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Introduction to Game Theory

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Presentation vs. exam

• You and your partner – Either study for the exam or prepare the presentation (not

both)

• Exam (50%) – If you study for the exam, your (expected) grade is 92 – If you don´t, your grade is 80

• Presentation (50%) – If both you and your partner prepare the presentation, your

(expected) joint grade is 100 – If one of you prepares the presentation and the other does not,

your joint grade is 92 – If none of you prepare the presentation, your joint grade is 84

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Presentation vs. exam: tabular form

Presentation Exam

Presentation

90, 90 86, 92

Exam 92, 86 88, 88

You

Your partner

• you decide independently of your partner, taking this knowledge into account • you want to maximize your grade

what do you do?

Game theory: set-up

• Finite set of players

• Set of strategies available to each player

• Payoff for each player as a function of the strategies selected by all players

• Everything a player cares about is reflected in her payoff, which she strives to maximize (rationality)

• Each player knows everything about the structure of the game (common knowledge)

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Back to presentation and exam

Presentation Exam

Presentation

90, 90 86, 92

Exam 92, 86 88, 88

You

Your partner

Presentation Exam

Presentation

90, 90 86, 92

Exam 92, 86 88, 88

You

Your partner

you should study for the exam

you should study for the exam

Presentation and exam: outcome

Presentation Exam

Presentation

90, 90 86, 92

Exam 92, 86 88, 88

You

Your partner

You both get a grade of 88, whereas you could both get a grade of 90, if you somehow cooperated

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Best response and dominant strategies

S¤1 is a best response of player 1 to strategy S2 of player two if

8S1 P1(S¤1 ; S2) ¸ P1(S1; S2)

S¤1 is a strict best response of player 1 to strategy S2 of player two if

8S1 6=S¤1 P1(S¤1 ; S2) > P1(S1; S2)

S¤1 is a dominant strategy for player 1 if it is a best response to every strategy

of player 2

8S1;S2 P1(S¤1 ; S2) ¸ P1(S1; S2)

S¤1 is a strictly dominant strategy for player 1 if it is a strict best response

to every strategy of player 2

8S1 6=S¤1 ;S2 P1(S¤1 ; S2) > P1(S1; S2)

Other considerations

• Finite games – finite set of players and finite set of strategies

– ordinal payoffs

• Simultaneity – each play makes her decision without knowing the

decisions of other players

S21 S2

2

S11 90, 90 86, 92

S12 92, 86 88, 88

S21 S2

2

S11 2, 2 0, 3

S12 3, 0 1, 1

same reasoning as

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Low-priced and upscale

• Two firms – Either produce a low-priced or upscale version of a product

(not both)

• Distribution of the population – 60% of the population buy low-priced

– 40% of the population buy upscale

• Popularity of the firms – If the two firms directly compete in a market segment, firm 1

gets 80% of the sales and firm 2 gets 20% of the sales

– If the two firms compete on different market segments, each gets 100% of the sales

Low-priced vs. upscale: tabular form

Low-priced Upscale

Low-priced

0.48, 0.12 0.60, 0.40

Upscale 0.40, 0.60 0.32, 0.08

Firm 1

Firm 2

Firm 1 has the strictly dominant strategy of producing low-priced items

Firm 2 has no strictly dominant strategy

Firm 1 will choose to produce low-priced items, then Firm 2’s best response to that will be to produce upscale items

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Three-client game

• Two firms – each can hope to do business with just one of three clients A, B,

or C

• Clients – The business of client A is 8 – The business of clients B and C is 2 each – Client A will either do business with the two firms

simultaneously, or with none

• Business approaches – If the two firms approach the same client, each gets half the

business of that client – If firm 1 approaches a client alone it gets nothing – If firm 2 approaches client B or C alone, it gets it full business

Three-client game: firm 1

A B C

A

4, 4 0, 2 0, 2

B 0, 0 1, 1 0, 2

C 0, 0 0, 2 1, 1

Firm 1

Firm 2

no dominant strategy for Firm 1

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Three-client game: firm 2

A B C

A

4, 4 0, 2 0, 2

B 0, 0 1, 1 0, 2

C 0, 0 0, 2 1, 1

Firm 1

Firm 2

no dominant strategy for Firm 2 either

(S¤1 ; S¤2) is a Nash equilibrium if S¤1 is a best response to S¤2 and S¤2 is a best

response to S¤1

8S1;S2 P1(S¤1 ; S

¤2) ¸ P1(S1; S

¤2) ^ P1(S

¤1 ; S

¤2) ¸ P1(S

¤1 ; S2)

Nash equilibrium

If one player has a dominant strategy, then that strategy together with the other player’s best response to it is a Nash equilibrium

Sometimes, there is no dominant strategy at all

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Three-client game: Nash equilibrium

A B C

A

4, 4 0, 2 0, 2

B 0, 0 1, 1 0, 2

C 0, 0 0, 2 1, 1

Firm 1

Firm 2

(A,A) is the unique Nash equilibrium

Looking for Nash equilibria

• Checking Nash equilibrium

– individual strategies are best responses to each other

• Finding Nash equilibria

– compute each player’s best responses to each strategy of the other player

– find the strategies that are mutual best responses

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Coordination game

• You and your partner

– Either prepare presentation in PowerPoint or in Keynote (not both)

• Coordination

– If both you and your partner choose the same application, each gets a payoff of 1

– Otherwise, each gets a payoff of 0

Coordination game: Nash equilibria

PowerPoint Keynote

PowerPoint

1, 1 0, 0

Keynote 0, 0 1, 1

You

Your partner

Two Nash equilibria

Focal point external to the game

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Unbalanced coordination game

PowerPoint Keynote

PowerPoint

2, 2 0, 0

Keynote 0, 0 1, 1

You

Your partner

Two Nash equilibria all the same

Focal point intrinsic to the game

Battle of sexes

PowerPoint Keynote

PowerPoint

1, 2 0, 0

Keynote 0, 0 2, 1

Your partner

Two Nash equilibria once more

You

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Stag and hare

Stag Hare

Stag

4, 4 0, 3

Hare 3, 0 3, 3

Hunter 2

Yet, two Nash equilibria

Hunter 1

Hawk-dove

• Two animals foraging food – Each animal can take either an aggressive (hawk)

or passive (dove) stance

• Payoffs – If the two animals behave passively, each gets a

payoff of 3

– If one behaves aggressively and the other passively, the aggressor gets a payoff of 5, while the passive gets a payoff of 1

– If both behave aggressively, each gets a payoff of 0

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Hawk-dove: Nash equilibria

Dove Hawk

Dove

3, 3 1, 5

Hawk 5, 1 0, 0

Animal 2

Two Nash equilibria

Animal 1

Matching cents

• You and your friend

– Either can choose heads or tails of its own one-cent coin

• Payoffs

– If coins match, you loose your cent to your friend

– If coins do not match, your friend looses her cent to you

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Matching cents: no Nash equilibria

Heads Tails

Heads

-1, +1 +1, -1

Tails +1, -1 -1, +1

Your friend

No Nash equilibria

You

zero-sum game: payoffs of the players sum to zero on every outcome

Pareto-optimality

(S¤1 ; S¤2) is a Pareto-optimal if

8S1;S2 P1(S¤1 ; S

¤2) > P1(S1; S2) _ P2(S

¤1 ; S

¤2) > P2(S1; S2)

_ P1(S¤1 ; S

¤2) = P1(S1; S2) ^ P2(S

¤1 ; S

¤2) = P2(S1; S2)

(S¤1 ; S¤2) is a Pareto-optimal if

:9S1;S2 P1(S¤1 ; S

¤2) < P1(S1; S2) ^ P2(S

¤1 ; S

¤2) · P2(S1; S2)

_ P1(S¤1 ; S

¤2) · P1(S1; S2) ^ P2(S

¤1 ; S

¤2) < P2(S1; S2)

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Presentation vs. exam again

Presentation Exam

Presentation

90, 90 86, 92

Exam 92, 86 88, 88

You

Your partner

• (Exam, Exam) is a Nash equilibrium that is not a Pareto-optimal outcome • (Presentation, Presentation), (Presentation, Exam), and (Exam,

Presentation) are Pareto-optimal outcomes, none of which is a Nash equilibrium

Social optimality

(S¤1 ; S¤2) is a social optimum if

8S1;S2 P1(S¤1 ; S

¤2) + P2(S

¤1 ; S

¤2) ¸ P1(S1; S2) + P2(S1; S2)

(Presentation, Presentation) is the only social optimum in the Presentation vs. Exam game

A social optimum is a Pareto-optimum, but not conversely

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Facility location

• Two firms – Firm 1 can open store in A, C, or E

– Firm 2 can open store in B, D, or F

• Payoffs – Customers go to the nearest store

– Same number of customers per town

– Payoffs proportional to number of customers

A B C D E F

Facility location: tabular form

B D F

A

1, 5 2, 4 3, 3

C 4, 2 3, 3 4, 2

E 3, 3 2, 4 5, 1

Firm 1

Firm 2

• if firm 2 chooses B, then firm 1’s best response is C • if firm 2 chooses D, then firm 1’s best response is C • if firm 2 chooses F, then firm 1’s best response is E

There is no strictly dominant strategy by firm 1, but there is a strictly dominated strategy by firm 1 (strategy A)

There is no strictly dominant strategy by firm 2, but there is a strictly dominated strategy by firm 2 (strategy F)

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Strictly dominated strategies

S¤1 is a strictly dominated strategy for player 1 if it is not a best response to

any strategy of player 2

8S29S1 6=S¤1 P1(S¤1 ; S2) < P1(S1; S2)

Iterated deletion of strictly dominated strategies

• find all strictly dominated strategies and delete them • consider the reduced game without the strictly dominated strategies • repeat the process

Facility location: reduced form

B D

C

4, 2 3, 3

E 3, 3 2, 4

Firm 1

Firm 2

A B C D E F

Firm 1 Firm 2

C is a strictly dominant strategy for firm 1 (E is strictly dominated) and D is a strictly dominant strategy for firm 2 (B is strictly dominated)

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Nash equilibrium and deletion of strictly dominated strategies - I

• A Nash equilibrium in the original game is a Nash equilibrium in the reduced game

Because (S¤1 ; S¤2) is a Nash equilibrium, for all S1,

P1(S¤1 ; S

¤2) ¸ P1(S1; S

¤2)

Because S¤1 is a strictly dominated strategy, there is S1 such that

P1(S¤1 ; S

¤2) < P1(S1; S

¤2)

Proof (Contradiction)

Suppose that (S¤1 ; S¤2) is a Nash equilibrium and S¤1 is a strictly dominated

strategy

Nash equilibrium and deletion of strictly dominated strategies - II

• A Nash equilibrium in the reduced game is a Nash equilibrium in the original game

Because (S¤1 ; S¤2) is a Nash equilibrium, for all S1 6= S01,

P1(S¤1 ; S

¤2) ¸ P1(S1; S

¤2)

Because S01 is a strictly dominated strategy, there is S1 6= S01 such that

P1(S01; S

¤2) < P1(S1; S

¤2)

Proof

Suppose that (S¤1 ; S¤2) is a Nash equilibrium in the reduced game and S01 is

a strictly dominated strategy in the original game

Therefore, there is S1 6= S01 such that

P1(S01; S

¤2) < P1(S1; S

¤2) · P1(S

¤1 ; S

¤2)

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Traffic at equilibrium

• 4,000 drivers • Each driver chooses either path ACB or ADB • Travel time in links AD e CB is 45 minutes • Travel time in links AC and DB is x/100 minutes, where x is

the number of drivers using the link • Payoff is the negative of the driver’s travel time

A B

C

D

x/100

x/100

45

45

Nash equilibria

• No driver has a dominant strategy

– ACB is the best response of a driver if all others opt for path ADB

– ADB is the best response of a driver if all others opt for path ACB

• Nash equilibria

– any set of 2,000 drivers that choose path ACB with the other 2,000 drivers choosing path ADB

– the travel time of each driver is 65 minutes

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Braess’s paradox

• Each of the 4,000 drivers has three choices – path ACB with travel time 45 + x/100 minutes

– path ADB with travel time 45 + x/100 minutes

– path ACDB with travel time 2x/100

A B

C

D

x/100

x/100

45

45

0

Nash equilibrium

• Choosing path ACDB is a strictly dominant strategy for each player

– 2x/100 < 45 + x/100 for all 0 ≤ x ≤ 4,000

• Unique Nash equilibrium

– each player chooses path ACDB

– travel time is 80 minutes; worse than travel time without link CD!

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General case

• Network • Travel time along link e

– Te(x) = ae x + be , where x is the number of drivers traversing the link

• n drivers – each with an origin and a destination – each with an arbitrary choice of paths from the origin to the

destination

• Questions – Is there a Nash equilibrium? – How much worse is the social cost of the equilibrium in relation

to the optimal social cost?

Traffic pattern and social cost

Tra±c pattern: P = fP1; : : : ; Png

Tra±c intensity on link e: xe = jfe 2 Pj j 1 · j · ngj

Travel time over link e: Te(xe)

Travel time of driver i: Ti(P) =P

e2Pi Te(xe)

Social cost:

S(P) =nX

i=1

Ti(P)

=

nX

i=1

X

e2Pi

Te(xe)

=X

e

xeTe(xe)

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Best response dynamics

while there is a driver whose best response to the paths chosen by other drivers is an alternate path to the one it is taking, switch to the alternate path

Tra±c pattern without driver i: P¡i = fP1; : : : ; Png¡fPig

Tra±c pattern with P 0i substituted for Pi: P0 = P¡i [fP 0ig

x¡ie =

(xe ¡ 1 if e 2 Pi

xe otherwise

x0e =

(x¡ie + 1 if e 2 P 0

i

x¡ie otherwise

Energy

Energy:

E(P) =X

e

xeX

y=1

Te(y)

=X

e

x¡ieX

y=1

Te(y) +X

e2Pi

Te(xe)

= E(P¡i) + Ti(P)

Best-response dynamics: Ti(P0) < Ti(P)

Energy decreases with best-response dynamics: E(P0) <E(P)

There is a traffic equilibrium

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Energy and social cost - I

A B

C

D

x 5

5 x

0 A B

C

D

x 5

5 x

0

e = 1 + 2 s = 2 + 2

e = 5 + 5 s = 5 + 5

e = 1 + 2 s = 2 + 2

e = 5 + 5 s = 5 + 5

e = 1 + 2 s = 2 + 2

e = 5 s = 5

e = 1 + 2 + 3 s = 3 + 3+ 3

e = 5 + 5 s = 5 + 5

e – energy s – social cost

e = 26 s = 28

e = 24 s = 28

Energy and social cost - II

A B

C

D

x 5

5 x

0 A B

C

D

x 5

5 x

0

e = 1 + 2 s = 2 + 2

e = 5 + 5 s = 5 + 5

e = 1 + 2 + 3 + 4 s = 4 + 4 + 4 + 4

e – energy s – social cost

e = 23 s = 30

e = 1 + 2 + 3 s = 3 + 3 + 3

e = 5 s = 5

e = 1 + 2 + 3 + 4 s = 4 + 4 + 4 + 4

e = 21 s = 30

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Energy and social cost - III

A B

C

D

x 5

5 x

0

e – energy s – social cost

e = 1 + 2 + 3 + 4 s = 4 + 4 + 4 + 4

e = 1 + 2 + 3 + 4 s = 4 + 4 + 4 + 4

e = 20 s = 32

Energy and social cost

Travel time over link e: Te(y) = aey+ be

xeTe(xe) ¸xeX

y=1

aey + be

= ae(xe + 1)xe

2+ xebe

¸ 1

2xeTe(xe)

S(P) ¸ E(P) ¸ 12S(P)

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Social cost of equilibrium

Tra±c pattern at optimal cost: P¤

Tra±c pattern at equilibrium: P

S(P¤) ¸ E(P¤) ¸ E(P) ¸ 12S(P)

social cost of equilibrium is at most twice the socially optimal cost

Sequential games

• Players take an order in choosing their strategies

• Finite and deterministic strategies

• Non-cooperative

• Players seek to maximize their payoff at every stage of the game (sub-game perfection)

• Perfect information

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Choosing regions for advertising

• Two firms trying to decide whether to focus their advertising on two possible regions A and B

• Firm 1 chooses first

• If firm 2 follows firm 1 into the same region then – firm 1 gets 2/3 of the market in that region

– firm 2 gets 1/3 of the market in that region

• If firm 2 chooses the other region then – each firm gets the full market in that region

• The market for region A is 12 and for region B is 6

Extensive form of the game

A

B

A

B

A

B

Firm 1

Firm 2

Firm 2 8, 4

12, 6

6, 12

4, 2

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Rollback equilibrium

A

B

A

B

A

B

Firm 1

Firm 2

Firm 2 8, 4

12, 6

6, 12

4, 2

A

B

A

B

A

B

Firm 1

Firm 2

Firm 2 8, 4

12, 6

6, 12

4, 2

To look forward, reason backward

Rollback equilibrium: firm 1 advertises in A and, then, firm 2 advertises in B

Normal form of the game

A if A, A if B A if A, B if B B if A, A if B B if A, B if B

A

8, 4 8, 4 12, 6 12, 6

B 6, 12 4, 2 6, 12 4, 2

Advertising in A is a stritcly dominant strategy for firm 1 Firm 1 advertises in A Firm 2 plays “B if A, A if B” or “B if A, B if B”; it advertises in B

Firm 1

Firm 2

A rollback equilibrium is a Nash equilibrium

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Market entry game

• Entry of firm 1 in a market where firm 2 is already established

• If firm 1 decides not to enter the market, then firm 1 has a payoff of 0 and firm 2 has a payoff of 2

• If firm 1 decides to enter the market, then

– if firm 2 cooperates, then both firms have a payoff of 1

– if firm 1 retaliates, then both firms have a payoff of -1

Extensive form of the game

Stay out

Enter

Retaliate

Cooperate

Firm 2

Firm 1

0, 2

-1, -1

1, 1

Rollback equilibrium: firm 2 enters and, then, firm 2 cooperates

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Normal form of the game

Retaliate if enter Cooperate if enter

Stay out

0, 2 0, 2

Enter -1, -1 1, 1

Firm 1

Firm 2

The rollback equilibrium (Enter, Cooperate if enter) is a Nash equilibrium

(Stay out, Retaliate if enter) is also a Nash equilibrium

Airbus, Boeing, EU, and US • Airbus and Boeing

– it costs Airbus 1000 ME to enter the markets

– Boeing is already in the markets

– monopoly in a market yields profits of 900 ME to the airline

– competition in a market yields a profit of 300 ME to the airline

• EU and US – EU and the US profits whatever the respective airlines profit plus

a 700 ME bonus if there is competition in the respective market

• Ordering of moves – EU decides whether or not to pass protective legislation (PL)

– US then decides whether or not it passes PL

– finally, airbus decides whether or not to enter the markets

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Computing payoffs: PL in both markets, Airbus builds

Airbus Boeing EU US

EU market 900 0 900 0

US market 0 900 0 900

entry -1000 0 0 0

competition 0 0 0 0

Payoffs -100 900 900 900

Computing payoffs: PL in EU only, Airbus builds

Airbus Boeing EU US

EU market 900 0 900 0

US market 300 300 300 300

entry -1000 0 0 0

competition 0 0 0 700

Payoffs 200 300 1200 1000

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Computing payoffs: No PL, Airbus builds

Airbus Boeing EU US

EU market 300 300 300 300

US market 300 300 300 300

entry -1000 0 0 0

competition 0 0 700 700

Payoffs -400 600 1300 1300

Computing payoffs: Airbus does not build

Airbus Boeing EU US

EU market 0 900 0 900

US market 0 900 0 900

entry 0 0 0 0

competition 0 0 0 700

Payoffs 0 1800 0 1800

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Extensive form of the game

PL

no PL

PL

no PL EU

US

Airbus

Airbus

build

not build

not build

build

not build

build

0, 1800, 0

0, 1800, 0

0, 1800, 0

900, 900, -100

1200, 1000, 200

1300, 1300, -400

Rollback equilibrium

PL

no PL

PL

no PL EU

US

Airbus

Airbus

build

not build

not build

build

not build

build

0, 1800, 0

0, 1800, 0

0, 1800, 0

900, 900, -100

1200, 1000, 200

1300, 1300, -400

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Airbus moves before US

PL

no PL

PL

EU

US

Airbus

Airbus

not build

build

not build

build

0, 1800, 0

0, 1800, 0

900, 900, -100

1200, 1000, 200

1300, 1300, -400

PL

no PL

no PL

0, 1800, 0

Airbus moves first of all

PL

EU

US

Airbus

not build

build

0, 1800, 0

900, 900, -100

1200, 1000, 200 no PL

PL

no PL 1300, 1300, -400

No first mover advantage

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Mixed strategies

• Strategies are randomized over the pure strategies

• Payoffs become average payoffs

• Concept of best response still holds; best set of probabilities chosen by a player given the sets of probabilities chosen by the other players

• It can be shown that a Nash equilibrium always exists

Matching cents mixed strategies

• You and your friend – If coins match, you loose your cent to your friend

– If coins do not match, your friend looses her cent to you

– You choose Heads with probability p

– Your friend chooses Heads with probability q

• Payoffs – If you play Heads, your average payoff is (1 – 2q)

– If you play Tails, your average payoff is (- 1 + 2q)

– Analogous reasoning for your friend

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Matching cents: mixed Nash equilibrium

• There are no pure strategies that yield a Nash equilibrium

• What is your best response to strategy q by your friend? – If (1 – 2q) > (-1 + 2q) you should play Heads

– If (1 – 2q) < (-1 + 2q) you should play Tails

– However, pure strategies do not have a Nash equilibrium

• For a Nash equilibrium we must have (1 – 2q)=(-1 + 2q), so that q = ½ (also p = ½)

Mixed Nash equilibrium

Expected payo® of player 1 is

PE1 (p; q) = p(qP1(S1; S2) + (1¡ q)P1(S1; ¹S2))

+ (1¡ p)(qP1( ¹S1; S2) + (1¡ q)P1( ¹S1; ¹S2))

Equilibrium probability q for player 2

qP1(S1; S2) + (1¡ q)P1(S1; ¹S2) = qP1( ¹S1; S2) + (1¡ q)P1( ¹S1; ¹S2)

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Penalty-Kick

• Kicker and goalie – Kicker can shoot left or right – Goalie can dive left or right

• Payoffs – fraction of scores when kicker shoots left and goalie dives

left is 0.58 – fraction of scores when kicker shoots left and goalie dives

right is 0.95 – fraction of scores when kicker shoots right and goalie dives

left is 0.90 – fraction of scores when kicker shoots right and goalie dives

right is 0.70

Penalty-kick

L (q)

R (1-q)

L (p)

0.58, -0.58 0.95, -0.95

R (1-p)

0.93, -0.93 0.70, -0.70

Goalie

• No pure Nash equilibrium • Mixed Nash equilibrium for q = 0.42 and p = 0.39

Kicker

p is the probability of the kicker shooting left and q the probability goalie diving left

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Unbalanced coordination game

PowerPoint (q)

Keynote (1-q)

PowerPoint (p)

1, 1 0, 0

Keynote (1-p)

0, 0 2, 2

You

Your partner

Two pure Nash equilibria all the same

One mixed Nash equilibrium at q = p = 2/3