Some thoughts about "equality" and "efficiency" by Jobst Heitzig

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)

               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.

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