In a recent home game, we ran into the following scenario:

Player 1 is all-in on the flop and called by Player 2 and Player 3.

We go to the turn with a dry sidepot, and P1 accidentally exposes his hand. Obviously this knowledge can have an impact on P2 and P3s actions.

This prompted an academic debate. Are there circumstances where this harms P1? Are there circumstances where this helps P1?

If yes to both, can we devise any general rules where it harms vs helps?

The only way that P1 can gain is for him to increase the aggression of the other two players.

Meaning, if he exposed his garbage hand (among the bottom 50% of the worst hands he can have, that still have some equity) it would induce the other two players to start fighting more for the pot, from which P1 would gain some extra fold equity, which would increase his overall equity and therefore EV.

For example, it would be a good "play" from him to expose his hand while having some sort of draw/overcards. This would make the other two players fight more for the pot and one might get the other one to fold, which would increase P1's equity.

It would harm him though, if he exposed a really strong hand, since that would make the other two players play super tight and would reduce his FE.

I am sort of confused by this statement. In what way would it harm him? Sure it would be better if the other players folded or if at least one of them folded but if they have both already called player 1 can do no worse than the equity he has vs the 2 players.

Now if it was before the players had called then it could certainly matter.

Imagine this situation for example:

Flop is J52o

P1 has 55 (all-in preflop)
P2 has AT
P3 has 88

If P1 doesn't show his hand, both players might think that they have some showdown value, so they might still fight for the main pot. For example, P2 could put P3 on a low pocket pair and start betting as a bluff to get him to fold 88, thinking that he will still win the main pot sometimes at the showdown with A high. This benefits P1, since he now doesn't lose when an 8 hits. A similar thing would happen if P3 folded a draw, which would benefit P1 even more.

If P1 shows his hand however, both players will know that there's no main pot to fight for, so they will play super tight-passive (like there's no pot at all). P3 will fold 88 even to a min bet, while AT still won't have a reason to bet. The hand will just go check-check to the river, unless someone hits the nuts.

I certainly understand how keeping his hand hidden the other 2 players may knock one another out raising player 1's equity.

What I specifically had issue with was how it could harm player 1 given both players had already called. The worse player 1 can do is achieve the equity of player 1's hand. Player 1 can only improve upon that result by exposing the cards. Player1 can never do worse than the equity only better.

Maybe I am just fussing over semantics.