Bayesian regret

In game theory, Bayesian regret is the average difference between the utility of a strategy and an ideal utility where desired outcomes are maximized.

The term Bayesian refers to Thomas Bayes (1702–1761), who proved a special case of what is now called Bayes' theorem, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference.

Social choice theory

In social choice theory, Bayesian regret is the average difference in social utility between the chosen candidate and the best candidate. It is only measurable if it is possible to know the voters' true numerical utility for each candidate – that is, in Monte Carlo simulations of virtual elections.

If Bayesian regret can be "measured only if it is possible to know the voters' true numerical utility for each candidate", it would seem to be irrelevant to two voting methods that instead grade the utility of each candidate: Majority Judgment (MJ) invites citizens to grade the suitability of each candidate for a single office: Excellent (ideal), Very Good, Good, Acceptable, Poor, or Reject (entirely unsuitable). The winner is the one who receives the highest median-grade. Since MJ allows each citizen honestly and fully to express their judgment of each candidate, it would seem to give each voter every appropriate reason to be pleased with the election. At least the voters who constitute the relevant absolute majority should be pleased. Similarly, Evaluative Proportional Representation (EPR) in Section 5.5.5 in Proportional Representation allows all the members of a legislature to be elected at the same time. Each elected member has a different weighted vote during its deliberations. At the same time, each citizen is assured that their honest vote proportionately increases the voting power of the member who received either their highest grade, remaining highest grade, or proxy vote. No vote is needlessly wasted quantitatively or qualitatively. Again, each citizen is given every appropriate reason to be please with an EPR election --no legitimate regret.

The term Bayesian is somewhat a misnomer, really meaning only "average probabilistic"; there is no standard or objective way to create distributions of voters and candidates.

The Bayesian regret concept was recognized as useful (and used) for comparing single-winner voting systems by Bordley and Merrill, and it also was invented independently by R. J. Weber. Bordley attributed it (and the whole idea of the usefulness of "social" utility, that is, summed over all people in the population) to John Harsanyi in 1955.

Economics

This term has been used to compare a random buy-and-hold strategy to professional traders' records. This same concept has received numerous different names, as the New York Times notes:

"In 1957, for example, a statistician named James Hanna called his theorem Bayesian Regret. He had been preceded by David Blackwell, also a statistician, who called his theorem Controlled Random Walks. Other, later papers had titles like 'On Pseudo Games', 'How to Play an Unknown Game', 'Universal Coding' and 'Universal Portfolios'".

See also

• Loss function
• Regret (decision theory)
• Social utility efficiency

References

Notes

• Robert F. Bordley: "A pragmatic method for evaluating election schemes through simulation", Amer. Polit. Sci. Rev. 77 (1983) 123–141.
• Samuel Merrill: Making multicandidate elections more democratic, Princeton Univ. Press 1988.
• Samuel Merrill: "A comparison of efficiency of multicandidate electoral systems", Amer. J. Polit. Sci. 28, 1 (1984) 23–48.