Game theory gives you a framework for analyzing situations where your best choice depends on what others do. Nash equilibrium and dominant strategies are the core tools for predicting outcomes in these strategic interactions. This section covers how to identify both, how to work through pure and mixed strategy equilibria, and why dominant strategies simplify the whole process.

Nash Equilibria in Games

Defining Nash Equilibrium

A Nash equilibrium is a strategy profile where no player can improve their payoff by unilaterally changing their strategy. Every player is already playing their best response to what everyone else is doing.

A few important properties:

Nash equilibrium applies across game types: simultaneous-move, sequential, and repeated games
Equilibria can be in pure strategies (each player picks one action with certainty) or mixed strategies (players randomize over actions with specific probabilities)
A single game can have multiple Nash equilibria, exactly one, or none in pure strategies
The existence of a Nash equilibrium does not mean the outcome is efficient or socially optimal

Identifying Nash Equilibria

The standard approach for normal-form (matrix) games:

Best response analysis. For each player, go through every possible strategy the opponent could play and identify which strategy gives that player the highest payoff. Underline or circle these best-response payoffs in the matrix.
Find mutual best responses. A cell where every player is simultaneously playing a best response is a Nash equilibrium. In a 2-player payoff matrix, that's a cell where both payoffs are circled.
Verify stability. Confirm that no player can profitably deviate. If a player is indifferent between two strategies, both could be part of an equilibrium.

For more complex games, you can also:

Use iterated elimination of dominated strategies (IEDS) to shrink the game before searching for equilibria
Construct best response functions for games with continuous strategy spaces (like Cournot duopoly), then find where they intersect
Apply backward induction for sequential games to find subgame perfect Nash equilibria

Defining Nash Equilibrium, Nash equilibrium - Simulace.info

Dominant Strategies and Game Outcomes

Types of Dominant Strategies

A dominant strategy gives a player the highest payoff no matter what the other players do. If you have one, you should always play it.

There are two flavors:

Strictly dominant: Yields a strictly higher payoff than every other strategy, for every possible combination of opponents' strategies. The classic example is "Defect" in the Prisoner's Dilemma: regardless of what the other player does, defecting always produces a higher payoff than cooperating.
Weakly dominant: Yields a payoff at least as high as every other strategy, and a strictly higher payoff for at least one opponent strategy profile. The distinction matters because eliminating weakly dominated strategies during IEDS can be order-dependent, meaning the sequence in which you eliminate them can affect which equilibria survive.

A rational player always plays a dominant strategy when one exists. Not every game has one, though. Most interesting games don't.

Impact on Game Outcomes

When every player has a dominant strategy, the resulting strategy profile is automatically a Nash equilibrium (called a dominant strategy equilibrium). This is the easiest type of equilibrium to find because you don't need to think about what others will do.

The catch: dominant strategy equilibria can be collectively terrible. In the Prisoner's Dilemma, both players defecting is the unique Nash equilibrium, but both would be better off cooperating. This gap between individual rationality and collective welfare is one of the most important insights in game theory. It shows up everywhere, from arms races to environmental policy to oligopoly pricing.

Beyond direct solutions, dominance reasoning helps simplify complex games. By iteratively eliminating strategies that are strictly dominated (IEDS), you can reduce a large payoff matrix to something manageable, sometimes all the way down to a single outcome. IEDS with strict dominance is order-independent: you'll reach the same reduced game no matter which dominated strategy you eliminate first.

Finding Nash Equilibria: Pure vs. Mixed Strategies

Pure Strategy Nash Equilibria

For a normal-form game presented as a payoff matrix, here's the step-by-step process:

List all strategy combinations. In a 2-player game with $m$ and $n$ strategies respectively, there are $m \times n$ cells to check.
Find Player 1's best responses. For each column (each of Player 2's strategies), find the row that gives Player 1 the highest payoff. Mark it.
Find Player 2's best responses. For each row (each of Player 1's strategies), find the column that gives Player 2 the highest payoff. Mark it.
Identify cells where both are marked. These are the pure strategy Nash equilibria.

Consider the Battle of the Sexes game. Both (Opera, Opera) and (Football, Football) are pure strategy Nash equilibria because in each case, neither player wants to deviate given the other's choice. The coordination problem is which equilibrium to play, since the two players have opposing preferences over which equilibrium is better.

For sequential games, use backward induction: start at the final decision nodes, determine optimal choices, and work backward to the root. This yields the subgame perfect Nash equilibrium, which rules out equilibria sustained by non-credible threats. A threat is non-credible if the player making it wouldn't actually follow through when the time came.

Mixed Strategy Nash Equilibria

When no pure strategy Nash equilibrium exists (like in Matching Pennies), or when you need to find all equilibria, mixed strategies come into play. Nash's existence theorem guarantees that every finite game has at least one Nash equilibrium when you allow mixed strategies.

The key principle: in a mixed strategy equilibrium, each player must be indifferent between all the pure strategies they play with positive probability. If one pure strategy gave a higher expected payoff, the player would shift all their probability to it, breaking the mix.

To solve for a mixed strategy equilibrium in a 2×2 game:

Assign probabilities. Let Player 1 play their first strategy with probability $p$ and their second with probability $1-p$ . Let Player 2 play their first strategy with probability $q$ and second with $1-q$ .
Write Player 2's indifference condition. Set Player 2's expected payoff from their first strategy equal to their expected payoff from their second strategy. This equation will be in terms of $p$ .
Solve for $p$ . This gives you Player 1's equilibrium mixing probability.
Write Player 1's indifference condition. Do the same in the other direction and solve for $q$ .
Calculate expected payoffs. Plug the equilibrium probabilities back in to find each player's expected payoff.

Step 3 trips people up: your mixing probability is pinned down by making the other player indifferent, not by your own payoffs. Think about why. If Player 1's mix didn't make Player 2 indifferent, then Player 2 would strictly prefer one pure strategy and wouldn't be willing to mix. So Player 1's probabilities must be exactly the ones that keep Player 2 on the knife's edge between their options.

For games with continuous strategy spaces (like Cournot duopoly where firms choose quantities), you derive each player's best response function by maximizing their payoff given the other's strategy choice. Concretely, take the first-order condition of Player 1's profit with respect to their own quantity, treating Player 2's quantity as given. Do the same for Player 2. Then solve the resulting system of equations simultaneously. The intersection point of the best response functions is the Nash equilibrium.