Subgames: Finding the Smaller Games Inside a Bigger Game 🎮

Welcome, students! In game theory, a dynamic game is a game played over time, where players make choices one after another instead of all at once. That means the order of decisions matters a lot. One of the most important ideas for studying these games is the idea of a subgame.

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

• Define a subgame.
• Locate subgames in a game tree.
• Explain why subgames matter for dynamic analysis.

Here’s the big idea: a subgame is like a smaller decision problem inside a larger game. If a game is a long road trip, a subgame is one section of the trip that starts at a point where everyone can see exactly what has happened so far. 🚗 Understanding subgames helps us check whether strategies are truly believable at every point in the game, not just at the start.

What Is a Subgame? 🧩

A subgame is a part of a game that begins at a decision node and includes everything that happens after that node. In simpler words, it is a smaller game that starts from a point that can actually be reached during play.

A node is a point in a game tree where a player must choose an action. A reachable node is a node that could happen if players follow the game rules and previous choices. A subgame must begin at such a node, not at a random place in the tree.

To count as a subgame, the decision problem must satisfy an important rule: once you start the subgame, you must include all later choices that follow from that point. You cannot cut off part of the game halfway through. That would leave out information and change the meaning of the game.

Think of a chess match ♟️. If we look at the position after $10$ moves, the rest of the game from that position onward is a smaller decision problem. If the position is truly reachable in the actual game, then that later part can be viewed as a subgame.

A useful way to remember this is:

• The subgame starts at a decision point.
• The starting point must be reachable.
• The subgame must include all future moves from that point.
• The subgame cannot split information sets in ways that break the structure of the game.

In many basic games with perfect information, every decision node can start a subgame. In games with imperfect information, the situation is more limited, because players may not know exactly where they are in the tree.

How to Locate Subgames in a Game Tree 🌳

A game tree shows all possible moves in a game. The root of the tree is the starting point, and the branches show the possible actions of the players. To find subgames, you need to inspect the tree carefully.

Here is a simple method students can use:

1. Find a decision node that could be reached during play.
2. Check whether the entire continuation after that node is included.
3. Make sure the subgame does not cut through any information set.
4. If all conditions are satisfied, you have found a subgame.

Let’s use a simple example. Suppose Player 1 moves first and chooses either $A$ or $B$. If Player 1 chooses $A$, then Player 2 moves. If Player 1 chooses $B$, then Player 3 moves. The game tree has a root node at the start, and two later decision nodes after $A$ and after $B$.

If those later nodes are reachable and each one leads to a complete continuation of the game, then each of them can be the start of a subgame. That means the game contains the full game itself as one subgame, plus smaller subgames that begin after $A$ or after $B$.

Now imagine a game where Player 2 has to move without knowing whether Player 1 chose $A$ or $B$. Those two decision nodes might be connected by an information set, meaning Player 2 cannot tell which node has been reached. In that case, you generally cannot start a subgame at just one of those nodes if doing so would cut that information set in half.

This is why subgames are easier to spot in games with perfect information. In a perfect-information game, each player knows the exact history of previous actions. So the tree can often be divided into many clear subgames.

A quick real-world example is a job interview process 👔. First, the employer decides whether to do a phone screen. If yes, there is an in-person interview later. The later interview stage can be thought of as a subgame because it starts after a reachable earlier decision and includes all later choices from that point.

Why Reachability Matters

The word reachable is very important. A subgame must begin at a node that can actually occur under the rules of the game.

Why does this matter? Because game theory studies real strategic possibilities, not imaginary ones. If a node can never be reached, then the decisions from that node are irrelevant for understanding actual play. You would be analyzing a situation that cannot happen.

For example, suppose a player could only reach a certain node if another player made a choice that the rules make impossible. That node is not part of a meaningful subgame in the usual sense. Subgame analysis focuses on decision problems that matter for actual strategic behavior.

This idea is closely connected to subgame perfection, which is a stronger way of checking whether strategies are good. A strategy profile is subgame perfect if it gives a Nash equilibrium in every subgame, not just in the original game. That means players’ planned actions must make sense at every point where future decisions might occur.

If students thinks of a game as a story, reachability asks: “Could the story really get to this chapter?” If the answer is yes, then that chapter may matter for subgame analysis. If not, it is not useful for checking credibility.

A Simple Example with an Entry Point

Consider this sequence:

• Player 1 chooses whether to enter a market.
• If Player 1 enters, Player 2 decides whether to compete aggressively or cooperate.

The second decision point is a subgame start, because it happens after a reachable action and includes the rest of the game.

Why is this useful? Because Player 2’s choice after entry helps us understand whether Player 1’s original choice is wise. If Player 2 would rationally compete aggressively after entry, then Player 1 may decide not to enter in the first place. This kind of backward thinking is exactly why subgames are important in dynamic analysis.

Now suppose the reverse. If Player 2’s threat to compete aggressively is not credible because it would hurt Player 2 too much, then we should not trust that threat when analyzing the whole game. Looking at the subgame tells us whether Player 2 would really do it if the time came.

This is one reason subgames are central in economics and business strategy 📈. A company may threaten to lower prices if a rival enters the market, but if lowering prices would also reduce the company’s own profit too much, the threat may not be believable. Studying the subgame after entry helps reveal that.

Why Subgames Matter for Dynamic Analysis

Subgames matter because dynamic games are solved by looking not only at the first move but also at later moves. Players can revise expectations as the game unfolds. A strategy that looks fine at the start may fail later when a player actually reaches a decision point.

Subgames help us ask a deeper question: “If we arrived here, would the planned action still be optimal?” This is crucial because players may say they will do something in advance, but once the moment arrives, the best action may be different.

This leads to the idea of credible behavior. A credible action is one a player would truly want to take when the time comes. A non-credible action is a threat or promise that is not actually in the player’s best interest once the subgame begins.

For example, imagine a movie theater says it will ban all customers who complain. That threat may not be credible if banning customers would reduce future business too much. In a game, other players notice these incentives. Subgame analysis helps us separate real incentives from empty threats.

By checking each subgame, we can rule out plans that only work because of unrealistic promises. This makes the analysis more realistic and powerful. It is especially important in bargaining, market entry, auctions, and negotiations.

How to Think About Subgames on an Exam ✍️

When you are asked to identify subgames, students, look for these clues:

• Is the node reachable?
• Does the node start a complete continuation of the game?
• Does the part after the node include every future decision that follows?
• Does starting there avoid breaking an information set?

If the answer to all of these is yes, then the node begins a subgame.

A common mistake is thinking that any later node is automatically a subgame. That is not true. The node must be a proper starting point for a full decision problem. Another mistake is forgetting that information matters. In games with imperfect information, not every node can start a subgame.

A good exam strategy is to draw a circle around the candidate node and then trace everything that follows. If the traced part forms a clean, complete tree and does not split an information set, you have likely found a subgame.

Conclusion

Subgames are the smaller decision problems inside a larger dynamic game. They begin at reachable decision nodes and include all future actions from that point onward. Finding subgames helps us analyze what players would really do if the game reached a later stage.

This matters because good dynamic analysis depends on credibility. A strategy is only convincing if it still makes sense inside every subgame. By locating subgames, students can better understand how rational players think through a game over time and why some threats or promises should not be believed. 🔍

Study Notes

• A subgame is a part of a game that starts at a decision node and includes all future play from that node.
• The starting node of a subgame must be reachable.
• A subgame cannot cut through an information set or leave out later decisions.
• In many games with perfect information, each decision node can start a subgame.
• In games with imperfect information, fewer subgames may exist.
• Subgames matter because they help test whether strategies are credible at every stage of the game.
• Subgame perfection means the strategy profile is a Nash equilibrium in every subgame.
• When locating a subgame, check reachability, completeness, and information sets.
• Subgames are useful for analyzing market entry, bargaining, pricing, and other dynamic strategic situations.