2. Simulation Selection Problems: Overview of an Economic Analysis Based On Paper By: Stephen E. Chick Noah Gans Presented By: Michael C. Jones MSIM 852 Fall 2007

3. Problem Statement: A manager must select a project to implement from a group of potential projects. The manager has a model of each project, and some idea of the statistics of the projects, but does not know the expected value of any project… The manager must select a project, or chose to simulate a project, or chose to abandon all of them and do nothing…

4. Problem Statement: Running a simulation costs money, and takes time. Recognize that taking time reduces the Net Present Value (NPV) of any project ultimately selected.

5. Net Present Value: How much is information worth?

6. Bayes’ Rule "The essence of the Bayesian approach is to provide a mathematical rule explaining how you should change your existing beliefs in the light of new evidence. In other words, it allows scientists to combine new data with their existing knowledge or expertise. The canonical example is to imagine that a precocious newborn observes his first sunset, and wonders whether the sun will rise again or not. He assigns equal prior probabilities to both possible outcomes, and represents this by placing one white and one black marble into a bag. The following day, when the sun rises, the child places another white marble in the bag. The probability that a marble plucked randomly from the bag will be white (ie, the child's degree of belief in future sunrises) has thus gone from a half to two-thirds. After sunrise the next day, the child adds another white marble, and the probability (and thus the degree of belief) goes from two-thirds to three-quarters. And so on. Gradually, the initial belief that the sun is just as likely as not to rise each morning is modified to become a near-certainty that the sun will always rise." * *”In Praise of Bayes,” The Economist, 30 Sept 2000

7. Bayes’ Rule An example will help…

8. Bayes’ Rule Suppose you have two bags of marbles to select from. One bag (a 1 ) has 75% white marbles and 25% black marbles, the other bag (a 2 ) has 25% white marbles and 75% black marbles. You pick a bag, but do not know which one. From the bag, you select 5 marbles (with replacement). They are 4 white and 1 black marbles. Call the experiment B. What bag do you have? What is P(a 1 |B)? What is P(a 2 |B)?

9. Bayes’ Rule but: and: so:

10. Bayes’ Rule but: and: so:

11. Multi-armed Bandit Problem A gambler has a slot machine with several arms (or several slot machines to select from) but does not know the expected payout. How can he maximize his probability of playing the machine with the highest payout?

12. Multi-armed Bandit Problem Classic Approach –Epsilon First –Epsilon Greedy –Epsilon Decreasing Indifference Zones… Gittins Index…

13. Multi-armed Bandit Problem Gittins Index… –Gittins found a dynamic allocation index based on Baye’s rule. –Does not follow Bayes directly. (Consider what could happen after the first pull if he followed Bayes exactly…) –Weights the results by the “regret” cost of incorrect selection, discounted to the horizon.

14. Gittins’ Index: Gittin’s Index is the value of the game: –For several machines, n= 1, 2, …N –machine n is in state i. If machine n is selected, a reward,, is earned. Machine n then transitions to state j, with some known probability.

15. Returning to the Manager: Gittin’s Index is difficult (and computationally expensive) to calculate. The Index may be approximated by the Optimal Expected Discounted Reward:

16. Returning to the Manager: The Index may be approximated by the Optimal Expected Discounted Reward. Select the most profitable of: –Do nothing. E[] = 0 –Simulate a project. E[] = –Implement a project now. E[] =

17. Procedure

18. Summary Strengths: –Provides a tools for accounting for cost (both opportunity and direct) of simulation. –May be automated so the business manager can apply the tool. Weaknesses: –Limiting assumptions (Known variance but unknown mean?) –Mathematically abstract.

19. Future Not applicable directly The authors state that the paper presents more questions than it answers. May provide stimulation for future research.

20. Future Not applicable directly The authors state that the paper presents more questions than it answers. May provide stimulation for future research. “If I have seen a little further it is by standing on the shoulders of Giants." Newton, 1676