In Bayesian statistics, "regret" is the difference between the maximum possible ideal utility minus the actual utility. (Depending on the application, regret can often be more convenient to deal with than utility. This terminology has been used in a large number of papers and is not new from us.) "Social" utility is the sum of utility over all the members of some human population. For voting systems purposes, "Bayesian regret" is the expectation value of social regret. It depends on both the voting system, the number of candidates, the number of voters, and the probabilistic models of utilities, candidates, and voter behaviors.
One reason I like BR is that for a "perfect" election method, BR=0, and for all other election methods, BR>0. Also, it often is pleasant to use a scaling of BR (multiply it by appropriate scale factor so that) that causes BR for "random winner" to be 1, which is sort of an estimate for "non-democracy."