POLS2130 · Week 4

Extensive Form Games & Subgame Perfect Equilibrium

Sequential games, backward induction, and why some Nash equilibria rely on threats that would never actually be carried out.

Extensive form games model strategic interactions where players move sequentially — one after another — rather than simultaneously. The game unfolds as a tree, with each branch representing a possible action. This structure lets us capture timing, information, and credibility in ways a payoff matrix cannot.

Structure

Game Tree

The full specification of choice nodes, branches (actions), and terminal nodes (payoffs). Visualises every possible path through the game.

Choice Nodes

Any point at which a player must select an action — labelled with the player who acts there.

The initial node is the root where the game begins. Terminal nodes are endpoints where payoffs are assigned to all players — listed as (Player 1, Player 2).

Key Concepts

Subgame

Any self-contained portion of the game tree that starts at a choice node and includes all successor nodes. A subgame could itself constitute a valid game.

Backward Induction

Reasoning from the end of the tree backwards to the start. Identify optimal actions at the final nodes, then use those to determine what is rational one step earlier — and so on.

Equilibrium

Nash Equilibrium

A strategy profile where no player can improve their payoff by unilaterally deviating. When derived from the normal form, it may include non-credible threats.

SPNE

Subgame Perfect Nash Equilibrium: a strategy profile that constitutes a Nash equilibrium in every subgame — even those never reached on the equilibrium path. Found via backward induction. Rules out non-credible threats.

How to Find the SPNE: Four Steps

Step 1. Identify all final decision nodes — the last choice nodes before payoffs are reached.

Step 2. Determine the optimal action at each final node — the action that maximises the payoff of the player acting there.

Step 3. Replace each final subgame with the payoff vector that results from that optimal action. Move one step back up the tree.

Step 4. Repeat Steps 1–3 until the initial node is reached. The resulting strategy profile is the SPNE.

Key Insight: The SPNE requires optimality at every node — including those never reached in equilibrium. This rules out Nash equilibria that rely on non-credible threats: threats a player would never rationally carry out if actually called upon to act.

The Regulation Game. A firm (Player 1) decides whether to Comply or Evade environmental regulations. A regulator (Player 2) then decides whether to Inspect or Ignore. Inspection is costly for the regulator but can catch violators. The firm prefers to evade if it expects no inspection; the regulator prefers to inspect only when evasion is likely.

Payoffs

Firm (Player 1): Evasion saves compliance cost (best payoff = 4). Complying is costly but safe (payoff = 3). Being caught evading is worst (payoff = 1). Complying while inspected is fine (payoff = 3).

Regulator (Player 2): Catching a violator is a big win (payoff = 4). Inspecting a compliant firm wastes resources (payoff = 2). Not inspecting a violator is an embarrassment (payoff = 1). Ignoring a compliant firm is fine (payoff = 3).

Payoffs are represented as (Firm, Regulator).

Game Tree

Figure: Regulation Game — Firm (P1) acts first; Regulator (P2) acts at each branch. Payoffs: (Firm, Regulator).

Step-Through: Backward Induction

Click a step above to walk through backward induction on this game.

Nash Equilibria vs SPNE

Converting to normal form, we can verify that this game has no pure-strategy Nash equilibrium. However, backward induction still yields a unique SPNE.

		Regulator (P2)
		Inspect	Ignore
Firm (P1)	Comply	3, 2	3, 3 SPNE
Firm (P1)	Evade	1, 4	4, 1

SPNE outcome

SPNE Result: The firm Complies. The regulator's strategy is: Ignore if Comply, Inspect if Evade.

The threat of inspection is credible — the regulator would genuinely inspect if evasion occurred (catching a violator yields payoff 4 vs. ignoring at payoff 1). Anticipating this, the firm has no incentive to evade. The off-path inspection branch is never reached in equilibrium, yet the credible threat of it is precisely what enforces compliance.

Note on SPNE and Nash Equilibrium. SPNE refines Nash equilibrium by imposing sequential rationality at every decision node — including those never reached on the equilibrium path. This rules out equilibria sustained by non-credible off-path threats. In some games this refinement has an even sharper bite: the Regulation Game has no pure-strategy Nash equilibrium in its normal form, yet backward induction yields a unique SPNE. The SPNE concept thus does more than filter implausible equilibria: it can identify a well-defined solution in games where the standard Nash criterion offers no pure-strategy prediction at all.