EC487 Advanced Microeconomics, Part I: Lecture 7 Leonardo Felli - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 7 Leonardo Felli 32L.LG.04 10 November, 2017

Nash Theorem Recall that we are focussing exclusively on finite games: A i is a finite set for every i ∈ N . Theorem (Nash Theorem) Every finite normal form game Γ Γ = { N ; A i , ∀ i ∈ N ; u i ( a ) , ∀ i ∈ N } has a mixed strategy Nash equilibrium. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 2 / 59

General Theorem We will prove the Nash Theorem as an immediate consequence of a somewhat more general/special existence theorem. Theorem Consider an N-players game Λ = { N ; D i , ∀ i ∈ N ; v i ( d ) , ∀ i ∈ N } where: ◮ D i is a compact, convex subset of an Euclidean space; ◮ v i ( d i , d − i ) is continuous and quasiconcave in d. Then Λ has a Nash equilibrium in pure strategies. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 3 / 59

Proof of General Theorem Recall that: Definition The function v i ( d i , d − i ) is quasiconcave if and only if the upper level set { d i | v i ( d i , d − i ) ≥ k } is a convex set for every scalar k . An alternative definition of quasiconcavity is: Definition The function v i ( d i , d − i ) is quasiconcave if and only if for every pair d i ∈ D i , ˆ d i ∈ D i and α ∈ [0 , 1] v i ( α d i + (1 − α ) ˆ d i , d − i ) ≥ min { v i ( d i , d − i ) , v i ( ˆ d i , d − i ) } . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 4 / 59

Proof of General Theorem (cont’d) The key tool we will use is. Theorem (Kakutani’s Fixed Point Theorem) Let X be a compact, convex and non-empty set in R n and F : X ⇒ X a correspondence that satisfies the following properties: ◮ non-empty; ◮ convex valued; ◮ upper-hemi-continuous. Then there exists a vector x ∗ ∈ X such that: x ∗ ∈ F ( x ∗ ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 5 / 59

Proof of General Theorem (cont’d) For each player i ≤ N define the best reply correspondence φ i as: d i ∈ D i | v i ( d i , d − i ) ≥ v i ( d ′ i , d − i ) , ∀ d ′ � � φ i ( d ) = i ∈ D i Notice that: ◮ φ i ( d ) is nonempty since D i is compact and v i ( d ) is continuous (Weierstrass Theorem). ◮ φ i ( d ) is convex-valued since D i is compact and v i ( d ) is quasiconcave in d i . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 6 / 59

Proof of General Theorem (cont’d) ◮ φ i ( d ) is upper-hemi-continuous: Consider the sequence { d h } where d h ∈ D 1 × · · · × D N and assume that d h → d Consider also the sequence { d h i } where d h i ∈ D i and assume that d h i → d i Assume that d h i ∈ φ i ( d h ) , ∀ h ≥ 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 7 / 59

Proof of General Theorem (cont’d) Notice that by definition of the correspondence φ i ( · ) for every ˆ d i ∈ D i we have − i ) ≥ v i ( ˆ v i ( d h i , d h d i , d h − i ) By continuity of v i ( d ) we then have v i ( d i , d − i ) ≥ v i ( ˆ d i , d − i ) In other words d i ∈ φ i ( d ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 8 / 59

Proof of General Theorem (cont’d) ◮ Now define the correspondence φ : D 1 × · · · × D N → D 1 × · · · × D N as φ ( d ) = φ 1 ( d ) × · · · × φ N ( d ) ◮ Notice then that φ ( d ) satisfies all the assumptions of Kakutani Fixed Point Theorem: The set D 1 × · · · × D N is a compact, convex and non-empty subset of an Euclidean space, since each D i is a compact, convex and non-empty subset of an euclidean space. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 9 / 59

Proof of General Theorem (cont’d) The correspondence φ ( d ) is non-empty, convex valued and upper-hemi-continuous since each φ i ( d ) is non-empty, convex valued and upper-hemi-continuous. ◮ Therefore there exists a fixed point d such that d ∈ φ ( d ) or there exists a strategy profile d such that each element of it is a best reply to the strategy profile itself. ◮ In other words, there exists a pure strategy Nash equilibrium of the game Λ. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 10 / 59

Proof of Nash Theorem Consider now the finite normal form game Γ: Γ = { N ; A i , ∀ i ∈ N ; u i ( a ) , ∀ i ∈ N } Its mixed extension is the game Γ ∆ : Γ ∆ = { N ; ∆( A i ) , ∀ i ∈ N ; U i ( σ ) , ∀ i ∈ N } where, if A i contains n strategies, ∆( A i ) is the ( n − 1)-dimensional simplex, σ i ∈ ∆( A i ) and � � � σ 1 ( a 1 ) · . . . · σ I ( a I ) U i ( σ ) = u i ( a 1 , . . . , a I ) a ∈ A Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 11 / 59

Proof of Nash Theorem (cont’d) Notice now that the mixed extension game Γ ∆ satisfies all the assumption of the general theorem considered above. ◮ ∆( A i ) is a compact, convex subset of an Euclidean space; ◮ U i ( σ ) is a continuous and quasiconcave (linear) function. Hence by the theorem above there exists a pure strategy Nash equilibrium of the game Γ ∆ that is a mixed strategy Nash equilibrium of the game Γ. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 12 / 59

A Voluntary Contribution Game Consider an economy in which each of N individuals is allocated a share of the endowment of the private good ω = ( ω 1 , . . . , ω N ) We assume that each individual voluntarily contributes an amount z i of his private good for the (collectively run) production of the public good:   N � z j y = g   j =1 We are going to take each individual’s contribution decision to be the one predicted by the Nash equilibrium of this symultaneous move contribution game . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 13 / 59

A Voluntary Contribution Game (cont’d) In other words, the amount of the private good z i that individual i contributes is the solution to the following best reply problem for i : U i ( x i , y ) max { x i , z i } x i + z i ≤ ω i s.t.   N � z j y = g   j =1 The first order conditions of this problem leads to the following marginal condition: ∂ U i /∂ y 1 ∂ U i /∂ x i = g ′ ( z ) , ∀ i = 1 , . . . , N Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 14 / 59

A Voluntary Contribution Game (cont’d) The Nash equilibrium of the voluntary contribution game is then defined by the following set of marginal conditions: ∂ U i /∂ y 1 ∀ i = 1 , . . . , N ∂ U i /∂ x i = g ′ ( z ) , Recall that the Pareto efficient allocation is the one defined by the Bowen-Lindahl-Samuelson condition: N ∂ U i /∂ y 1 � ∂ U i /∂ x i = g ′ ( z ) i =1 Clearly, the allocation generated by this contribution game is Pareto inefficient: agent i contributes up to the point where the marginal cost of the public good (MRT) is equal to i ’s marginal rate of substitution between y and z . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 15 / 59

A Voluntary Contribution Game (cont’d) To highlight the nature of the inefficiency consider the following parametric characterization: U i ( x i , y ) = x i + v ( y ) , v ′ ( · ) > 0 , v ′′ ( · ) < 0 , v (0) = 0 , v ′ (0) = + ∞ g ( z ) = z . Player i ’s best reply is then the solution to the problem: x i + v ( y ) max { x i , z i } x i + z i ≤ ω i s.t. N � z j y = j =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 16 / 59

A Voluntary Contribution Game (cont’d) Notice first that monotonicity of each player’s utility function implies x i + z i = ω i Player i ’s best reply is then the solution to the problem:   N ω i − z i + v � z j max   z i j =1 The first order conditions are then:   N v ′ �  = 1 z j  j =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 17 / 59

A Voluntary Contribution Game (cont’d) There exist potentially lots of Nash Equilibria of this game. Assume that ω i = ω and consider the symmetric Nash equilibrium such that z i = z for every i ≤ N . Each player Nash equilibrium contribution z ∗ is then such that: v ′ ( N z ∗ ) = 1 Consider now the symmetric Pareto efficient allocation in this environment. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 18 / 59

A Voluntary Contribution Game (cont’d) The general central planner problem, for λ i > 0 for all i ≤ N , is: N λ i [ x i + v ( y )] � max { x i , z i } i =1 N ( x i + z i ) ≤ N ω � s.t. i =1 N � z i y = i =1 or N � � N �� x i + v � � x i max λ i N ω − { x i } i =1 i =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 19 / 59

A Voluntary Contribution Game (cont’d) The first order conditions are, for every i ≤ N : N � N � � λ j v ′ � x i N ω − = λ i j =1 i =1 or � N � λ i � v ′ z i = � N j =1 λ j i =1 In the symmetric case case λ i = λ and z i = z ∗∗ the Pareto efficient allocation is such that: v ′ ( N z ∗∗ ) = 1 N of course in the case z ∗∗ is such that v ( N z ∗∗ ) − z ∗∗ ≥ 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 10 November, 2017 20 / 59