Yet Even More on Three-Person Games

The optimized-out game which resulted after my last rumination on three-person games wound up being quite complicated and non-intuitive. Since that game has two distinct things which can happen in each round, it makes sense to remove the weirder one of them, which results in the following game:

A spinner picks one of the three players to be the selecter. That player picks one of the two other players to get a point. The selecter obviously knows who did the selecting and who got the point. The non-selected player is told this information as well. However, the player who got the point isn't told which other player selected them. Play is repeated for a number of rounds.

The scoring can then be restated in a way which leaves play mostly unchanged as follows:

A spinner picks one of the three players to be the selecter. The selecter picks one of the other two players to be the victim, and the victim loses a point (scores in this game go down, but they never go up). The selecter and victim know who did the selecting, but the other player is left in the dark.

Come to think of it, this randomness and being left in the dark doesn't add much to the game. We can get things back to being zero sum as follows:

Each round each player gets one point, then each player points to one of the other two players. Each player then loses one point for every player pointing at them.

This game has has the interesting situation in which two players are pointing at each other and a question of how to get out of that. But it also has the unfortunate simple equilibrium where each player points to the player on their right, then they each point to the player on their left, then right, etc. Perhaps there is something to the slightly randomized version.