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7/23/2019 Advanced Microeconomic
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Advanced Microeconomics
Harald Wiese
University of Leipzig
Winter term 2012/2013
Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 1 / 70
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What I like about economic theory and microeconomics
Not just talk, but theory with formal models (> sociology)
The core of neoclassical economics: Walras model of perfectcompetition (chapter XIX)
interdependence of markets andprices to bring markets into equilibrium
and also and beyond:decisions under uncertainty (chapters III and IV)detailed analysis of consumer behavior (chapters VI and VII)interactive decisions = game theory (chapters X to XIII)simple concepts like Pareto eciency with wide ramications (chapter
XIV)auction theory (chapters XVII and XVIII)adverse selection (chapter XXII) and hidden action (XXIII)
However: hard work ahead
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Technical stu
3 class meetings
Monday 09:15 - 10:45 in SR 13 (I 274)Wednesday 09:15 - 10:45 in SR 17 (I 374)Friday 09:15 - 10:45 in SR 13 (I 274)
mixing lectures and exercises
written exam 120 min. at two datesmidterm on 07.12.2012 (60 min.)nal in February (60 min.)
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Literature
Harald Wiese: Advanced microeconomics (which is more detailed
than the lecture)Andreu Mas-Colell, Michael D. Whinston, Jerry R. Green:Microeconomic Theory
Hugh Gravell, Ray Rees: Microeconomics
Georey Jehle, Philip Reny: Advanced Microeconomic TheorySamuel Bowles: Microeconomics Behavior, Institutions, andEvolution
Thomas Hnscheid: Schwester Helga
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Course overview
Part A. Basic decision and preference theory:
Decisions in strategic formDecisions in extensive form
Ordinal preference theory
Decisions under risk
Chapter IIDecisions
in strategic form
Decision theory Game theory
Chapter IIIDecisions
in extensive form
Chapter XGames
in strategic form
Chapter XIIGames
in extensive form
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Course overview
Part B. Household theory and theory of the rm:
The household optimumComparative statics and duality theory
Production theory
Cost minimization and prot maximization
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Course overview
Part C. Games and industrial organization:
Games in strategic formPrice and quantity competition
Games in extensive form
Repeated games
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Course overview
Part D. Bargaining theory and Pareto optimality:
Pareto optimality in microeconomicsCooperative game theory
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Course overview
Part E. Bayesian games and mechanism design:
Static Bayesian gamesThe revelation principle and mechanism design
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Course overview
Part F. Perfect competition and competition policy:
General equilibrium theory I: the main resultsGeneral equilibrium theory II: criticism and applications
Introduction to competition policy and regulation
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Course overview
Part G. Contracts and principal-agent theories:
Adverse selectionHidden action
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Part A. Basic decision and preference theory
1 Decisions in strategic (static) form
2 Decisions in extensive (dynamic) form3 Ordinal preference theory
4 Decisions under risk
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Decisions in strategic formOverview
1 Introduction
2 Sets, functions, and real numbers
3 Dominance and best responses
4 Mixed strategies and beliefs
5
Rationalizability
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Introduction
Decision in strategic form:
helps to ease into game theory;concerns one-time (once-and-for-all) decisions where payos dependon:
strategy;
state of the world.
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Introduction
Examplestate of the world
bad weather good weather
strategy
production
of umbrellas100 81
productionof sunshades
64 121
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IntroductionExample 1: production of umbrellas or sunshades
DenitionA decision situation in strategic form is a triple
= (S, W, u) ,
withS (decision makers strategy set),
W (set of states of the world), and
u : S
W
!R (payo function).
= (S, u : S! R) is called a decision situation in strategic form withoutuncertainty.
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I d i
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IntroductionExample 1: production of umbrellas or sunshades
In this example:
S = fumbrella, sunshadeg;W = fbad weather, good weatherg;payo function u given by
u(umbrella, bad weather) = 100,
u(umbrella, good weather) = 81,
u(sunshade, bad weather) = 64,
u(sunshade, good weather) = 121.
Note that:
decision maker can choose only one strategy from S;
only one state of the world from W can actually happen.
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I d i
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IntroductionExample 1: production of umbrellas or sunshades
DenitionDecision situation without uncertainty:
u(s, w1) = u(s, w2)
for all s2 S and all w1, w2 2 W.Trivial case: jWj = 1.
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I d i
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IntroductionExample 2: Newcombs problem
Assumptions:
2 boxes:
1000 e in box 1;0 e or 1 mio. e in box 2.
strategies: open box 2,
only, or both boxes;Higher Being put the money into the boxes depending on Hisprediction of your choice;
Higher Beings predictions can be correct (She knows the books youread) or wrong.
Problem
What would you do?
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I t d ti
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IntroductionExample 2: Newcombs problem, version 1
prediction:box 2, only
prediction:both boxes
you open
box 2, only
1 000 000 Euros 0 Euro
you openboth boxes
1 001 000 Euros 1 000 Euro
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I t d ti
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IntroductionExample 2: Newcombs problem, version 2
predictionis correct
predictionis wrong
you open
box 2, only
1 000 000 Euros 0 Euro
you openboth boxes
1 000 Euro 1 001 000 Euros
We come back to Newcombs problem later ...
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Introduction
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IntroductionExample 3: Cournot monopoly
Denition = (S, ), where:
S = [0,) (set of output decisions),
: S! R (payo function) dened by (s) = p(s) s C(s) withp : S! [0,) (inverse demand function);C : S! [0,) (cost function).
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Decisions in strategic form
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Decisions in strategic formSets, functions, and real numbers
1 Introduction
2 Sets, functions, and real numbers
3 Dominance and best responses
4 Mixed strategies and beliefs
5 Rationalizability
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Sets
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SetsSet, element, and subset
Denition (set and elements)Set any collection of elements that can be distinguished from eachother. Set can be empty: ?.
Denition (subset)
Let M be a nonempty set. A set N is called a subset of M (N M) ifevery element from N is contained in M. fg are used to indicate sets.M1 = M2 i M1 M2 and M2 M1.
ProblemTrue?
f1, 2g = f2, 1g = f1, 2, 2g
f1, 2, 3
g f1, 2
gHarald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 24 / 70
Sets
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SetsTuples
DenitionLet M be a nonempty set. A tuple on M is an ordered list of elementsfrom M. Elements can appear several times. A tuple consisting of nentries is called n-tuple. () are used to denote tuples.(a1 , ..., an ) = (b1, ..., bm ) if n = m and ai = bi for all i = 1, ..., n.
Problem
True?
(1, 2, 3) = (2, 1, 3)
(1, 2, 2) = (1, 2)
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Sets
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SetsCartesian product
DenitionLet M1 and M2 be nonempty sets. The Cartesian product of M1 and M2(M1 M2) is dened by
M1
M2 :=
f(m1, m2) : m1
2M1 , m2
2M2
g.
Example
In a decision situation in strategic form, SW is the set of tuples (s, w).
ProblemLet M := f1, 2, 3g and N := f2, 3g. Find MN and depict this set in atwo-dimensional gure where M is associated with the abscissa (x-axis)and N with the ordinate (y-axis).
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Functions
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FunctionsInjective and surjective functions
DenitionLet M and N be nonempty sets.
A function f : M! N associates with every m 2 M an element fromN, denoted by f (m) and called the value of f at m.
SetsM domain (of f)N range (of f)
f (M) := ff (m) : m 2 Mg =[
m2Mff (m)g image (of f).
Injective function if f (m) = f (m0) implies m = m0 for allm, m0 2 M.Surjective function if f (M) = N holds.
Bijective function both injective and surjective.
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Functions
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Functionsvalue versus image
mx
( )mf
M
( )Mf
f (M) := ff (m) : m 2 Mg =[
m
2M
ff (m)g
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Functions
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FunctionsExercise
ProblemLet M := f1, 2, 3g and N := fa, b, cg.Dene f : M! N by
f (1) = a,
f (2) = a and
f (3) = c.
Is f surjective or injective?
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Functions
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FunctionsTwo dierent sorts of arrows
1 f : M
!N (domain is left; range is right).
2 m 7! f (m) on the level of individual elements of M and N.
Example
A quadratic function may be written as
f : R ! R,x 7! x2 .
or in a shorter form: f : x
7!x2
or very short (and in an incorrect manner): f (x) = x2
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Functions
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FunctionsInverse function
DenitionLet f : M! N be an injective function. The function f1 : f (M) ! Mdened by
f1 (n) = m , f (m) = n
is called fs inverse function.
Problem
Let M := f1, 2g and N := f6, 7, 8, 9g. Dene f : M! N byf (m) = 10
2m. Dene f s inverse function.
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Cardinality
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y
Denition
Let M and N be nonempty sets and let f : M! N be a bijective function.=) M and N have the same cardinality (denoted by jMj = jNj).If a bijective function f : M! f1, 2, ..., ng exists, M is nite and containsn elements. Otherwise M is innite.
ProblemLet M := f1, 2, 3g and N := fa, b, cg. Show jMj = jNj.
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Real numbers
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Theorem
Theorem (cardinality)The setsN,Z, andQ are innite and we have
jNj = jZj = jQj .
However: jQj < jRjand even
jQj < jfx2 R : a x bgjfor any numbers a, b with a < b.
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Real numbers
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Intervals
[a, b] : = fx2 R : a x bg ,[a, b) : = fx2 R : a x< bg ,(a, b] : = fx2 R : a < x bg ,(a, b) : = fx2 R : a < x< bg ,
[a,) : = fx2 R : a xg and(, b] : = fx2 R : x bg .
Problem
Given the above denition for intervals, can you nd an alternativeexpression forR?
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Convex combination
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Denition
Let x and y be elements ofR. )kx+ (1 k) y, k2 [0, 1]
the convex combination (the linear combination) of x and y.
Problem
0 x+ (1 0) y =?
Problem
Where is 14 1 + 34 2, closer to 1 or closer to 2?
[a, b] = fx2 R : a x bg= fka + (1 k) b : k2 [0, 1]gHarald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 36 / 70
Decisions in strategic form:
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Dominance and best responses
1 Introduction
2 Sets, functions, and real numbers
3 Dominance and best responses
4 Mixed strategies and beliefs
5 Rationalizability
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Weak and strict dominance
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Denition
= (S, W, u) decision situation in strategic form.Strategy s2 S (weakly) dominates strategy s0 2 S i
u(s, w) u(s0, w) holds for all w2 W andu(s, w) > u(s0, w) is true for at least one w2 W.
Strategy s2 S strictly dominates strategy s0 2 S iu(s, w) > u(s0, w) holds for all w2 W.Dominant strategy a strategy that dominates every other strategy(weakly or strictly).
s0 is called (weakly) dominated or strictly dominated.
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Weak and strict dominance
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Newcombs problem, version 1
prediction:box 2, only
prediction:both boxes
you openbox 2, only 1 000 000 Euros 0 Euro
you openboth boxes
1 001 000 Euros R 1 000 Euro R
Which strategy is best
for the rst state of the world,
for the second state of the world?
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Weak and strict dominance
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Newcombs problem, version 2
predictionis correct
predictionis wrong
you openbox 2, only 1 000 000 Euros R 0 Euro
you openboth boxes
1 000 Euro 1 001 000 Euros R
Which strategy is best
for the rst state of the world,
for the second state of the world?
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Power set
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Denition
Let M be any set. The set of all subsets of M is called the power set of Mand is denoted by 2M.
Examples
M :=f
1, 2, 3g
has the power set
2M = f, f1g , f2g , f3g , f1, 2g , f1, 3g , f2, 3g , f1, 2, 3gg .
Note that the empty set also belongs to the power set of M !
Problem
Can you derive a general rule to calculate the number of elements of anypower set?
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arg max
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Example
Consider a rm that tries to maximize its prot by choosing the outputx optimally. The output x is taken from a set X (e.g. the interval [0,))and the prot is a real (Euro) number.
(x) 2 R : prot resulting from the output x,max
x (x) 2 R : maximal prot by choosing x optimally,
argmaxx
(x) X : set of outputs that lead to the maximal prot
Of course,
maxx (x) = (x) for all x from argmax
x (x) .
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arg max
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The following cases are possible:
argmaxx (x) contains several elements.argmaxx (x) contains just one element.
argmaxx (x) = .
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Best response
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Denition
= (S, W, u) .The function sR : W ! 2S a best-response function (a best response, abest answer) if sR is given by
sR (w) := arg maxs2
Su(s, w) .
Problem
Use best-response functions to characterize s as a dominant strategy.Hint: characterization means that you are to nd a statement that isequivalent to the denition.
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Decisions in strategic form:Mi d t t i d b li f
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Mixed strategies and beliefs
1 Introduction
2 Sets, functions, and real numbers
3 Dominance and best responses
4 Mixed strategies and beliefs
5 Rationalizability
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Probability distribution
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Denition
Probability distribution on M : prob : 2M
! [0, 1] withprob() = 0,
prob(A[B) = prob(A) + prob(B) for all A, B2 2M obeyingA\ B = andprob(M) = 1 (summing condition).
Subsets of M are also called events.
Problem
Probability for the eventsA : "the number of pips (spots) is 2",
B : "the number of pips is odd", and
A
[B
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Mixed strategy
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Denition
Let S be a nite strategy set. A mixed strategy
is a probabilitydistribution on S :
(s) 0 for all s2 S and s2S
(s) = 1 (summing condition)
(set of mixed strategies) > also stands for sum, as above
(s) = 1 for one s2 S > identify s with (s) > 0 and (s0) > 0 for s6= s0 > properly mixed strategy
Notation:
(s1) ,
(s2) , ...,
sjSj
.
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Mixed strategyDecision situation in strategic form with mixed strategies
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Decision situation in strategic form with mixed strategies
Denition
= (S, W, u)
Example
Choosing umbrella with probability 13
and sunshade with probability 23
is amixed strategy.
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Belief
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Denition
Let W be a set of states of the world. (set of probability distributions on W);
2 (belief);Notation: (w1) , ..., wjWj .
Example
Bad weather with probability 14 and good weather with probability34
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Extending the payo function
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So far,u : S
W
!R
Two extensions:
probability distribution on Wrather than a specic state of the world w2 W> lottery
mixed strategy on Srather than a specic strategy s2 S
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Extending the payo functionbeliefs and lotteries
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beliefs and lotteries
state of the worldbad weather, 14 good weather,
34
strategy
productionof umbrellas
100 81
productionof sunshades
64 121
For example,
Lumbrella =
100, 81;
1
4,
3
4
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Lottery and expected value
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Denition
A tupleL = [x; p] := [x1 , ..., x ; p1 , ..., p ]
is called a lottery where xj 2 R is the payo accruing with probability pjL (set of simple lotteries).` = 1
L is called a trivial lotteryidentify L = [x; 1] with x.
Denition
Assume a simple L = [x1 , ..., x ; p1, ..., p ] . Its expected value is denotedby E (L) and given by
E (L) =`
j=1
pjxj.
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Extending the payo functionmixed strategies
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g
Denition
The payo under and w is dened by
u(, w) := s2S
(s) u(s, w)
Problem
Calculate the expected payo in the umbrella-sunshade decision situationif the rm chooses umbrella with probability 13 and sunshade with
23 .
Dierentiate between w =bad weather and w ="good weather".
Thus, the payo for a mixed strategy is the mean of the payos forthe pure strategies.
How are best pure and best mixed strategies related?
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Best pure and best mixed strategies
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Lemma
Best pure and best mixed strategies are related by the following two claims:
Any mixed strategy that puts positive probabilities on best purestrategies, only, is a best strategy.
If a mixed strategy is a best strategy, every pure strategy with
positive probability is a best strategy.
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Mixing strategies and states of the worldExercise
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Problem
Consider again the umbrella-sunshade decision situation in which
the rm chooses umbrella with 13 and sunshade with probability23 and
the weather is bad with probability 14 and good with probability34 .
Calculate u
13 ,
23
,
14 ,
34
!
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Extending the payo functionsummary
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The payo function u : SW ! R has been extended:beliefs:
u : S! R(s, ) 7! u(s, ) =
w2W (w) u(s, w) = E (Ls)
mixed strategies
u : W ! R(, w) 7! u(, w) =
s2S(s) u(s, w)
both beliefs and mixed strategies
u : ! R(, ) 7! u(, ) =
s2S
w2W(s) (w) u(s, w)
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Best-response functions
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Denition
Given
= (S,
W,
u),
there are four best-response functions:
sR,W : W ! 2S, given by sR,W (w) := arg maxs2S
u(s, w) ,
R,W : W ! 2, given by R,W (w) := arg max
2u(, w) ,
sR,
: ! 2S
, given by sR,
() := arg maxs2S u(s, ) , and
R, : ! 2, given by R, () := arg max2
u(, )
sR or R instead of sR,W, etc. (if there is no danger of confusion)
Problem
Complete the sentence: 2 R,W (w) implies(s) = 0 for all ... .
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Best-response functionsTheorem
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Theorem
= (S, W, u) . We have
2 ands2sR,() (s) = 1 imply2 R, () and2 R, () implies s2 sR, () for all s2 S with (s) > 0.
These implications continue to hold for W and w rather than and.
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Best-response functions
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Example
w1 w2s1 4 1s2 1 2
Let := (w1) be the probability of w1. We have s1 2 sR, () in thecase of
4 + (1) 1 1 + (1) 2,i.e., if 14 holds.
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Best-response functions
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Best-reply function is given by R, : ! 2.For
6= 14 , there is exactly one best strategy,
= 0 ( = (0, 1) = s2) or = 1 ( = (1, 0) = s1).
= 14 implies that every pure strategy (> every mixed strategy) isbest.
R, () =
8 14[0, 1] , = 140, < 14
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Best-response functions
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4
1 1
1
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Best-response functionsExercise
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Problem
Sketch the best-reply function R, for
w1 w2s1 1 3
s2 2 1
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Decisions in strategic form:Rationalizability
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1 Introduction
2 Sets, functions, and real numbers3 Dominance and best responses
4 Mixed strategies and beliefs
5 Rationalizability
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Rationalizable strategies
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Example
w1 w2s1 4 4s2 1 5s3 5 1
Can a rational decider choose s1 although it is not a best response tow1 and not a best response to w2?
Yes: the rational decider may entertain the belief on W with (w1) = (w2) =
12 .
Given this belief, s1 is a perfectly reasonable strategy.
Problem
Show s1 2 sR,
12 ,
12
!
Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 64 / 70
Rationalizable strategies
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Denition
= (S
,
W,
u).
A mixed strategy 2 is called rationalizable with respect to W if aw2 W exists such that 2 R,W (w).Strategy 2 is called rationalizable with respect to if a belief
2 exists such that
2
R,
(
).
Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 65 / 70
Nobel price 2010
I 2010 th S i Rik b k P i i E i S i i M f
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In 2010, the Sveriges Riksbank Prize in Economic Sciences in Memory ofAlfred Nobel was awarded to the economists
1/3 Peter A. Diamond (Massachusetts Institute of Technology,Cambridge, MA),
1/3 Dale T. Mortensen (Northwestern University, Evanston, IL), and
1/3 Christopher A. Pissarides (London School of Economics and Political
Science)
for their analysis of markets with search frictions.
Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 66 / 70
Further exercises: Problem 1
(a) If strategy s 2 S strictly dominates strategy s 0 2 S and strategy s 0
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(a) If strategy s2 S strictly dominates strategy s0 2 S and strategy s0strictly dominates strategy s00 2 S, is it always true that strategy sstrictly dominates strategy s00?
(b) If strategy s2 S weakly dominates strategy s0 2 S and strategy s0weakly dominates strategy s00 2 S, is it always true that strategy sweakly dominates strategy s00?
Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 67 / 70
Further exercises: Problem 2
Prove the following assertions or give a counter example!
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Prove the following assertions or give a counter-example!
(a) If
2 is rationalizable with respect to W, then is rationalizable
with respect to .
(b) If s2 S is a weakly dominant strategy, then it is rationalizable withrespect to W.
(c) If s
2S is rationalizable with respect to W, then s is a weakly
dominant strategy.
Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 68 / 70
Further exercises: Problem 3 (without gure)
Consider the problem of a monopolist faced with the inverse demand
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Consider the problem of a monopolist faced with the inverse demandfunction p(q) = a b q, in which a can either be high, ah, or low, al.The monopolist produces with constant marginal and average cost c.Assume that ah > al > c and b> 0. Think of the monopolist as settingthe quantity, q, and not the price, p.
(a) Formulate this monopolists problem as a decision problem instrategic form. Determine sR,W!
(b) Assume ah = 6, al = 4, b = 2, c = 1 so that you obtain the plot given
in the gure. Show that any strategy q /2h
alc2b ,
ahc2b
iis dominated
by either sR,W
ah
or sR,W
al
. Show also that no strategy
q2 h
alc2b
,ahc
2b i dominates any other strategy q0 2 halc
2b,
ahc2b i .
(c) Determine all rationalizable strategies with respect to W.(d) Dicult: Determine all rationalizable strategies with respect to .
Hint: Show that the optimal output is a convex combination ofsR,W a
h
and sR,W
al
.Harald Wiese (University of Leipzig) Advanced Microeconomics Winter term 2012/2013 69 / 70
Further exercises: Problem 3 (gure)
4
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0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
quantity
profit
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