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Bayesian regret is gotten via this procedure:
Each voter has a personal "utility" value for the election of each candidate. (E.g., if Nixon is elected, then voter Dan Cooper will acquire -55 extra lifetime happiness units.) In a computer simulation, the "voters" and "candidates" are artificial, and the utility numbers are generated by some randomized "utility generator" and assigned artificially to each candidate-voter pair.
Now the voters vote, based both on their private utility values, and (if they are strategic voters) on their perception from "pre-election polls" (also generated artificially within the simulation, e.g. from a random subsample of "people") of how the other voters are going to act.
(Note. Some people here have gotten the wrong impression that this is assuming that voters will be "honest" or that we are assuming that honest range voters will use candidate-utilities as their candidate-scores. Other people thought we insisted on i.i.d. normal random numbers as utility values [or that some other specific randomized utility generator was insisted upon]. All those impressions are incorrect; these assumptions are not made.)
The election system E elects some winning candidate W.
The sum over all voters V of their utility for W, is the "achieved societal utility."
The sum over all voters V of their utility for X, maximized over all candidates X, is the "optimum societal utility" which would have been achieved if the election system had magically chosen the societally best candidate.
The difference between 5 and 4 is the "Bayesian Regret" of the election system E, at least in this experiment. It might be zero, but if E was bad or if this election was unlucky for E, then it will be positive because W and X will be different candidates.
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