sequential game and the game tree

Not every contest happens in a single blind instant. Often one player moves, the other sees it and responds, then perhaps the first responds again — like chess, or a business deciding whether to enter a market after watching the incumbent. When the order of moves matters and later players can react to what they've seen, we call it a sequential game. The natural way to picture it is not a table but a branching diagram called a game tree, which lays out who moves when, what they can choose, and where each path of choices finally ends up.

A game tree starts at a single point (the first decision) and splits into branches, one per available move. Each branch leads to the next player's decision, which splits again, and so on, until the branches reach the leaves — the final outcomes, each labeled with the payoffs to every player. Reading a tree, you trace a path from root to leaf as a sequence of moves and replies. Because a later player can see what came before, the same actions written as a simultaneous payoff matrix and as a tree can yield different predictions: knowing who goes first, and who can react, genuinely changes the game.

Sequential games and trees are how economists model timing-rich situations: an incumbent firm deciding whether to threaten a price war if a rival enters; a country making a first move in a negotiation; a chain store building a reputation across towns. The tree's great payoff is that it can be solved by reasoning backward from the end — backward induction — which often exposes that a fierce-sounding threat is empty because the threatener would never actually carry it out once the moment came.