[ZetaZero]

picking 1 would still dominate over any other strategy, in a head-to-head.

However, if you are trying to maximize your H2H score against a pool of contestants, a cooperation strategy, where you both alternated taking the lower number would be close to optimal.

[StevenWaterman]

For a short intuitive proof of that 1st sentence, imagine both players pick 1 every time. You both lose 1 point. If you're tied for score, you're still tied after the round.

Now imagine you change your number to stop ties. Well now you've picked a number bigger than 1, and you lose every time. If you were tied, now you're losing.

[HWR_14]

This is only true if your goal is to have a score higher than your opponent. It's not true if your goal is to have the highest possible score.

If ties are with 0, you always choose 1 in either situation. If ties are worth -1, choosing 1 always achieves a greater than or equal to score but the number will be much lower than if you some percentage of the time choose another number.