What is the relationship between the maximin payoff and the set of NE payoffs in a finite game?

In a finite game, the maximin payoff refers to the largest payoff a player can guarantee for themselves regardless of the strategies chosen by other players. On the other hand, the set of Nash equilibrium (NE) payoffs refers to all the payoffs that can be achieved when each player chooses their strategy in a way that no player would have an incentive to unilaterally deviate.

The relationship between the maximin payoff and the set of NE payoffs can be understood as follows:

1. Inclusion of maximin payoff in the set of NE payoffs: If the maximin payoff for a player is included in the set of NE payoffs, it means that there exists a strategy profile where the player achieves their maximin payoff, and this strategy profile is also a Nash equilibrium. In other words, the player can guarantee their highest payoff by playing a strategy that is part of a Nash equilibrium.

2. Exclusion of maximin payoff from the set of NE payoffs: If the maximin payoff is not included in the set of NE payoffs, it means that the player cannot achieve their maximin payoff in any Nash equilibrium. This indicates that playing a more aggressive strategy might lead to higher payoffs, but it also opens the possibility for other players to deviate and potentially harm the player's payoff.

Overall, the relationship between the maximin payoff and the set of NE payoffs is that the maximin payoff serves as a benchmark or lower bound for what a player can achieve, and the set of NE payoffs represents all the possible payoffs that can be achieved when players choose their strategies rationally.