In strategic situations, your best move often depends on what others do. Game theory gives us a formal way to think about this dependency. Instead of just guessing, we can map out a player's optimal choice for every possible action their opponent might take. This map is called a Best Response Function.

Let's say Player 1's strategy is $s_1$ and Player 2's is $s_2$. Player 1's payoff is $u_1(s_1, s_2)$. The Best Response Function for Player 1, denoted $BR_1(s_2)$, tells us which strategy $s_1$ maximizes their payoff, given Player 2's choice of $s_2$. It's the answer to the question: "If they do that, what's the best thing for me to do?"

So, if every player has a function that tells them their best response, where does the game settle? It settles at a Nash Equilibrium. A Pure Strategy Nash Equilibrium (PSNE) is a set of strategies, one for each player, where every player's strategy is a best response to everyone else's. No one has a reason to change their plan on their own. It's a stable point of no regrets.

Formally, a strategy profile $(s_1^*, s_2^*, ..., s_n^*)$ is a Pure Strategy Nash Equilibrium if for every player $i$, $s_i^*$ is a best response to the other players' strategies $s_{-i}^*$. This means no player can do better by unilaterally deviating.

Let's see this in action with the Cournot Duopoly model an economic model where two competing firms decide how much of a product to produce. They sell the same product and make their quantity decisions at the same time. The total quantity on the market determines the price.

Assume the market demand curve is given by $P(Q) = a - Q$, where $Q$ is the total quantity, $Q = q_1 + q_2$. Let's also assume the cost to produce for each firm is $c$ per unit, so the total cost for firm $i$ is $C_i(q_i) = c \cdot q_i$.

Firm 1's profit, $\Pi_1$, is its revenue minus its cost.

To find Firm 1's best response, we find the quantity $q_1$ that maximizes its profit, treating $q_2$ as a fixed number. We do this by taking the derivative of the profit function with respect to $q_1$ and setting it to zero.

Solving for $q_1$ gives us Firm 1's Best Response Function:

Since the firms are identical, Firm 2's Best Response Function is symmetric:

The Nash Equilibrium occurs where these two functions intersect. Each firm is playing a best response to the other's action. We find it by substituting one equation into the other.

Solving this equation for $q_1^*$ gives us the equilibrium quantity for Firm 1. Since the firms are identical, $q_2^*$ will be the same.

Do stable outcomes always exist? In his groundbreaking 1951 paper, John Nash proved that for any game with a finite number of players and a finite number of strategies, at least one Nash Equilibrium exists. This was a monumental result for economics and social sciences.

However, this proof does not guarantee a pure strategy equilibrium. Some games, like Rock, Paper, Scissors, don't have a stable outcome where players stick to one choice. In these cases, the equilibrium involves players randomizing their strategies, a concept known as a mixed strategy Nash Equilibrium.