When to stop consulting. (English) Zbl 1541.90204
Jones, Michael A. (ed.) et al., Mathematics of decisions, elections and games. AMS special session, virtual, April 8, 2022. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 795, 1-14 (2024).
Summary: Consider a decision-maker whose objective is to maximize the probability of making a correct binary (Yes-No) decision. To gain information, the decision-maker can consult independent experts. But how many? When should consultation stop?
We model the decision-maker’s current state of knowledge using the number of recommendations for each alternative. After every consultation, the decision-maker decides whether to make a decision immediately, based on the advice already received, or to consult again. We impose a ceiling on the number of consultations and, possibly, a cost for each consultation.
Surprisingly, we find that even when consultation is free, it can be optimal not to consult. Less surprising, consultation may be uniquely optimal when evidence is scarce, especially if it is approximately balanced between Yes and No. For a decision-maker who, initially, is maximally uncertain, we obtain complete optimal strategies when consultation is free, and we show that there is a fundamental limit on expected utility (\(\frac{3}{4}\), where the minimum utility is 0 and the maximum is 1), even when consultation is unlimited. We then explore the strategic effects of a positive cost, linking them to the zero-cost case. We also compare our results on consultation to strategies in the Secretary Problem, another decision problem in which information is collected sequentially until a decision is made.
For the entire collection see [Zbl 1536.91003].
We model the decision-maker’s current state of knowledge using the number of recommendations for each alternative. After every consultation, the decision-maker decides whether to make a decision immediately, based on the advice already received, or to consult again. We impose a ceiling on the number of consultations and, possibly, a cost for each consultation.
Surprisingly, we find that even when consultation is free, it can be optimal not to consult. Less surprising, consultation may be uniquely optimal when evidence is scarce, especially if it is approximately balanced between Yes and No. For a decision-maker who, initially, is maximally uncertain, we obtain complete optimal strategies when consultation is free, and we show that there is a fundamental limit on expected utility (\(\frac{3}{4}\), where the minimum utility is 0 and the maximum is 1), even when consultation is unlimited. We then explore the strategic effects of a positive cost, linking them to the zero-cost case. We also compare our results on consultation to strategies in the Secretary Problem, another decision problem in which information is collected sequentially until a decision is made.
For the entire collection see [Zbl 1536.91003].
MSC:
| 90B50 | Management decision making, including multiple objectives |
| 62C05 | General considerations in statistical decision theory |
| 91B06 | Decision theory |
References:
| [1] | Beta-binomial Distribution, Wikipedia, Beta-binomial_distribution, accessed 2022-04-28. |
| [2] | Steven J. Brams, Theory of Moves, Cambridge University Press, 1993, pp. 130-38. |
| [3] | Steven J. Brams and D. Marc Kilgour, Revisiting the Secretary Problem, Preprint, New York University. |
| [4] | Bruss, F. Thomas, A unified approach to a class of best choice problems with an unknown number of options, Ann. Probab., 882-889, 1984 · Zbl 0553.60047 |
| [5] | Robert F. Kennedy, Thirteen Days, W. W. Norton, 1969. |
| [6] | Jeanne Pavy, The Role of the Executive Committee in the Cuban Missile Crisis, http://people.loyno.edu/ history/journal/1983-4/1983-4.htm/, accessed 2022-04-02. |
| [7] | Secretary Problem, Wikipedia, Secretary_problem, accessed 2022-04-28. |
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