In a game involving both mixed strategy equilibria and cooperative games, how can players ensure that they achieve a Nash Equilibrium while also maximizing their collective payoff?

In a game, players can reach a Nash Equilibrium by choosing strategies where no player can gain by changing their choice alone, while still maximizing their collective payoff through cooperation. This means that each player must find a balance between their own interests and the benefits of working together. For example, imagine two friends deciding whether to play soccer or basketball. If they both choose soccer, they enjoy the game together, resulting in a better experience than if one chooses soccer and the other chooses basketball, which would lead to frustration. By agreeing to play soccer together, they reach a Nash Equilibrium because neither wants to switch to basketball alone, and they also maximize their enjoyment.