In our last lesson, we established the formal definition of a strategic game, breaking it down into its three essential components: players, their strategy sets, and their payoff functions. This provided the mathematical vocabulary needed to describe any strategic interaction.
Today, we will build directly on that foundation by learning how to represent these games in a structured and analyzable format. This lesson focuses on the normal form (or strategic form) representation, which uses a payoff matrix to systematically map every possible combination of strategies to its corresponding outcome. Mastering this representation is crucial, as it is the standard canvas upon which we will analyze and solve games throughout this course.
The Normal Form for Two-Player Games
For games with two players, the normal form is typically represented by a two-dimensional grid, often called a bimatrix. This structure provides a clear and immediate visualization of the entire game.
The fundamental layout is as follows:
- Players: One player is designated as the "row player" (Player 1) and the other as the "column player" (Player 2).
- Strategies: The rows of the matrix correspond to the strategies available to Player 1, and the columns correspond to the strategies available to Player 2.
- Payoffs: Each cell in the matrix represents a unique strategy profile—one strategy choice for each player. Inside the cell, we write a pair of numbers: the payoff to the row player (Player 1) followed by the payoff to the column player (Player 2).
The following image illustrates this conventional structure for a generic 2x2 game.
This matrix is a compact representation of the two payoff functions, and . For example, if Player 1 chooses Strategy B () and Player 2 chooses Strategy A (), the strategy profile is . The corresponding payoffs are and , which are found in the bottom-left cell.
For a more formal definition of this bimatrix representation, let's turn to a set of lecture slides.
[PDF] Evolutionarily stable strategies in nonlinear and multiplayer games
This document provides a rigorous definition of the payoff matrix structure.
In the section "Payoff matrices I," please read the paragraph that formally defines the bimatrix representation. Focus on how the two matrices, A and B, are used to represent the payoffs for Player 1 and Player 2, respectively. Note the convention that a_{ij} is the payoff to Player 1 when they choose strategy S_i and Player 2 chooses strategy T_j.
Extending to Multiplayer Games
The bimatrix is elegant for two players, but how do we represent a game with three or more participants? A two-dimensional grid is no longer sufficient to capture the payoffs for every strategy profile.
The formal mathematical answer is that the representation generalizes from a 2D matrix to an n-dimensional tensor (or multi-dimensional array). In an -player game, each of the dimensions corresponds to a player's strategy set. Each entry in this tensor holds an -tuple of payoffs, one for each player. Given your mathematical background, you can think of this as a direct extension of the matrix concept.
This article explains how the normal form representation scales to the general n-player case.
Read the paragraph that begins "In game theory...". Pay close attention to the statement that for an n-player game, the representation generalizes to an n-dimensional tensor. This provides the formal structure for representing multiplayer games.
While an n-dimensional tensor is the correct abstraction, it can be difficult to visualize for . For the common case of a three-player game, a more intuitive approach is to use a "stack" of matrices.
Imagine Player 1 still chooses the row and Player 2 still chooses the column of a matrix. Player 3's role is to choose which matrix in a series is being played. Each of Player 3's strategies corresponds to a different 2D payoff matrix.
The following video provides a quick, clear illustration of this concept.
Nash Equilibrium with 3 Players
This video on 3-player Nash Equilibrium begins by setting up the normal form representation for a three-player game.
Please watch the first 30 seconds. Notice how the game is depicted as two separate 2x2 matrices, one for each of Player 3's possible strategies (E or F).
To find the outcome for a strategy profile like (Player 1 chooses A, Player 2 chooses D, Player 3 chooses E), you would first go to the matrix corresponding to Player 3's choice (E), and then find the cell at the intersection of row A and column D. That cell would contain a triplet of payoffs for Players 1, 2, and 3.
This structure can be extended to more players (e.g., a 4-player game could be a matrix of matrices), but it quickly becomes cumbersome. The stacked-matrix visualization is most practical for .
Let's look at a more detailed breakdown of this multiplayer structure.
[PDF] Evolutionarily stable strategies in nonlinear and multiplayer games
This document also discusses the structure of multiplayer games formally.
First, read the subsection "Introducing multi-player matrix games" to understand the general concept. Then, read the subsection "Payoffs for two strategies, three and four players I". This part explicitly shows the payoff structure for a 3-player, 2-strategy game as two 2x2 matrices, formalizing the visualization from the video.
The key insight is that regardless of the number of players, the normal form is a comprehensive mapping from the space of all possible strategy profiles to the corresponding payoff vectors . The payoff matrix, or its higher-dimensional generalization, is simply the tool we use to write down this mapping explicitly.
Conclusion
In this lesson, we have learned how to construct the normal form representation for any finite strategic game. This is the foundational skill for analyzing simultaneous-move interactions.
The key takeaways are:
- The normal (or strategic) form represents a game by explicitly listing the payoffs for every player for every possible strategy profile.
- For two-player games, this is achieved with a bimatrix, where rows and columns represent the players' strategies and cells contain their payoff pairs.
- For multiplayer games, the representation generalizes to an n-dimensional tensor. For a three-player game, this is commonly visualized as a stack of matrices, where the third player's strategy choice determines which matrix is in play.
Now that we can formally represent a game, our next step is to begin analyzing it. In the upcoming lesson, we will explore our first solution concept: identifying and eliminating strictly dominated strategies. This is a powerful first step in simplifying a game and predicting the behavior of rational players.