I thought some while about "equality" and "efficiency" as a basic principles of voting. It seems that both terms can mean quite different things that we should perhaps try to distinguish. First, there is "formal equality" (=anonymity), "equality of outcomes", "equality of expected outcomes", and "equality of power". On the efficiency side, there is "total utility efficiency", "social welfare efficiency", "Condorcet efficiency" and so on.
To start a discussion, I have devised a simple example which I use to distinguish these kinds of equality and efficiency, and which therefore also leads to quite different results under various voting methods.
Example preferences:
-------------------- mean utility
1 voter: A> > > > > >C>G|E>T>M>B 26/7=3.7
1 voter: B> > > >M>T>G| >E> >C>A 33/7=4.7
1 voter: C>T> >G>M> | > >E> >B>A 40/7=5.7
| | | | | | | | | | |
utility 11 10 8 7 6 5 4 3 2 1 0
(> designates a utility gap of 1,
| shows the approval cutoff under above-mean strategy)
Analysis:
---------
Approval Borda median total Gini social
option beats score score utility utility welfare
C all 2 12 ! 5 17 31/9
T ABEMG 2 11 6 18 ! 38/9
G ABEM 3 ! 11 5 17 43/9 !
M ABE 2 9 7 ! 15 33/9
E AB 0 7 3 9 27/9
B A 1 7 1 12 14/9
A - 1 6 0 11 11/9
C is both the Borda and the beats-all (Condorcet) winner. The beats are transitively C>T>G>M>E>B>A.
M maximizes median utility.
T maximizes total utility.
G maximizes Gini social welfare (=expected minimum utility of two randomly drawn voters) and is the Approval winner with above-mean approval strategies. In a more realistic scenario, the third voter would rather approve of C and T only since that would change the Approval winning set to {C,T,G}.
If we want equal utilities for all voters, we must elect E.
If we want equal expected utilities for all voters, one solution is to elect A,M,E with probabilities 6/s,11/s,(s-17)/s, for some arbitrary s≥17. (There also exist other solutions with three possible winners.)
If we give each voter equal voting power, i.e. let her distribute 1/3 of the winning probability, the winner is A,B, or C, each with probability 1/3. For the B and C voters, it would increase their expected utility when they gave their probability share of 2/3 all to M rather than B and C. However, under Random Ballot, this is a prisoner's dilemma since, given the other's strategy, it is always best to vote sincerely A or B instead of M. Therefore, such a cooperation won't probably occur with Random Ballot. In this respect, Random Ballot is quite inefficient in terms of total utility and/or social welfare.