Contingent Plans in Extensive-Form Games

students, imagine planning a road trip when you do not know yet whether it will rain, whether traffic will be heavy, or whether your friend will call with a different idea 🚗🌦️. In game theory, people face similar situations all the time: they make a choice now, then later respond to what happens next. That is what extensive-form games help us study.

In this lesson, you will learn how players use contingent plans—also called strategies—to describe what they will do in every possible situation. By the end, you should be able to:

Define a contingent strategy.
List the contingencies that must be included in a game tree.
Distinguish a strategy from a move.

A key idea is this: in an extensive-form game, a player does not just choose one action. A player chooses a full plan for every future situation they might face. That complete plan is what game theorists call a strategy.

What Is a Contingent Strategy?

A contingent strategy is a complete plan that tells a player what action to take at every decision point they might reach. “Contingent” means “depending on what happens.” So instead of saying only what a player does today, a contingent strategy says what the player will do if each possible event occurs.

This matters because many games are sequential. One player moves, then another player observes the result and makes a choice, and then the first player may move again. In that kind of game, the player must be ready for multiple possible histories.

For example, suppose students is playing a simple shopping game. First, students decides whether to buy a phone now or wait. If students waits, the price may go down or up later. A contingent strategy is not just “wait.” It is something like:

If the price goes down, buy the phone.
If the price goes up, do not buy.

That is a full plan for both possible future situations.

In game theory, a strategy is not about what seems likely; it is about what a player would do at every decision point they could possibly face. This makes strategies useful for predicting behavior and solving games.

Strategy vs. Move

A common mistake is to confuse a move with a strategy.

A move is a single action taken at one moment in the game. For example, choosing “left” or “right” in one turn is a move.

A strategy is the entire rule for the player’s behavior in the whole game. It includes what the player will do now and what the player will do later under every possible contingency.

Here is a simple way to remember the difference:

A move is one step.
A strategy is the full route map 🗺️

Suppose a player has two decision points. At the first point, they can choose $A$ or $B$. Later, if a certain event happens, they can choose $C$ or $D$. A strategy must specify choices at both points. For example:

At the first decision point: choose $A$.
At the second decision point: choose $D$ if that point is reached.

That pair of instructions is one strategy.

The move at the first node is just $A$. The strategy is the complete plan: first $A$, then $D$ if needed.

This distinction is important because two different strategies can lead to the same move at the beginning but differ later. In game theory, those are still different strategies, because they tell the player to behave differently in future contingencies.

Contingencies in a Game Tree

An extensive-form game is often drawn as a game tree. A game tree shows decision nodes, branches, and possible outcomes. Each branch represents a choice or chance event, and each path represents a possible history of play.

To write a contingent strategy, students must list what the player will do at each decision node they may reach. These are the required contingencies.

A player’s strategy must include:

Every decision node that belongs to that player.
Every possible action at each such node.
A choice for each node, even if that node is not reached in a particular play.

That last point is very important. A strategy must be complete even for situations that may never happen.

For example, imagine a two-stage game:

First, Player 1 chooses $L$ or $R$.
If $L$ is chosen, Player 2 then chooses $U$ or $D$.
If $R$ is chosen, Player 2 also chooses $U$ or $D$.

Player 2 must give a plan for both possible nodes, even though only one node will be reached in any actual play. A valid strategy for Player 2 could be:

Choose $U$ after $L$.
Choose $D$ after $R$.

This is one contingent strategy because it tells Player 2 what to do in both possible contingencies.

If the game tree has information sets, the same idea still applies, but the player may not be able to tell exactly which node they are at. In that case, a strategy still specifies an action for the information set. The strategy remains a complete plan based on what the player knows.

A Real-World Example of Contingent Planning

Think about a student, students, studying for a test 📚. students decides in the afternoon whether to study now or go play basketball. If students studies, then later students may review notes or do practice questions. If students does not study now, then later students may need to study late at night.

A strategy in this situation is not just “study” or “play basketball.” It is a full plan, such as:

If there is free time now, study.
If tired later, do only review notes.
If energetic later, do practice questions.

This is the same logic as a contingent strategy in game theory. The plan depends on what happens earlier.

Game theory uses this idea to model decision-making in economics, politics, business, and everyday life. A company may plan how to price a product depending on whether a competitor enters the market. A sports coach may choose a defensive play depending on how the opponent lines up. In each case, the decision depends on the earlier state of the game.

Why Complete Plans Matter

Complete plans matter because sequential choices affect future behavior. If a strategy were only a single move, it would leave out what happens later. But in extensive-form games, the future matters from the start.

A complete strategy allows us to:

Predict how a player will act in every possible path.
Compare different plans fairly.
Find best responses to other players’ plans.
Analyze equilibrium outcomes in sequential settings.

For example, suppose Player 1 knows that Player 2 will respond aggressively if challenged and calmly if not challenged. Player 1 must consider Player 2’s full strategy, not just one possible action. That is why game theory models strategies as complete contingent plans.

If students were asked, “What is your strategy?” in a sequential game, the right answer should not be just one move. It should describe what students would do at each decision point that might arise.

Example Game Tree and Strategy Listing

Consider this simple game tree:

Player 1 chooses $A$ or $B$.
If Player 1 chooses $A$, then Player 2 chooses $C$ or $D$.
If Player 1 chooses $B$, then Player 2 chooses $E$ or $F$.

What are Player 2’s required contingencies?

Player 2 must choose:

One action after $A$.
One action after $B$.

So Player 2’s strategy is a pair of choices, one for each possible history. Examples include:

After $A$, choose $C$; after $B$, choose $E$.
After $A$, choose $D$; after $B$, choose $F$.
After $A$, choose $C$; after $B$, choose $F$.

Each of these is a different strategy, even though only one of the two decision points will be reached in an actual play.

Now compare that with a move. If the current node is the one after $A$, then choosing $C$ is a move. But the strategy is the entire pair of instructions for both possible nodes.

This is one of the most important lessons in extensive-form games: the player’s strategy is defined before play unfolds, and it covers all future contingencies.

Conclusion

Contingent plans are the heart of extensive-form game analysis. A contingent strategy is a complete plan for what a player will do at every decision point they might face. A move is just one action at one node, but a strategy includes the full set of instructions for all possible histories.

students, when you study a game tree, always ask:

What decision nodes belong to each player?
What action must be chosen at each node?
What would the player do if a different path were reached?

If you can answer those questions, you can write strategies correctly and understand how sequential games work. This is the foundation for deeper topics like backward induction, subgame perfection, and equilibrium in dynamic games.

Study Notes

A contingent strategy is a complete plan that tells a player what to do in every possible situation they may face.
A move is one action at one point in the game; a strategy is the full plan across all possible contingencies.
In a game tree, a player’s strategy must specify an action at every decision node the player might reach.
A strategy is complete even if some nodes are never reached in actual play.
Different strategies can share the same first move but differ later.
Extensive-form games model sequential interaction, so future choices matter from the start.
Strategy listings often look like pairs or lists of actions, one for each possible history.
Thinking in terms of “if this happens, then do that” helps convert a game tree into a strategy.
Complete strategies are essential for predicting behavior and analyzing equilibrium in dynamic games.