# Dynamic Games and Subgame Perfection

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 10

# Golden Balls

Split

Split

**1**

**2**

Steal

Steal

1

1

2

0

0

2

0

0

# Golden Balls

Split

Split

**1**

**2**

Steal

Steal

1

1

2

0

0

2

0

0

## Big Ideas

**Credibility**: can you credibly threaten some retaliation, or promise

some reward, to get the other player to do something you want?

**Finite vs. Infinite Time Horizon**: Does the game end?

**Subgames**: games will have more than one move; we can break them

up and examine *subgames* which are like games-within-games.

## What's wrong with Nash Equilibrium?

## Today's Agenda

Part 1: Discrete Strategies

Part 2: Continuous Strategies

Review: Extensive Form Games

Backward Induction

Strategies in Extensive Form Games

Subgame Perfect Nash Equilibrium

Example: Entry Deterrence

Example: Ultimatum Game

Example: Stackelberg Duopoly

# Review: Extensive Form Games

**Normal-Form vs. Extensive-Form Representations**

The **extensive-form** representation

of a game specifies:

The **normal-form** representation

of a game specifies:

The strategies available to each player

The player's payoffs for each combination of strategies

The players in the game

**When each player moves**

The actions available to each player each time it's their move

The players in the game

The player's payoffs for each combination of actions

## Circle the best responses. What's wrong?

## One way to solve this game: **backwards induction**.

Backwards induction is a method of determining the outcome(s) of a game by starting at the end and working backwards.

Finite games: start from terminal nodes

Infinite games: a bit more complicated

# Subgame Perfect Nash Equilibrium

1

2

X

Y

X

Y

A

B

3

2

1

0

2

0

1

3

C

D

1

2

2

Create the normal-form representation of this game. What are the Nash Equilibria?

### Definition: Subgame Perfect Nash Equilibrium

In an extensive-form game of complete and perfect information,

a **subgame** in consists of a **decision node** and all subsequent nodes.

A Nash equilibrium is **subgame perfect **if the players' strategies

constitute a Nash equilibrium in every subgame.

(We call such an equilibrium a Subgame Perfect Nash Equilibrium, or SPNE.)

Informally: a SPNE doesn't involve any non-credible threats or promises.

A Subgame Perfect Nash Equilibrium must specify a NE in every subgame!

# Continuous Strategies

How do we represent a **continuous strategy** in an extensive-form game?

(For example, the quantity chosen by a firm in a Cournot-like game?)

Player 1: Offers a split of $100 to player 2.

Player 2: Accepts or rejects the offer:

if accepts, split as player 1 said

if rejects, nobody gets any

# Example: Ultimatum Game

**Players**: Two firms, Firm 1 and Firm 2

**Payoffs**:

Market price is determined by total output produced

Profit to Firm 2:

Profit to Firm 1:

New twist:

Firm 1 chooses \(q_1\) first;

Firm 2 observes \(q_1\) and chooses \(q_2\)

What are the strategy spaces?

# Example: Stackelberg Duopoly

Profit to Firm 2:

Profit to Firm 1:

## Comparing Outcomes

#### Econ 51 | Spring 22 | Dynamic Games and Subgame Perfection

By Chris Makler

# Econ 51 | Spring 22 | Dynamic Games and Subgame Perfection

Sequential and Repeated Games of Perfect Information

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