We know that strategic interactions involve players, actions, and payoffs. But to analyze a situation, we need a formal structure. For simultaneous-move games, where everyone chooses their action at the same time without knowing the others' choices, we use the normal form, or strategic form. This is most often represented by a payoff matrix.

Imagine two coffee shops, A and B, located across the street from each other. They each need to decide whether to set a 'High' price or a 'Low' price for their coffee. Their profits depend on the combination of their choices.

This is a payoff matrix. By convention, Player 1's (Shop A) strategies are the rows, and Player 2's (Shop B) strategies are the columns. Each cell shows the outcome, with the first number being the payoff for Player 1 and the second for Player 2. For instance, if Shop A chooses 'High' and Shop B chooses 'Low', Shop A gets $20 and Shop B gets $120.

With the game laid out, how can we predict what rational players will do? One of the most powerful tools is identifying dominated strategies. A strategy is dominated if there's another strategy that always performs better, or at least as well, regardless of what other players do.

There are two flavors of dominance. A strategy $s_i$ is strictly dominated by another strategy $s'_i$ if, for every possible action by the other players, the payoff from playing $s'_i$ is strictly greater than the payoff from playing $s_i$.

A rational player will never play a strictly dominated strategy. Why would you choose an option that is guaranteed to be worse than another available option?

In the game above, look at Player 1's options. If Player 2 plays 'C', Player 1 gets 4 from playing 'C' and 6 from playing 'D'. If Player 2 plays 'D', Player 1 gets -2 from 'C' and 0 from 'D'. In every case, Player 1's payoff from 'D' is greater than from 'C'. Therefore, for Player 1, 'C' is strictly dominated by 'D'. We can eliminate it.

Weak dominance is a bit looser. A strategy $s_i$ is weakly dominated by $s'_i$ if, for every action by the other players, the payoff from $s'_i$ is greater than or equal to the payoff from $s_i$, and there is at least one case where it is strictly greater.

Eliminating weakly dominated strategies can also simplify a game, but it's a slightly more controversial assumption. It's possible to eliminate your way to a solution that feels less intuitive, or where the order of elimination matters.

The real power of dominance comes from a process called the Iterated Elimination of Strictly Dominated Strategies (IESDS). It works like this:

1. Identify any strictly dominated strategies for any player.
2. Remove them from the game. This creates a smaller, simpler game.
3. Look at the new, smaller game and repeat the process. Are there any new strictly dominated strategies?
4. Continue until no more strategies can be eliminated.

If this process leaves you with a single strategy for each player, you've found the game's unique solution.

This method is powerful because it relies on a very basic assumption of rationality: players won't choose an option that's demonstrably worse than another. It's a way to reason not just about your own best moves, but about how other rational players will reason, and how their reasoning affects your best move.

Not all games are created equal. Some are symmetric, while others are asymmetric.

A symmetric game is one where all players are in the same situation. They have the same set of strategies, and the payoffs for playing a particular strategy depend only on the strategies being played, not on who is playing them. The classic Prisoner's Dilemma is a symmetric game. If you and your opponent could switch places, the game would look exactly the same from their new perspective.

An asymmetric game is one where players have different strategy sets or different payoff structures. Think of a game between a startup and an established market leader. The startup might have strategies like 'Disrupt' or 'Seek Acquisition', while the incumbent has strategies like 'Price War' or 'Acquire Competitor'. Their goals, resources, and potential payoffs are fundamentally different.

Recognizing whether a game is symmetric or asymmetric is a crucial first step. It tells you whether you can look for a single, optimal strategy profile that applies to everyone, or if you need to analyze each player's unique position and incentives separately.

By building these matrices and analyzing them for dominance and symmetry, we transform complex strategic situations into a clear, analyzable format. This is the foundation upon which more advanced game theory concepts are built.