advanced microeconomics · advanced microeconomics extensive form games extensive form de–nition...

Advanced Microeconomics


ECON5200 - Fall 2014


Introduction

I What you have done:

- consumers maximize their utility subject to budget constraintsand firms maximize their profits given technology and marketprices;

- no strategic behavior.

I What we will do:

- in many interesting situations, agents’optimal behaviordepends on the other agents’behavior;

- strategic behavior.

I Game theory provides a language to analyze such strategicsituations;

I Countless number of examples! Auctions, Bargaining, Pricecompetition, Civil Conflicts. . .


Introduction

Road map

I Static Game:

1. With Complete Information (I);

2. With Incomplete Information (II).

I Dynamic Game:

1. With Complete Information (II-III);

2. With Incomplete Information (III).


Strategic Games with Complete Information

Strategic Game with Pure Strategies

I N players with i ∈ I ;

I s ∈ S ≡ ∏i=1,..,N

Si pure strategy profile, si ∈ Si ;

I ui (s) payoff;

I G ≡ 〈I , {Si}i , {ui (s)}i 〉 strategic formof finite game withpure strategy.



Strategic Game with Mixed Strategies

I σ ∈ ∆ (S) ≡ ∏i=1,..,N

∆ (Si ) mixed strategyprofile, σi ∈ ∆ (Si );

I ui (σ) = ∑s∈S

∏j=1,..,N

σj (sj ) ui (sj ) expected utility;

I G ≡ 〈I , {∆ (Si )}i , {ui (σ)}i 〉 strategic form of finite gamewith mixed strategy;

I Interpreting mixed strategies:

- as object of choice;

- as pure strategies of a perturbed game (see later in BayesianGames);

- as beliefs.



Equilibrium Concepts

I Nash Equilibrium⇒ it is assumed that each player holds thecorrect expectation about the other players’behavior and actrationally (steady state equilibrium notion);

I Rationalizability⇒ players’beliefs about each other’s actionsare not assumed to be correct, but are constrained byconsideration of rationality;

I Every Nash equilibrium is rationalizable.



Rationalizability

DefinitionIn G , si is rationalizableif there exists Zj ⊂ Sj for each j ∈ I suchthat:

1. si ∈ Zi ;

2. every sj ∈ Zj is a best response to some belief µj ∈ ∆ (Z−j ).

I Common knowledge of rationality;

I An action is rationalizable if and only if it can be rationalizedby an infinite sequence of actions and beliefs.



Example (1 - Rationalizability - See notes!)...



Strictly Dominance

Definitionsi is not strictly dominatedif it does not exist a strategy σi :

ui (σi , s−i ) > ui (si , s−i ) , ∀s−i ∈ S−i



Strictly Dominance

I A unique strictly dominant strategy equilibrium (D,D):

I It is Pareto dominated by (C ,C ). Does it really occur??



Iterative Elimination of Strictly Dominated Strategies

DefinitionSet S0 = S , then for any m > 0 si ∈ Smi iff there does not existany σi such that:

ui (σi , s−i ) > ui (si , s−i ) , ∀s−i ∈ Sm−1−i

DefinitionFor any player i , a strategy is said to be rationalizable if and only ifsi ∈ S∞

i ≡⋂m≥0

Smi .



Example (2 - Beauty Contest - See notes!)...



Iterated Weak Dominance

I There can be more that one answer for iterated weakdominance;

I Not for iterated strong dominance.



Example (3 - Cournot vs Bertrand Competition - Proposedas exercise)

Examplen profit-maximizer-firms produce qi quantity of consumption goodat a marginal cost equal to c > 0;

I demand function is P = max {1−Q, 0} with Q ∈ ∑i=1...n

qi ;

I Find:

1. The rationalizable equilibria when n = 2;

2. The rationalizable equilibria when n > 2;

3. Compare your results with the Bertrand competition outcome.



Nash Equilibrium

Definitionσi ∈ ∆ (Si ) is a best responseto σ−i ∈ ∆ (S−i ) if:

ui (σi , σ−i ) ≥ ui (si , σ−i ) for all si ∈ Si

Let Bi (σ−i ) ⊂ ∆ (Si ) be the set of player’i best response.

Definitionσ is a Nash equilibriumprofile if for each i ∈ I .

σi ∈ Bi (σ−i )



Nash Theorem

Theorem (Nash (1950))A Nash equilibrium exists in a finite game.

Theorem (Kakutani Fixed Point Theorem)Let X be a compact, convex and non-empty subset of Rn, acorrespondence f : X → X has a fixed point if:

1. f is non-empty for all x ∈ X;

2. f is convex for all x ∈ X;

3. f is upper hemi-continuous (closed graph).



Best Response Correspondence Example



The Kitty Genovese Problem/Bystander EffectI n identical people;

I x > 1 benefits if someone calls the police;

I 1 cost of calling the police;

What is the symmetric mixed strategy equilibriumwith p equalto the probability of calling the policy?

I In equilibrium each player must be indifferent between callingor not the police;

I If i calls the police, gets x − 1 for sure;I If i doesn’t, gets:

0 with Pr (1− p)n−1

x with Pr 1− (1− p)n−1



The Kitty Genovese Problem/Bystander EffectI Indifference when:

x − 1 = x(1− (1− p)n−1

)I Equilibrium symmetric mixed strategy is p = 1− (1/x)1/(n−1)

I http://en.wikipedia.org/wiki/Murder_of_Kitty_Genovese



Zero-Sum Game

DefinitionA N-player game G is a zero-sum game(a strictly competitivegame) if ∑

i=1,..,N

ui (s) = K for every s ∈ S .



Zero-Sum Game

Definitionσi ∈ ∆ (Si ) is maxminimizerfor player i if:

minσ−i∈∆(S−i )

ui (σi , σ−i ) ≥ minσ−i∈∆(S−i )

ui(σ′i , σ−i

)for each σi ∈ ∆ (Si )

A maxminimizer maximizes the payoff in the worst case scenario

TheoremLet G be a zero-sum game. Then σ ∈ ∆ (S) is a Nash Equilibriumiff, for each i , σ is a maxminimizer.



Example (4 - All-Pay Auction - Proposed as exercise)

Two players submit a bid for an object of worth k ;

I bi ∈ [0, k ] individual strategy space where bi is the bid;

I The winner is the player with the highest bid;

I If tie each player gets half the object, k/2;

I Each player pays her bid regardless of whether she wins;

I Find that:

1. No pure Nash equilibria exist;

2. The mixed strategy equilibrium is equal to the one representedhere below.



Example (4 - All-Pay Auction - Proposed as exercise)


Extensive Form Games

Representation of a Game

I Normal or strategic form;

I Extensive form.

The Extensive form contains all the information about a game:

I who moves when;

I what each player knows when he moves;

I what moves are available to him;

I where each move leads.

whereas a normal form is a ‘summary’representation.



Extensive Form

DefinitionA treeis a set of nodes and directed edges connecting these nodessuch that:

1. for each node, there is at most one incoming edge;

2. for any two nodes, there is a unique path that connect thesetwo nodes.

DefinitionAn extensive form game consists of i) a set of players (includingpossibly Nature), ii) a tree, iii) an information set for each player,iv) an informational partition, and v) payoffs for each player ateach end node (except Nature).



Extensive Form

DefinitionAn information setis a collection of points (nodes) such that:

1. the same player i is to move at each of these nodes;

2. the same moves are available at each of these nodes.

DefinitionAn information partitionis an allocation of each node of the tree(except the starting and end-nodes) to an information set.

DefinitionA (behavioral) strategyof a player is a complete contingent-plandetermining which action he will take at each information set he isto move.



Extensive Form vs Normal Form

advanced microeconomics · advanced microeconomics extensive form games extensive form de–nition...

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