The other day, I was thinking about how to define a method having exit
or entry incentives, and thus being vulnerable to strategic nomination.

Strategic nomination incentives are subsets of IIA failures. For
single-candidate incentives, it seems pretty straightforward to define:

Suppose we have a c-candidate election in a spatial model. Candidates
prefer those who are closer to themselves in the spatial model to those
further away, just like the voters do.

Then, if there exists some candidate X (at some position in issue space)
who, by entering, could change the winner from A to B in the resulting
(c+1)-candidate election, and B is closer to X than A is, then there is
an entry incentive.

And if there exists some current non-winning candidate C who, were he
not to run, could change the winner from A to B in the resulting
(c-1)-candidate election, and B is closer to C than A is, then there is
an exit incentive.

So far so good.

But when discussing general IIA failures, it's usually a good idea to
consider coalitions (groups of voters). This is what makes Condorcet
methods have no more IIA failures than non-Condorcet methods. If there
is a CW, eliminating anybody will still let that CW be the CW. On the
other hand, if there is a cycle and the winner is X, or if the method
fails to elect the Condorcet winner, we can eliminate everybody but X
and some Y with Y>X to demonstrate IIA failure.

For exit incentive, this is still quite easy: if there exist a group of
candidates, all of whom prefer B to the current winner A, and
eliminating them all makes B win, then there's an exit incentive.

But for entry incentive, things get much harder, and kind of weird, too.
The coalitional version would be "if there exist k points in opinion
space closer to B than A, and inserting a candidate at each of these
points makes B win, then there's an entry incentive".

The latter is harder to calculate, but worse is that it produces extreme
results. For instance, in Borda, one can always "clone one's way to
victory". Any loser B can win just by inserting enough clones of himself
into the election. So Borda's coalitional entry incentive would be 100%,
all the time no matter the pre-cloning number of candidates.

What do you think? Does Borda's 100% coalitional entry incentive suggest
that extending entry incentive to coalitions doesn't really work, or
does it just show that Borda is that bad?

-km

I think it's a bit of a myth that one can always clone one's way to victory in Borda. It requires co-ordinating your voters to put the candidate you want to win of the clones at the top while at the same time hoping that the opposition don't do the reverse.
Toby
On Wednesday 16 April 2025 at 10:44:06 BST, Kristofer Munsterhjelm via Election-Methods election-methods@lists.electorama.com wrote:

The other day, I was thinking about how to define a method having exit
or entry incentives, and thus being vulnerable to strategic nomination.

Strategic nomination incentives are subsets of IIA failures. For
single-candidate incentives, it seems pretty straightforward to define:

Suppose we have a c-candidate election in a spatial model. Candidates
prefer those who are closer to themselves in the spatial model to those
further away, just like the voters do.

Then, if there exists some candidate X (at some position in issue space)
who, by entering, could change the winner from A to B in the resulting
(c+1)-candidate election, and B is closer to X than A is, then there is
an entry incentive.

And if there exists some current non-winning candidate C who, were he
not to run, could change the winner from A to B in the resulting
(c-1)-candidate election, and B is closer to C than A is, then there is
an exit incentive.

So far so good.

But when discussing general IIA failures, it's usually a good idea to
consider coalitions (groups of voters). This is what makes Condorcet
methods have no more IIA failures than non-Condorcet methods. If there
is a CW, eliminating anybody will still let that CW be the CW. On the
other hand, if there is a cycle and the winner is X, or if the method
fails to elect the Condorcet winner, we can eliminate everybody but X
and some Y with Y>X to demonstrate IIA failure.

For exit incentive, this is still quite easy: if there exist a group of
candidates, all of whom prefer B to the current winner A, and
eliminating them all makes B win, then there's an exit incentive.

But for entry incentive, things get much harder, and kind of weird, too.
The coalitional version would be "if there exist k points in opinion
space closer to B than A, and inserting a candidate at each of these
points makes B win, then there's an entry incentive".

The latter is harder to calculate, but worse is that it produces extreme
results. For instance, in Borda, one can always "clone one's way to
victory". Any loser B can win just by inserting enough clones of himself
into the election. So Borda's coalitional entry incentive would be 100%,
all the time no matter the pre-cloning number of candidates.

What do you think? Does Borda's 100% coalitional entry incentive suggest
that extending entry incentive to coalitions doesn't really work, or
does it just show that Borda is that bad?

-km

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