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Traditional Decision . . . Utility Is Defined . . . What If We Only Have . . . Two Envelopes Problem It Is Important to Take All How Realistic Is This . . . Available Information into At First Glance, This . . . This Is Not Really a . . . Account When Making a Idealized Formulation: . . . So What Should We Do? Decision: Case of the Two Home Page Envelopes Problem Title Page ◭◭ ◮◮ Laxman Bokati, Olga Kosheleva, and ◭ ◮ Vladik Kreinovich Page 1 of 36 University of Texas at El Paso 500 W. University Go Back El Paso, TX 79968, USA Full Screen lbokati@miners.utep.edu, olgak@utep.edu vladik@utep.edu Close Quit

Traditional Decision . . . Utility Is Defined . . . 1. Traditional Decision Theory: Reminder What If We Only Have . . . • Decision theory is based on the notion of utility . Two Envelopes Problem How Realistic Is This . . . • To describe this meaning, we need to select two alter- At First Glance, This . . . natives: This Is Not Really a . . . – a very bad alternative A − which is worse that any- Idealized Formulation: . . . thing that we will actually encounter, and So What Should We Do? – a very good alternative A + which is better than Home Page anything that we will actually encounter. Title Page • Then, for each number p from the interval [0 , 1], we ◭◭ ◮◮ can form a lottery L ( p ) in which: ◭ ◮ – we get A + with probability p and Page 2 of 36 – we get A − with the remaining probability 1 − p . Go Back Full Screen Close Quit

Traditional Decision . . . Utility Is Defined . . . 2. Traditional Decision Theory (cont-d) What If We Only Have . . . • For p = 0, the lottery L ( p ) coincides with the very bad Two Envelopes Problem alternative A − . How Realistic Is This . . . At First Glance, This . . . • Thus, L (0) is worse that any actual alternative A ; we This Is Not Really a . . . will denote this by L (0) = A − < A . Idealized Formulation: . . . • For p = 1, the lottery L ( p ) coincides with the very So What Should We Do? good alternative A + . Home Page • Thus, L (1) is better than any actual alternative A : Title Page A < A + . ◭◭ ◮◮ • We can ask the user to compare the alternative A with ◭ ◮ the lotteries L ( p ) corresponding to different p . Page 3 of 36 Go Back Full Screen Close Quit

Traditional Decision . . . Utility Is Defined . . . 3. Traditional Decision Theory (cont-d) What If We Only Have . . . • We assume that for every two alternatives A and B , Two Envelopes Problem the user always decides: How Realistic Is This . . . At First Glance, This . . . – whether A is better (i.e., B < A ) This Is Not Really a . . . – or whether B is better (i.e., A < B ), Idealized Formulation: . . . – or whether A and B are of the same quality to this So What Should We Do? user; we will denote this by A ∼ B. Home Page • We also assume that the user’s decisions are consistent: Title Page – that the preference relation < is transitive and ◭◭ ◮◮ – that p < p ′ implies L ( p ) < L ( p ′ ). ◭ ◮ • One can see that under these assumptions, there is a Page 4 of 36 threshold value p 0 such that: Go Back – for all p < p 0 , we have L ( p ) < A , and Full Screen – for all p < p 0 , we have A < L ( p ). Close Quit

Traditional Decision . . . Utility Is Defined . . . 4. Traditional Decision Theory (cont-d) What If We Only Have . . . • This threshold value is called the utility of the alterna- Two Envelopes Problem tive A . The utility is usually denoted by u ( A ). How Realistic Is This . . . At First Glance, This . . . • By definition of the utility, for every small value ε > 0, This Is Not Really a . . . we have L ( u ( A ) − ε ) < A < L ( u ( A ) + ε ) . Idealized Formulation: . . . • For very small ε , lotteries with probabilities u ( A ) and So What Should We Do? u ( A ) ± ε are practically indistinguishable. Home Page • So we can say that the alternative A is equivalent to Title Page the lottery L ( u ( A )). ◭◭ ◮◮ • We will denote this equivalence by A ≡ L ( u ( A )) . ◭ ◮ • Suppose now that for some action a , we have conse- Page 5 of 36 quences A 1 , . . . , A n with probabilities p 1 , . . . , p n . Go Back • This means that the action a is equivalent to a lottery Full Screen in which we get each A i with probability p i . Close Quit

Traditional Decision . . . Utility Is Defined . . . 5. Traditional Decision Theory (cont-d) What If We Only Have . . . • Each alternative A i , in its turn, is equivalent to the Two Envelopes Problem lottery L ( u ( A i )) in which: How Realistic Is This . . . At First Glance, This . . . – we get A + with probability u ( A i ) and This Is Not Really a . . . – we get A − with the remaining probability 1 − u ( A i ). Idealized Formulation: . . . • Thus, the action a is equivalent to a two-stage lottery, So What Should We Do? in which: Home Page – first, we select A i with probability p i , and Title Page – then, depending on A i , we select A + with probabil- ◭◭ ◮◮ ity u ( A i ) and A − with probability 1 − u ( A i ). ◭ ◮ • As a result of this two-stage lottery, we get either A + Page 6 of 36 or A − , and the probability of getting A + is equal to Go Back n � Full Screen p = p i · u ( A i ) . i =1 Close Quit

Traditional Decision . . . Utility Is Defined . . . 6. Traditional Decision Theory (cont-d) What If We Only Have . . . • So, the action a is equivalent to the lottery L ( p ). Two Envelopes Problem How Realistic Is This . . . • By definition of utility, this means that the utility u ( a ) At First Glance, This . . . of the action a is equal to this probability p , i.e., that This Is Not Really a . . . n Idealized Formulation: . . . � u ( a ) = p i · u ( A i ) . So What Should We Do? i =1 Home Page • By definition of utility, we select the action with the Title Page largest possible value of utility. ◭◭ ◮◮ • The right-hand side of the above formula is the ex- ◭ ◮ pected value of the utility. Page 7 of 36 • So, a rational person should select the alternative with the largest possible value of expected utility. Go Back Full Screen Close Quit

Traditional Decision . . . Utility Is Defined . . . 7. Utility Is Defined Modulo a Linear Transfor- What If We Only Have . . . mation Two Envelopes Problem • Numerical values of utility depend on the selection of How Realistic Is This . . . the very bad and very good alternatives A − and A + . At First Glance, This . . . This Is Not Really a . . . • What if we select a different pair A ′ − and A ′ + ? Idealized Formulation: . . . • Let us consider the case A − < A ′ − < A ′ + < A + . So What Should We Do? • Every alternative A is equivalent to a lottery L ′ ( u ′ ( A )) Home Page in which: Title Page – we get A ′ + with probability u ′ ( A + ) and ◭◭ ◮◮ – we get A ′ − with probability 1 − u ′ ( A ). ◭ ◮ • A ′ − is equivalent to a lottery u ( A ′ − ) in which: Page 8 of 36 – we get A + with probability u ( A ′ − ) and Go Back – we get A − with the remaining probability 1 − u ( A ′ − ). Full Screen Close Quit

Traditional Decision . . . Utility Is Defined . . . 8. Linear Transformation (cont-d) What If We Only Have . . . • A ′ + is equivalent to a lottery u ( A ′ + ) in which: Two Envelopes Problem How Realistic Is This . . . – we get A + with probability u ( A ′ + ) and At First Glance, This . . . – we get A − with the remaining probability 1 − u ( A ′ + ). This Is Not Really a . . . • Thus, the original alternative A is equivalent to a two- Idealized Formulation: . . . stage lottery in which: So What Should We Do? – we first select A ′ − or A ′ + , and Home Page – then, depending on what we selected on the first Title Page stage, select A + or A − . ◭◭ ◮◮ • As a result of this two-stage lottery, we get either A + ◭ ◮ or A − ; the probability p of selecting A + is equal to Page 9 of 36 p = u ′ ( A ) · u ( A ′ + ) + (1 − u ′ ( A )) · u ( A ′ − ) = Go Back u ( A ′ − ) + u ′ ( A ) · ( u ( A + ) − u ( A − )) . Full Screen • By definition of utility, this probability p is the utility u ( A ) of the alternative A in terms of A − and A + . Close Quit

Traditional Decision . . . Utility Is Defined . . . 9. Linear Transformation (cont-d) What If We Only Have . . . • Thus, u ( A ) = u ( A ′ − ) + u ′ ( A ) · ( u ( A + ) − u ( A − )) . Two Envelopes Problem How Realistic Is This . . . • In other words, the utility is defined modulo a linear At First Glance, This . . . transformation. This Is Not Really a . . . • This is similar to measuring quantities like time or tem- Idealized Formulation: . . . perature, where the numerical value depends: So What Should We Do? Home Page – on the selection of the starting point and – on the selection of the measuring unit. Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 36 Go Back Full Screen Close Quit