Explain the formula when the Nash equilibrium does not hold

The Nash equilibrium is a concept in game theory that describes a situation where each player in a game chooses a strategy that is optimal given the strategies chosen by the other players. In other words, no player can improve their outcome by changing their strategy, assuming that the other players do not change their strategies.

However, there are situations where the Nash equilibrium does not hold. This can occur when one or more players have dominant strategies, meaning that there is a strategy that is always optimal for them regardless of the strategies chosen by the other players. In this case, the other players may not be able to achieve their optimal outcomes, even if they coordinate their strategies.

Another situation where the Nash equilibrium may not hold is when there are multiple equilibria, meaning that there are multiple sets of strategies that satisfy the conditions for a Nash equilibrium. In this case, the players may not be able to agree on which equilibrium to choose, leading to a situation where no equilibrium is reached.

In summary, the formula for when the Nash equilibrium does not hold is when one or more players have dominant strategies or when there are multiple equilibria