Decision Making Privacy-Motivated . . . under Uncertainty: - PowerPoint PPT Presentation

Quantitative . . . The Notion of Utility Traditional Approach: . . . Need for Distributed . . . Decision Making Privacy-Motivated . . . under Uncertainty: Uncertainty Leads to . . . Uncertainty in . . . Algorithmic Approach Uncertainty in System . . . Symmetry Approach . . . (brief overview of related Home Page UTEP research) Title Page ◭◭ ◮◮ Vladik Kreinovich ◭ ◮ Department of Computer Science Page 1 of 62 University of Texas at El Paso El Paso, TX 79968, USA Go Back vladik@utep.edu Full Screen http://www.cs.utep.edu/vladik Close Quit

Quantitative . . . The Notion of Utility 1. Quantitative Approach to Decision Making: Traditional Approach: . . . Misunderstandings Need for Distributed . . . • Researchers and practitioners in computer science usu- Privacy-Motivated . . . ally start with the utility-based approach. Uncertainty Leads to . . . Uncertainty in . . . • Many humanities researchers believe that the utility- Uncertainty in System . . . based approach is oversimplified and long discredited . Symmetry Approach . . . • Main reason: they consider an easy-to-dismiss carica- Home Page ture instead of the actual utility approach. Title Page • In view of this widely spread misunderstanding, we first ◭◭ ◮◮ start by explaining the actual utility-based approach. ◭ ◮ • Our main area of research is how to add uncertainty to the traditional approach. Page 2 of 62 • We concentrate on interval and fuzzy uncert., empha- Go Back sizing that “fuzzy” has a very precise meaning in CS. Full Screen • In this process, we provide examples of applications . Close Quit

Quantitative . . . The Notion of Utility 2. Decision Making: General Need and Traditional Traditional Approach: . . . Approach Need for Distributed . . . • To make a decision, we must: Privacy-Motivated . . . Uncertainty Leads to . . . – find out the user’s preference, and Uncertainty in . . . – help the user select an alternative which is the best Uncertainty in System . . . – according to these preferences. Symmetry Approach . . . • Traditional approach is based on an assumption that Home Page for each two alternatives A ′ and A ′′ , a user can tell: Title Page – whether the first alternative is better for him/her; ◭◭ ◮◮ we will denote this by A ′′ < A ′ ; ◭ ◮ – or the second alternative is better; we will denote this by A ′ < A ′′ ; Page 3 of 62 Go Back – or the two given alternatives are of equal value to the user; we will denote this by A ′ = A ′′ . Full Screen Close Quit

Quantitative . . . The Notion of Utility 3. The Notion of Utility Traditional Approach: . . . • Under the above assumption, we can form a natural Need for Distributed . . . numerical scale for describing preferences. Privacy-Motivated . . . Uncertainty Leads to . . . • Let us select a very bad alternative A 0 and a very good Uncertainty in . . . alternative A 1 . Uncertainty in System . . . • Then, most other alternatives are better than A 0 but Symmetry Approach . . . worse than A 1 . Home Page • For every prob. p ∈ [0 , 1], we can form a lottery L ( p ) Title Page in which we get A 1 w/prob. p and A 0 w/prob. 1 − p . ◭◭ ◮◮ • When p = 0, this lottery simply coincides with the ◭ ◮ alternative A 0 : L (0) = A 0 . Page 4 of 62 • The larger the probability p of the positive outcome Go Back increases, the better the result: Full Screen p ′ < p ′′ implies L ( p ′ ) < L ( p ′′ ) . Close Quit

Quantitative . . . The Notion of Utility 4. The Notion of Utility (cont-d) Traditional Approach: . . . • Finally, for p = 1, the lottery coincides with the alter- Need for Distributed . . . native A 1 : L (1) = A 1 . Privacy-Motivated . . . Uncertainty Leads to . . . • Thus, we have a continuous scale of alternatives L ( p ) Uncertainty in . . . that monotonically goes from L (0) = A 0 to L (1) = A 1 . Uncertainty in System . . . • Due to monotonicity, when p increases, we first have Symmetry Approach . . . 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 : ◭◭ ◮◮ def u ( A ) = sup { p : L ( p ) < A } = inf { p : L ( p ) > A } . ◭ ◮ • Then, for every ε > 0, we have Page 5 of 62 L ( u ( A ) − ε ) < A < L ( u ( A ) + ε ) . Go Back • We will describe such (almost) equivalence by ≡ , i.e., Full Screen we will write that A ≡ L ( u ( A )). Close Quit

Quantitative . . . The Notion of Utility 5. Fast Iterative Process for Determining u ( A ) Traditional Approach: . . . • Initially: we know the values u = 0 and u = 1 such Need for Distributed . . . that A ≡ L ( u ( A )) for some u ( A ) ∈ [ u, u ]. Privacy-Motivated . . . Uncertainty Leads to . . . • What we do: we compute the midpoint u mid of the Uncertainty in . . . interval [ u, u ] and compare A with L ( u mid ). Uncertainty in System . . . • Possibilities: A ≤ L ( u mid ) and L ( u mid ) ≤ A . Symmetry Approach . . . • Case 1: if A ≤ L ( u mid ), then u ( A ) ≤ u mid , so Home Page u ∈ [ u, u mid ] . Title Page ◭◭ ◮◮ • Case 2: if L ( u mid ) ≤ A , then u mid ≤ u ( A ), so ◭ ◮ u ∈ [ u mid , u ] . Page 6 of 62 • After each iteration, we decrease the width of the in- Go Back terval [ u, u ] by half. • After k iterations, we get an interval of width 2 − k which Full Screen contains u ( A ) – i.e., we get u ( A ) w/accuracy 2 − k . Close Quit

Quantitative . . . The Notion of Utility 6. How to Make a Decision Based on Utility Val- Traditional Approach: . . . ues Need for Distributed . . . • Suppose that we have found the utilities u ( A ′ ), u ( A ′′ ), Privacy-Motivated . . . . . . , of the alternatives A ′ , A ′′ , . . . Uncertainty Leads to . . . Uncertainty in . . . • Which of these alternatives should we choose? Uncertainty in System . . . • By definition of utility, we have: Symmetry Approach . . . Home Page • A ≡ L ( u ( A )) for every alternative A , and • L ( p ′ ) < L ( p ′′ ) if and only if p ′ < p ′′ . Title Page • We can thus conclude that A ′ is preferable to A ′′ if and ◭◭ ◮◮ only if u ( A ′ ) > u ( A ′′ ). ◭ ◮ • In other words, we should always select an alternative Page 7 of 62 with the largest possible value of utility. Go Back • So, to find the best solution, we must solve the corre- Full Screen sponding optimization problem. Close Quit

Quantitative . . . The Notion of Utility 7. Before We Go Further: Caution Traditional Approach: . . . • We are not claiming that people estimate probabilities Need for Distributed . . . when they make decisions: we know they often don’t. Privacy-Motivated . . . Uncertainty Leads to . . . • Our claim: when people make definite and consistent Uncertainty in . . . choices, these choices can be described by probabilities. Uncertainty in System . . . • Example: a falling rock does not solve equations but Symmetry Approach . . . follows Newton’s equations ma = md 2 x dt 2 = − mg. Home Page Title Page • In practice, decisions are often not definite (uncertain) and not consistent. ◭◭ ◮◮ ◭ ◮ • Inconsistency is one of the reasons why people make bad decisions (drugs, health hazards, speeding). Page 8 of 62 • People do choose A > B > C > A ; we need psycholo- Go Back gists and sociologists to study and solve this problem. Full Screen • Uncertainty is what we concentrate on; see below. Close Quit

Quantitative . . . The Notion of Utility 8. How to Estimate Utility of an Action Traditional Approach: . . . • For each action, we usually know possible outcomes Need for Distributed . . . S 1 , . . . , S n . Privacy-Motivated . . . Uncertainty Leads to . . . • We can often estimate the prob. p 1 , . . . , p n of these out- Uncertainty in . . . comes. Uncertainty in System . . . • By definition of utility, each situation S i is equiv. to a Symmetry Approach . . . lottery L ( u ( S i )) in which we get: Home Page • A 1 with probability u ( S i ) and Title Page • A 0 with the remaining probability 1 − u ( S i ). ◭◭ ◮◮ • Thus, the action is equivalent to a complex lottery in ◭ ◮ which: Page 9 of 62 • first, we select one of the situations S i with proba- Go Back bility p i : P ( S i ) = p i ; • then, depending on S i , we get A 1 with probability Full Screen P ( A 1 | S i ) = u ( S i ) and A 0 w/probability 1 − u ( S i ). Close Quit

Quantitative . . . The Notion of Utility 9. How to Estimate Utility of an Action (cont-d) Traditional Approach: . . . • Reminder: Need for Distributed . . . Privacy-Motivated . . . • first, we select one of the situations S i with proba- Uncertainty Leads to . . . bility p i : P ( S i ) = p i ; Uncertainty in . . . • then, depending on S i , we get A 1 with probability Uncertainty in System . . . P ( A 1 | S i ) = u ( S i ) and A 0 w/probability 1 − u ( S i ). Symmetry Approach . . . • The prob. of getting A 1 in this complex lottery is: Home Page n n � � Title Page P ( A 1 ) = P ( A 1 | S i ) · P ( S i ) = u ( S i ) · p i . i =1 i =1 ◭◭ ◮◮ • In the complex lottery, we get: ◭ ◮ � n • A 1 with prob. u = p i · u ( S i ), and Page 10 of 62 i =1 Go Back • A 0 w/prob. 1 − u . Full Screen • So, we should select the action with the largest value of expected utility u = � p i · u ( S i ). Close Quit