Use of Symmetries in Additivity How Utility u Depends . . . - PowerPoint PPT Presentation

How Do People Make . . . Symmetries Play a . . . Scaling Shift Use of Symmetries in Additivity How Utility u Depends . . . Economics Probabilistic Choice Vladik Kreinovich 1 , Olga Kosheleva 1 , Decision Making . . . Nguyen Ngoc Thach 2 , and Nguyen Duc Trung 2 Taking Future Effects . . . Home Page 1 University of Texas at El Paso, El Paso, Texas 79968, USA olgak@utep.edu, vladik@utep.edu Title Page ◭◭ ◮◮ 2 Banking University of Ho Chi Minh City Ho Chi Minh City, Vietnam, Thachnn@buh.edu.vn ◭ ◮ Page 1 of 19 Go Back Full Screen Close Quit

How Do People Make . . . Symmetries Play a . . . 1. A Brief Overview Scaling • Many semi-heuristic econometric formulas can be de- Shift rived from the natural symmetry requirements. Additivity How Utility u Depends . . . • The list of such formulas includes many famous formu- Probabilistic Choice las provided by Nobel-prize winners, such as: Decision Making . . . – Hurwicz optimism-pessimism criterion for decision Taking Future Effects . . . making under uncertainty, Home Page – McFadden’s formula for probabilistic decision mak- Title Page ing, ◭◭ ◮◮ – Nash’s formula for bargaining solution. ◭ ◮ • It also includes Cobb-Douglas formula for production, Page 2 of 19 gravity model for trade, etc. Go Back Full Screen Close Quit

How Do People Make . . . Symmetries Play a . . . 2. How Do People Make Predictions? Scaling • How do people make predictions? How did people Shift know that the Sun will rise in the morning? Additivity How Utility u Depends . . . • Because in the past, the sun was always rising. Probabilistic Choice • In all these cases, to make a prediction, we look at Decision Making . . . similar situations in the past. Taking Future Effects . . . Home Page • We then make predictions based on what happened in such situations. Title Page • Some predictions are more complicated than that – ◭◭ ◮◮ they are based on using physical laws. ◭ ◮ • But how do we know that a law – e.g., Ohm’s law – is Page 3 of 19 valid? Go Back • Because in several previous similar situations, this for- Full Screen mula was true. Close Quit

How Do People Make . . . Symmetries Play a . . . 3. How to Describe This Idea in Precise Terms? Scaling • We can shift or rotate the lab, Ohm’s law will not Shift change. Additivity How Utility u Depends . . . • In general, we have some phenomenon p depending on Probabilistic Choice the situation s . Decision Making . . . • We replace the original situation s by the changed sit- Taking Future Effects . . . uation T ( s ). Home Page • Invariance means that the phenomenon remains the Title Page same after the change: p ( T ( s )) = p ( s ) . ◭◭ ◮◮ • A particular case of an invariance is when we have, e.g., ◭ ◮ a spherically symmetric object. Page 4 of 19 • If we rotate this object, it will remain the same – this is exactly what symmetry means in geometry. Go Back Full Screen • Because of this example, physicists call each invariance symmetry . Close Quit

How Do People Make . . . Symmetries Play a . . . 4. Symmetries Play a Fundamental Role in Physics Scaling • In the past, physical theories – e.g., Newton’s mechan- Shift ics – were formulated in terms of diff. eq. Additivity How Utility u Depends . . . • Nowadays theories are usually formulated in terms of Probabilistic Choice their symmetries, and equations can be derived. Decision Making . . . • Traditional physical equations can also be derived from Taking Future Effects . . . their symmetries. Home Page • Predictions in economics are also based on similarity. Title Page • So, let us see if we can derive economic equations from ◭◭ ◮◮ the corresponding symmetries. ◭ ◮ Page 5 of 19 Go Back Full Screen Close Quit

How Do People Make . . . Symmetries Play a . . . 5. Scaling Scaling • Physical equations – like Ohm’s law V = I · R – deal Shift with numerical values of different physical quantities. Additivity How Utility u Depends . . . • These numerical values depend on the measuring unit. Probabilistic Choice • If we replace the original measuring unit with a λ times Decision Making . . . smaller one, then x → x ′ = λ · x . Taking Future Effects . . . • Fundamental equations y = f ( x ) should not change if Home Page we change the measuring unit (e.g., dollars or pesos). Title Page • We can’t require f ( λ · x ) = f ( x ), then f ( x ) = const. ◭◭ ◮◮ • We can require that for each λ , there is a C ( λ ) s.t. if ◭ ◮ y = f ( x ), then y ′ = f ( x ′ ), where y ′ = C ( λ ) · y . Page 6 of 19 • For continuous f ( x ), this implies f ( x ) = A · x c . Go Back • The requirement y = f ( x 1 , . . . , x n ) ⇒ y ′ = f ( x ′ 1 , . . . , x ′ n ), i = λ i · x i and y ′ = C ( λ 1 , . . . ) · y , implies where x ′ Full Screen f = A · x c 1 1 · . . . · x c n Close n . Quit

How Do People Make . . . Symmetries Play a . . . 6. Shift Scaling • For some quantities (e.g., time or temperature), the Shift numerical value also depends on the starting point. Additivity How Utility u Depends . . . • If we replace the original starting point measuring unit with an earlier one, we get x ′ = x + x 0 . Probabilistic Choice Decision Making . . . • Fundamental equations y = f ( x ) should not change if Taking Future Effects . . . we change the starting point. Home Page • Example: salary itself? salary + social benefits? Title Page • It’s reasonable to require that for each x 0 , there is a ◭◭ ◮◮ C ( x 0 ) s.t. y = f ( x ) ⇒ y ′ = f ( x ′ ), with y ′ def = C ( λ ) · y . ◭ ◮ • For continuous f ( x ), this implies f ( x ) = A · exp( c · x ). Page 7 of 19 Go Back Full Screen Close Quit

How Do People Make . . . Symmetries Play a . . . 7. Additivity Scaling • How trade y depends on the GDP x : y = f ( x )? Shift Additivity • We can apply f ( x ) to the whole country’s GDP, or to regions whose GDPs are x ′ and x ′′ : x = x ′ + x ′′ . How Utility u Depends . . . Probabilistic Choice • The result should be the same: Decision Making . . . f ( x ′ + x ′′ ) = f ( x ′ ) + f ( x ′′ ) . Taking Future Effects . . . Home Page • For continuous f ( x ), this implies f ( x ) = c · x . Title Page • In multi-D case, we have f ( x ′ 1 + x ′′ 1 , . . . , x ′ n + x ′′ n ) = ◭◭ ◮◮ f ( x ′ 1 , . . . , x ′ n ) + f ( x ′′ 1 , . . . , x ′′ n ). ◭ ◮ • This implies that f ( x 1 , . . . , x n ) = c 1 · x 1 + . . . + c n · x n . Page 8 of 19 Go Back Full Screen Close Quit

How Do People Make . . . Symmetries Play a . . . 8. How Can We Describe Human Preferences? Scaling • We select a very good alternative A + and a very bad Shift alternative A − . Additivity How Utility u Depends . . . • For each p ∈ [0 , 1], L ( p ) is a lottery in which we get Probabilistic Choice A + with probability p , else A − . Decision Making . . . • For each realistic alternative A , it is better than L (0) = Taking Future Effects . . . A − and worse than L (1) = A + : L (0) < A < L (1). Home Page • Of course, if L ( p ) < A and p ′ < p , then L ( p ′ ) < A . Title Page Similarly, if A < L ( p ) and p < p ′ , then A < L ( p ′ ). ◭◭ ◮◮ • Thus, one can show that there exists a threshold value ◭ ◮ u ( A ) = sup { p : L ( p ) < A } (called utility ) such that: Page 9 of 19 – for p < u ( A ), we have L ( p ) < A , and Go Back – for p > u ( A ), we have A < L ( p ). Full Screen • The alternative A is equivalent to the lottery L ( u ( A )), in the sense that L ( u − ε ) < A < L ( u + ε ) for all ε > 0. Close Quit

How Do People Make . . . Symmetries Play a . . . What If We Select Different A + and A − ? 9. Scaling • Let us consider the case when A ′ − < A − < A + < A ′ + . Shift Additivity • Then, A ∼ L ( u ( A )), i.e., A + with prob. u ( A ) else A − . How Utility u Depends . . . • A + ∼ L ′ ( u ′ ( A + )), i.e., A ′ + with prob. u ′ ( A + ) else A ′ − . Probabilistic Choice • A − ∼ L ′ ( u ′ ( A − )), i.e., A ′ + with prob. u ′ ( A − ) else A ′ − . Decision Making . . . Taking Future Effects . . . • Thus, A is equivalent to a 2-step lottery in which we Home Page get A ′ + with probability Title Page u ′ ( A ) = u ( A ) · u ′ ( A + ) + (1 − u ( A )) · u ′ ( A − ) . ◭◭ ◮◮ • Otherwise, we get A ′ − . ◭ ◮ • Thus, changing a pair follows the same formulas as Page 10 of 19 when we change the starting point and the meas. unit. Go Back • Laws should not depend on the choice of a pair. Full Screen • So, we get scale- and shift-invariance. Close Quit