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lemma
proved
Common Guarantee Gives The Value
If player I and player II both guarantee the same number $w$, then $w$ is the unique value of the game, and the strategies witnessing the two guarantees are optimal.
Proof
The two guarantees and weak duality give the chain $$ w \le \underline v \le \overline v \le w. $$ Thus $\underline v=\overline v=w$, and the witnessing strategies attain the value. The minimax theorem (Von Neumann Minimax Theorem) closes the inner two inequalities, the player-guarantee bound is the outer pair (Player Guarantee).
References
- [MFoGT, Lem. 2.2.8] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. A common guarantee is unique and equal to the value.