In game theory, every interaction is a game. To understand it, we first need to map it out. The primary tool for this is the payoff matrix. It's a simple table that shows the players, their possible strategies, and the resulting outcomes, or 'payoffs', for every possible combination of choices.

Let's use the classic Prisoner's Dilemma. Two partners in crime are arrested and held in separate cells, with no way to communicate. The prosecutor offers each the same deal. Their payoffs are measured in years of prison time, so lower numbers are better.

Here’s how their situation looks in a payoff matrix:

To read the matrix, pick a row for Prisoner A's choice and a column for Prisoner B's. The cell where they intersect shows the outcome. For example, if Prisoner A stays silent but Prisoner B confesses, the payoff is (-10, 0). This means A gets 10 years, and B walks free.

With the game mapped out, we can analyse the strategies. The simplest solution to find is a dominant strategy: a choice that yields the best outcome for a player, no matter what the other player does.

Let's look at the Prisoner's Dilemma again from Prisoner A's perspective:

• If B confesses, A is better off confessing too (-5 years is better than -10).
• If B stays silent, A is still better off confessing (0 years is better than -1).

Confessing is Prisoner A’s dominant strategy. Since the game is symmetrical, it’s also Prisoner B’s dominant strategy. Both players, acting rationally in their own self-interest, will choose to confess.

Not all games have a clear dominant strategy. Sometimes, the best move depends entirely on the other player's choice. In these cases, we look for a Nash Equilibrium. This is a set of strategies where no player can improve their payoff by unilaterally changing their own strategy, assuming the other players don't change theirs.

Consider a different game, the Battle of the Sexes. A couple wants to spend the evening together, but they forgot to agree on where to meet. One prefers the Opera, the other prefers the Football match. They'd rather go to their less preferred event together than go to their favorite one alone.

The payoff matrix might look like this, where higher numbers are better:

Here, there's no dominant strategy. If Partner 1 thinks Partner 2 is going to the Opera, they should go to the Opera. But if they think Partner 2 is going to Football, they should go to Football.

This game has two Nash Equilibria:

1. Both go to the Opera.
2. Both go to Football.

In either of these outcomes, neither person can improve their situation by changing their mind alone. If they're both at the Opera, Partner 1 switching to Football would change their payoff from 3 to 0. It’s a stable, though not necessarily ideal, outcome.

The final key distinction is between non-cooperative and cooperative games.

In non-cooperative games, players make decisions independently. They cannot form binding agreements, even if they can communicate. The Prisoner's Dilemma is a classic example. Even if the prisoners promised each other to stay silent, the incentive to betray that promise for a better personal outcome is too strong.

In cooperative games, players can form alliances and make binding commitments. The focus shifts from individual strategy to what groups can achieve together. The analysis involves how to form coalitions and fairly distribute the joint gains. Think of business partners negotiating a contract or countries forming a trade agreement. The rules of the game allow for trust and shared goals.

With these core concepts, you can start to frame strategic interactions, from personal habits to professional negotiations, through the lens of game theory.