Normal-form game

Imagine playing a game with a friend.

You can use a chart to show your moves. This chart is called a normal-form game. It lists every plan a player can make. It also shows the prize each person gets.

Some charts use two numbers in each box. The first number is for the first player. The second number is for the second player. This helps you see the best plan to pick. It makes the game easy to study.

Imagine you are playing a game with a friend. You both make your moves at the same time. To study this, you can use a special chart. This chart is called a normal-form game.

A normal-form game uses a grid called a matrix. This grid shows every plan a player can make. These plans are called strategies. A strategy is a full plan of action. It covers every part of the game.

The chart also shows the prize each person gets. These prizes are called payoffs. In many charts, each box has two numbers. The first number is the payoff for the first player. The second number is for the second player.

One famous example is the Prisoner's Dilemma. In this game, players can cooperate or defect. Defect means to break a deal. By looking at the numbers, players can find the best choice. This helps them see which moves are not good to make. This tool helps us find a Nash equilibrium. That is a point where no player wants to change their plan.

Imagine you are playing a game with a friend. You both make your moves at the same time without seeing what the other does. To study these types of games, experts use a special description called a normal-form game.

This method uses a grid called a matrix to show every possible choice. It helps us see the best ways to play. It is very useful for finding a special balance called a Nash equilibrium. This is a point where no player wants to change their plan.

To build this grid, we need to know two main things. First, we need the strategy space for every player. A strategy space is the list of all choices available to a person. A strategy is more than just one move. It is a complete plan for every single part of the game. This plan must cover what a player will do even if things change.

Second, we need the payoff function. This function shows the prize or result for every combination of moves. These results are usually written as real numbers. The numbers tell us the value of the outcome for each player.

One famous way to use these grids is to find dominated strategies. A dominated strategy is a choice that is always worse than another choice. We can see this in a game called the Prisoner's Dilemma. In this game, players can either cooperate or defect. If they both defect, they both get a certain result.

By looking at the numbers in the matrix, we can compare them. In the Prisoner's Dilemma, defecting is better than cooperating no matter what the other person does. This makes the choice to defect very clear to the players.

Sometimes games are not played at the same time. In a sequential game, one person moves and then the other person moves. This is different from the simple simultaneous games we first discussed.

To use a normal-form matrix here, the second player needs a very big plan. Their strategy must say what to do for every possible move by the first player. This includes moves that might never actually happen in the real game. Even though it seems like a lot of work, it keeps the math organized. It allows us to see every possible outcome in one single chart.

Math experts have studied these ideas for a long time. John von Neumann and Oskar Morgenstern wrote a famous book about this. Their book is called Theory of Games and Economic Behavior. It was first published in 1944 by Princeton University Press.

This work helped people understand how players make choices. Today, we use these matrices to understand many different kinds of games. Whether the game is simple or very hard, the matrix helps us see the truth. It turns a messy game into a clear map of choices.

In the field of game theory, a normal-form game is a specific way to describe a game. While other methods, like the extensive form, use graphs to show how a game unfolds, the normal form uses a mathematical grid called a matrix. This representation is extremely useful for identifying certain outcomes. Specifically, it helps researchers find strictly dominated strategies and Nash equilibria. A Nash equilibrium is a state where no player can benefit by changing their strategy while others keep theirs unchanged.

A normal-form representation must include every possible strategy and every corresponding payoff for every player. To understand this, we must define two core components: the strategy space and the payoff function. The strategy space for a player is the complete set of all possible strategies available to them. It is important to note that a strategy is not just a single move. Instead, a strategy is a complete plan of action for every single stage of a game. This plan must account for every possibility, even if a specific stage never actually occurs during play.

The second component is the payoff function. This function acts as a mapping from the cross-product of all players' strategy spaces to their individual sets of payoffs. In most normal-form representations, these payoffs are expressed as real numbers. These numbers represent either cardinal or ordinal utility, which tells us the value or preference of an outcome. When players choose a specific combination of strategies, known as a strategy profile, the payoff function takes that profile as an input. The output of this function is the specific representation of the payoff for each player.

One of the primary uses of the payoff matrix is to identify dominated strategies. A strategy is considered strictly dominated if there is another strategy that always provides a better payoff, regardless of what the other players do. A classic example of this is the Prisoner's Dilemma. In this game, players choose to either "cooperate" or "defect." By examining the matrix, we can see that defecting provides a better result than cooperating in every scenario. For example, if the column player chooses to cooperate, the row player gets 0 by defecting but only -1 by cooperating. Because defecting is always better, the unique Nash equilibrium for this game is for both players to defect.

While normal-form matrices are often used for simultaneous games, they can also represent sequential games. In a sequential game, one player moves first and is observed by the next player. To represent this in a matrix, the second player must have a much more complex strategy space. Their strategies must be contingent on the first player's actions. For instance, a player might have a strategy that says, "If player one plays Top, I will play Left; otherwise, I will play Right." Even if certain combinations of moves are impossible in actual play, the normal form requires these contingencies to be listed to remain complete.

In some cases, games are symmetric, meaning the payoffs do not change based on which player chooses which action. In these symmetric games, the matrix can be simplified. Instead of listing two numbers in each cell, the matrix might only show a single payoff for the row player. This can make the mathematical representation much cleaner. Furthermore, researchers can map the topological space of different games. This allows them to see how small, incremental changes in incentives can actually change the entire nature of a game.

The formal mathematical structure of a normal-form game requires a finite set of players, denoted as $I$. Each player $i$ within that set has a finite number of pure strategies, $k$. There is also an association of these strategies to the players, which is an $I$-tuple. To fully specify the game, a payoff function must be provided for every single player in the set. This rigorous framework allows mathematicians to turn complex human interactions into precise, solvable models.

The study of these structures has a deep history in economic and mathematical thought. Much of our modern understanding comes from the work of John von Neumann and Oskar Morgenstern. They published a foundational text titled "Theory of Games and Economic Behavior." While it was originally published in 1944 by Princeton University Press, it remains a central reference for the field. Their work helped establish how we use matrices to understand decision-making and strategic interaction in the world today.