18.11: Multiple Equilibria

Multiple equilibria occur when strategic interactions between players result in several potential stable outcomes, each being a Nash equilibrium. This happens when a player's best response changes depending on the other's choice, leading to various combinations where neither player has an incentive to deviate from their strategy.

Imagine two streaming services, StreamNow and ViewPrime, deciding when to release a new show—spring, winter, or not at all. Both benefit most when they release shows in different seasons, maximizing viewership by avoiding direct competition.

First, consider StreamNow's response: if ViewPrime releases in spring, StreamNow's best option is winter. If ViewPrime chooses winter, StreamNow's best move is spring. If ViewPrime doesn't release anything, StreamNow's optimal choice is spring, facing no competition.

Next, examine ViewPrime's response: if StreamNow releases in spring, ViewPrime's best choice is winter. If StreamNow picks winter, ViewPrime's best response is spring. If StreamNow does not release, ViewPrime's best move is also spring, dominating that period.

In this game, two Nash equilibria exist: one where StreamNow releases in spring and ViewPrime in winter, and another where StreamNow releases in winter and ViewPrime in spring. Both outcomes are stable, as each company maximizes its payoff by avoiding competition. However, which equilibrium will occur cannot be predicted solely from this information, as either configuration is equally beneficial. External factors like coordination or pre-release announcements may influence the final outcome.

Burger Queen (BQ) and King's Burger (KB) are deciding whether to launch a new spicy burger during the summer, fall, or not at all. Both chains benefit the most by launching in different seasons, as launching in the same season would split the market.

The payoff matrix shows the outcomes.

First, consider the payoffs of BQ.

If KB chooses summer, BQ's best response is fall.

If KB opts for fall, BQ's best response is summer.

If KB picks none, BQ's still has the best response as summer.

Now, consider the payoffs of KB.

If BQ chooses summer, KB's best response is fall.

If BQ opts for fall, KB's best response is summer.

If BQ picks none, KB's best response is summer.

Now, identify the cell with two tick marks to find the equilibrium. This cell represents the choice that is most likely to produce the highest payout for each player, given the other player's best responses.

Here, the game has two Nash equilibria.

BQ launching in summer while KB opts for fall, or BQ launching in fall while KB opts for summer. However, with the given information, it cannot be determined which company chooses which season.

• Multiple Equilibria - Several potential stable outcomes in strategic interactions.
• Nash Equilibrium - A stable state of a system involving interacting participants.
• Best Response - Optimal strategy for a player based on other participants' actions.
• Stable Outcome - An outcome where none have incentive to deviate their strategy.
• Strategic Interactions - Actions between players in a system that affect each other's outcomes.

• Define Multiple equilibria – Explain when & why it occurs (e.g., multiple equilibria).
• Contrast different Nash Equilibria – Outline key differences (e.g., spring release vs. winter release).
• Explore Examples – Describe a scenario with streaming services (e.g., StreamNow vs. ViewPrime).
• Explain Best Response Mechanism – how a player's choice changes based on the other's decision.
• Apply Multiple Equilibria in Context – Understand potential outcomes in a market-based scenario.

• [Question 1] What is multiple equilibria and how does it occur in strategic interactions?
• [Question 2] What is the best response mechanism and how does it affect player's choices?
• [Question 3] Can you identify the Nash Equilibria in the example of StreamNow and ViewPrime?

• Students – Exploring multiple equilibria improves strategic thinking.
• Educators – Use this as a real-world example for teaching game theory.
• Researchers – Insight for studying decision-making in competitive scenarios.
• Economy and Business Enthusiasts – Useful for understanding strategies in competitive markets.