Introduction to
Game Theory

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 7

pollev.com/chrismakler

Name a company.

  • Motivation: why game theory?
  • Overview of the next 6 weeks
  • Components of a game
  • The extensive form (ch. 2)
  • Strategies and the normal form (ch. 3)
  • Beliefs, mixed strategies, and expected payoffs (ch. 4)

Today's Agenda

  • Up until now: agents only (really) interact with "the market" via prices
  • In real life, people, firms, countries ("players") interact with each other.
  • Our economic lives are interconnected: our well-being doesn't depend only on our own actions, but on the actions taken by others
  • Questions:
    • OPTMIZATION: How do you operate in a world like this?
    • EQUILIBRIUM: What is our notion of "equilibrium" in a world like this, and how is it different from competitive equilibrium?
    • POLICY: Given how people behave in strategic settings, how can we design "mechanisms" to achieve policy goals?

Motivation

  • The branch of economics that studies strategic interactions between economic agents.
  • Everyone's payoffs depend on the actions chosen by all agents
  • To "play the game," each agent thinks strategically about how the other agents are playing

Game Theory

  • Industrial organization: situations where a few firms dominate the market,
    and each firm's decisions affect others
  • Political economy: campaigning, governing, international diplomacy,
    provision of public goods
  • Contract negotiations: incentive structures, credible threats, negotiating over price
  • Interpersonal relationships: team dynamics, division of chores within a family

Applications

  • Today: Setup of the Model
    • Lecture: Notation and terminology [Watson Ch. 1-5]
    • Section: Review of monopoly profit maximization (HW ex. 4.3, 4.4)
  • Week 5: Equilibrium in a One-Shot Game
    • Tuesday: Dominance, best response, rationalizability [Watson Ch. 6-7]
    • Thursday: Nash Equilibrium [Watson Ch. 9-11]
  • Week 6: Dynamic Games
    • Tuesday: Dynamic games & subgame perfection [Watson Ch. 14-15]
    • Thursday: Repeated games & collusion [Watson Ch. 22-23]
  • Week 7: Applications, Review, and Midterm II

Next Three Weeks:
Games of Complete & Perfect Information

  • Players: who is playing the game?
  • Actions: what can the players do at different points in the game?
  • Information: what do the players know when they act?
  • Outcomes: what happens, as a function of all players' choices?
  • Payoffs: what are players' preferences over outcomes?

Components of a Game

  • Outcomes: what happens, as a function of all players' choices?
  • Payoffs: what are players' preferences over outcomes?

1

2

1

1

,

0

0

,

1

1

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0

0

,

Left

Right

Left

Right

1

2

Left

Right

Left

Right

Both OK

Both OK

Crash

Crash

Outcomes

Two bikers approach on an unmarked bike path.

Payoffs

i = \text{Player }i
-i = \text{Player(s) other than }i
\text{Example: Consider a strategy profile for four players }s = (s_1, s_2, s_3, s_4)
\text{If we consider player 2, then }s_i = s_2 \text{ and } s_{-i}=(s_1, s_3, s_4)
u_i(s_i,s_{-i}) = \text{Player $i$'s utility from playing $s_i$ when others play $s_{-i}$}

Notation Convention

The Extensive Form

(Watson, Chapter 2)

Extensive Form

Nodes:

Branches:

Initial node: where the game begins

Decision nodes: where a player makes a choice; specifies player

Terminal nodes: where the game ends; specifies outcome

Individual actions taken by players; try to use unique names for the same action (e.g. "left") taken at different times in the game

Information sets:

Sets of decision nodes at which the decider and branches are the same, and the decider doesn't know for sure where they are.

A "tree" representation of a game.

Example: Gift-Giving

She chooses to give one of three gifts:
X, Y, or Z.

1

X

Y

Z

Player 1 makes the first move.

Initial node

Player 1's actions at her decision node

(and decision node)

Example: Gift-Giving

Twist: Gift X is unwrapped,
but Gifts Y and Z are wrapped.
(Player 1 knows what they are,
but player 2 does not.)

After each of player 1's moves,
player 2 has the move: she can either accept the gift or reject it.

2

Accept X

Reject X

2

1

X

Y

Z

We represent this by having an information set connecting
player 2's decision nodes
after player 1 chooses Y or Z.

2

2

Player 2's actions

Player 2's decision nodes

Information set

Accept Y

Reject Y

Accept Z

Reject Z

Also: player 2 cannot make her action contingent on Y or Z; her actions must be "accept wrapped" or "reject wrapped"

Accept Wrapped

Reject Wrapped

Accept Wrapped

Reject Wrapped

Example: Gift-Giving

After player 2 accepts or rejects the gift, the game ends (terminal nodes) and payoffs are realized.

1

0

1

0

2

0

2

0

3

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–1

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1

X

Y

Z

,

,

,

,

,

,

Accept X

Reject X

Accept Wrapped

Reject Wrapped

Accept Wrapped

Reject Wrapped

Terminal Nodes

Player 1's payoffs

Player 2's payoffs

In this game, both players get a payoff of
0 if any gift is rejected,
1 if gift X is accepted, and
2 if gift Y is accepted.

 

If gift Z is accepted, player 1 gets a payoff of 3, but player 2 gets a payoff of –1.

Strategies and the Normal Form

(Watson, Chapter 3)

Strategies and Strategy Spaces

A strategy is a  complete, contingent plan  of action for a player in a game.

This means that every player
must specify what action to take
at every decision node in the game tree!

A strategy space is the set of all strategies available to a player.

Strategies & Strategy Spaces

Player 1 has a single decision:
which gift to give (X, Y, or Z).

1

0

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3

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–1

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X

Y

Z

,

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,

Accept X

Reject X

Accept Wrapped

Reject Wrapped

Accept Wrapped

Reject Wrapped

Player 2 might have to make one of two decisions: accept or reject gift X,
and accept or reject a wrapped gift.

Let's abbreviate these as A/R and A'/R'.

A

R

S_1 = \{X,Y,Z\}
S_2 = \{AA', AR', RA', RR'\}

A'

R'

A'

R'

Then player 2's strategy space is

Therefore player 1's strategy space is

Strategy Profiles

A strategy profile \(s = (s_1,s_2)\) is a vector showing which strategy from their strategy space is chosen by each player.

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–1

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X

Y

Z

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A

R

S_1 = \{X,Y,Z\}
S_2 = \{AA', AR', RA', RR'\}

A'

R'

A'

R'

s_1 = Z
s_2 = AR'

The outcome of this is that gift Z is given and rejected, and both players receive a payoff of 0.

\text{Example: }s = (Z, AR')

Note: the strategy profile specifies which action is taken at every decision node!

Notation

Strategy for player \(i\):

Strategy space for player \(i\):

S_1 = \{X,Y,Z\}
s_1 = Z

Strategy profile:

s_i
S_i
(s_1, s_2, ..., s_n)
S_2 = \{AA', AR', RA', RR'\}
s_2 = AR'
(\ \ \ ,\ \ \ \ \ \ \ )
Z
AR'

(a complete, contingent plan for how player \(i\) will move)

(set of all possible strategies for player \(i\))

(list of strategies chosen by each player \(i = 1,2,...,n\))

Player 1's Strategy Space:

Player 2's Strategy Space:

\{
\}
\{
\}
A,P,O
A,P

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How many strategies does player 1 have
in her strategy space?

Player 1's Strategy Space:

Player 2's Strategy Space:

Continuous Strategies

Strategy for player \(i\):

Strategy space for player \(i\):

S_1 = \mathbb{R}_+
q_1 = 6

Strategy profile:

s_i
S_i
(q_1, q_2)
S_2 = \mathbb{R}_+
q_2 = 3
(\ \ \ ,\ \ \ )
6
3

(set of all possible strategies for player \(i\))

(list of strategies chosen by each player \(i\))

Payoffs for both players, as a function of what strategies are played

\pi_1(q_1,q_2) = 12q_1 - q_1^2 - q_1q_2
\pi_2(q_1,q_2) = 12q_2 - q_2^2 - q_1q_2

Suppose two firms each simultaneously choose a quantity \(q_i\) to produce.

Normal-Form Game

List of players: \(i = 1, 2, ..., n\)

Strategy spaces for each player, \(S_i\)

Payoff functions for each player \(i: u_i(s)\),
where \(s = (s_1, s_2, ..., s_n)\) is a strategy profile 
listing each player's chosen strategy.

1

0

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0

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3

0

–1

0

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1

X

Y

Z

,

,

,

,

,

,

A

R

S_1 = \{X,Y,Z\}
S_2 = \{AA', AR', RA', RR'\}

A'

R'

A'

R'

\(X\)

\(AA'\)

1

2

\(AR'\)

\(RA'\)

\(RR'\)

\(Y\)

\(Z\)

Normal Form Representation

1

0

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0

2

0

3

0

–1

0

2

2

1

X

Y

Z

,

,

,

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,

,

A

R

S_1 = \{X,Y,Z\}
S_2 = \{AA', AR', RA', RR'\}

A'

R'

A'

R'

\(X\)

\(AA'\)

1

2

\(AR'\)

\(RA'\)

\(RR'\)

\(Y\)

\(Z\)

0

0

,

Normal Form Representation

1

0

1

0

2

0

2

0

3

0

–1

0

2

2

1

X

Y

Z

,

,

,

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,

,

A

R

S_1 = \{X,Y,Z\}
S_2 = \{AA', AR', RA', RR'\}

A'

R'

A'

R'

\(X\)

\(AA'\)

1

2

\(AR'\)

\(RA'\)

\(RR'\)

\(Y\)

\(Z\)

0

0

,

3

–1

,

Normal Form Representation

1

0

1

0

2

0

2

0

3

0

–1

0

2

2

1

X

Y

Z

,

,

,

,

,

,

A

R

S_1 = \{X,Y,Z\}
S_2 = \{AA', AR', RA', RR'\}

A'

R'

A'

R'

\(X\)

\(AA'\)

1

2

\(AR'\)

\(RA'\)

\(RR'\)

\(Y\)

\(Z\)

1

1

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2

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–1

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–1

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0

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Normal Form Representation

Normal Form Representation

\(OA\)

\(I\)

1

2

\(O\)

\(OB\)

\(IA\)

\(IB\)

2

2

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2

2

,

4

2

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4

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2

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1

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3

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Normal Form Representation

\(A\)

\(C\)

1

2

\(D\)

\(B\)

\(A\)

\(C\)

1

2

\(D\)

\(B\)

1

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Relationship between Normal-Form
and Extensive-Form Games

Both include the players, strategies, and payoffs.

The extensive form also includes information
on timing and information.

We usually use the normal form for
static (simultaneous-move) games of complete information.

Different extensive forms might have the same normal form.

Mixed Strategies, Beliefs, and Expected Payoffs

(Watson, Chapter 4)

Mixed Strategy

  • Play one element of your strategy space
    with probability 1, others with probability 0
  • Example: "Play heads" or "play tails"

Pure Strategy

  • Place positive probability on more than one element of your strategy space
  • Example: "Flip a coin and play whatever comes up on top."

Equilibria with mixed strategies are sometimes the only equilibrium!

  • Your probability distribution over another player's strategies
  • Represents the probability you believe they'll play each strategy (for whatever reason)

Beliefs

  • Your probability distribution over your own strategies.
  • Represents the probability with which you intend to play each strategy

Mixed Strategies

A

B

X

Y

1

2

5

4

5

0

0

4

4

4

Mixed strategy for player 1:
probability distribution
over {A, B}

Belief for player 1:
probability distribution over {X, Y}

2

Expected Payoffs

\({1 \over 6}\)

\({1 \over 3}\)

\({1 \over 2}\)

\(0\)

Player 1's beliefs

\({1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 7\)

\(=3\)

Player 1's expected payoffs from each of their strategies

\(X\)

\(A\)

1

\(B\)

\(C\)

\(D\)

\(Y\)

\(Z\)

6

6

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3

6

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2

8

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0

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12

6

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9

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6

8

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11

4

,

Your expected payoff from playing one of your strategies
is the weighted average of the payoffs, weighted by your beliefs about what the other person is playing

2

\({1 \over 6}\)

\({1 \over 3}\)

\({1 \over 2}\)

\(0\)

Player 1's beliefs

\(X\)

\(A\)

1

\(B\)

\(C\)

\(D\)

\(Y\)

\(Z\)

6

6

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6

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2

8

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7

0

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12

6

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9

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6

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11

4

,

pollev.com/chrismakler

Given these beliefs, what is player 1's expected payoff from playing Y?

Expected Payoffs

2

Your expected payoff from playing one of your strategies
is the weighted average of the payoffs, weighted by your beliefs about what the other person is playing

Expected Payoffs

\({1 \over 6}\)

\({1 \over 3}\)

\({1 \over 2}\)

\(0\)

Player 1's beliefs

\({1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 7\)

\(=3\)

\({1 \over 6} \times 12 + {1 \over 3} \times 6 + {1 \over 2} \times 0 + 0 \times 5\)

\(=4\)

\({1 \over 6} \times 6 + {1 \over 3} \times 0 + {1 \over 2} \times 6 + 0 \times 11\)

\(=4\)

Player 1's expected payoffs from each of their strategies

\(X\)

\(A\)

1

\(B\)

\(C\)

\(D\)

\(Y\)

\(Z\)

6

6

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3

6

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2

8

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7

0

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12

6

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9

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11

4

,

\(X\)

\(A\)

1

2

\(B\)

\(C\)

\(D\)

\(Y\)

\(Z\)

6

6

,

3

6

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2

8

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0

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12

6

,

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3

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11

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,

If  you are playing a mixed strategy, and the other player is  playing a pure strategy, your expected payoff is the weighted average given the way you are mixing.

Expected Payoffs

\({1 \over 6}\)

\({1 \over 3}\)

\({1 \over 2}\)

\(0\)

Player 2's strategy

\({1 \over 6} \times 6 + {1 \over 3} \times 6 + {1 \over 2} \times 8 + 0 \times 0\)

\(=7\)

\({1 \over 6} \times 6 + {1 \over 3} \times 3 + {1 \over 2} \times 2 + 0 \times 0\)

\(=3\)

\({1 \over 6} \times 0 + {1 \over 3} \times 9 + {1 \over 2} \times 8 + 0 \times 4\)

\(=7\)

Player 2's expected payoffs given each of 1's strategies

Section Preview

Simple case: linear demand, constant MC, no fixed costs

P(Q) = 14 - Q
c(Q) = 2Q
\text{No fixed costs }\Rightarrow MC = AC = 2

Baseline Example: Monopoly

P(Q) = 14 - Q
c(Q) = 2Q
\pi(Q) = P(Q)Q - c(Q)
= (14 - Q)Q - 2Q
= 14Q - Q^2 - 2Q
\text{total revenue}
\text{total cost}

14

2

units

$/unit

P(Q) = 14 - Q
MC = AC = 2
\text{No fixed costs }\Rightarrow MC = AC = 2

14

P

Q

Baseline Example: Monopoly

P(Q) = 14 - Q
c(Q) = 2Q
\pi(Q) = P(Q)Q - c(Q)
= (14 - Q)Q - 2Q
= 14Q - Q^2 - 2Q
\text{total revenue}
\text{total cost}

14

2

units

$/unit

P(Q) = 14 - Q
MC = AC = 2
\text{No fixed costs }\Rightarrow MC = AC = 2

14

P

Q

Profit

Baseline Example: Monopoly

P(Q) = 14 - Q
c(Q) = 2Q
\pi(Q) = P(Q)Q - c(Q)
= (14 - Q)Q - 2Q
= 14Q - Q^2 - 2Q
\pi'(Q) = 14 - 2Q - 2 = 0
Q^* = 6
\text{total revenue}
\text{total cost}
\text{marginal revenue}
\text{marginal cost}
P^* = 14 - 6 = 8
\pi^* = 8\times 6 - 2 \times 6 = 36

14

8

2

6

Q

P

P(Q) = 14 - Q
MR(Q) = 14 - 2Q
MC = AC = 2

36

Next Steps

  • Section: review of the monopoly model
  • Tuesday: Analyzing a single player's optimal behavior in a static game; introduction to Cournot duopoly
  • Thursday: Analyzing equilibrium in a static game, and in the Cournot model