Questions: Extensive Form Games and Game Trees

Question 1

In a game tree, player 2 moves after player 1 but cannot observe which action player 1 took. How is this represented in the extensive form?

• A. Player 2's decision node is removed from the tree to indicate their uncertainty
• B. A separate chance node is inserted before player 2's decision
• C. Player 2's nodes are grouped into an information set — a dashed oval indicating they cannot be distinguished
• D. The two nodes are merged into a single node with more branches

Answer: (C)

An information set is the standard representation of imperfect information. When player 2 cannot distinguish two nodes — because they don't observe which branch player 1 took — those nodes are connected by a dashed oval. Player 2 must choose the same action at all nodes within an information set, since they have no way to tell them apart. A game with all singleton information sets is a game of perfect information; any multi-node information set indicates imperfect information.

Question 2

A dominant firm threatens to wage an aggressive price war if a smaller rival enters the market. Backward induction reveals that the price war would harm the dominant firm more than tolerating entry. What does this imply for the subgame-perfect equilibrium?

• A. The threat is credible because the dominant firm would carry it out to protect its reputation
• B. The threat is incredible — the rival correctly predicts the firm would accommodate entry, so entry occurs in equilibrium
• C. The rival will not enter because any price war is too risky, regardless of its credibility
• D. The Nash equilibrium requires the rival to stay out because the threat is stated explicitly in the strategic form

Answer: (B)

Backward induction evaluates whether threats are credible by asking: 'If this decision node were actually reached, what would the player do?' If waging the price war is worse for the dominant firm than accommodating entry, a rational firm would not follow through. The rival, reasoning backward, correctly predicts accommodation — so the threat does not deter entry. This is an incredible threat, eliminated by subgame-perfect equilibrium. The normal form might support staying-out as a Nash equilibrium if the rival assumes the threat will be carried out, but backward induction removes it.

Question 3

In a game of perfect information, nearly every information set contains exactly one node, so the extensive form and normal form yield identical Nash equilibria.

• T. True
• F. False

Answer: False

Perfect information means each player always knows which node they are at (singleton information sets), but this does NOT mean the extensive and normal forms yield the same predictions. The normal form can support Nash equilibria based on incredible threats that backward induction eliminates. In the normal form, a threat 'I will start a price war' is just a strategy, and if the rival believes it, not-entering is a Nash equilibrium. In the extensive form, backward induction reveals the threat would never be executed, so the only subgame-perfect equilibrium has the rival entering. Perfect information games have the richest scope for backward induction precisely because every node is reachable and testable.

Question 4

Backward induction eliminates Nash equilibria that rely on threats the threatening player would not actually carry out at the relevant decision node.

• T. True
• F. False

Answer: True

This is the core purpose of backward induction and subgame-perfect equilibrium. Starting from terminal nodes and working backward, each player's optimal action at each decision node is determined given what will happen downstream. A strategy that involves a threat that would be suboptimal to execute — 'I'll fight if you enter, even though fighting costs me more than accommodating' — cannot survive backward induction. Only threats that are best responses at the node where they would be executed remain, ensuring the equilibrium is self-enforcing at every subgame.

Question 5

Why does the extensive form reveal strategic possibilities that the normal (strategic) form obscures? Illustrate with an example involving the credibility of threats.

Model answer: The normal form lists strategy profiles and payoffs in a matrix, suppressing the order of moves and what players observe. It can support Nash equilibria where a threat deters an opponent even if the threat would never be executed. The extensive form represents the game as a tree with decision nodes and information sets, allowing backward induction: we can ask whether a threat would actually be carried out if the relevant node were reached. Example: in entry deterrence, a monopolist threatens a price war if a rival enters. In the normal form, 'stay out / threaten price war' can be a Nash equilibrium because neither party is deviating. Backward induction in the extensive form reveals the monopolist prefers accommodating entry over a costly price war, so the threat is incredible — the unique subgame-perfect equilibrium has the rival entering.

The extensive form adds information about timing and observability. Timing lets backward induction separate credible from incredible threats. The normal form collapses this structure, allowing equilibria to rest on off-path predictions that would never be tested — precisely the kind of incredible threat that the extensive form and subgame perfection are designed to eliminate.