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D2MAC: Difference between revisions
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Whatever two voters are drawn, both approve of C, hence C is the certain winner (i.e. has a winning probability of 1).
''This shows that D2MAC does not necessarily elect the favourite of a majority when there is a strong compromise option.''
===Two factions with a compromise option and unilateral cooperation===
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B wins if the first voter is amoung the 45 B-voters and the second voter is amoung the 54 "non-cooperative" A-voters. This has a probability of 45%*54% = 24.30%.
''This shows that under D2MAC a majority (here the 54 A-voters) cannot necessarily make sure their favourite (here A) wins with certainty. Rather every group of voters who favour the same option can make sure their favourite gets at least a winning probability of
===Two factions with a compromise option and bilateral partial cooperation===
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B wins if (a) the first voter is amoung the 15 "non-cooperative" B-voters or (b) the first voter is amoung the 30 "cooperative" B-voters but the second voter is not amoung the 60 "cooperative" voters. This has a probability of 15% + 30%*40% = 27%.
===Two factions with a strong compromise option: strategic considerations===
Assume these voters have the following cardinal utility functions:
: 55 voters: A 100, C 80, B 0
: 45 voters: B 100, C 75, A 0
Then it is quite probable that voters will behave like this:
: first 55 voters: A favourite, C also approved.
: other 45 voters: B favourite, C also approved.
This is because this voting behaviour of "full cooperation" is a group strategic equilibrium, which means that no group of voters would wish to have voted differently. To see this, note that with the above behaviour, C is the certain winner and the expected utilities for the voters are
: first 55 voters: 80
: other 45 voters: 75
Had some ''x'' of the last 45 voters voted no approval for C instead, they would have ended up with a smaller expected utility than 75, namely
: (''x'' % + (45 - ''x'')% * ''x'' %) * 100 + (100 - ''x'')% * (100 - ''x'')% * 75 = 75 - 0.05 ''x'' - 0.002 ''x'' ² < 75.
Analogously, had some x of the other 55 voters voted no approval for C instead, they would have ended up with a smaller expected utility than 80.
Note that the resulting total (expected) utility is approx. 78. If A had been declared the winner (as majoritarian methods do), it had only been 55.
''This shows that D2MAC can be more efficient in maximing total utility than majoritarian methods.''
==Variants==
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These two variants differ from D2MSC and D2MGC in that the ratings of each voters are first normalized by an affine transformation so that the voter's favourite receives a rating of 1 and so that 0 is the expected value of the voter's rating of the candidate that a randomly chosen ballot assigned the highest rating to.
[[Category:Single-winner voting methods]]
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