Con 1) Voting for a third party is throwing your vote away.

We have all heard this argument before, and it is true. [...] If you vote for a third party, the same candidate would be elected as if you did not vote at all.

Note that this argument also goes through for voting for a second party. Or voting for any party that doesn't tie for first place or win by exactly one vote. So this is less an argument against voting for third parties specifically and more of an argument against voting at all.

Con 2) Actually if you use a reflexive decision theory, you are much more likely to change the election, so you should vote like it matters.

Looking at the problem like a timeless decision agent, you see that your choice on voting is probably correlated with that of many other people. You voting for a primary party is logically linked with other people voting for a primary party, and those people whose votes are logically linked with yours are more likely to agree with you politically.

That looks like magical thinking. In practice, if I silently decide to vote for party Y instead of party X, that has no effect on anyone else's vote (at least at that election).

Does this mean it's a clean sweep for the four Pros mentioned in the post? I wouldn't say so, because I disagree with at least two of them. Still, I agree with Eliezer on the basic answer to your original question: if you think a third party is the most meritorious party, vote for it.

If you always vote for one of the two most popular parties instead, you're at risk of experiencing a ratchet effect. Suppose you prefer the unpopular party X to the popular party Y, and the popular party Y to the popular party Z. For fear of handing an advantage to Z, you vote for Y instead of X. Now suppose Y & Z both shift a little away from where you stand politically. You end up voting for Y again because it's still a bit closer to you than Z. Now suppose Y & Z both shift a little further away again. You still vote for Y because it's still a little closer than Z. This can repeat indefinitely, and if so you wind up voting for Y even as it becomes arbitrarily awful to you.