I have been pondering this for some time, and have been unable to find the relevant game theory material to understand this and learn. I’m very noob to this, so it may be something quite obvious that I am missing.

The situation is this:

Given one round of game play, the game favors one of the players. Lets say V1=1/14

When adding 1/2 round of additional play, the game favors P2, lets say V1=-1/13

When adding a complete round of additional play, the game once again favors player 1, but less so than originally, say V1=1/12

When infinite extensions are programmed into a computer, the value of the game converges on V1=x, where the absolute value of the game to either player converges on a number, lets say 1/10.

If it can be shown by provable valid optimal strategy (not just a computer sim) that the infinite value of the game is exactly x...

Then could not player 2 alter her strategy during the first turn of an infinite game and offer player 1 this value, and remove the option of player 1 to continue the game, and player 1 must accept this strategy, since continuing any number of rounds yields strictly less value?

Thanks for any insights.