The Value of AAP

Imagine the following two player game between A and B[1],

  • All that a player is allowed to do is play or not. Feel free to imagine whatever “play” might mean. Let’s say in our case it means rolling a die. For simplicity, let’s assume the die has only side to it. Marked, say, 5.
  • If a players plays, then let’s give that player the number he rolls. So, if A throws, A gets 5 points. The game’s quirk is, if A throws the die and gets 5, then B gets -10. If A decides not to throw, the game stops there.
  • Now B gets to play he can choose to play or not. And rules are similar — B either gets +5 and A gets -10. Or, both get 0 and the game stops there.
  • The game can thus continue till both players keep throwing the die. If one of them stops throwing, the game reaches ends.

Now the first question one would ask is, what’s even the point of this game. Let’s assume there is. Now how does one play the game?

If you are A, it appears, the best way is to not throw the die. Because if you do, B will ‘retaliate’ and the -2x of the other person’s turn will certainly leave you worse off regardless of whether it’s your first turn or nth turn. So might as well not throw in the first.

The same logic holds for B as well, does it not? So it appears the best strategy is to not play the game at all.

Except, herein lies a complexity. It’s anyway worse off option at any position for B to throw the die. Which means, if you as A throw the die and get 5 points, and make B get -10, it still makes it the best thing for B to not throw the die. But that makes it sensible for B to act in retaliation and make you, A, lose 10 points. Which means you wouldn’t gain points now. In sum, we get the liar’s paradox. And in this variety it’s regressing to infinity. That is, it makes sense to throw the die if and only if it makes sense not to throw the die.

In our case, assume A & B have already played ‘n’ turns and therefore are regressing to infinity anyway. Now for fun, let’s introduce a third player C. But add the caveat that he needs to throw at least once to be considered part of the game. Doesn’t C have a finite number throws and walk away to win as an option?

One is tempted to think, A and B are INC and BJP. And C is AAP. At least when the game is restricted to areas where there is a bipolar contest between the INC and BJP.

 

[1] – I read some version of this game somewhere and I am not able to remember enough to cite. If you are bored and have access, do dig away and let me know. Or if you already know and this is more famous than I thought as a game, do let me know. I’m merely recollecting from most likely incorrect memory.