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Subgame Perfect Equilibrium

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Subgame Perfect Equilibrium

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Subgame Perfect Equilibrium

🎯 Learning Objectives

Apply backward induction to solve perfect-information extensive-form games
Define subgame perfect Nash equilibrium (SPNE)
Distinguish credible from non-credible threats
Solve entry deterrence and Stackelberg competition games
Represent SPNE strategies as a complete plan of action

5.1 Intuition: The Logic of Foresight

In sequential games, what will happen depends on what players expect to happen later. The key insight: look forward, reason backward.

🔑 Key Insight: A threat is only credible if it would be rational to carry out when the moment arrives. SPNE eliminates empty threats.

Example: The Ultimatum Game

Player 1 proposes a split of $10 (e.g.,$ 6 for self, $4 for P2)
Player 2 accepts (both get the split) or rejects (both get $0) Backward induction:
If P2 gets a positive offer, accepting gives > $0, so P2 should accept
Knowing this, P1 offers the minimum positive amount ($1)
SPNE: P1 offers $1, P2 accepts But in experiments, P2 often rejects "unfair" offers! This shows that fairness preferences matter beyond pure self-interest.

5.2 Formal Definition

Subgame

A subgame is a subset of an extensive-form game that:

Begins at a single decision node (singleton information set)
Contains all subsequent nodes
Does not cut across information sets

Subgame Perfect Nash Equilibrium (SPNE)

A strategy profile is a subgame perfect Nash equilibrium if it specifies a Nash equilibrium in every subgame of the original game. (Diagram)

5.3 Backward Induction Algorithm

Algorithm:

Start at the last decision nodes (just before terminal payoffs)
At each node, the player chooses the action that maximises their payoff
Replace that node with the resulting payoff
Move up the tree, repeating until the root

Example: Entry Deterrence

(Diagram) Step 1 (last node): If Entrant enters, Incumbent chooses between Fight (payoff 2) and Accommodate (payoff 4). Accommodate is chosen. Step 2 (root): Entrant chooses between:

Enter → (2, 4) [knowing Incumbent will accommodate]
Stay Out → (0, 8) Enter gives Entrant 2 > 0. Enter is chosen. SPNE: Entrant enters, Incumbent accommodates. Payoffs: (2, 4). Note: "If you enter, I'll fight!" is a non-credible threat — once entry occurs, fighting hurts the incumbent too.

5.4 Stackelberg Competition

Two firms: Leader chooses quantity

q_1

first, then Follower chooses

q_2

. Market price:

P(Q) = a - Q

where

Q = q_1 + q_2

Costs:

C_i(q_i) = c q_i

Backward induction: Step 1 (Follower): Given

q_1

, Follower maximises:

\pi_2 = (a - q_1 - q_2 - c) q_2

FOC:

a - q_1 - 2q_2 - c = 0 \implies q_2^*(q_1) = \frac{a - q_1 - c}{2}

This is the Follower's reaction function. Step 2 (Leader): Knowing

q_2^*(q_1)

, Leader maximises:

\pi_1 = (a - q_1 - q_2^*(q_1) - c) q_1 = (a - q_1 - \frac{a - q_1 - c}{2} - c) q_1

= \frac{(a - q_1 - c)}{2} q_1

FOC:

\frac{a - 2q_1 - c}{2} = 0 \implies q_1^* = \frac{a - c}{2}

Then

q_2^* = \frac{a - \frac{a-c}{2} - c}{2} = \frac{a - c}{4}

Stackelberg outcome:

q_1 = \frac{a-c}{2}

q_2 = \frac{a-c}{4}

Q = \frac{3(a-c)}{4}

Compare to Cournot (simultaneous):

q_1 = q_2 = \frac{a-c}{3}

Q = \frac{2(a-c)}{3}

The first-mover advantage: Stackelberg leader produces more and earns higher profit than in Cournot.

5.5 Credible vs. Non-Credible Threats

Type	Definition	Example
Credible threat	Rational to carry out when the time comes	"I'll lower price if you enter" (if it's profitable)
Non-credible threat	Would harm the threatener if carried out	"I'll fight to the death" (if fighting is costly)

SPNE eliminates non-credible threats by requiring optimal play in every subgame.

5.6 Limitations of Backward Induction

Limitation	Description
Perfect information required	Cannot apply directly with information sets
Uniqueness	Ties may produce multiple equilibria
Computational complexity	Large game trees are unwieldy
Behavioural violations	Humans don't always backward-induct (Ultimatum Game)

📊 Formula Summary

Concept	Definition
SPNE	Nash equilibrium in every subgame
Backward induction	Solve from terminal nodes to root
Stackelberg leader quantity	$q_1^* = \frac{a-c}{2}$
Stackelberg follower quantity	$q_2^* = \frac{a-c}{4}$

✅ Practice Questions

Q1: Find the SPNE: (Diagram)

Solution

At P2's node: B gives 2 > 0 from A → P2 chooses B. P1 anticipates this: L gives 3, R gives 2 → P1 chooses L. SPNE: P1 plays L, P2 plays B (if reached). Payoffs: (3, 1). Q2: In entry deterrence, what happens if the incumbent's payoffs change to: Fight (0, 5), Accommodate (2, 4)? Solution

At the last node: Incumbent chooses Fight (payoff 5) over Accommodate (payoff 4). Now the threat is credible! Entrant anticipates: Enter gives (-1, 5), Stay Out gives (0, 8). Entrant chooses Stay Out. SPNE: Entrant stays out, Incumbent fights if entered. Q3: Stackelberg with $a=100$ , $c=20$ . Find quantities and price. Solution

$q_1^* = \frac{100-20}{2} = 40$ $q_2^* = \frac{100-40-20}{2} = 20$ $Q = 60$ $P = 100 - 60 = 40$ Profits: $\pi_1 = (40-20)40 = 800$ , $\pi_2 = (40-20)20 = 400$ The leader earns twice the follower's profit. Join Discord PreviousExtensive Form AdvancedNextRepeated Games