Dynamic Games and Subgame Perfection

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 10

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Golden Balls

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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.

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

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

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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:

\pi_2(q_1,q_2) = P(q_1,q_2)q_2 - c_2(q_2)

Profit to Firm 1:

\pi_1(q_1,q_2) = P(q_1,q_2)q_1 - c_1(q_1)
P(q_1,q_2) = 14 - (q_1 + q_2)
c_1(q_1) = 2q_1
c_2(q_2) = 2q_2
\text{Market Demand: }
\text{Firm 1's Costs: }
\text{Firm 2's Costs: }

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:

\pi_2(q_1,q_2) = P(q_1,q_2)q_2 - c_2(q_2)

Profit to Firm 1:

\pi_1(q_1,q_2) = P(q_1,q_2)q_1 - c_1(q_1)
P(q_1,q_2) = 14 - (q_1 + q_2)
c_1(q_1) = 2q_1
c_2(q_2) = 2q_2
\text{Market Demand: }
\text{Firm 1's Costs: }
\text{Firm 2's Costs: }

Comparing Outcomes

P = 8, Q = 6
CS = 32
\pi_1 = \pi_2 = 16
P = 5, Q = 9
CS = 40.5
\pi_1 = 18, \pi_2 = 9

By Chris Makler

Econ 51 | Spring 22 | Dynamic Games and Subgame Perfection

Sequential and Repeated Games of Perfect Information

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