# [EM] expectation with sincere voting, table.

Bart Ingles bartman at netgate.net
Sat Apr 1 14:30:34 PST 2000

```
MIKE OSSIPOFF wrote:
>
> B  Approval  Plurality  IRV  Borda
> .4  1.6       1.6        4/3   3
> .6  1.6       1.4        4/3   3
> .25 1.75      1.75       4/3   3
> .75 1.75      1.25       4/3   3
> .1  1.9       1.9        4/3   3
> .9  1.9       1.1        4/3   3

I don't know about the IRV figures, but the Borda figures don't seem
right to me.  For example, in the u(B) = 0.9 case, the Borda figure
should probably be 2, if calculated in the same way as the others.

I am assuming the probability of a tie between AB, AC, and BC are equal,
and that the probability of a three-way tie is negligible.  For u(A) =
1.0, u(B) = 0.9, and u(C) = 0.0, the utilities in the case of a tie
(broken randomly) if the voter doesn't participate are shown in the
first column.

---------------------------------------------------------
Outcome utility for voter with u(A)=1, u(B)=.9, u(C)=.1
where participation breaks a two-way tie:

Non-participation   Plurality   Approval   Borda
AB       .95           1.00       .95       1.00
AC       .50           1.00      1.00       1.00
BC       .45            .45       .90        .90

mean(x3) 1.9           2.45      2.85       2.90
---------------------------------------------------------
Utility gains for ideal strategies:

Non-participation   Plurality   Approval   Borda
AB        0           .05         0         .05
AC        0           .50         .50       .50
BC        0            0          .45       .45

mean(x3)  0           .55         .95      1.00
---------------------------------------------------------

***

Calculating the strategy for a tie-breaking voter is one way of
analyzing strategy characteristics of a method, but I don't think it's
the only way.  This approach only looks at a single voter in isolation,
who only has the power to make or break a tie, and not to put a favorite
into front-runner territory to begin with.

Another approach would be to assume that the faction to which the voter
belongs will behave as a unit (albeit an average-sized one).  Since the
faction has a common goal, there is no need for one member to know who
the others are or to communicate with these members in order for them to
act in unison (but must he must assume the others are using the same
logic).  "Faction" in this case would mean a group of voters who have
incentives to use the same strategy in a given election.

I believe that the tie-breaker approach would fail to show any incentive
for truncation or order-reversal under Borda or pairwise methods, while
the second approach would at times.

For example, one faction might be identified as:
[  ABC;  u(B) < 0.5  ]
Voters belonging to this faction might have incentive to truncate under
pairwise or Borda using this "faction analysis" approach, but not using
the "tie-breaker" approach.

```