Payoff Matrix vs Sequential Game
Payoff Matrix and Sequential Game are two Game Theory & Information concepts in AP Economics that students often mix up. A payoff matrix is a table listing every combination of the players' strategies and the payoff each one earns, written as (row player, column player). A sequential game is one where players move in turns and later movers see what came before, so it is drawn as a game tree and solved backward from the end. Here is how they compare side by side.
Rows are one player's strategies, columns are the other player's, and each cell holds a pair of numbers. The convention is that the first number in the pair belongs to the row player and the second to the column player, though a well-labeled matrix says so directly. To solve one, find each player's best response: fix the opponent's choice, then compare only the numbers that belong to the player you are studying. A cell where both players are already playing a best response is a Nash equilibrium, and there can be none, one, or several. A payoff matrix assumes the players move at the same time or without seeing each other's move; a game where one side moves first belongs in a game tree instead.
Because the second player observes the first player's move before choosing, a sequential game is drawn as a tree of decision nodes rather than a table. You solve it by backward induction: start at the last decision, pick the move that pays that player most, cross out the branches they would never take, then step back one node and repeat with those later choices treated as settled. The result is a path through the tree that no player wants to leave, which is why order of play can change the outcome completely. This is the difference from a simultaneous game shown in a payoff matrix, where neither side knows the other's move; here the first mover can act on the knowledge that their choice will be seen.
Payoff Matrix vs Game Tree: Two Ways to Write Down the Same Strategic Problem
| Payoff Matrix | Sequential Game (Game Tree) | |
|---|---|---|
| What it puts on the page | Every combination of moves with the payoff to each player | The order of moves and what each player knows when choosing |
| Assumption about timing | Players choose without seeing the other's move | The later player sees the earlier move first |
| Shape | A grid of cells, payoffs written as (row player, column player) | Branches running out from an opening decision node |
| How it is solved | Test each cell for a profitable one player switch, or cross off dominated strategies | Work backwards from the final decisions |
| What it displays well | Dominance and the full set of equilibria | Whether a threat or a promise is worth believing when the moment comes |
| Where first mover advantage appears | Nowhere, since nobody moves first | Directly, because the opening choice limits the reply |
| Typical exam instruction | Identify the dominant strategy and circle the equilibrium cell | Complete the tree and state what each player does |
The same numbers give different answers depending on which form the question uses
Two firms each choose design A or design B, and being on the same design is what matters. Payoffs as (row player, column player) are (4, 2) if both take A, (2, 4) if both take B, and (0, 0) for either mismatch. Written as a grid, this has two equilibria. At A with A, either firm switching alone earns 0 instead of its 4 or 2, so neither moves. At B with B the same check holds. The grid gives no way to say which happens, and that is an honest answer rather than a missing step. Now redraw it as a tree with Row choosing first and Column seeing that choice. Column's reply to A is A, worth 2 against 0, and its reply to B is B, worth 4 against 0. Row works one step back, sees that choosing A leads to 4 and choosing B leads to 2, and takes A. The outcome is (4, 2), and the second equilibrium has vanished. Nothing about the payoffs changed. Moving first turned a tie into a /glossary/first-mover-advantage, so read the timing in the stem before you pick a method.
A grid can hide a threat that a tree exposes
The two forms are not simply alternatives, because a sequential game can be squeezed into a grid by listing each player's full plan, one entry for every situation they might face. Doing that keeps the payoffs and loses the timing, and grids built this way carry equilibria that survive the one player switch test yet rest on a player promising to do something they would never actually do. The tree shows the promise being tested at the point it would have to be carried out, and /glossary/backward-induction discards any branch a player would not choose there. This is the reason entry deterrence, price wars and bargaining are drawn as trees in class rather than as grids. Practical advice for the exam follows from that. If the stem says the players choose at the same time, or that neither observes the other, build a grid and run dominance. If it describes one player moving after another, or reacting to what has already happened, draw the tree, label who decides at each node, and solve from the last decision back. Choosing the wrong form is a bigger loss than an arithmetic slip, because every later step inherits the error.
Frequently asked questions
When should I use a payoff matrix instead of a game tree?
Use a matrix when the players choose at the same time or neither can see the other's move before deciding, and use a tree when one player moves first and the other observes it. The matrix is built for dominance and for scanning every cell, while the tree is built for order and for testing whether a later move would really be made. The stem almost always says which situation you are in.
Can the same game be shown both ways?
Yes, a sequential game can be rewritten as a matrix by treating each player's complete plan of action as a single strategy. The translation keeps the payoffs but drops the visible timing, so equilibria can appear in the grid that rely on moves a player would never actually make. That is why the tree is the safer form whenever order matters.
How are payoffs written in a payoff matrix?
Each cell shows two numbers in the order (row player, column player), so the first number belongs to the player choosing between the rows. Reading them in the wrong order flips the whole analysis and is one of the easiest ways to lose marks on an otherwise correct answer. Check the labels on the grid before you start, since some questions put the row player's payoff second.
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