Building Payoff Matrices for Oligopoly Games

The Formal Elements of a Strategic Game

Before we can build a matrix, we need to understand the formal components of a game. Any strategic game, whether it's between competing firms, countries, or individuals in a relationship, can be broken down into three essential elements:

1. Players: The decision-makers in the game. In our context, these will typically be the firms in an oligopoly.
2. Strategies (or Actions): A complete set of possible actions a player can take.
3. Payoffs: The outcome or utility each player receives for each possible combination of strategies chosen by all players. In business economics, payoffs are usually measured in profit.

A resource from a Master's course in Microeconomics provides a very clear and formal definition of these elements, which should align well with your analytical background.

The key takeaway is that we can represent a complex strategic interaction with a simple, formal structure. For the common case of a two-player game where each player has a finite number of actions, this structure is most easily visualized as a matrix.

From Description to Matrix: The Prisoner's Dilemma

The most famous example in game theory is the Prisoner's Dilemma. It provides a powerful illustration of how to translate a narrative into a payoff matrix. While not an economic example on its surface, the logic it reveals is directly applicable to oligopoly behavior.

The following resource describes the classic scenario and shows its corresponding payoff matrix.

The construction of the matrix follows a clear logic:

• Players: Bob and Rob.
• Strategies: Each player has two strategies: Confess (C) or Stay Quiet (Q).
• Matrix Structure: We assign one player to the rows (e.g., Bob) and the other to the columns (e.g., Rob). This creates a 2x2 grid.
• Payoffs: Each cell in the grid represents one possible outcome. For example, the top-left cell is where Bob stays Quiet and Rob stays Quiet. We fill this cell with the payoff tuple (Bob's Payoff, Rob's Payoff), which in this case is (3, 3). We do this for all four cells, mapping each story outcome to a cell in the matrix.

An Oligopoly Game: Output Decisions

This same logic applies directly to firms in an oligopoly. Consider two firms, A and B, who are the only producers of a certain good. Each must decide whether to "Hold down output" (cooperating to keep prices high) or "Increase output" (competing for market share).

Let's deconstruct this matrix just as we did for the Prisoner's Dilemma:

• Players: Firm A (the row player) and Firm B (the column player).
• Strategies: Each firm can either "Hold down output" or "Increase output".
• Payoffs: The numbers represent the profits for each firm. Let's examine the top-right cell:
  • Firm A chooses to "Hold down output".
  • Firm B chooses to "Increase output".
  • The outcome is that Firm A gets a profit of $200, while Firm B gets a profit of $1,500. This represents a scenario where Firm B takes advantage of Firm A's restraint to capture a larger share of the market.

This matrix perfectly illustrates oligopolistic interdependence: Firm A's profit doesn't just depend on its own output decision; it is critically dependent on Firm B's decision as well. For example, if Firm A chooses to "Increase output", its profit could be either $1,500 (if B holds down) or $400 (if B also increases). To make a decision, Firm A must anticipate what Firm B will do.

Your Turn: Construct a Payoff Matrix

Now, let's apply this method to a new scenario.

Scenario: Two coffee shops, "The Grind" and "Bean There," are the only specialty coffee providers in a town. They must independently and simultaneously decide whether to run a major advertising campaign.

The expected weekly profits for each firm are as follows:

• If neither advertises, they share the market peacefully and each earns $10,000.
• If both advertise, the campaigns largely cancel each other out. They still share the market, but both must pay for the ads, so their profits fall to $7,000 each.
• If The Grind advertises and Bean There does not, The Grind attracts many of Bean There's customers. The Grind's profit becomes $15,000, while Bean There's profit slumps to $4,000.
• If Bean There advertises and The Grind does not, the reverse happens. Bean There earns $15,000, and The Grind earns $4,000.

Your task is to translate this story into a payoff matrix. Take a moment to identify the players and strategies, and draw the 2x2 matrix, filling in the profit payoffs for each of the four outcomes. Let's set The Grind as the row player and Bean There as the column player.

...

Here is the completed payoff matrix for this advertising game:

As you can see, each narrative outcome maps to one cell in the matrix, with the payoffs written in the conventional order: (Row Player's Payoff, Column Player's Payoff).

Conclusion

In this lesson, we established the foundational skill for analyzing strategic interactions in an oligopoly: constructing a payoff matrix. This process provides a structured way to visualize the interdependence between firms.

Key Takeaways:

• A strategic game is formally defined by its players, strategies, and payoffs.
• A payoff matrix is the standard representation for a two-player, simultaneous-move game.
• To construct a matrix, you must identify the players, list their strategies for the rows and columns, and fill each cell with the corresponding payoff tuple (Row Player, Column Player) for that specific combination of strategies.
• This matrix makes the concept of strategic interdependence explicit, showing how one player's outcome is contingent on the other's choice.

Now that we know how to build the game board, our next lesson will focus on how to play the game. We will learn how to analyze a payoff matrix to predict the most likely outcome by identifying dominant strategies and finding the Nash equilibrium.