What is player 2's strategy space?

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What is player 1's strategy space?

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Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 5

How did Nick get Ibrahim to avoid getting stuck in the "prisoners' dilemma"?

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

**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.

**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

Today: extend notion of best-response (Nash) equilibrium to the class of **dynamic games** with **complete** & **perfect** information.

"dynamic"

takes place over time;

not simultaneous/one-shot

"complete information"

all players know all relevant aspects of the game (especially payoffs)

"perfect information"

all players observe all moves

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Consider the following game.

Player 1 starts.

She chooses X or Y.

Player 2 observes player 1's choice.

If player 1 chose X, player 2 chooses A or B.

If player 1 chose Y, player 2 chooses C or D.

After player 2 makes his choice, the payoffs to each player are realized.

"Play X."

"Play Y."

What is player 1's **strategy space**?

What is player 2's **strategy space**?

"Play A after X, and C after Y."

"Play A after X, and D after Y."

"Play B after X, and C after Y."

"Play B after X, and D after Y."

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What is player 1's **strategy space**?

What is player 2's **strategy space**?

AC

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BD

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We can create the normal-form representation of this game.

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We can create the normal-form representation of this game.

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What are the Nash equilibria of this normal-form game?

Do any of the Nash equilibria __not__ make sense?

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AC

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Think about this: after player 1 makes her move, we are in one of two **subgames**.

What should player 2 do in each subgame?

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AC

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Think about this: after player 1 makes her move, we are in one of two **subgames**.

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What should player 2 do in each subgame?

Anticipating how player 2 will react, therefore, what will player 1 choose?

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Think about the other Nash equilibrium.

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This is better for player 2.

It's kind of like player 2 is __threatening__ to play D if player 1 chooses Y.

But this threat is not **credible**.

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.

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

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

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?)

- Two firms ("duo" in duopoly)
- Each chooses how much to produce
- Market price depends on

the**total amount produced** - Each firm faces a
**residual demand curve**

based on the other firm's choice

**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?

- Firm 1 chooses how much to produce first \(q_1\),

at a cost of $2 per unit - Firm 2 observes firm 1's choice,

and**then**chooses how much to produce \(q_2\),

also at a cost of $2 per unit - The market price is determined by \(P(q_1,q_2) = 14 - (q_1+q_2)\)
- Profits are realized
- \(\pi_1(q_1,q_2) = (14 - [q_1 + q_2]) \times q_1 - 2q_1\)
- \(\pi_2(q_1,q_2) = (14 - [q_1 + q_2]) \times q_2 - 2q_2\)

What is firm 2's optimal strategy,

having observed \(q_1\)?

P(q_1,q_2) = 14 - (q_1+q_2)

c_2(q_2) = 2q_2

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

= (14 - q_1 - q_2)q_2 - 2q_2

= 14q_2 - q_1q_2 - q_2^2 - 2q_2

\pi'_2(q_2) = 14-q_1 - 2q_2 - 2 = 0

q_2^*(q_1) = 6-\frac{1}{2}q_1

\text{total revenue}

\text{total cost}

\text{marginal revenue}

\text{marginal cost}

2

P

P(q_2|q_1) = 14-q_1 - q_2

MR(Q) = 14-q_1 - 2q_2

MC(q_2) = 2

6-\frac{1}{2}q_1

14-q_1

q_2

"Firm 2's Residual Demand Curve"

This **function** is firm 2's strategy.

It specifies what firm 2 will do after every possible move of firm 1.

What is firm 1's best response to firm 2's strategy of \(q_2^*(q_1)\)?

P(q_1,q_2) = 14 - (q_1+q_2)

c_1(q_1) = 2q_1

q_2^*(q_1) = 6-\frac{1}{2}q_1

Firm 2's strategy:

\pi_1(q_1|q_2^*(q_1)) = P(q_1 + q_2^*(q_1))q_1 - c_1(q_1)

= (14 - q_1 - [6 - {1 \over 2}q_1])q_1 - 2q_1

= 14q_1 - q_1^2 - [6q_1 - {1 \over 2}q_1^2] - 2q_1

\pi'_1(q_1|q_2^*(q_1)) = 14 - 2q_1 - (6 - q_1) - 2 = 0

q_1^* = 6

**Firm 2's strategy**: whatever \(q_1\) firm 1 produces, produce \(6 -{1 \over 2} q_1\) (or 0)

**Firm 1's strategy**: produce 6 units of output

Given what the **other firm** is doing, does either firm

have any incentive to change its strategy?

In equilibrium, firm 2 produces 3 units.

**Why don't we just say that its strategy is "produce 3 units of output"?**

Consider the following strategy for firm 2:

"You should produce the Cournot quantity of 4.

If you do, I'll also produce 4.

If you don't, I'll produce so much that the price is 2,

so we both sell at marginal cost and nobody makes a profit."

What is firm 1's best response to this strategy?

- Today we saw that some
**Nash Equilibria**are not subgame perfect. - Next time, we'll look at a special kind of dynamic game: repeated games.
- We'll show that playing a repeated game enables us to have outcomes

that are*not*Nash equilibria of the one-shot game - In particular, we'll show that we can sustain cooperation in

Prisoners' Dilemma type games if players have an**ongoing relationship**

(are playing an infinitely repeated game)