Backward Induction in Game Trees

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

In our last lesson, we built a powerful visual tool: the game tree. We used it to map out your sequential market entry decision, creating a clear schematic of your choices and your potential competitor's reactions. You now have the "map" of the strategic landscape.

Today, we'll learn how to find the best route on that map. The learning outcome for this lesson is to use backward induction on a game tree to find the optimal sequence of moves. This method, often described as "looking ahead and reasoning back," is a cornerstone of strategic thinking. It will allow you to anticipate how a rational competitor will react to your moves, enabling you to make the best initial decision for your woodworking business.

The Core Idea: Look Ahead and Reason Back

Before diving into the mechanics, let's understand the concept. Backward induction works by starting at the end of the game and working your way back to the beginning. At each decision point, you figure out what a rational player would do, assuming they want to maximize their own payoff.

Think of it like planning a complex project with multiple dependent steps, something you might have encountered in engineering. To determine the best way to start, you first need to understand the optimal way to complete the final step. Knowing that, you can figure out the best way to approach the second-to-last step, and so on, until you've traced the optimal path all the way back to your initial choice.

Let's apply this to the market entry game tree we created for your business in the last lesson.

This is the game tree we built previously. You (the Entrant) move first. If you enter, "Incumbent Cabinets" reacts. The payoffs represent potential annual profits in thousands of dollars for (You, Incumbent).

Solving Your Market Entry Game with Backward Induction

We will now "solve" this game to find the optimal path. The key assumption is that both you and the Incumbent are rational and will always choose the action that leads to your best personal outcome.

Step 1: Go to the Last Decision in the Game
The final decision is made by the Incumbent after you have chosen to Enter. This is the Incumbent's decision node.

Step 2: Determine the Last Player's Best Move
From their node, the Incumbent looks at their possible payoffs:

• If they choose to Fight, their payoff is $30k.
• If they choose to Accommodate, their payoff is $70k.

Since $70k is greater than $30k, a rational Incumbent will always choose to Accommodate. They would be acting against their own interest to start a costly price war.

Step 3: "Prune" the Tree
Because the Incumbent will never rationally choose to Fight, we can effectively remove that branch from our consideration. The threat of a price war is what we call a non-credible threat. You can confidently predict that if you enter, the outcome will be that the Incumbent accommodates.

The game tree now looks much simpler in your mind:

• The Stay Out path leads to a $0 payoff for you.
• The Enter path now leads to a predictable outcome where the Incumbent accommodates, giving you a $50k payoff.

Step 4: Move Back to the First Decision
Now, let's return to your initial decision at the start of the game. With your newfound foresight, your choice is clear:

• Choose Stay Out and get $0.
• Choose Enter and get $50k.

Step 5: Determine the First Player's Best Move
Since $50k is greater than $0, your optimal strategy is to Enter the market.

The solution, found through backward induction, is: You enter, and the Incumbent accommodates. This sequence of optimal choices is called the Subgame Perfect Equilibrium.

Seeing It in Action

The best way to solidify this process is to see it applied to a few examples. The following video provides a clear, step-by-step demonstration of backward induction on two different game trees.

Formalizing the Concepts

You've now successfully used backward induction to solve strategic problems. As we wrap up, let's read a short text that formalizes the key ideas and places them in an entrepreneurial context. This will reinforce what you've learned and introduce the standard terminology.

Conclusion

Today you've learned one of the most practical tools in game theory. By mastering backward induction, you've moved from simply mapping out a decision to finding the single most rational path through it. You can now anticipate competitor reactions and preemptively understand which threats are real and which are bluffs.

Key Takeaways:

• Backward induction is the process of solving a sequential game by starting at the final decision and working backward.
• It operates on the assumption that all players are rational, meaning they will always choose the action that maximizes their own payoff.
• This method allows you to identify and ignore non-credible threats—actions that a competitor might threaten but would not be in their interest to carry out.
• The optimal path found through backward induction is called the Subgame Perfect Equilibrium.

Preview of the Next Lesson:

We have focused on sequential games, where players move one after another. But what about situations where you and a competitor must act at the same time, without knowing the other's choice? A classic example is setting your prices for the season. In the next lesson, we will shift focus to these simultaneous-move games and learn how to model them using a new tool: the payoff matrix.