Modeling Decisions with Payoff Matrices

Hello! Welcome to the fourth lesson in our module on Core Models for Competitive Analysis.

In our last two lessons, we explored sequential games, where players make decisions one after another. You learned how to map these situations using a game tree and how to solve them using backward induction. This is perfect for analyzing decisions like market entry, where you act and a competitor reacts.

However, many critical business decisions aren't sequential. When you set your prices for your new custom doors, you'll likely be doing it at the same time as your competitors, without knowing their final decision. This is a simultaneous-move game.

Today's learning outcome is to model a simultaneous decision, such as pricing, using a payoff matrix. This tool is the standard way to represent and analyze situations where your best choice depends on what another player does, and you both have to act in the dark.

From Sequential to Simultaneous Games

Let's quickly contrast the two types of games we're discussing:

• Sequential Game (Last Lesson): You move, the other player sees your move, then they move. The tool is a game tree. You solve it with backward induction.
• Simultaneous Game (This Lesson): You and the other player move at the same time, or without observing the other's action. The tool is a payoff matrix. We will learn how to build one today and how to solve it in the next lesson.

Understanding the Payoff Matrix

A payoff matrix is a grid that lays out all possible outcomes of a simultaneous game. It has three core components:

1. Players: The decision-makers in the game.
2. Actions (or Strategies): The set of possible choices each player can make.
3. Payoffs: The outcome (e.g., profit, market share, or just a preference ranking) each player receives for every possible combination of actions.

The following video offers a quick, clear introduction to the structure of a payoff matrix and how to read the payoffs for each player.

To reinforce this, let's look at a typical pricing game matrix.

For example, if Firm B chooses 'High' and Firm A chooses 'Low', we look at the top-right cell. The outcome is (£12, £2), meaning Firm B earns £12 and Firm A earns £2.

Building a Payoff Matrix for Your Woodworking Business

Now, let's build a matrix for a realistic decision you face: setting the initial pricing for your premium cabinets. This is a classic simultaneous decision because you have to set your prices without knowing for certain how your main competitor will price their products.

We will follow a systematic, 4-step process. The resource below, from Profit Analytics for Entrepreneurs, provides an excellent framework for this, which we will adapt for your specific situation.

Inspired by that process, let's model your pricing decision.

Step 1: Identify the Players
The game is between you and your most direct competitor.

• Player 1 (You): Your new woodworking business. You'll be the "row player".
• Player 2 (Competitor): An established local business, let's call them "Artisan Cabinets". They'll be the "column player".

Step 2: Outline Possible Actions
For simplicity, let's assume you both have two main pricing strategies.

• Premium: Price high to signal top quality and maximize profit per unit.
• Competitive: Price lower to attract more customers and gain market share.

Step 3: Determine the Payoffs (Using Ordinal Ranks)
Calculating exact profit outcomes is difficult. Instead, we can use ordinal payoffs, as discussed in the reading. We'll simply rank the four possible outcomes for each player from best (4) to worst (1).

Let's think through the outcomes from your perspective:

• Best Outcome (4): You price Competitive while Artisan prices Premium. You capture a large chunk of the market with a quality product at a better price.
• Good Outcome (3): You both price Premium. There's no price war, and both firms enjoy high margins.
• Bad Outcome (2): You both price Competitive. A price war begins, eroding profits for both of you.
• Worst Outcome (1): You price Premium while Artisan prices Competitive. You lose potential customers who are drawn to the lower price, making your launch difficult.

Artisan Cabinets will have a similar, symmetrical set of preferences.

Step 4: Construct the Payoff Matrix
Now we can assemble these pieces into our matrix. The payoffs in each cell are listed as (Your Payoff, Artisan's Payoff).

You \ Artisan	Premium	Competitive
Premium	(3, 3)	(1, 4)
Competitive	(4, 1)	(2, 2)

This matrix now provides a clear, one-page summary of the strategic pricing landscape. It forces you to think not just about your own actions, but about how your results are intertwined with your competitor's choices.

Test your understanding!

Imagine you're deciding on your product focus: specializing only in high-end Doors or offering a broader line of Doors & Cabinets. Your competitor has the same choice.

Let's say your payoffs are ranked as follows:

• You both specialize in different things (e.g., you do Doors, they do Cabinets): Best for both (Payoff: 4).
• You both go broad (Doors & Cabinets): You compete everywhere, hurting profits (Payoff: 2).
• You go broad, but they specialize: Good for you, you cover more ground (Payoff: 3).
• You specialize, but they go broad: Bad for you, they are a one-stop-shop (Payoff: 1).

Construct the 2x2 payoff matrix for this game. Assume the competitor has symmetrical payoffs.

Show answer

Here is the payoff matrix for the product focus game. Your business is the row player. Payoffs are (You, Competitor).

You \ Competitor	Doors	Doors & Cabinets
Doors	(2, 2)	(1, 3)
Doors & Cabinets	(3, 1)	(2, 2)

(Note: We assume a competitor focused on 'Doors' is symmetrical to one focused on 'Cabinets' for this problem. If they both choose the same specialty, it's like a price war - head-to-head competition, so payoff is 2. If they choose different specialties, it's not modeled here, but that would be a high-payoff scenario of market segmentation.)

Analyzing the Matrix: Finding Your Best Response

Building the matrix is the first step. The next is to use it to analyze the strategic incentives. While we will fully explore the "solution" in the next lesson, we can start by asking a simple question: "Given what my competitor might do, what is my best response?"

The rest of the video you started earlier demonstrates a fantastic visual method for this analysis by circling the best payoffs.

Conclusion

In this lesson, you've learned how to translate a complex, simultaneous business decision into a structured and analyzable format. The payoff matrix is an essential tool for any strategist, helping to clarify the interdependencies that define competitive landscapes.

Key Takeaways:

• Simultaneous games are when players act without knowing their opponents' current move. Pricing is a classic example.
• A payoff matrix is the tool used to model these games, consisting of players, actions, and payoffs.
• Ordinal payoffs (ranking outcomes) are a powerful way to build a matrix when exact financial data is unavailable, keeping the focus on strategic preferences.
• The process involves four steps: Identify Players, Outline Actions, Determine Payoffs, and Construct the Matrix.

Preview of the Next Lesson:

We ended today's lesson by analyzing "best responses." You might have noticed in our pricing game that regardless of what Artisan Cabinets does, your "best response" is always the same. This leads us to a powerful concept. In our next lesson, we will formalize this analysis to find the Nash Equilibrium—the stable outcome of the game where no player has an incentive to change their strategy. This will help you predict the most likely result of your pricing interactions.