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Assume that a new Chess-variant designed to increase the level of competition throughout the game, provide additional excitement at the finish: "A Final Target Point will be set." The Final Target Score will be determined by an agreement. The first player to reach the Final Target Score will win. In game, you can add points for your own score by capturing and getting approximations for piece values. For instance, capturing a Rook, you score 5; capturing a Queen, you score 9; capturing a King, you score ∞ (it's a win). Don't mind about this set of game rules, I know here it is Economics Stack Exchange. There are still difficulties in assessing the consequences of individual changes on game dynamics and appeal.

My question to you is—
The player going second (is given 2 points) wants a Final Target Score $x.$
The player going first wants a Final Target Score $y$ far from $x.$
So how do we have Pareto–optimality for the Final Target Score (according to your economic perspective)?

I find something maybe helpful here, which are—
"Sharing a small number of items. This is done, for example, in the adjusted winner procedure."
But I don't know how to continue then. I hope to see nice proofs. Any comments are welcome and appreciated.

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    $\begingroup$ I don't understand the question. One player wins, and one player loses. The latter is going to wish they'd rather be the winner; there is no envy-free allocation here. $\endgroup$ Apr 24 at 17:01
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    $\begingroup$ What is the payoff function? $\endgroup$ Apr 27 at 9:08

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