"BAYESIAN REGRET FOR DUMMIES" Q. I was asked to explain "Bayesian regret" and why (at least in my view) it is the "gold standard" for comparing single-winner election methods. OVERSIMPLIFIED INTO A NUTSHELL: The "Bayesian regret" of an election method E is the "expected avoidable human unhappiness" caused by using E. MORE PRECISE ANSWER: Bayesian regret is gotten via this PROCEDURE: 1. 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 sim, 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. 2. 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 in the sim, e.g. from a random subsample of "people") of how the other voters are going to act. 3. The election system E elects some winning candidate W. 4. The sum over all voters V of their utility for W, is the "achieved societal utility." 5. 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. 6. 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 the election was unlucky for E, then it will be positive. We now redo steps 1-6 a zillion times (i.e. running a zillion simulated elections) to find the average Bayesian regret of election system E. COMMENTS: Bayesian regret of an election system E may differ if 1. vary the number of voters, 2. vary the number of candidates, 3. vary the kind of "utility generator", 4. use different kind of assumed "voter strategy", or 5. put different amount of "voter ignorance". To describe the latter, we can put in voter ignorance by artificially adding random noise to the voter's private utility values, and then having the voter act based on those distorted values. The higher the amplitude of the noise, the more ignorance there is. So there are at least 5 different "knobs" we can "turn" on our machine for measuring the Bayesian Regret of an election method E. RESULTS OF MY COMPUTER STUDY: from http://math.temple.edu/~wds/homepage/works.html #56 This study measured Bayesian regrets for about 30 different election methods. 144 different combinations of "knob settings" were tried. The amazing result is that, in all 144 scenarios, range voting was the best (had lowest Bayesian regret, up to statistically insignificant noise) in EVERY SINGLE ONE of those 144 with either honest voters, or with strategic voters. HERE IS A SIMPLIFIED TABLE of results (from only 2 of the 144 scenarios, and only 10 of the voting systems). (Read in constant-width font. Each tabulated Bayesian regret value is an average over a million or more randomized simulated elections.) column A: 5 candidates, 20 voters, random utilities. column B: 5 candidates, 50 voters, utilities based on two "issues". A B magic optimum winner 0 0 honest range .04941 .05368 honest borda .13055 .10079 honest IRV .32314 .23786 honest plurality .48628 .37884 random winner 1.50218 1.00462 strategic range=approval .31554 .23101 strategic borda .70219 .48438 strategic plurality .91522 .61072 strategic IRV .91522 .61072 Incidentally, note that with strategic voters (at least using the strategy I assumed here) strategic plurality and strategic IRV seem to be the same! That is because of the devastating THEOREM: Generically (i.e. if no ties), IRV and Plurality voting with strategic voters will yield the same winner in a large election: Namely the most popular among the two pre-election poll "frontrunners" will always win... PROOF SKETCH: For plurality voting, this was well known: strtaegic voters always vote for one of the two perceived frontrunners since other votes are extremely likely to be "wasted". For IRV: the two poll-frontrunners will garner all the top-rankings from strategic voters, thus never being eliminated until the final round, whereupon the most popular one will win. This is assuming a strategy where voters always rank one of the frontrunners top and the other bottom to "maximally exaggerate" and try to maximize their chances of affecting the election. They consider it pointless to top-rank anybody else because that person (they figure) has far tinier chance of winning. (Note: the optimum strategy for IRV voting is not known, so my computer sim was using this, non-optimum, IRV strategy, which however is usually a lot better than honesty.) QED.