Game Tree

A game tree is a diagram of a sequential game in Intermediate Microeconomic Theory. It maps each player's possible moves, the order of play, and the payoffs at the end.

What is Game Tree?

A game tree is the main way Intermediate Microeconomic Theory shows a sequential game. Instead of listing choices in a table, the tree lays out who moves first, what each player can do next, and what happens after every path.

The structure is simple but powerful. Decision nodes show where a player chooses an action, branches show the available actions, and terminal nodes, often called leaves, show the final outcomes or payoffs. If player 1 moves first and player 2 responds after seeing that choice, the tree makes that timing visible.

This matters because order changes strategy. In a simultaneous game, players choose without seeing the other side's move. In a sequential game, later players can react to earlier actions, so a move can shape what comes next. The tree helps you see not just what each player could do, but what they would actually do once the game reaches their turn.

A strategy in a game tree is broader than a single move. It is a complete plan for every decision point a player might face. That means you are not just naming one choice at the top of the tree, you are specifying what the player would do in every possible branch, even ones that may never happen.

The big advantage of the tree is that it sets up backward induction. You solve the game from the final decisions back to the start, asking what each player would do at the last node, then the node before that, and so on. That process leads to subgame perfect equilibrium, which is the version of equilibrium that survives after you check every part of the tree for rational play.

A common classroom example is a simple entry or bargaining game. One firm moves first, another observes and responds, and the tree shows whether an early action creates a threat, a response, or a credible promise. Once you can read the tree, you can see why some strategies work and others fall apart.

Why Game Tree matters in Intermediate Microeconomic Theory

Game trees are how you turn a word problem about timing into a solvable economics problem. In Intermediate Microeconomic Theory, that means you can analyze entry decisions, bargaining, deterrence, pricing moves, and other situations where one action changes the options later on.

The tree is also the bridge between the story and the equilibrium concept. A payoff matrix works well when choices happen at the same time, but it hides the order of moves. A game tree shows timing directly, which is what you need before you can apply backward induction or check whether a strategy is actually credible.

This is why game trees come up in topics like subgame perfect equilibrium and Centipede Game. They show whether a threat really would be carried out, whether a promise is believable, and how one player's move changes the other player's best response. If you can read the tree cleanly, you can explain strategic behavior instead of just naming it.

How Game Tree connects across the course

Backward Induction

Backward induction is the method you use on a game tree to solve from the end of the game back to the start. You look at the last decision node, pick the best action there, then move one step earlier and repeat. In sequential games, this is how you find the path of rational play instead of guessing from the top of the tree.

Subgame Perfect Equilibrium

Subgame perfect equilibrium is the equilibrium concept that comes out of solving a game tree correctly. A strategy profile is subgame perfect only if it gives a Nash equilibrium in every subgame, not just on the whole game. That is why game trees matter so much, they let you check whether threats and promises still make sense after each branch.

Payoff Matrix

A payoff matrix summarizes simultaneous-move games, while a game tree is built for sequential play. If order matters and players observe earlier actions, the matrix usually hides the timing you need. The tree lets you see the decision sequence, then you can convert the end outcomes into payoffs and solve the game step by step.

Information Sets

Information sets show what a player knows when making a choice in a game tree. If a player cannot tell exactly which node they are at, the tree will group those nodes together. That difference matters because perfect information games have clear, observed moves, while imperfect information changes what actions are rational.

Is Game Tree on the Intermediate Microeconomic Theory exam?

A problem set or quiz question will often give you a tree and ask you to trace the best response at each node, then name the equilibrium outcome. You might also be asked to draw the tree from a word problem, label the branches, and explain why one threat is not credible. When the question is about sequential bargaining, entry deterrence, or a centipede-style game, the move is usually to work backward from the final decision nodes. A good answer shows the path of play, not just the final payoff.

Game Tree vs Payoff Matrix

A payoff matrix and a game tree both represent strategic games, but they are not the same tool. A payoff matrix fits simultaneous choices, while a game tree fits sequential choices where timing and observation matter. If the order of moves changes the logic of the game, use the tree.

Key things to remember about Game Tree

• A game tree is the diagram economists use to show a sequential game, including who moves first, what options each player has, and what payoffs end each path.

• Branches show possible actions, decision nodes show choice points, and leaves show outcomes, so the whole strategic path is visible at once.

• Strategies in a game tree are complete plans, not single moves, because they specify what a player would do at every possible decision point.

• Backward induction is the standard way to solve a game tree, and it leads to subgame perfect equilibrium when you check every subgame for rational play.

• If timing and observation matter in the problem, a game tree usually gives you the cleanest way to analyze the game.

Frequently asked questions about Game Tree

What is a game tree in Intermediate Microeconomic Theory?

A game tree is a diagram of a sequential game that shows the order of play, each player's possible actions, and the final payoffs. It is the standard way to represent strategic decisions when one player can respond after seeing another player's move. That makes it easier to solve the game with backward induction.

How do you read a game tree?

Start at the first decision node and follow each branch for the available actions. Then move through the next nodes in order, watching who gets to choose and what information they have. The terminal nodes at the end of each path give the payoffs, which you use to work backward.

What is the difference between a game tree and a payoff matrix?

A payoff matrix is best for simultaneous games, where players choose without seeing each other's actions. A game tree is for sequential games, where the order of moves matters and later players react to earlier choices. If you need to show timing or credibility, the tree is usually the better tool.

Why do professors use game trees for subgame perfect equilibrium?

Because subgame perfect equilibrium checks whether the strategy is rational at every point in the game, not just at the beginning. The tree lets you solve each branch and each subgame separately. That is how you spot threats that sound strong but would not actually be carried out.