Why Unexpectedly Positive A New Reformulation . . . Experiences - PowerPoint PPT Presentation

Formulation of the . . . What Does Optimism . . . α Can Be Interpreted . . . Why Unexpectedly Positive A New Reformulation . . . Experiences Make Decision Resulting Explanation Acknowledgments Makers More Optimistic: Home Page An Explanation Title Page ◭◭ ◮◮ Andrzej Pownuk and Vladik Kreinovich ◭ ◮ Computational Science Program Page 1 of 12 University of Texas at El Paso 500 W. University Go Back El Paso, Texas 79968, USA ampownuk@utep.edu, vladik@utep.edu Full Screen Close Quit

1. Formulation of the Problem Formulation of the . . . What Does Optimism . . . • Experiments show that unexpectedly positive experi- α Can Be Interpreted . . . ences make decision makers more optimistic. A New Reformulation . . . • This was first observed on rats: rats like being tickled, Resulting Explanation and tickled rats became more optimistic. Acknowledgments • Several later papers showed that the same phenomenon Home Page holds for other decision making situations as well. Title Page • Similarly, decision makers who had an unexpectedly ◭◭ ◮◮ negative experiences became more pessimistic. ◭ ◮ • There seems to be no convincing explanation for this Page 2 of 12 experimental fact. Go Back • We show that this phenomenon can be explained in the Full Screen traditional utility-based decision theory. Close Quit

2. What Does Optimism Mean? Formulation of the . . . What Does Optimism . . . • Traditional decision theory assumes that we know the α Can Be Interpreted . . . probabilities of all possible consequences of each action. A New Reformulation . . . • Then, a rational decision maker maximizes the ex- Resulting Explanation pected value u ( a ) of a special function called utility . Acknowledgments • In this case, there is no such thing as optimism or pes- Home Page simism: we just select the best alternative a . Title Page • In practice, we often have only partial information ◭◭ ◮◮ about these probabilities. ◭ ◮ • In such situations, there are several possible probability Page 3 of 12 distributions consistent with our knowledge. Go Back • For different distributions, we have, in general, differ- ent values of the expected utility. Full Screen • As a result, for each alternative a , we have an interval Close [ u ( a ) , u ( a )] of possible values of u ( a ). Quit

3. What Does Optimism Mean (cont-d) Formulation of the . . . What Does Optimism . . . • In this case, we should select an alternative a that max- α Can Be Interpreted . . . imizes u ( a ) = α · u ( a ) + (1 − α ) · u ( a ). A New Reformulation . . . • This idea was proposed by the Nobelist Leo Hurwicz. Resulting Explanation Acknowledgments • The selection of α , depends on the person. Home Page • The value α = 1 means that the decision maker only takes into account the best possible consequences. Title Page • In other words, the values α = 1 corresponds to com- ◭◭ ◮◮ plete optimism. ◭ ◮ • Similarly, the value α = 0 corresponds to complete Page 4 of 12 pessimism. Go Back • The larger α , the close this decision maker to complete Full Screen optimism. Close • The optimism-pessimism index α is a numerical mea- sure of the decision maker’s optimism. Quit

4. What Does Optimism Mean (cont-d) Formulation of the . . . What Does Optimism . . . • The optimism-pessimism index α is a numerical mea- α Can Be Interpreted . . . sure of the decision maker’s optimism. A New Reformulation . . . Resulting Explanation • Thus, the phenomenon to-be-explained takes the fol- lowing precise meaning: Acknowledgments – if a decision maker has unexpectedly positive expe- Home Page riences, then this decision maker’s α increases; Title Page – if a decision maker has unexpectedly negative ex- ◭◭ ◮◮ periences, then this decision maker’s α decreases. ◭ ◮ Page 5 of 12 Go Back Full Screen Close Quit

5. α Can Be Interpreted as the Subjective Prob- Formulation of the . . . ability of Positive Outcome What Does Optimism . . . α Can Be Interpreted . . . • The decision maker selects an alternative a that max- A New Reformulation . . . imizes α · u ( a ) + (1 − α ) · u ( a ). Resulting Explanation • Here, u ( a ) corresponds to the positive outcome, and Acknowledgments u ( a ) corresponds to negative outcome. Home Page • For simplicity, let us consider the situation when we Title Page have only two possible outcomes: ◭◭ ◮◮ – the positive outcome, with utility u ( a ), and ◭ ◮ – the negative outcome, with utility u ( a ). Page 6 of 12 • A traditional approach to decision making assumes Go Back that we know the probabilities of different outcomes. Full Screen • In this case of uncertainty, we do not know the actual Close (objective) probabilities. Quit

6. α Can Be Interpreted as the Subjective Prob- Formulation of the . . . ability of Positive Outcome (cont-d) What Does Optimism . . . α Can Be Interpreted . . . • In the case of uncertainty, we do not know the actual A New Reformulation . . . (objective) probabilities. Resulting Explanation • However, we can always come up with estimated (sub- Acknowledgments jective) ones. Home Page • Let us denote the subjective probability of the positive Title Page outcome by p + . ◭◭ ◮◮ • Then, the subjective probability of the negative out- ◭ ◮ come is equal to 1 − p + . Page 7 of 12 • The expected utility is equal to p + · u ( a )+(1 − p + ) · u ( a ). Go Back • This is exactly what we optimize when we use Hur- wicz’s approach, with α = p + . Full Screen • Thus, the value α can be interpreted as the subjective Close probability of the positive outcome. Quit

7. A New Reformulation of Our Problem Formulation of the . . . What Does Optimism . . . • Unexpectedly positive experiences increase the subjec- α Can Be Interpreted . . . tive probability of a positive outcome. A New Reformulation . . . • Unexpectedly negative experiences decrease the sub- Resulting Explanation jective probability of a positive outcome. Acknowledgments • To explain this phenomenon, let us recall where sub- Home Page jective probabilities come from. Title Page • If we observe an event in n out of N cases, our estimate ◭◭ ◮◮ is n/N . ◭ ◮ • Example: if a coin fell heads 6 times out of 10, we Page 8 of 12 estimate the probability of it falling heads as 6/10. Go Back • Let us show that this leads to the desired explanation. Full Screen Close Quit

8. Resulting Explanation Formulation of the . . . What Does Optimism . . . • Suppose that a decision maker had n positive experi- α Can Be Interpreted . . . ences in the past N situations. A New Reformulation . . . • Then, the decision maker’s subjective probability of a Resulting Explanation positive outcome is p + = n/N . Acknowledgments • Unexpectedly positive experiences means that: Home Page Title Page • we have a series of new experiments, • in which the fraction of positive outcomes was ◭◭ ◮◮ higher than the expected frequency p + . ◭ ◮ • In other words, unexpectedly positive experiences Page 9 of 12 means that n ′ /N ′ > p , where: Go Back • N ′ is the overall number of new experiences, and Full Screen • n ′ is the number of those new experiences in which Close the outcome turned out to be positive. Quit

9. Resulting Explanation (cont-d) Formulation of the . . . What Does Optimism . . . • The new subjective probability p ′ + is equal to the new α Can Be Interpreted . . . + = n + n ′ A New Reformulation . . . ratio p ′ N + N ′ . Resulting Explanation • Here, by definition of p + , we have n = p + · N . Acknowledgments • Due to unexpected positiveness of new experiences, we Home Page have n ′ > p + · N ′ . Title Page • By adding this inequality and the previous equality, we ◭◭ ◮◮ conclude that n + n ′ > p + · ( N + N ′ ), i.e., that ◭ ◮ + = n + n ′ p ′ N + N ′ > p + . Page 10 of 12 Go Back • In other words, unexpectedly positive experiences in- crease the subjective probability of a positive outcome. Full Screen Close Quit

10. Resulting Explanation (final) Formulation of the . . . What Does Optimism . . . • The subjective probability of the positive outcome is α Can Be Interpreted . . . exactly the optimism-pessimism coefficient α . A New Reformulation . . . + > p + means that α ′ > α . Resulting Explanation • Thus, p ′ Acknowledgments • So, unexpectedly positive experiences make the deci- sion maker more optimistic. Home Page Title Page • Similarly, if we had unexpectedly negative experiences, + = n + n ′ i.e., n ′ < p + · N ′ , then p ′ N + N ′ < p + and α ′ < α . ◭◭ ◮◮ ◭ ◮ • So, we conclude that unexpectedly negative experi- Page 11 of 12 ences make the decision maker less optimistic. Go Back • This is also exactly what we observe. Full Screen • So, we have the desired explanation. Close Quit

11. Acknowledgments Formulation of the . . . What Does Optimism . . . This work was supported in part: α Can Be Interpreted . . . A New Reformulation . . . • by the National Science Foundation grants: Resulting Explanation Acknowledgments • HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and Home Page • DUE-0926721, and Title Page • by an award from Prudential Foundation. ◭◭ ◮◮ ◭ ◮ Page 12 of 12 Go Back Full Screen Close Quit