In one-period games, players make their decisions simultaneously without knowing what the other will choose. The Nash equilibrium represents a point where no player can improve their outcome by changing their decision, assuming the other player's choice remains the same. This concept applies directly to situations like mobile catering services deciding on market locations without coordination.

Consider two mobile food vendors choosing between setting up in a busy plaza or a quieter business district. If one vendor goes to the plaza alone, it captures all the customers there. If both go to the plaza, they share the customer base. However, the shares would not be equal, as one vendor potentially has an advantage due to a stronger reputation. Meanwhile, if one vendor opts for the quieter business district, it monopolizes that area's smaller customer pool.

The Nash equilibrium emerges when each vendor's choice maximizes their outcome based on the other's decision, and neither would benefit from switching locations alone. For instance, if Vendor A finds that staying in the plaza offers the best results regardless of Vendor B's choice, while Vendor B finds that choosing the business district is optimal when Vendor A goes to the plaza, then this setup is stable. Neither vendor gains by changing their location as it would lead to lower profits.

In this equilibrium, both vendors have adapted their strategies to minimize competition and maximize their respective profits, given the other's choice. This shows how Nash equilibrium in one-period games leads to predictable and stable outcomes where no competitor has an incentive to change their decision, thus ensuring the balance of strategies in competitive settings.