Backward Induction in Extensive-Form Games

students, imagine playing a game where every move changes what happens next 🎮. In some games, players choose at the same time, but in extensive-form games, players move one after another, and each move can depend on what happened before. This lesson is about backward induction, a method for solving finite sequential games by starting at the end and working backward.

What you will learn

By the end of this lesson, students, you should be able to:

Apply backward induction to a game tree.
Determine the best action at the final decision stage.
Predict the equilibrium path in simple sequential games.

Backward induction is powerful because it helps players think ahead. Instead of guessing randomly, a player asks, “If I choose this action now, what will the next player do later?” That idea is used in economics, business, sports strategy, and negotiations. For example, a store might decide whether to lower prices first, knowing that a rival may respond later by matching the price or keeping it high 🛒.

What is backward induction?

Backward induction is a method for solving finite sequential games, meaning games with a limited number of steps. The idea is simple:

Start with the last decision in the game.
Find the best choice for the player who moves last.
Move one step backward and use that choice to decide what earlier players should do.
Repeat until you reach the first move.

At each step, the player assumes that later players will act rationally and choose their best available action. This is important because earlier decisions depend on what later decisions will be.

A game tree is a picture of the game. It shows decision points, possible actions, and payoffs at the end. Each path from the root of the tree to a terminal node represents one complete history of play.

Key idea

Backward induction finds the subgame perfect equilibrium in games where each subgame can be solved by rational choice at every stage. In simple terms, the solution is stable because each player’s action is best not only in the whole game but also in every smaller part of the game.

Solving from the end of the tree

Let’s use a simple example. Suppose Player 1 moves first and chooses either $A$ or $B$. Then Player 2 moves and chooses either $L$ or $R$. The payoffs are:

If Player 1 chooses $A$ and Player 2 chooses $L$, the payoff is $(2,1)$.
If Player 1 chooses $A$ and Player 2 chooses $R$, the payoff is $(0,3)$.
If Player 1 chooses $B$ and Player 2 chooses $L$, the payoff is $(1,0)$.
If Player 1 chooses $B$ and Player 2 chooses $R$, the payoff is $(4,2)$.

To solve this game by backward induction, students, start with Player 2’s decision after each possible move by Player 1.

Step 1: Player 2’s choice after $A$

If Player 1 chose $A$, Player 2 compares her own payoffs:

Choosing $L$ gives Player 2 a payoff of $1$.
Choosing $R$ gives Player 2 a payoff of $3$.

Since $3>1$, Player 2 chooses $R$.

Step 2: Player 2’s choice after $B$

If Player 1 chose $B$, Player 2 compares:

Choosing $L$ gives Player 2 a payoff of $0$.
Choosing $R$ gives Player 2 a payoff of $2$.

Since $2>0$, Player 2 chooses $R$ again.

Step 3: Player 1’s choice

Now Player 1 anticipates Player 2’s response:

If Player 1 chooses $A$, the outcome will be $(0,3)$ because Player 2 will choose $R$.
If Player 1 chooses $B$, the outcome will be $(4,2)$ because Player 2 will choose $R$.

Player 1 compares her own payoffs:

From $A$, Player 1 gets $0$.
From $B$, Player 1 gets $4$.

So Player 1 chooses $B$.

The equilibrium path is therefore $B \rightarrow R$, with payoff $(4,2)$.

This is the essence of backward induction: solve the last move first, then use that result to solve the earlier move.

A more realistic example: business competition

Backward induction often appears in business. Suppose a new company enters a market, and an established company must decide whether to fight by cutting prices or accommodate by staying calm.

A simple sequential game might look like this:

Player 1: the new company decides whether to enter or stay out.
Player 2: if entry happens, the existing company chooses fight or accommodate.

Payoffs could reflect profits. For example:

If the newcomer stays out, both companies earn moderate profits.
If the newcomer enters and the incumbent accommodates, both earn positive profits.
If the newcomer enters and the incumbent fights, both may earn low profits because price wars are costly 💼.

To solve the game, students, ask:

What would the incumbent do after entry?
Knowing that, should the newcomer enter?

If accommodation gives the incumbent a higher payoff than fighting, then the incumbent will accommodate. The newcomer then predicts this and may choose to enter. This shows how backward induction helps forecast behavior in real-world strategic settings.

How to apply backward induction carefully

Backward induction is reliable when the game has a clear end and players can see the relevant choices. Follow these steps:

1. Identify the final decision nodes

Look for the last player to act in each branch of the game tree. These are the terminal decision points before payoffs are assigned.

2. Compare payoffs for the player moving last

At each final node, the last mover chooses the action that gives the highest payoff to that player. Remember, each player cares about their own payoff, not the other player’s.

3. Replace that decision with the chosen action

Once the best final action is known, treat it as fixed and move one step backward in the tree.

4. Continue backward to the beginning

Repeat the same reasoning until the first player’s move is solved.

5. Read off the equilibrium path

The sequence of actions chosen by rational players is the equilibrium path.

Example with three stages

Let’s solve a game with three decisions.

Player 1 chooses $X$ or $Y$.
If $X$ is chosen, Player 2 chooses $M$ or $N$.
If $Y$ is chosen, Player 3 chooses $P$ or $Q$.

Suppose the terminal payoffs are:

$X \rightarrow M$ gives $(1,4)$
$X \rightarrow N$ gives $(3,2)$
$Y \rightarrow P$ gives $(2,1)$
$Y \rightarrow Q$ gives $(0,5)$

Now solve from the end:

After $X$

Player 2 chooses between $4$ and $2$ for her own payoff. Since $4>2$, Player 2 chooses $M$.

After $Y$

Player 3 chooses between $1$ and $5$ for her own payoff. Since $5>1$, Player 3 chooses $Q$.

Player 1’s decision

Now Player 1 knows:

If $X$ is chosen, the outcome is $(1,4)$.
If $Y$ is chosen, the outcome is $(0,5)$.

Player 1 compares $1$ and $0$, so Player 1 chooses $X$.

The equilibrium path is $X \rightarrow M$.

This example shows that backward induction works even when different branches lead to different players later in the game tree.

Common mistakes to avoid

students, here are mistakes students often make:

Looking forward instead of backward first: The last player’s choice must be solved before earlier choices.
Using the wrong player’s payoff: Each player chooses the action that maximizes their own payoff.
Ignoring later responses: Earlier players must anticipate future actions.
Forgetting that the game must be finite: Backward induction is used for games with a definite end.
Confusing the equilibrium path with every possible path: The equilibrium path is the one predicted by rational play, not every branch of the tree.

A good habit is to mark each final node with the best action for the last player, then work step by step toward the start of the game tree.

Why backward induction matters

Backward induction is more than a classroom method. It explains why people make certain strategic choices in everyday life. A chess player may think several moves ahead ♟️. A company may choose a pricing strategy while predicting the competitor’s response. A person negotiating a deal may decide whether to make a first offer based on how the other side is likely to react.

The logic is always the same: if later decisions are predictable, earlier decisions should take them into account. That is the heart of dynamic strategic reasoning.

Conclusion

Backward induction is a systematic way to solve finite sequential games. students, you begin at the final decision, find the optimal move for the last player, and then step backward through the game tree until the first move is solved. This method helps identify the equilibrium path and predict how rational players will behave in games with sequential moves. When you practice on simple trees, the process becomes easier: solve from the end, move backward, and read the outcome.

Study Notes

Extensive-form games show sequential decisions using a game tree.
Backward induction solves a finite game by reasoning from the last move back to the first.
At each decision node, the player chooses the action that gives the highest payoff to that player.
The equilibrium path is the sequence of actions predicted by backward induction.
Backward induction often leads to a subgame perfect equilibrium.
Always start by solving the final decision nodes first.
Earlier players anticipate the rational choices of later players.
In business, sports, and negotiation, backward induction helps predict strategic responses.
The method works best in finite games with clear terminal payoffs.