humans Use . lottery L(u(Si)) in which we get: Home Page • A1 with probability u(Si) and Title Page • A0 with the remaining probability 1 − u(Si). JJ II • Thus, the action is equivalent to a complex lottery in J I which: Page8 of 29 • first, we select one of the situations Si with proba- Go Back bility pi: P (Si) = pi; Full Screen • then, depending on Si, we get A1 with probability P (A1 j Si) = u(Si) and A0 w/probability 1 − u(Si). Close Quit Outline Decision Making: . 8. How to Estimate Utility of an Action (cont-d) The Notion of Utility • Reminder: Subjective Probabilities Example of Seemingly . • first, we select one of the situations Si with proba- Relation to Quantum . bility pi: P (Si) = pi; From Individual to . • then, depending on Si, we get A1 with probability Alternative (Non- . P (A1 j Si) = u(Si) and A0 w/probability 1 − u(Si). Why Humans Use . • The prob. of getting A1 in this complex lottery is: Home Page n n X X Title Page P (A1) = P (A1 j Si) · P (Si) = u(Si) · pi: i=1 i=1 JJ II • In the complex lottery, we get: J I n P • A1 with prob. u = pi · u(Si), and Page9 of 29 i=1 Go Back • A0 w/prob. 1 − u. • So, we should select the action with the largest value Full Screen P of expected utility u = pi · u(Si). Close Quit Outline Decision Making: . 9. Non-Uniqueness of Utility The Notion of Utility Subjective Probabilities • The above definition of utility u depends on A0, A1. 0 0 Example of Seemingly . • What if we use different alternatives A0 and A1? Relation to Quantum . • Every A is equivalent to a lottery L(u(A)) in which we From Individual to . get A w/prob. u(A) and A w/prob. 1 − u(A). 1 0 Alternative (Non- . 0 0 • For simplicity, let us assume that A0 < A0 < A1 < A1. Why Humans Use . 0 0 0 0 Home Page • Then, A0 ≡ L (u (A0)) and A1 ≡ L (u (A1)). • So, A is equivalent to a complex lottery in which: Title Page JJ II 1) we select A1 w/prob. u(A) and A0 w/prob. 1−u(A); 0 0 0 2) depending on Ai, we get A1 w/prob. u (Ai) and A0 J I 0 w/prob. 1 − u (Ai). Page 10 of 29 0 • In this complex lottery, we get A1 with probability Go Back 0 0 0 0 u (A) = u(A) · (u (A1) − u (A0)) + u (A0): Full Screen • So, in general, utility is defined modulo an (increasing) Close linear transformation u0 = a · u + b, with a > 0. Quit Outline Decision Making: . 10. Subjective Probabilities The Notion of Utility Subjective Probabilities • In practice, we often do not know the probabilities pi of different outcomes. Example of Seemingly . • For each event E, a natural way to estimate its subjec- Relation to Quantum . tive probability is to fix a prize (e.g., $1) and compare: From Individual to . Alternative (Non- . { the lottery `E in which we get the fixed prize if the Why Humans Use . event E occurs and 0 is it does not occur, with Home Page { a lottery `(p) in which we get the same amount Title Page with probability p. JJ II • Here, similarly to the utility case, we get a value ps(E) for which, for every " > 0: J I Page 11 of 29 `(ps(E) − ") < `E < `(ps(E) + "): Go Back • Then, the utility of an action with possible outcomes n Full Screen P S1;:::;Sn is equal to u = ps(Ei) · u(Si).