18.10: Nash Equilibrium in One-Period Games

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.

Consider two rival mobile catering services, T-Truck and B-Van. They must independently choose to serve lunch either downtown or in the industrial area without knowing the other's choice.

If one truck operates alone at a location, it captures all the customers there. If both select the same location, they split the customers, but T-Truck, with its better reputation, always gets a larger share.

This payoff matrix shows their choices and outcomes.

First, consider the payoffs of T-Truck.

If B-Van chooses Downtown, T-Truck's best response is Downtown. If B-Van opts for the Industrial Area, T-Truck will still favor Downtown.

Now, consider the payoffs of B-van.

If T-Truck chooses Downtown, B-Van's better choice is the Industrial Area. If T-Truck picks the Industrial Area, B-Van opts for Downtown.

Here, T-Truck has a dominant strategy. Regardless of B-Van's choice, T-truck will choose Downtown.

Now, B-van's best response to T-truck choosing Downtown is the Industrial Area.

As a result, the Nash equilibrium occurs with the T-Truck downtown and the B-Van in the Industrial Area. Here, neither would switch locations after learning the other's choice, as any change would reduce their profits.

• Nash Equilibrium - A solution concept in game theory where no player can benefit from unilaterally changing their strategy.
• One-Period Games - Games in which players make decisions simultaneously, unaware of each other's choices.
• Business District - A quieter area where a vendor can monopolize a smaller customer pool.
• Plaza - A busier location where vendors may need to share customers if they select the same spot.
• Vendor Advantage - The potential benefit a vendor has over another due to reputation or other factors.

• Define Nash Equilibrium – Describe this solution concept in game theory (e.g., Nash Equilibrium).
• Contrast Plaza vs Business District – Discuss the potential benefits for mobile vendors choosing between these two areas (e.g., customer base size).
• Explore Vendors' choices – Illustrate scenario where vendors' choices lead to Nash Equilibrium (e.g., selection of location).
• Explain Vendor Advantage – Detail how reputation or other factors might affect a vendor’s customer share.
• Apply in Context – Consider how Nash Equilibrium could play out in different one-period games.

• What is the Nash Equilibrium and how does it apply to one-period games?
• How does the location choice between a Plaza and a Business District impact vendors' profits?
• How does a vendor's advantage influence the Nash Equilibrium in this context?

• Economics Students – Understand how the concept of Nash Equilibrium is applied in real-world scenarios.
• Educators – Provides a relevant example to demonstrate Nash Equilibrium in game theory.
• Researcher in Economic Theory – Application of Nash equilibrium for understanding market dynamics.
• Game Theory Enthusiasts – Offer insights into how Nash Equilibrium plays out in one-period games.