Game Theory — Nash Equilibrium, Strategies & Payoff Matrix

Game theory studies decisions where your best move depends on what other people do. Three ideas carry most introductory problems: a strategy is what you choose, a payoff is what you get, and a Nash equilibrium is a set of choices where no player can do better by changing alone.

When this way of thinking applies

In an ordinary optimization problem, you pick the best option in a fixed situation. In game theory the situation can change, because other players are choosing too, either at the same time or in response to you. The question shifts from "What is my best move?" to "What is my best move given what others might do?" Whenever outcomes depend on other agents' choices, this strategic-interaction lens is the right tool.

A strategy is a player's available choice or rule for acting; in a one-shot game it may be a single action such as cooperate or defect. A payoff is the result from a particular combination of choices, representing money, points, utility, or any ranking where larger is better for that player.

The procedure, step by step

Step 1: List the players and the choices each one can make.

Step 2: Build the payoff matrix. In a two-player game, each cell matches one strategy from Player A with one from Player B. Here is a standard Prisoner's Dilemma style matrix; the first number in each cell is Player A's payoff, the second is Player B's:

Read each cell as one complete outcome: if both cooperate, each gets $3$; if one defects while the other cooperates, the defector gets $5$ and the cooperator gets $0$; if both defect, each gets $1$. The exact numbers are not a law of game theory, just one payoff pattern. What matters is the incentive structure: each player is tempted to defect, even though both would rather reach mutual cooperation than mutual defection.

Step 3: Find best responses. For each possible choice by the other player, mark which strategy gives the higher payoff.

Step 4: Locate the equilibrium, a cell where both players are already playing best responses. A Nash equilibrium is a strategy profile where no player can improve their own payoff by switching alone while the others keep their strategies. Equivalently, each player's choice is a best response to the others'. It does not mean the outcome is best for everyone; it means no one has a one-sided incentive to move.

Step 5: Interpret the result. Check whether the equilibrium is stable, efficient, or in tension with the best joint outcome.

Worked example: finding the Nash equilibrium

Use the matrix above.

If Player B cooperates, Player A compares cooperate for $3$ with defect for $5$. Defect is better.

If Player B defects, Player A compares cooperate for $0$ with defect for $1$. Defect is still better.

So for Player A, defect is the best response in either case, and by symmetry the same holds for Player B. That makes $(\text{Defect}, \text{Defect})$ a Nash equilibrium: once both players are there, neither can improve by changing alone.

But it is not the best joint outcome. The total payoff at $(\text{Cooperate}, \text{Cooperate})$ is $3+3=6$, while at $(\text{Defect}, \text{Defect})$ it is only $1+1=2$. The key insight: a Nash equilibrium can be stable without being collectively best.

Where students get stuck, and how to self-check

Each step has a typical slip:

• Interpreting the equilibrium. Nash equilibrium does not mean the best possible outcome for everyone; it only means no player benefits from changing alone.
• Finding best responses. Reading the matrix from only one player's perspective fails. Every cell must be checked from each player's point of view.
• Building the matrix. The model depends on the payoff structure; if the payoffs change, the best responses and equilibrium can change too.

To run the whole procedure again yourself, change one payoff and recompute the best responses, for example asking what happens if mutual cooperation pays $(4,4)$ or mutual defection pays $(2,2)$. That quickly shows equilibrium depends on incentives, not on the labels attached to the strategies. Comparing this setup with a coordination game, where players benefit from matching each other's choices, makes Nash equilibrium even easier to recognize.

When game theory is used

Game theory is used in economics, auctions, pricing, negotiation, voting, network design, and evolutionary biology. The details differ by field, but the same core question keeps returning: how should one agent act when others are choosing too? More advanced settings add mixed strategies, repeated games, and games with more than two players, but for a first pass, pure strategies and a payoff matrix are enough to build the main intuition.