Understanding Nash Equilibrium: A Best-Response Approach

Introduction

Hello! In our last lesson, we saw how the powerful logic of Iterated Elimination of Strictly Dominated Strategies (IESDS) can predict game outcomes by assuming that rational players will not choose obviously inferior options. We used this to solve games like the Median Voter model, where candidates converge to the center.

However, IESDS has a major limitation: what happens when no strategy is strictly dominated? In many games, like the "Battle of the Sexes" where two people want to meet up but have different preferences for the venue, every choice is a potential best response to something the other player might do. IESDS has no "bite" in these cases and cannot help us.

This reveals the need for a more universal concept of a stable outcome. This lesson introduces the most celebrated solution concept in game theory: the Nash Equilibrium. Our goal is to define Nash Equilibrium (NE) as a profile of mutually best-response strategies. We will unpack what a "best response" is, build the formal definition of NE, and explore the powerful intuitions that make it the bedrock of modern strategic analysis.

1. From Dominance to Best Response

The idea of a dominated strategy was about finding a strategy that is never the best choice. Let's flip this around and ask: given what we believe our opponent will do, what is the best choice? This is the concept of a best response.

A player's best response is the strategy (or set of strategies) that yields the highest possible payoff, given a particular strategy chosen by the other players.

Formally, for a two-player game, we say that player 1's strategy $s_{1}^{*}$ is a best response to player 2's strategy $s_{2}$ if player 1's payoff from playing $s_{1}^{*}$ is at least as large as their payoff from playing any of their other available strategies $s_{1}^{'}$ . We write this as:

u_{1} (s_{1}^{*}, s_{2}) \geq u_{1} (s_{1}^{'}, s_{2}) for all s_{1}^{'} \in S_{1}

where $S_{1}$ is the set of all strategies available to player 1.

This concept shifts our focus from eliminating bad strategies to identifying optimal ones in a specific context.

2. The Definition of Nash Equilibrium

Now, let's take the crucial leap. What if, in a given strategy profile, every single player is simultaneously playing a best response to what all the other players are doing? This special state of stability is a Nash Equilibrium.

A Nash Equilibrium is a strategy profile—a combination of one strategy for each player—where no player can improve their payoff by unilaterally changing their own strategy, assuming everyone else sticks to their chosen strategy. It is a profile of mutual best responses.

For a game with $n$ players, a strategy profile $s^{*} = (s_{1}^{*}, s_{2}^{*}, \dots, s_{n}^{*})$ is a Nash Equilibrium if, for every player $i$ , their strategy $s_{i}^{*}$ is a best response to the strategies of all other players, $s_{- i}^{*}$ . Mathematically:

u_{i} (s_{i}^{*}, s_{- i}^{*}) \geq u_{i} (s_{i}^{'}, s_{- i}^{*}) for all s_{i}^{'} \in S_{i} and for all players i

To explore the formal definition and the intuition behind it, watch the following two short video clips. The first is a very concise and formal presentation, while the second, from Professor Ben Polak at Yale, offers a slightly different phrasing and rich examples.

2. Nash Equilibrium: Formal Definition and Intuition (Game Theory Playlist 3)

This first video, from Selcuk Ozyurt, provides a clear, formal definition of Nash Equilibrium and introduces two key ways to think about it: as a 'regret-free' outcome and as a state where players' beliefs about each other are correct.

Watch the video from the beginning to 03:43 and then from 05:58 to 08:41. Focus on the mathematical definition and the two core interpretations: 'stability'/'no regrets' and 'correct beliefs'.

5. Nash equilibrium: bad fashion and bank runs

This second video, from YaleCourses, provides a complementary perspective on the same core ideas, reinforcing the definition and motivations for studying Nash Equilibrium.

Watch from 00:31 to 06:14. Pay attention to how Professor Polak defines a Nash Equilibrium as a profile of 'mutual best responses' and explains the 'no regrets' and 'self-fulfilling beliefs' motivations.

3. The Two Pillars of Nash Equilibrium: Stability and Beliefs

As you saw in the videos, there are two powerful ways to interpret what a Nash Equilibrium represents.

Pillar 1: No Regrets (Stability)

Imagine a game is played and the outcome is a Nash Equilibrium. After the fact, each player looks at what their opponents did and asks themselves, "Given what they played, could I have done better by choosing a different strategy?" In a Nash Equilibrium, the answer for every player is "No." No player has an incentive to unilaterally deviate from their strategy. This gives the equilibrium a sense of stability or self-enforcement. It's a "regret-free" outcome.

Pillar 2: Self-Fulfilling Beliefs

Alternatively, think about the game before it's played. Suppose there is a common understanding that a particular Nash Equilibrium will be the outcome. For instance, if Player 1 believes that Player 2 will play their part of the NE, then Player 1's best response is... to play their own part of that same NE. This holds true for all players. If everyone believes the NE will be played, it becomes rational for everyone to play it, thus fulfilling the initial beliefs. This is a very strong assumption about coordination and belief, as the presenters noted.

For a more rigorous look at these ideas, the following text formalizes the concept of mutual best response.

[PDF] Lecture 2: Mixed Strategy and Nash Equilibrium

This document, from a course on algorithmic game theory, provides a mathematically dense definition of Nash Equilibrium. It emphasizes that NE is a 'mutual best response' and discusses the fundamental questions that arise from this solution concept.

Please read Section 3, 'Nash Equilibrium — “Mutual Best Response”' (up to Example 3.3) and Section 5, 'Solution Concept'. Focus on Definition 3.1, which is the formal statement we've been discussing. Section 5 provides excellent context, framing NE within the key questions of existence, uniqueness, and computation that drive research in this field.

4. Visualizing Mutual Best Response

Your background in mathematics makes you comfortable with functions. We can visualize the "mutual best response" condition as the intersection of players' best-response correspondences.

A best-response correspondence, $B R_{i} (s_{- i})$ , is a function that maps the other players' strategies to player $i$ 's set of best responses. A Nash Equilibrium is a strategy profile $s^{*}$ where every player's strategy is in their best-response set: $s_{i}^{*} \in B R_{i} (s_{- i}^{*})$ for all $i$ .

Consider the classic "Battle of the Sexes" game. Two people want to go out together. One prefers the Movie, the other prefers the Ballet, but both would rather go to their less-preferred event together than go to their favorite event alone.

Best-Response Correspondences in the Battle of the Sexes Game — This image shows the payoff matrix for the Battle of the Sexes game, along with the best-response correspondences for each player. 'p' is the probability the Lecturer chooses Movie, and 'q' is the probability the Catalan chooses Movie. The graphs show each player's best choice of 'p' or 'q' for any given choice by their opponent. The intersections of these graphs represent the Nash Equilibria of the game.

Look at the graphs on the right. The red line shows the Lecturer's best response $p (q)$ for any given $q$ by the Catalan. If the Catalan is very likely to choose Movie ( $q > 2/3$ ), the Lecturer's best response is to also choose Movie for sure ( $p = 1$ ). The blue line shows the Catalan's best response.

A Nash Equilibrium occurs where these two best-response correspondences intersect. At these points, both players are simultaneously playing a best response to each other. This game has three such intersections, which means it has three Nash Equilibria (two pure, one mixed).

This is precisely what Professor Polak illustrates with his crossing lines in the video you watched earlier (LINK, from 06:14 to 08:02). The point where the best response functions cross is the point of mutual best response—the Nash Equilibrium.

Conclusion

Today, we've introduced the central solution concept in non-cooperative game theory. It provides a notion of equilibrium for any game, resolving the main limitation of the iterated elimination method we studied previously.

Here are the key takeaways from this lesson:

A best response is a strategy that maximizes a player's payoff, given the strategies chosen by the other players.
A Nash Equilibrium is a strategy profile where every player's strategy is a best response to the strategies of all the others. It is a state of mutual best response.
This concept is powerful because it represents a stable outcome where no player has a unilateral incentive to deviate (the "no regrets" principle).
It can also be interpreted as a set of self-fulfilling beliefs; if players expect a Nash Equilibrium, it is rational for them to play it.

We have defined what a Nash Equilibrium is and why it's important. The next logical step is to figure out how to find these equilibria in a game. In our next lesson, we will develop a simple, systematic algorithm to identify all pure-strategy Nash Equilibria in matrix games, putting this powerful definition into practice.

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