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Price Theory

Lecture 8: Game Theory

Topics for today’s lecture . . .

1. The anatomy of a game

2. Sequential moves games

3. Simultaneous moves games

4. Repeated games

A flat tyre . . .

Instead of studying, two friends spend the night before an exam partying.

Turning up late to the exam, the two friends tell their professor that they are late because of

a flat tyre.

The professor allows each student to take a makeup exam; the exam consists of one question:

Which tyre?

Exercise: What if it were you?

Which tyre would you select as your answer?

(Write down your answer, we will use them again later today.)

Definition: Game theory

The study of strategic interaction; decision making when the payoff of each individual

decision-maker depends on the actions of all decision-makers.

Game theory is more complex than individual optimisation as an individual’s optimal action

typically varies with the actions of the other players in the game.

The anatomy of a game

Players

A player is any party (individual or organisation) that may be faced with a strategic choice in

the game.

It is typically assumed that all players in a game are:

• Intelligent: The players have knowledge of the structure (rules) of the game, and understand the potential consequences of their choices.

• Rational: Each player has complete and transitive preferences over the possible outcomes of the game, which satisfy the independence axiom.

Note: Both of these assumptions can be relaxed. However, doing so adds substantial

complexity to the game, and is beyond the scope of this course.

Actions

Actions are the choices (or moves) available to a player within a game.

A game may require each player to select a single action, plan a sequence of (possibly

situation-contingent) actions, or randomise between the available actions.

A plan of action that describes how a player will act in every conceivable situation is referred

to as a strategy.

Timing

The timing of a game captures the order in which players act:

• In a sequential moves game, players take turns choosing their actions.

• In a simultaneous moves game, players take their actions at the same time.

The timing also dictates the time horizon over which players interact:

• Finite horizon games have an end.

• Infinite horizon games are repeated without end.

Payoffs

Payoffs represent the preferences of each player, over the possible outcomes of the game.

A player’s payoffs completely capture the player’s objectives within the game.

• In a properly specified game a player has no objective other than to maximise their own (expected) payoff.

If a player is an individual, the payoffs will typically be the utilities created by the outcomes of

the game.

If a player is a firm, the payoff is typical the profit it receives from the game.

Discussion: Thinking about competition as a game

Competition in the market for smart-phones can be thought of as a game.

• Who do you think are the players in this game?

• What actions are available to each of these players?

• How would you describe the timing of the game?

• What are the objectives of each player, and how might you characterise their payoffs?

Sequential moves games

A market entry game

Suppose that Psi-Pharma’s patent over an anti-cancer drug is about to expire.

Alpha-Biotech is preparing to enter the market to compete with Psi. It has two options:

• Construct a small-capacity plant, that will have little impact on Psi.

• Construct a large-capacity plant, that will create strong competition for Psi.

Psi-Pharma can respond to Alpha’s entry into the market in two ways:

• It can accomodate Alpha’s entry, conceding market share.

• It can initiate a price war, aggressively discounting its price.

The game tree

Alpha

small Psi

large Psi

accomodate 4, 20

price war 1, 16

accomodate 8, 10

price war 2, 12

The game tree illustrates the timing of

the game:

• Alpha moves first, choosing the capacity of its production facilities.

• Psi then chooses how to respond.

The payoffs, in millions of dollars, are

listed for each possible outcome:

• The first number is the first-mover’s (Alpha’s) payoff.

• The second number is the second-mover’s (Psi’s) payoff.

Definition: Best response

The strategy (or strategies) that deliver a player the highest (expected) payoff, given the

strategies of the remaining players in the game.

In order for a player to rationalise employing a strategy, it must be a best-response to the

strategies that the remaining players have been observed playing, or are expected to play.

Backward induction

Alpha

small Psi

large Psi

price war 1, 16

accomodate 4, 20

accomodate 8, 10

price war 2, 12

Sequential moves games can be analysed

by backward induction.

Starting with the final decisions of the

game, find the player’s best-response to

the previous actions.

• If Alpha builds a small plant, Psi’s best-response is to accomodate.

• If Alpha builds a large plant, Psi’s best-response is to initiate a price war.

Backward induction continued

large Psi

Alpha

small Psi

price war 1, 16

accomodate 4, 20

accomodate 8, 10

price war 2, 12

Having determined how Psi will respond

to each of Alpha’s actions, we see that,

• Alpha’s payoff to building a small plant will be $4M.

• Alpha’s payoff to building a large plant will be $2M.

Alpha Biotech’s optimal entry strategy is

to build a small production facility.

Exercise: The cutting the cake game

100%, 0%

Billy

0%, 100%

choose

the split

Sara

large piece

small piece

Billy and Sara are arguing over who

should get the last piece of cake.

• They both think more cake is better.

Their mother tells Billy to cut the cake

into two pieces, in any way he wants,

however Sara will get to choose which

piece she eats.

Use backward induction to solve this

game.

Exercise solutions

100%, 0%

Billy

0%, 100%

choose

the split

Sara small piece

large piece

Regardless of how Billy cuts the cake,

Sara will choose the largest piece.

Billy’s problem is, therefore, to maximise

the size of the smallest piece.

• It follows that Billy will split the cake into equal halves.

Note: In this game, Sara enjoys a second

mover advantage.

Exercise: The ultimatum game

IG = $100,

IH = $0

Greg

IG = $0,

IH = $100

choose

the split

Harry

accept (IG, IH)

reject ($0,$0)

Greg and Harry are engaged in a take-it

or leave-it negotiation over $100.

• Greg must propose how the $100 will be split between the two players.

• Harry can either accept the proposal, or reject it. In the event of a rejection

neither player receives anything.

Use backward induction to solve this

game. (Assume that the players care only

about the monetary payoffs they receive.)

Exercise solutions

IG = $100,

IH = $0

Greg

IG = $0,

IH = $100

choose

the split

Harry

accept (IG, IH)

reject ($0,$0)

Because Harry cares only about his

monetary payoff, he will accept any offer

that is greater than $0.

• He will be indifferent between the two actions if the offer is $0.

Therefore Greg will offer Harry the

smallest possible amount ($1 or possibly

less).

Note: In this game, Greg enjoys a first

mover advantage.

Discussion: The ultimatum game

IG = $100,

IH = $0

Greg

IG = $0,

IH = $100

choose

the split

Harry

accept (IG, IH)

reject ($0,$0)

What offer would you make in this game

if you were Greg?

What offer would you accept in this game

if you were Harry?

Is your reasoning influenced by,

• fairness?

• heuristics?

• reputation?

Simultaneous moves games

An advertising game

Suppose that Alpha-Biotech and Psi-Pharma compete in the pharmaceuticals market.

Each firm must choose an advertising strategy:

• A firm can choose a large campaign, a small campaign, or no campaign at all.

• Advertising is costly, but helps a firm to capture market share.

The two firms must select their strategies simultaneously, and without knowing what their

rival is doing.

The payoff matrix

Large

Alpha Small

None

Large Small None

Psi

0 6 9

4 8 10

5 7 9

0 8 9

12 16 15

18 20 18

The payoff matrix illustrates how each

possible outcome of the game affects the

two players.

• Alpha’s choice of strategy determines the row.

• Psi’s choice of Strategy determines the column.

• The corresponding payoffs (in millions of dollars) can be found at the

intersection of these strategies.

Definition: Nash equilibrium

A situation in which each player in chooses their best response, given the strategies chosen by

the other players.

A Nash equilibrium describes play in a game that is stable in the sense that no player can

improve their (expected) payoff by altering their strategy.

Finding Alpha’s best responses

Large

Alpha Small

None

Large Small None

Psi

0 6 9

4 8 10

5 7 9

0 8 9

12 16 15

18 20 18

The first step in identifying a Nash

equilibrium is to find each firm’s best

responses.

• If Psi chooses a large campaign, Alpha’s best-response is no campaign.

• If Psi chooses a small campaign, Alpha’s best-response is a small

campaign.

• If Psi chooses a no campaign, Alpha’s best-response is a small campaign.

Finding Psi’s best responses

Large

Alpha Small

None

Large Small None

Psi

0 6 9

4 8 10

5 7 9

0 8 9

12 16 15

18 20 18

• If Alpha chooses a large campaign, Psi’s best-response is no campaign.

• If Alpha chooses a small campaign, Psi’s best-response is a small

campaign.

• If Alpha chooses a no campaign, Psi’s best-response is a small campaign.

The pure-strategy Nash equilibrium is for

each firm to choose a small campaign, as

this is the mutual best response.

The game of chicken

Swerve

Straight

Billy

Swerve Straight

Sara

0 -2

2 -10

0 2

-2 -10

Billy and Sara are playing a game of

chicken; riding their bikes towards each

other.

• Whoever swerves loses.

• Crashing is even worse.

Both players’ best-responses are to do the

opposite of her/his rival.

There are two pure-strategy Nash

equilibria to this game.

Exercise: What if it were you? (continued)

Recall the story of the two friends who were late for an exam.

In the makeup exam, each friend can choose between: Front left, front right, rear left, and

rear right.

1. If the two friends take the exam at the same time, and cannot communicate, what are

the possible pure-strategy equilibria of the game? (You do not need to construct a payoff

matrix.)

2. Compare the answer you wrote down earlier with your neighbours. Did your choices

constitute an equilibrium?

Exercise solutions

1. There are four pure-strategy equilibria:

• Both choose front left.

• Both choose front right.

• Both choose rear left.

• Both choose rear right.

Rock, paper, scissors

Rock

Billy Paper

Scissors

Rock Paper Scissors

Sara

0 -1 1

1 0 -1

-1 1 0

0 1 -1

-1 0 1

1 -1 0

This is the payoff matrix for the game

Rock, Paper, Scissors.

Each player’s best-response is to choose

the strategy that defeats her/his rival’s

choice.

There is no cell in which the two players’

best-responses intersect.

• If a player is predictable she/he loses.

• The solution is to be unpredictable.

Definition: Mixed strategy

A probability distribution by which a player randomly selects a pure strategy.

Every game that can be characterised by a payoff matrix has at least one (possibly

mixed-strategy) Nash equilibrium.

Mixed strategy Nash equilibrium

Rock

Billy Paper

Scissors

Rock Paper Scissors

Sara

0 -1 1

1 0 -1

-1 1 0

0 1 -1

-1 0 1

1 -1 0

If Billy (or Sara) selects each pure

strategy with probability 1/3, Sara (or

Billy) can do no better than to pick a

strategy at random.

It follows that each player choosing each

pure strategy with probability 1/3, is a

mixed strategy Nash equilibrium.

Note: You will not be asked to find

mixed-strategy Nash equilibria in this

course.

Quiz 1

Billy Middle

Down

Left Centre Right

Sara

10 9 3

5 1 8

12 7 5

20 32 40

5 1 6

35 40 40

Sara’s best-response to Billy playing

Down is,

(a) Left.

(b) Centre.

(d) Both Centre and Right.

Quiz 2

Billy Middle

Down

Left Centre Right

Sara

10 9 3

5 1 8

12 7 5

20 32 40

5 1 6

35 40 40

How many pure-strategy Nash equilibria

does this game possess?

(a) 0.

(b) 1.

(d) 3.

Repeated games

Household chores

Clean

Shirk

Billy

Clean Shirk

Sara

15 5

20 10

15 20

5 10

Sara and Billy both have chores to do

around the house.

• Cleaning the house makes it more pleasant for them both.

• Housework is tedious and they would each rather be doing something else.

Billy’s best-responses are to shirk.

Sara’s best-responses are likewise to shirk.

Definition: Dominant strategy

A strategy that is a best-response regardless of which strategies are employed by the

remaining players.

If all players have a dominant strategy, every player selecting their dominant strategy is a

Nash equilibrium.

Prisoners’ dilemma

Clean

Shirk

Billy

Clean Shirk

Sara

15 5

20 10

15 20

5 10

This is an example of a prisoners’

dilemma:

• A game in which each player has a dominant strategy.

• But in which the Nash equilibrium delivers the worst possible collective

outcome.

The outcome of the game would be

improved if Billy and Sara could find a

way to cooperate.

Cooperation in games

In many games, cooperation will require individual players to act against their self interest

(choose a strategy that is not a best response), in order to maximise the collective welfare of

all players.

Cooperation can be facilitated in repeated games if the value of the future relationship is

sufficient to motivate players to take cooperative actions today.

• Cheating on the agreement increases a players payoff today.

• But results in the player losing the benefits of cooperation in the future.

Cooperation and cheating

Clean

Shirk

Billy

Clean Shirk

Sara

15 5

20 10

15 20

5 10

Suppose that Billy and Sara play this

game every week.

• If both players do their share of the cleaning, they both receive a payoff of

15.

• If Billy shirks and doesn’t do his share of the cleaning, then his payoff is 20.

• If neither player cleans, they both receive a payoff of 10.

Discounting the future

ρ = 0.2

ρ = 0.5

0 t

Decision-makers typically discount future

payoffs.

Suppose that a decision-maker discounts

the future at a rate ρ > 0.

A payoff received at a time t in the future

will be discounted by the factor,

(1 + ρ)t

Note: You will not be asked to calculate

discounted payoffs in this course.

Grim trigger strategy

gain

loss

cooperate

cheat

0 t

The grim trigger strategy uses the

following rule:

• Cooperate so as long as the other player does the same.

• If either player ever cheats, choose the dominant strategy thereafter.

This is an equilibrium so long as the

once-off gain from cheating is less than

the discounted cost of lost future

cooperation.

Temporary punishments

gain

loss 10

cooperate

cheat

0 t

Under the grim trigger strategy a single

mistake ends cooperation forever.

An alternative is temporary punishments:

• Cooperate so as long as the other player does the same.

• If either player cheats, choose the dominant strategy for the specified

punishment period.

• After a punishment ends, restore cooperation.

Factors that undermine cooperation

0 t

• Players discount the future heavily (they are impatient).

• Players interact infrequently.

• Cheating is hard to detect.

• The one-time gain from cheating is large relative to the gains from

cooperation.

Exercise: Unravelling

Clean

Shirk

Billy

Clean Shirk

Sara

15 5

20 10

15 20

5 10

Suppose that Sara and Billy know they

will play this game twice.

1. What action will each player choose

the second time they play?

2. What action will each player choose

the first time they play?

3. Now suppose that Billy and Sara know

that the game will be repeated 20

times. What will be the outcome of

this game?

Exercise solutions

1. The second time Billy and Sara play the game is also the final time. There can be no

gains from future cooperation because the game ends.

It follows that the only considerations motivating Billy and Sara are the payoffs they

receive in the present. Therefore both Billy and Sara will choose their dominant strategy,

shirk.

2. Realising that both players will shirk the second time they play the game, Billy and Sara

both realise that their choices the first time they play the game will have no impact on

their future payoffs.

Therefore both Billy and Sara will choose their dominant strategy, shirk, the first time

they play the game as well.

Exercise solutions

3. If the game is repeated 20 times, the same dynamic will occur.

• The last time the game is played, both players will shirk.

• Therefore, the second-last time the game is played, both players will shirk.

• And so on . . .

Thus the players will not cooperate at any point in the game.

The prospect of future interaction

In order to sustain cooperation, there must be, at every point in the game, the prospect for

future interaction.

• This does not mean that the game must continue infinitely.

• Rather, at every point in time there must be some possibility that the game will be played again in the future.

If the game has a known end, then cooperation unravels in the manner of the previous

exercise.

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Game theory

• Sequential moves

• Best response

• Backward induction

• First/second mover advantage

• Simultaneous moves

• Nash equilibrium

• Mixed strategy

• Repeated games

• Dominant strategy

• Cooperation & cheating

• Unravelling