## Bayesian Analysis

### Toward Rational Social Decisions: A Review and Some Results

#### Abstract

Bayesian decision theory is profoundly personalistic. It prescribes the decision $d$ that minimizes the expectation of the decision-maker’s loss function $L(d,\theta)$ with respect to that person’s opinion $\pi(\theta)$. Attempts to extend this paradigm to more than one decision-maker have generally been unsuccessful, as shown in Part A of this paper. Part B of this paper explores a different decision set-up, in which Bayesians make choices knowing that later Bayesians will make decisions that matter to the earlier Bayesians. We explore conditions under which they together can be modeled as a single Bayesian. There are three reasons for doing so:

1. To understand the common structure of various examples, in some of which the reduction to a single Bayesian is possible, and in some of which it is not. In particular, it helps to deepen our understanding of the desirability of randomization to Bayesians.

2. As a possible computational simplification. When such reduction is possible, standard expected loss minimization software can be used to find optimal actions.

3. As a start toward a better understanding of social decision-making.

#### Article information

Source
Bayesian Anal., Volume 9, Number 3 (2014), 685-698.

Dates
First available in Project Euclid: 5 September 2014

https://projecteuclid.org/euclid.ba/1409921110

Digital Object Identifier
doi:10.1214/14-BA876

Mathematical Reviews number (MathSciNet)
MR3256060

Zentralblatt MATH identifier
1327.62039

#### Citation

Kadane, Joseph B.; MacEachern, Steven N. Toward Rational Social Decisions: A Review and Some Results. Bayesian Anal. 9 (2014), no. 3, 685--698. doi:10.1214/14-BA876. https://projecteuclid.org/euclid.ba/1409921110

#### References

• Berry, S. and Kadane, J. (1997). “Optimal Bayesian randomization.” Journal of the Royal Statistical Society: Series B, 59(4): 813–819.
• deFinetti, B. (1937). “La Prèvision: ses lois logique, se sources subjectives,” Annales de L’Institut Henri Poincarè. Translated in “Studies in Subjective Probability”. John Wiley and Sons.
• DeGroot, M. and Kadane, J. (1980). “Optimal challenges for selection.” Operations Research, 28: 952–968.
• Etzioni, R. and Kadane, J. (1993). “Optimal experimental design for another’s analysis.” Journal of the American Statistical Association, 88: 1404–1411.
• Harsanyi, J. (1967/1968). “Games with incomplete information played by Bayesian players.” Management Science, 14(3), 159-183 (Part I), 14(5), 320-344 (Part II), 14(7), 486-502 (Part III).
• (1982). “Subjective probability and the theory of games: Comments on Kadane and Larkey’s paper.” Managment Science, 28: 120–124.
• Kadane, J. (2011). Principles of Uncertainty. Boca Raton: Chapman and Hall.
• Kadane, J. and Larkey, P. (1982). “Subjective probability and the theory of games.” Management Science, 28: 113–120.
• Kadane, J., Stone, C., and Wallstrom, G. (1999). “The donation paradox for peremptory challenges.” Theory and Decision, 47: 139–151.
• Lindley, D. (1972). Bayesian Statistics. Society for Industrial and Applied Mathematics.
• Lindley, D. and Singpurwalla, N. (1991). “On the evidence needed to reach action between adversaries, with application to acceptance sampling.” Journal of the American Statistical Association, 86: 933–937.
• Ramsey, F. (1931). The Foundations of Mathematics and Other Logical Essays, chapter Truth and Probability, Chapter VII. Routledge.
• Roth, A., Kadane, J., and DeGroot, M. (1977). “Optimal peremptory challenges in trial by juries: A bilateral sequential process.” Operations Research, 25: 901–919.
• Savage, L. (1954). Foundations of Statistics. New York: John Wiley and Sons.
• Seidenfeld, T., Kadane, J., and Schervish, M. (1989). “On the shared preferences of two Bayesian decision-makers (reprinted in The Philosopher’s Annual, Vol. XII (1989).” Journal of Philosophy, 5: 225–244.
• Von Neumann, J. and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton: Princeton University Press.