Finding Nash Equilibria in Simultaneous-Move Games

What is a Nash Equilibrium?

At its core, a Nash Equilibrium describes a state of "no regrets." It's a combination of strategies, one for each player, where no single player can get a better outcome by unilaterally changing their own strategy, assuming everyone else sticks to theirs. It represents a stable, self-enforcing agreement or outcome.

Given your formal background, you'll appreciate the precision of the definition. Let's ground our understanding in the formal language of the field before moving to application.

As the text highlights, the key is that each player's action is a best response to the other players' equilibrium actions. Unlike a dominant strategy, which is best regardless of what others do, a Nash Equilibrium strategy is only guaranteed to be optimal if others are also playing their part of the equilibrium. This requires players to reason about what their partners will do.

A Classic Coordination Problem: The Battle of the Sexes

Many relationship negotiations are not zero-sum; both partners often prefer coordinating on a shared activity over going their separate ways. However, they may have different preferences over which shared activity to choose. This creates a "mixed-motive" game of both cooperation and conflict. The classic model for this is the "Battle of the Sexes."

Let's adapt this to a scenario relevant to our course. Imagine Sarah (the wife) and David (the husband) are deciding on their Saturday night plan.

• Players: Sarah (Wife), David (Husband)
• Actions: {Hotwife Party, Quiet Night In}
• Preferences: Both want to spend the evening together, but Sarah prefers the excitement of the party, while David prefers a quiet night at home. The worst outcome for both is to do different things and end up alone and resentful.

We can assign utility values to these preferences to create a payoff matrix. Let's say coordinating on Sarah's preferred activity gives her a payoff of 10 and David a payoff of 5 (he enjoys her happiness via compersion, but it's not his ideal night). Coordinating on David's preference is the reverse: (5, 10). Miscoordination is disastrous for both: (0, 0).

The payoff matrix is as follows, with Sarah as the row player and David as the column player. The payoffs are (Sarah, David).

So, what is the predicted outcome? This is where finding the Nash Equilibria becomes essential.

Finding Pure-Strategy Nash Equilibria

There are two common ways to find pure-strategy Nash Equilibria (PSNE).

1. Cell-by-Cell Inspection (The "No Regrets" Test)

You can go through each of the four cells in the matrix and ask: "Does either player have a profitable unilateral deviation?" If the answer is "no" for both players, that cell is a Nash Equilibrium.

The following video demonstrates this method clearly.

Applying this logic to our scenario:

• At (Hotwife Party, Hotwife Party): Sarah is getting 10. If she switches to Quiet Night In, she gets 0. No regret for her. David is getting 5. If he switches, he gets 0. No regret for him. This is a Nash Equilibrium.
• At (Quiet Night In, Quiet Night In): Sarah is getting 5. If she switches, she gets 0. No regret. David is getting 10. If he switches, he gets 0. No regret. This is also a Nash Equilibrium.
• At the two miscoordination cells (0,0): Both players have a profitable deviation. For example, at (Hotwife Party, Quiet Night In), Sarah could switch to Quiet Night In to get 5 instead of 0, and David could switch to Hotwife Party to get 5 instead of 0. These are not stable.

2. Best-Response Analysis (The "Underlining Method")

A more systematic and less error-prone method, especially for larger games, is to find each player's best response to every possible strategy of the other player. A cell where every player is playing a best response is a Nash Equilibrium. A simple way to track this is by underlining the best-response payoffs.

This video provides an excellent walkthrough of this technique.

Let's apply this method to our scenario:

1. Assume David plays Hotwife Party (left column). What is Sarah's best response? Hotwife Party gives her 10, Quiet Night In gives her 0. Her best response is Hotwife Party. We underline her payoff of 10.
2. Assume David plays Quiet Night In (right column). What is Sarah's best response? Hotwife Party gives 0, Quiet Night In gives 5. Her best response is Quiet Night In. We underline her payoff of 5.
3. Assume Sarah plays Hotwife Party (top row). What is David's best response? Hotwife Party gives 5, Quiet Night In gives 0. His best response is Hotwife Party. We underline his payoff of 5.
4. Assume Sarah plays Quiet Night In (bottom row). What is David's best response? Hotwife Party gives 0, Quiet Night In gives 10. His best response is Quiet Night In. We underline his payoff of 10.

Our final matrix looks like this:

The two cells where both payoffs are underlined are our pure-strategy Nash Equilibria: (Hotwife Party, Hotwife Party) and (Quiet Night In, Quiet Night In).

The Strategic Importance of Multiple Equilibria

The model doesn't give us a single answer. It tells us there are two stable outcomes. This indeterminacy is, in itself, a crucial strategic insight. It means that while the couple is likely to coordinate, the game theory model alone cannot tell us which equilibrium they will land on. One equilibrium strongly favors Sarah's preferences, and the other strongly favors David's.

This is where the real "battle" occurs. Factors outside the simple matrix—such as social convention ("we did what you wanted last week"), who makes the first move in the conversation, or who is more stubborn—will determine which equilibrium is selected.

Furthermore, the very structure of the payoffs dictates the equilibria. What if David's sense of compersion was so high that he actually preferred the Hotwife Party to a quiet night in? The payoffs might change, and with them, the equilibria.

This reading demonstrates a vital point for your work: accurately quantifying the emotional payoffs is paramount. If one partner's preference for their desired activity outweighs their preference for togetherness, the equilibrium where they go out separately could become stable. Your expertise in analyzing relationships is key to estimating these utilities, which in turn determines the predicted outcome of the negotiation.

A quick note on a more advanced topic: when a game has no pure-strategy Nash Equilibrium (like Rock-Paper-Scissors), or in addition to pure ones, there can also exist mixed-strategy Nash Equilibria, where players randomize their actions. The video by William Spaniel introduces this concept. We will explore this in more detail later, but it's useful to know that randomization itself can be a strategic choice.

Conclusion

Today we moved from building game models to solving them. We have seen how to identify the stable outcomes of a simultaneous-move negotiation by finding the Nash Equilibria.

Key Takeaways:

• A Nash Equilibrium is a strategy profile where no player can benefit by unilaterally changing their strategy. It represents a stable outcome with "no regrets."
• We can find pure-strategy Nash Equilibria by either checking each outcome for profitable deviations or, more systematically, by using best-response analysis (the underlining method).
• Games like the Battle of the Sexes often have multiple equilibria. This indicates a coordination problem where the final outcome is determined by factors beyond the simple game matrix, such as communication or sequential moves.
• The number and nature of equilibria are highly sensitive to the players' payoff structures. Small changes in how much a player values an outcome can fundamentally alter the predicted result of the game.

Preview of the Next Lesson:
The indeterminacy of the Battle of the Sexes begs the question: what happens if one person can act first? If Sarah can credibly commit to going to the party, David's best response is to follow. This moves us from the world of simultaneous games to sequential ones. In our next lesson, we will introduce Subgame Perfect Equilibrium, a powerful refinement of the Nash concept used to analyze sequential negotiations, credible threats, and promises.