To analyze a competitive situation, we first need a map. In game theory, this map comes in two main flavors: normal form and extensive form. The choice depends on what you need to capture about the interaction.

Normal form, also called strategic form, provides a bird's-eye view. It simultaneously displays all players, their possible strategies, and the resulting payoffs for every combination of choices. This is most useful for games where players act at the same time, or at least without knowing the other's move. Think of it as a spreadsheet for strategy.

The table above shows a simple market entry game. Each cell contains the payoffs for Firm A and Firm B, respectively. For instance, if both enter, they both lose 5 units (perhaps due to a price war). If A enters and B stays out, A captures the market and gets a payoff of 10.

When the timing of moves and the flow of information matter, we turn to the extensive form. This model uses a game tree to map out the sequence of decisions. It's perfect for analyzing situations like chess, negotiations, or any scenario where one player's move is a direct response to another's.

Payoffs are the numbers we use to represent the outcome of a game, but they aren't always about money. A payoff represents utility—a measure of preference or satisfaction an agent gets from a particular outcome. Mapping real-world goals to a utility function is one of the most critical steps in strategic modeling.

For a business, utility might be profit, market share, or brand reputation. In sports, it could be points scored or the probability of winning a championship. The key is to assign numerical values that accurately reflect the desirability of each possible end state. A higher number means a better outcome for that player. This allows us to formally analyze choices, even when the stakes are qualitative.

Once we've modeled the game and its payoffs, the goal is to predict behavior. The most famous concept for this is the Nash Equilibrium. A set of strategies forms a Nash Equilibrium if no player can improve their outcome by unilaterally changing their own strategy, assuming all other players stick to their choices. It’s a point of stability, a state of mutual best response.

Think of two companies setting prices. If they've both chosen a price that constitutes a Nash Equilibrium, neither company has an incentive to raise or lower its price on its own. Any change would make them worse off, given what their competitor is doing. This idea was formalized by John Nash, but its foundations were laid by pioneers like John von Neumann decades earlier.

Finding a Nash Equilibrium can be complex, but sometimes we can simplify the game by looking for dominant and dominated strategies.

A dominant strategy is an action that yields the highest payoff for a player, no matter what the other players do. It's the obvious best move. Conversely, a dominated strategy is one that is always worse than some other specific strategy, regardless of what others choose. Rational players will never play a dominated strategy, so we can often eliminate them to make the game easier to solve.

A Nash Equilibrium isn't always the best overall outcome. Sometimes, when every player makes the best decision for themselves, the group as a whole ends up in a bad spot. The Price of Anarchy (PoA) is a concept from computer science and economics that measures how much the efficiency of a system degrades due to this selfish behavior.

It's defined as the ratio of the system's performance in the worst possible Nash Equilibrium to the performance of the optimal, centrally-controlled solution. A PoA of 1 means that the equilibrium is perfectly efficient. A PoA of 2 means the selfish outcome is half as good as the optimal one. A famous example of this is Braess's paradox in traffic networks, where adding a road can actually make congestion worse because drivers, acting selfishly, flock to the new, seemingly faster route.

Understanding concepts like PoA is crucial for engineering better systems. By analyzing the incentives, you can design rules or structures—whether it's a road network, an auction platform, or a corporate bonus system—where an individual's dominant strategy aligns with a better outcome for the group. The goal is to make doing the right thing for the system also the best thing for yourself.

By modeling the players, strategies, and payoffs, we create a formal framework to dissect any competitive landscape. This allows us to anticipate actions, find stable outcomes, and even design better games.