For example, the Abreu-Matsushima mechanism implements essentially any implementable property at the unique rationalizable Nash equilibrium (a much stronger guarantee than anything preserved by the revelation principle). If you actually use the Abreu-Matsushima mechanism you will find that it basically never works unless the players want it to work (and often, not even then) as has been verified empirically.
The VCG mechanism maximizes social welfare whenever the players play undominated strategies. In practice, the possibility of even very weak collusion between even two players destroys this guarantee.
In general, the claim that rational players will always choose a Nash equilibrium has little empirical support, and in fact doesn’t have very good theoretical support either outside of two-player zero sum games (of course in the OP the game is two-player and zero sum; there my complaint is that common knowledge of rationality is a bad assumption).
The Nash equilibrium suggests playing randomly in matching pennies—yet you can do much better than that if facing an irrational opponent—such as a typical unmodified human. The Nash equilibrium is for when both players play rationally.
Can you give examples of a mathematical proof leading to an ineffective mechanism?
For example, the Abreu-Matsushima mechanism implements essentially any implementable property at the unique rationalizable Nash equilibrium (a much stronger guarantee than anything preserved by the revelation principle). If you actually use the Abreu-Matsushima mechanism you will find that it basically never works unless the players want it to work (and often, not even then) as has been verified empirically.
The VCG mechanism maximizes social welfare whenever the players play undominated strategies. In practice, the possibility of even very weak collusion between even two players destroys this guarantee.
In general, the claim that rational players will always choose a Nash equilibrium has little empirical support, and in fact doesn’t have very good theoretical support either outside of two-player zero sum games (of course in the OP the game is two-player and zero sum; there my complaint is that common knowledge of rationality is a bad assumption).
The Nash equilibrium suggests playing randomly in matching pennies—yet you can do much better than that if facing an irrational opponent—such as a typical unmodified human. The Nash equilibrium is for when both players play rationally.