We have two players playing in turns. Each player can decide to stop in which case the game is stopped and the two players can go on with their lives, or to act.
The player that acts gains and the other player loose twice as much.
So in the first round: player I can stop the game, if he acts he gets 1 and the other player II gets -2.
If the games continues in the second turn player II can stop the game, if he acts he gets 1 and player I gets -2.
If the games continues in the third turn player I can stop the game, if he acts he gets +1 and player II gets -2.
How to play this game?
Combinatorics and more 
Isn’t this obvious? You play to win.
Player 1 acts first and wins.
There HAS to be some other constraint.
Perhaps this is the following standard game (though I couldn’t find it on Wikipedia):
Two players playing in turns. A pot of $100 is currently “up for grabs”. At each turn, a player can either take $1, and have the game continue, or he can grab $2 and have the game end immediately. It’s not possible to take more money than is left, and neither player gets unclaimed, remaining money.
A cooperative solution would be to alternately take $1. Each player thus gets $50.
However, the optimal “rational” solution (if your utility function depends only on your own wealth) is to immediately grab $2 and end the game. This can be seen through backtracking:
Suppose there is only $2 left. Then clearly you should grab $2.
Suppose there is $3 left. Then if you take $1, the other player will grab $2 and end the game. Alternatively, you could just take $2. The latter dominates and so it is the optimal strategy.
By induction, regardless of how much money is left, you should take $2.
However, the game I mentioned doesn’t really seem like much of a “retaliation” game. There’s probably something else going on.
“A strange game. The only winning move is not to play. How about a nice game of chess?”
For a twist on the game:
-suppose we are mid-game, i.e. it is given that both players have already been acting for several rounds,
– suppose further that the players have different views of what each has received: player 1 thinks he has -10 and his opponent has -7, but player 2, who just acted, thinks that both of them have -15.
It’s player 1’s move; “how to play this game?”
The different subjective views of the score shouldn’t affect rational players but humans often value fairness above their own self-interest.
I think there is no incentive to play. Suppose player 1 acts in the first move. The scores of (player1, player2) now are (+1, -2). The only rational move for player 2 is to act and get the scores (-1, -1). This makes the difference in score between the two players as 0. In case player 1 acts again, then he can only expect a repeat of previous rounds, reducing the scores of each player by 1 (after every two rounds). So, the only rational move for player 1 is to not act and end the game in round 1.
Assuming that the strategies of each player doesn’t change from round to round,
the following three situations are all Nash equilibria.
(a) Player 1 decides to stop, and player 2 wants to play. The expected value of each player is 0.
(b) Player 1 decides to play, and player 2 stops. The expected value of player 1 is 1 and the expected value of player 2 is -2.
(c) Both players flip a coin each time it is their turn, and decide to stop w/prob. 1/2 and play w/prob. 1/2. The expected value of player 1 is 0, and the expected value of player 2 is -1.
The question of how to act relies on a piece of information not given. What is the objective of the game? Is it to end with as many points as possible? Is it to end with more points than the other player? It is to end with at least as many points as the other player?
Strategy is pointless without a goal. Motivation precedes action.
Well, I fully identify with reactions of the form “something must be wrong.” “there HAS to be a way around it,” “it is pointless,” “it is not interesting,” etc., which reflect my own emotions regarding the retaliation reality that this little game pretends to model.
I like the Centipede Game
http://en.wikipedia.org/wiki/Centipede_game
as a model of retaliation vs cooperation since its theoretical solution (its subgame perfect equilibrium) is more surprising but still contrasts from what’s done in practice.
I don’t pretend to have a full analysis of this game, but I presume the kind of reasoning that players are supposed to have is along the following lines.
If I act, then probably the other player will retaliate and I’ll end up in a worse position than I’m in now. So the sensible thing to do (regardless of what has happened up to now) is not to act. But that means that if I do act, it will be sensible for the other player not to act. But that means I gain by acting. So the sensible thing is after all to act. But that means it will be sensible for the other player to act after I’ve acted. But that means I won’t gain by acting.
We seem to get an infinite regress reminiscent of the liar paradox: it makes sense to act if and only if it makes sense not to act.
A very nice post — I hadn’t heard of this game, and certainly don’t find it pointless or uninteresting (though I see that you subtly refer not to the game but to the reality it models, about which no comment is necessary).
Thanks, Tim; Indeed the post meant to express (in a strange way, I admit) my sadness from a reality of retaliation. I thought about various variants of the game, (e.g., the case were the stakes are doubled each time, or increase linearly), asked around a bit (so for some cases the answer depended on the parity of the number of rounds, and was not well-posed for the infinite case, and in some other cases a mixed strategy seemed to lead to equilibrium), but at the end I did not try to analyze any of the variants.
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