Preserving truth under submodel transitions does not guarantee equilibrium existence, since equilibria require strategic best-response consistency, not merely semantic monotonicity.

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Equilibrium (in game theory)(in game theory and strategic analysis): A stable situation where no player can improve their outcome by changing their strategy alone, so everyone has settled into their best choice given what others are doing.
Equilibrium existence(in game theory): Whether or not a stable, balanced state actually exists in a particular strategic situation.
Semantic monotonicity(as used in logic and model theory): A property where adding more information or conditions to a logical statement doesn't contradict or reverse what was already established—think of it like a promise that remains true even with added details.
Strategic best-response consistency(in game theory): A requirement that each player's chosen strategy must be their smartest move given what they expect the other players to do.
Submodel transitions(in logic and game theory): Moving from one simplified version of a system to another simplified version, like switching between different ways of representing the same problem.
Truth preservation(in logic and philosophy of language): Keeping statements or facts true and valid as you move from one framework or system to another.

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