23506 Chap.7 Extensive-Form Games, Game Trees and Subgame Perfect Equilibrium
Extensive-Form Games, Game Trees and Subgame Perfect Equilibrium
An extensive-form tree records chronological decisions, available actions, information and terminal payoffs. Each decision node belongs to a player; each branch carries an action; each terminal node keeps one declared payoff order. Information sets join nodes the acting player cannot distinguish, so the player must choose the same action across those nodes. Perfect information means every decision information set is a singleton.
A dynamic-game strategy is a complete contingency plan. It assigns an action at every information set belonging to the player, including information sets that are not reached on the equilibrium path. Pure strategy counts multiply the action counts across a player’s information sets. Converting a tree to normal form preserves these off-path components, so several strategies can generate the same observed path while supporting different threats.
A proper subgame begins at a singleton decision node and includes every successor without cutting an information set; the whole game also counts as a subgame. Subgame-perfect equilibrium requires a Nash equilibrium in every subgame. In a finite perfect-information game, backward induction solves last decisions and carries their continuation values toward the root. Threats or promises fail when their prescribed action would not be optimal at the node where they would be carried out. Always report the complete profile, equilibrium path and terminal payoff separately.
What this chapter covers
- 01Nodes, branches and terminal payoff order
- 02Information sets and observation
- 03Strategies as complete contingency plans
- 04Normal-form conversion
- 05Proper subgames and nesting
- 06Credibility, backward induction and SPE
Sequential Chicken by backward induction
- 2Identify the proper continuation subgame after Player 1 continues.
- 2Solve Player 2’s choice at that node.
- 2Carry the resulting continuation payoff to Player 1.
- 2Compare Player 1’s initial actions.
- 2Report complete strategy profile, path and payoff.
Key terms
- Decision node
- A point where a specified player chooses among available actions.
- Information set
- Nodes the acting player cannot distinguish when selecting one common action.
- Proper subgame
- A continuation beginning at a singleton decision node and containing whole information sets.
- Contingent strategy
- An action plan for every information set a player could face.
- Non-credible threat
- A threatened continuation action that would not be optimal if its node were reached.
- SPE
- A strategy profile that is Nash in the whole game and every proper subgame.
Extensive-Form Games, Game Trees and Subgame Perfect Equilibrium FAQ
Does every decision node start a subgame?
No. The node must be a singleton information set and its continuation cannot cut another information set.
Does the whole game count as a subgame?
Yes, so every SPE is also a Nash equilibrium of the whole game.
Why include off-path actions?
They determine continuation payoffs after deviations and whether threats are credible.
How many pure strategies does a player have?
Multiply its available action counts across all of its information sets.
Can linked nodes receive different strategy actions?
No. One information set receives one action because the player cannot tell which node occurred.
Is perfect information the same as certainty?
No. Perfect information concerns observation of prior play; a game can still contain a Nature move.
How does backward induction work?
Solve terminal decision problems, replace them with continuation values and repeat toward the root.
What if a last mover ties?
Retain every best response and carry every associated continuation value backward.
Can a Nash equilibrium fail SPE?
Yes, when it depends on a non-optimal action in a proper off-path subgame.
What should a final answer report?
The complete contingent profile, the path it generates and the terminal or expected payoff.
How do I package a complete extensive-form solution?
Begin by writing the player order used at terminal payoffs, then identify every information set and which prior actions are observed. Count the whole game and each proper subgame using the singleton-start and no-cut rules. List each player’s pure strategies as complete plans, defining the tuple order; multiply available actions across information sets rather than counting terminal paths. Solve the smallest terminal subgames first. At a sequential node compare the acting player’s continuation payoff; at a simultaneous continuation solve its normal-form Nash equilibria; at an unresolved chance branch calculate expected value. Retain ties and carry every equilibrium continuation backward. Report the final SPE as a complete contingent profile, not merely the actions observed. Then trace from the root to state the equilibrium path and terminal or expected payoff separately. Finish by explaining why each off-path action is optimal if its subgame is reached. This presentation earns the distinct marks usually hidden in subgame count, strategy completeness, credibility, root optimality and outcome.
Exam move
Draw small trees from prose and label payoff order before solving. Circle information sets, count subgames using the no-cut rule, and list each player’s complete strategies in a declared tuple order. Convert at least one tree to normal form so you can see how off-path plans produce duplicate paths and Nash threats. Solve backward using an action/payoff ledger at each node, retaining ties. After obtaining SPE, trace forward once to verify the path and payoff. Practise explaining why every SPE is Nash and constructing a Nash profile that fails SPE because its off-path threat is not optimal.
Working through Extensive-Form Games, Game Trees and Subgame Perfect Equilibrium in 23506? Sia is AskSia’s AI Economics tutor — ask any 23506 Extensive-Form Games, Game Trees and Subgame Perfect Equilibrium question and get a clear, step-by-step explanation grounded in how 23506 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
