gt brno

An introduction to simple Game Theory

based on ”Thinking Strategically”

by Dixit and Nalebuff

by Kjetil K. Haugen

Molde University College

Servicebox 8, N-6405 Molde, Norway

E-mail: [email protected]

These overheads are meant to be a simple aidfor students in Brno who follow the course

”Game Theory and soccer” held in Brno, may 2005

1

GT – An optimization view (1)

An optimization problem:

max{f(x1)|x1 ∈ S1

}(1)

where x1 =[x11, . . . , x1

n

]is a vector of decision

variables, f() is the objective and x1 ∈ S1 is a

set of constraints

2


An m-player game:

max{fj(x

1, . . . , xm)|x1 ∈ S1, . . . xm ∈ Sm}

(2)

where xj =[x

j1, . . . , x

jn

]is a vector of deci-

sion variables for player j, fj() is the objective

(Pay-off function) for player j and xj ∈ Sj is a

set of constraints for player j, j ∈ {1, . . . , m}.

3


Consequences:

i) Parametric optimization problems. That

is, each player must solve an optimization

problem conditional on the other players

decision variables.

ii) Solving these problems returns Best Re-

ply functions.

iii) Rj(x1, . . . , xj−1, xj+1, . . . xm) solve (2) ∀j.

iv) What next? Nash Equilibrium (J.F Nash

1951) – any ”intersection” of all Rj’s.

4

An intuitive approach

The Newspaper front page example:

2 Newspapers NP1 and NP2 must decide onwhich front page to choose the next day.Each can choose (simultaneously) between 2possible (equal) front page stories:

1) Budget problems (BP)

2) Revolutionary new cure for AIDS (AIDS)

We assume further that if one the newspapers(alone) choose AIDS they get 70% of themarket while a choice of BP yields 30% ofthe market.

If both papers choose same story, the marketsplits in 2 – assumption of ”equal” newspa-pers.

5

A Normal form game

A ”game matrix” or a game in Normal form

6

Best replies

Assumption of rational (optimizing) behavior:

Player NP1:

RNP1(AIDS) = max{35,30} = 35 ⇒ AIDS

RNP1(BP ) = max{70,15} = 70 ⇒ AIDS

Player NP2:


RNP2(BP ) = max{70,15} = 70 ⇒ AIDS

7

Dominant Strategies

Both players choose the AIDS strategy what-

ever their opponent do, hence the solution

(equilibrium) is straightforward – both choose

the AIDS front page.

This is refererred to as a bf Dominant Strat-

egy Equilibrium

8

Dominant Strategy for one player

Assume now that one of the papers is larger

than the other; NP1 is read by 60% of the

market and NP2 is read by 40% of the market

if they have the same front page. No change

with different front pages.

9

Solution still straightforward

Player NP1:


RNP1(BP ) = max{70,18} = 70 ⇒ AIDS

Player NP2:

RNP2(AIDS) = max{28,30} = 30 ⇒ BP

RNP2(BP ) = max{70,12} = 70 ⇒ AIDS

As player NP1 chooses the AIDS strategy nomatter what player NP2 does, Player NP2 maysimpy optimize (solve a decision problem) andobtain BP as his equilibrium strategy.

Consequently, One dominant strategy (in a 2player game) is enough.

10

Dominated Strategies (1)

Zero-sum game:

Player 2’s pay-off = (−1)· Player 1’s pay-off

11


Player 2’s pay-offs for strategies t2 and t3:

As −7 > −15 and −8 > −10, t2 � t3 and

strategy t3 may be removed.

12


Player 2’s payoffs for remaining game:

As 9 > 3 and 8 > 7, s2 � s1 and strategy s1may be removed.

13


Remaining game:

Consequently, as −8 > −9, Player 2 chooses

t2 while Player 1 (”must”) choose s2 and the

equilibrium soultion yields 8 for Player 1 and

-8 for Player 2, simply obtained by succsessive

elimination of dominated strategies.

14

Nash Equilibrium (1)

No Dominant/Dominated strategies:

Stragey combination (s1, t1); Both players op-

timize ”at the same time”; Nash equilibrium

at points where best replies intersect.

15


Continuous Strategy Space – Cournot model:

Given 2 producers facing a market demandp = M − q, q = q1 + q2 and c(q) = cq, p isprice, q is market sales, q1, q2 production byproducers 1 and 2 respectively and c is unitproduction cost.

Assuming profit maximization, Pay-off func-tions for each player:

Pi(q1, q2) = (p − c) qi = (M − c − q1 − q2) qi(3)

Best reply functions:

∂Pi(q1, q2)

∂qi= 0 ⇒ M − c − qj − 2qi = 0

⇒ Ri(qj) = q∗i (qj) =1

2

(M − c − qj

)(4)

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Cournot model – graphically:

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Sequential games

Game trees and Zermello’s algorithm:

Assume that Player 1 chooses to make his

decision before Player 2.

Zermello’s algorithm: ”Dynamic Program-

ming” (Backward recursion) on the game

tree. The Nash equilibrium changes.

18

Mixed Strategies (1)

Assume that all players allow randomized

strategies ie. Player i choses strategy j with

probability pji and

∑j p

ji = 1

No Nash equilibria in ”pure strategies”

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Using expected Pay-off yields:

P1(p, q) = 1 − p − q + 2pq (5)

P2(p, q) = p + q − 2pq

∂P1(p, q)

∂p= −1 + 2q (6)

∂P2(p, q)

∂q= 1 − 2p

R1(q) =

⎧⎪⎪⎨⎪⎪⎩

p∗ = 1 q > 12

p∗ ∈ [0,1] q = 12

p∗ = 0 q < 12

R2(p) =

⎧⎪⎪⎨⎪⎪⎩

q∗ = 0 p > 12

q∗ ∈ [0,1] p = 12

q∗ = 1 p < 12

20


Or graphically:

Theorem: (Nash 1951) Every finite game

has at least one Nash equilibrium if mixed

strategies are allowed.

21

gt brno

Economy & Finance