WORKED NUMERICAL EXAMPLE OF A BAYESIAN REGRET CALCULATION
================Warren D. Smith Nov 2008=================
Here is a simple example. 3-voter 2-candidate plurality vote.
Table of utilities:
candA candB
-----------------
voter1 1 9
voter2 4 3
voter3 5 2
CandA wins by 2 votes to 1. Utility=1+4+5=10.
If CandB had won, utility would be 9+3+2=14.
Because 14>10 there is nonzero regret=14-10=4.
Now do it again but with a new table of 6 utility values
generated by your random number generator.
Table of utilities:
candA candB
-----------------
voter1 4 6
voter2 1 3
voter3 5 3
CandB wins, utility=6+3+3=12. If CandA had won utility would be 4+1+5=10.
Since 12>10 we see regret=0.
MEAN regret = (0+4)/2 = 2 (in these two elections)
Keep doing it, a zillion times,
keeping track of the EXPECTED regret in
all zillion election experiments.
Result is the mean regret of 2-candidate 3-voter honest plurality voting
using whatever utility generator we have.