# 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

Save this PDF as:

Size: px
Start display at page:

Download "6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium"

## Transcription

1 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18,

2 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed Strategy Nash Equilibrium in Finite Games Existence in Games with Infinite Strategy Spaces Reading: Fudenberg and Tirole, Chapter 1. 2

3 Example Introduction In this lecture, we study the question of existence of a Nash equilibrium in both games with finite and infinite pure strategy spaces. We start with an example, pricing-congestion game, where players have infinitely many pure strategies. We consider two instances of this game, one of which has a unique pure Nash equilibrium, and the other does not have any pure Nash equilibria. 3

4 Example Pricing-Congestion Game Consider a price competition model studied in [Acemoglu and Ozdaglar 07]. 1 unit of traffic Reservation utility R Consider a parallel link network with I links. Assume that d units of flow is to be routed through this network. We assume that this flow is the aggregate flow of many infinitesimal users. Let l i (x i ) denote the latency function of link i, which represents the delay or congestion costs as a function of the total flow x i on link i. Assume that the links are owned by independent providers. Provider i sets a price p i per unit of flow on link i. The effective cost of using link i is p i + l i (x i ). Users have a reservation utility equal to R, i.e., if p i + l i (x i ) > R, then no traffic will be routed on link i. 4

5 Example 1 Game Theory: Lecture 5 Example We consider an example with two links and latency functions l 1 (x 1 ) = 0 and l 2 (x 2 ) = 3 x 2 2. For simplicity, we assume that R = 1 and d = 1. Given the prices (p 1, p 2 ), we assume that the flow is allocated according to Wardrop equilibrium, i.e., the flows are routed along minimum effective cost paths and the effective cost cannot exceed the reservation utility. Definition A flow vector x = [x i ] i=1,...,i is a Wardrop equilibrium if I i=1 x i d and p i + l i (x i ) = min{p j + l j (x j )}, for all i with x i > 0, j p i + l i (x i ) R, for all i with x i > 0, with I = d if min j {p j + l j (x j )} < R. i=1 x i 5

6 Example 1 (Continued) Example We use the preceding characterization to determine the flow allocation on each link given prices 0 p 1, p 2 1: x 2 (p 1, p 2 ) = { 2 3 (p 1 p 2 ), 0, p 1 p 2, otherwise, and x 1 (p 1, p 2 ) = 1 x 2 (p 1, p 2 ). The payoffs for the providers are given by: u 1 (p 1, p 2 ) = p 1 x 1 (p 1, p 2 ) u 2 (p 1, p 2 ) = p 2 x 2 (p 1, p 2 ) We find the pure strategy Nash equilibria of this game by characterizing the best response correspondences, B i (p i ) for each player. The following analysis assumes that at the Nash equilibria (p 1, p 2 ) of the game, the corresponding Wardrop equilibria x satisfies x 1 > 0, x 2 > 0, and x 1 + x 2 = 1. For the proofs of these statements, see [Acemoglu and Ozdaglar 07]. 6

7 Example 1 (Continued) Example In particular, for a given p 2, B 1 (p 2 ) is the optimal solution set of the following optimization problem maximize 0 p1 1, 0 x 1 1 p 1 x 1 3 subject to p 1 = p 2 + (1 x 1 ) 2 Solving the preceding optimization problem, we find that { } B 1 (p 2 ) = min 1, 3 + p Similarly, B 2 (p 1 ) = p

8 Example 1 (Continued) Example 1 Best Response Functions p p 1 The figure illustrates the best response correspondences as a function of p 1 and p 2. The correspondences intersect at the unique point (p 1, p 2 ) = (1, 1 2 ), which is the unique pure strategy equilibrium. 8

9 Example 2 Game Theory: Lecture 5 Example We next consider a similar example with latency functions given by { 0 if 0 x 1/2 l 1 (x) = 0, l 2 (x) = x 1/2 x 1/2, for some sufficiently small ɛ > 0. The following list considers all candidate Nash equilibria (p 1, p 2 ) and profitable unilateral deviations for ɛ sufficiently small, thus establishing the nonexistence of a pure strategy Nash equilibrium: 1 p 1 = p 2 = 0: A small increase in the price of provider 1 will generate positive profits, thus provider 1 has an incentive to deviate. 2 p 1 = p 2 > 0: Let x be the corresponding flow allocation. If x 1 = 1, then provider 2 has an incentive to decrease its price. If x 1 < 1, then provider 1 has an incentive to decrease its price. 3 0 p 1 < p 2 : Player 1 has an incentive to increase its price since its flow allocation remains the same. 4 0 p 2 < p 1 : For ɛ sufficiently small, the profit function of player 2, given p 1, is strictly increasing as a function of p 2, showing that provider 2 has an incentive to increase its price. 9 ɛ

10 We start by analyzing existence of a Nash equilibrium in finite (strategic form) games, i.e., games with finite strategy sets. Theorem (Nash) Every finite game has a mixed strategy Nash equilibrium. Implication: matching pennies game necessarily has a mixed strategy equilibrium. Why is this important? Without knowing the existence of an equilibrium, it is difficult (perhaps meaningless) to try to understand its properties. Armed with this theorem, we also know that every finite game has an equilibrium, and thus we can simply try to locate the equilibria. 10

11 Approach Game Theory: Lecture 5 Recall that a mixed strategy profile σ is a NE if u i (σ i, σ i ) u i (σ i, σ i ), for all σ i Σ i. In other words, σ is a NE if and only if σi B (σ ) for all i, (σ i i where B i i ) is the best response of player i, given that the other players strategies are σ i. We define the correspondence B : Σ Σ such that for all σ Σ, we have B(σ) = [B i (σ i )] i I (1) The existence of a Nash equilibrium is then equivalent to the existence of a mixed strategy σ such that σ B(σ): i.e., existence of a fixed point of the mapping B. We will establish existence of a Nash equilibrium in finite games using a fixed point theorem. 11

12 Definitions Game Theory: Lecture 5 A set in a Euclidean space is compact if and only if it is bounded and closed. A set S is convex if for any x, y S and any λ [0, 1], λx + (1 λ)y S. convex set not a convex set 12

13 Weierstrass s Theorem Theorem (Weierstrass) Let A be a nonempty compact subset of a finite dimensional Euclidean space and let f : A R be a continuous function. Then there exists an optimal solution to the optimization problem minimize f (x) subject to x A. There exists no optimal that attains it 13

14 Kakutani s Fixed Point Theorem Theorem (Kakutani) Let A be a non-empty subset of a finite dimensional Euclidean space. Let f : A A be a correspondence, with x A f (x) A, satisfying the following conditions: A is a compact and convex set. f (x) is non-empty for all x A. f (x) is a convex-valued correspondence: for all x A, f (x) is a convex set. f (x) has a closed graph: that is, if {x, y } {x, y } with y n f (x n ), then y f (x). Then, f has a fixed point, that is, there exists some x A, such that x f (x). n n 14

15 Kakutani s Fixed Point Theorem Graphical Illustration is not convex-valued does not have a closed graph 15

16 Proof of Nash s Theorem 1 2 The idea is to apply Kakutani s theorem to the best response correspondence B : Σ Σ. We show that B(σ) satisfies the conditions of Kakutani s theorem. Σ is compact, convex, and non-empty. By definition Σ = Σ i i I where each Σ i = {x j x j = 1} is a simplex of dimension S i 1, thus each Σ i is closed and bounded, and thus compact. Their product set is also compact. B(σ) is non-empty. By definition, B i (σ i ) = arg max u i (x, σ i ) x Σ i where Σ i is non-empty and compact, and u i is linear in x. Hence, u i is continuous, and by Weirstrass s theorem B(σ) is non-empty. 16

17 Proof (continued) Game Theory: Lecture 5 3. B(σ) is a convex-valued correspondence. Equivalently, B(σ) Σ is convex if and only if B i (σ i ) is convex for all i. Let σ i, σ i B i (σ i ). Then, for all λ [0, 1] B i (σ i ), we have u i (σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i, u i (σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i. The preceding relations imply that for all λ [0, 1], we have λu i (σ i, σ i ) + (1 λ)u i (σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i. By the linearity of u i, u i (λσ i + (1 λ)σ i, σ i ) u i (τ i, σ i ) for all τ i Σ i. Therefore, λσ i + (1 λ)σ i B i (σ i ), showing that B(σ) is convex-valued. 17

18 Proof (continued) Game Theory: Lecture 5 4. B(σ) has a closed graph. Suppose to obtain a contradiction, that B(σ) does not have a closed graph. Then, there exists a sequence (σ n, σˆ n) (σ, σˆ ) with σˆ n B(σ n ), but σˆ / B(σ), i.e., there exists some i such that σˆ i / B i (σ i ). This implies that there exists some σ i Σ i and some ɛ > 0 such that u i (σ i, σ i ) > u i (σˆ i, σ i ) + 3ɛ. By the continuity of u i and the fact that σ n i σ i, we have for sufficiently large n, u i (σ i, σn i ) u i (σ i, σ i ) ɛ. 18

19 Proof (continued) Game Theory: Lecture 5 [step 4 continued] Combining the preceding two relations, we obtain u i (σ i, σ n i ) > u i (σˆ i, σ i ) + 2ɛ u i (σˆ in, σ n i ) + ɛ, where the second relation follows from the continuity of u i. This contradicts the assumption that σˆ n i B i (σ n i ), and completes the proof. The existence of the fixed point then follows from Kakutani s theorem. If σ B (σ ), then by definition σ is a mixed strategy equilibrium. 19

20 Existence of Equilibria for Infinite Games A similar theorem to Nash s existence theorem applies for pure strategy existence in infinite games. Theorem (Debreu, Glicksberg, Fan) Consider a strategic form game I, (S i ) i I, (u i ) i I such that for each i I S i is compact and convex; u i (s i, s i ) is continuous in s i ; u i (s i, s i ) is continuous and concave in s i [in fact quasi-concavity suffices]. Then a pure strategy Nash equilibrium exists. 20

21 Definitions Game Theory: Lecture 5 Suppose S is a convex set. Then a function f : S R is concave if for any x, y S and any λ [0, 1], we have f (λx + (1 λ)y ) λf (x) + (1 λ)f (y). concave function not a concave function 21

22 Proof Game Theory: Lecture 5 Now define the best response correspondence for player i, B i : S i S i, { } B i (s i ) = s i S i u i (s i, s i ) u i (s i, s i ) for all s i S i. Thus restriction to pure strategies. Define the set of best response correspondences as B (s) = [B i (s i )] i I. and B : S S. 22

23 Proof (continued) We will again apply Kakutani s theorem to the best response correspondence B : S S by showing that B(s) satisfies the conditions of Kakutani s theorem. 1 2 S is compact, convex, and non-empty. By definition S = S i i I since each S i is compact [convex, nonempty] and finite product of compact [convex, nonempty] sets is compact [convex, nonempty]. B(s) is non-empty. By definition, B i (s i ) = arg max u i (s, s i ) s S i where S i is non-empty and compact, and u i is continuous in s by assumption. Then by Weirstrass s theorem B(s) is non-empty. 23

24 Proof (continued) Game Theory: Lecture 5 3. B(s) is a convex-valued correspondence. This follows from the fact that u i (s i, s i ) is concave [or quasi-concave] in s i. Suppose not, then there exists some i and some s i S i such that B i (s i ) arg max s Si u i (s, s i ) is not convex. This implies that there exists s i, s i S i such that s i, s i B i (s i ) and λs i + (1 λ)s i / B i (s i ). In other words, λu i (s i, s i ) + (1 λ)u i (s i, s i ) > u i (λs i + (1 λ) s i, s i ). But this violates the concavity of u i (s i, s i ) in s i [recall that for a concave function f (λx + (1 λ)y ) λf (x) + (1 λ)f (y )]. Therefore B(s) is convex-valued. 4. The proof that B(s) has a closed graph is identical to the previous proof. 24

25 Remarks Nash s theorem is a special case of this theorem: Strategy spaces are simplices and utilities are linear in (mixed) strategies, hence they are concave functions of (mixed) strategies. Continuity properties of the Nash equilibrium set : Consider strategic form games with finite pure strategy sets S i and utilities u i (s, λ), where u i is a continuous function of λ. Let G (λ) = I, (S i ), (u i (s, λ)) and let E (λ) denote the Nash correspondence that associates with each λ, the set of (mixed) Nash equilibria of G (λ). Proposition Assume that λ Λ, where Λ is a compact set. Then E (λ) has a closed graph. Proof similar to the proof of closedness of B(σ) in Nash s theorem. This does not imply continuity of the Nash equilibrium set E (λ)!! 25

26 Existence of Nash Equilibria Can we relax (quasi)concavity? Example: Consider the game where two players pick a location s 1, s 2 R 2 on the circle. The payoffs are u 1 (s 1, s 2 ) = u 2 (s 1, s 2 ) = d(s 1, s 2 ), where d(s 1, s 2 ) denotes the Euclidean distance between s 1, s 2 R 2. No pure Nash equilibrium. However, it can be shown that the strategy profile where both mix uniformly on the circle is a mixed Nash equilibrium. 26

27 A More Powerful Theorem Theorem (Glicksberg) Consider a strategic form game I, (S i ) i I, (u i ) i I such that for each i I 1 S i is a nonempty and compact metric space; 2 u i (s i, s i ) is continuous in s. Then a mixed strategy Nash equilibrium exists. With continuous strategy spaces, space of mixed strategies infinite dimensional! We will prove this theorem in the next lecture. 27

### 6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation

6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma

### 6.254 : Game Theory with Engineering Applications Lecture 14: Nash Bargaining Solution

6.254 : Game Theory with Engineering Applications Lecture 14: Nash Bargaining Solution Asu Ozdaglar MIT March 30, 2010 1 Introduction Outline Rubinstein Bargaining Model with Alternating Offers Nash Bargaining

### Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31

Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).

### 6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games

6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### GAME THEORY. Department of Economics, MIT, A normal (or strategic) form game is a triplet (N, S, U) with the following properties

14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Normal-Form Games A normal (or strategic) form game is a triplet (N, S, U) with the following properties N = {1, 2,..., n} is a finite set

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### Convexity in R N Main Notes 1

John Nachbar Washington University December 16, 2016 Convexity in R N Main Notes 1 1 Introduction. These notes establish basic versions of the Supporting Hyperplane Theorem (Theorem 5) and the Separating

### Lecture 8: Market Equilibria

Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecturer: Bruno Codenotti Lecture 8: Market Equilibria The market setting transcends the scenario of games. The decentralizing effect of

### MATHEMATICAL BACKGROUND

Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

### Lecture 5: Mixed strategies and expected payoffs

Lecture 5: Mixed strategies and expected payoffs As we have seen for example for the Matching pennies game or the Rock-Paper-scissor game, sometimes game have no Nash equilibrium. Actually we will see

### Weighted Congestion Games

Players have different weights. A weighted congestion game is a tuple Γ = (N, R, (Σ i ) i N, (d r ) r R, (w i ) i inn ) with N = {1,..., n}, set of players R = {1,..., m}, set of resources Σ i 2 R, strategy

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### Introduction to Game Theory Lecture Note 3: Mixed Strategies

Introduction to Game Theory Lecture Note 3: Mixed Strategies Haifeng Huang University of California, Merced Mixed strategies and von Neumann-Morgenstern preferences So far what we have considered are pure

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### 6.254 : Game Theory with Engineering Applications Lecture 10: Evolution and Learning in Games

6.254 : Game Theory with Engineering Applications Lecture 10: and Learning in Games Asu Ozdaglar MIT March 9, 2010 1 Introduction Outline Myopic and Rule of Thumb Behavior arily Stable Strategies Replicator

### Mathematical Economics - Part I - Optimization. Filomena Garcia. Fall Optimization. Filomena Garcia. Optimization. Existence of Solutions

of Mathematical Economics - Part I - Fall 2009 of 1 in R n Problems in Problems: Some of 2 3 4 5 6 7 in 8 Continuity: The Maximum Theorem 9 Supermodularity and Monotonicity of An optimization problem in

### Mila Stojaković 1. Finite pure exchange economy, Equilibrium,

Novi Sad J. Math. Vol. 35, No. 1, 2005, 103-112 FUZZY RANDOM VARIABLE IN MATHEMATICAL ECONOMICS Mila Stojaković 1 Abstract. This paper deals with a new concept introducing notion of fuzzy set to one mathematical

### 6.254 : Game Theory with Engineering Applications Lecture 1: Introduction

6.254 : Game Theory with Engineering Applications Lecture 1: Introduction Asu Ozdaglar MIT February 2, 2010 1 Introduction Optimization Theory: Optimize a single objective over a decision variable x R

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

### Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17

Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### Jeux finiment répétés avec signaux semi-standards

Jeux finiment répétés avec signaux semi-standards P. Contou-Carrère 1, T. Tomala 2 CEPN-LAGA, Université Paris 13 7 décembre 2010 1 Université Paris 1, Panthéon Sorbonne 2 HEC, Paris Introduction Repeated

### Math 230a HW3 Extra Credit

Math 230a HW3 Extra Credit Erik Lewis November 15, 2006 1 Extra Credit 16. Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = p q. Let E be the set of all p Q such that 2 < p

### Game Theory: Supermodular Games 1

Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution

### Graphical method. plane. (for max) and down (for min) until it touches the set of feasible solutions. Graphical method

The graphical method of solving linear programming problems can be applied to models with two decision variables. This method consists of two steps (see also the first lecture): 1 Draw the set of feasible

### ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

### Economics 200C, Spring 2012 Practice Midterm Answer Notes

Economics 200C, Spring 2012 Practice Midterm Answer Notes I tried to write questions that are in the same form, the same length, and the same difficulty as the actual exam questions I failed I think that

### Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that

### Prof. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy.

Homework 13 Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y. 13.1. Prove or give a counterexample. f preserves convergent sequences if {x n } n=1 is a convergent sequence in X then

### Convergence of Feller Processes

Chapter 15 Convergence of Feller Processes This chapter looks at the convergence of sequences of Feller processes to a iting process. Section 15.1 lays some ground work concerning weak convergence of processes

### Lecture 3: Convex Sets and Functions

EE 227A: Convex Optimization and Applications January 24, 2012 Lecture 3: Convex Sets and Functions Lecturer: Laurent El Ghaoui Reading assignment: Chapters 2 (except 2.6) and sections 3.1, 3.2, 3.3 of

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### MATH 304 Linear Algebra Lecture 16: Basis and dimension.

MATH 304 Linear Algebra Lecture 16: Basis and dimension. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis. Equivalently, a subset S V is a basis for

### AM 221: Advanced Optimization Spring Prof. Yaron Singer Lecture 8 February 24th, 2014

AM 221: Advanced Optimization Spring 2014 Prof. Yaron Singer Lecture 8 February 24th, 2014 1 Overview Last week we talked about the Simplex algorithm. Today we ll broaden the scope of the objectives we

### ORF 523 Lecture 8 Spring 2016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Tuesday, March 8, 2016

ORF 523 Lecture 8 Spring 2016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Tuesday, March 8, 2016 When in doubt on the accuracy of these notes, please cross check with the instructor s

### NET Institute*

NET Institute* www.netinst.org Working Paper #07-21 Revised January 2008 Competition and Contracting in Service Industries Ramesh Johari Stanford University Gabriel Weintraub Columbia Business School *

### Advanced Topics in Machine Learning (Part II)

Advanced Topics in Machine Learning (Part II) 3. Convexity and Optimisation February 6, 2009 Andreas Argyriou 1 Today s Plan Convex sets and functions Types of convex programs Algorithms Convex learning

### CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES

CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES DEFINITION A Banach space is a real normed linear space that is a complete metric space in the metric defined by its norm. A complex Banach space is a

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Preference and Utility

Preference and Utility Ichiro Obara UCLA October 2, 2012 Obara (UCLA) Preference and Utility October 2, 2012 1 / 20 Preference Relation Preference Relation Obara (UCLA) Preference and Utility October 2,

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

### Lecture V: Mixed Strategies

Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.3-1.3.A 1 The Advantage of Mixed Strategies Consider the following Rock-Paper-Scissors game: Note that

### UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory

UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.

### Limits and convergence.

Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

### 18.312: Algebraic Combinatorics Lionel Levine. Lecture 8

18.312: Algebraic Combinatorics Lionel Levine Lecture date: March 1, 2011 Lecture 8 Notes by: Christopher Policastro Remark: In the last lecture, someone asked whether all posets could be constructed from

### 14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

### Computational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2

Computational Learning Theory Spring Semester, 2003/4 Lecture 1: March 2 Lecturer: Yishay Mansour Scribe: Gur Yaari, Idan Szpektor 1.1 Introduction Several fields in computer science and economics are

### Persuasion by Cheap Talk - Online Appendix

Persuasion by Cheap Talk - Online Appendix By ARCHISHMAN CHAKRABORTY AND RICK HARBAUGH Online appendix to Persuasion by Cheap Talk, American Economic Review Our results in the main text concern the case

### First Welfare Theorem

First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

### Level Sets of Arbitrary Dimension Polynomials with Positive Coefficients and Real Exponents

Level Sets of Arbitrary Dimension Polynomials with Positive Coefficients and Real Exponents Spencer Greenberg April 20, 2006 Abstract In this paper we consider the set of positive points at which a polynomial

### Labor Economics, 14.661. Lecture 3: Education, Selection, and Signaling

Labor Economics, 14.661. Lecture 3: Education, Selection, and Signaling Daron Acemoglu MIT November 3, 2011. Daron Acemoglu (MIT) Education, Selection, and Signaling November 3, 2011. 1 / 31 Introduction

### 6.207/14.15: Networks Lecture 16: Cooperation and Trust in Networks

6.207/14.15: Networks Lecture 16: Cooperation and Trust in Networks Daron Acemoglu and Asu Ozdaglar MIT November 4, 2009 1 Introduction Outline The role of networks in cooperation A model of social norms

### Introduction to Linear Programming.

Chapter 1 Introduction to Linear Programming. This chapter introduces notations, terminologies and formulations of linear programming. Examples will be given to show how real-life problems can be modeled

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition

Span. Econ. Rev. 2, 65 71 (2000) c Springer-Verlag 2000 The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition Todd R. Kaplan, David Wettstein

### Lecture 12: Sequences. 4. If E X and x is a limit point of E then there is a sequence (x n ) in E converging to x.

Lecture 12: Sequences We now recall some basic properties of limits. Theorem 0.1 (Rudin, Theorem 3.2). Let (x n ) be a sequence in a metric space X. 1. (x n ) converges to x X if and only if every neighborhood

### Chapter 1. Sigma-Algebras. 1.1 Definition

Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is

### Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania

Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically

### Preference Relations and Choice Rules

Preference Relations and Choice Rules Econ 2100, Fall 2015 Lecture 1, 31 August Outline 1 Logistics 2 Binary Relations 1 Definition 2 Properties 3 Upper and Lower Contour Sets 3 Preferences 4 Choice Correspondences

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### HEX. 1. Introduction

HEX SP.268 SPRING 2011 1. Introduction The game of Hex was first invented in 1942 by Piet Hein, a Danish scientist, mathematician, writer, and poet. In 1948, John Nash at Princeton re-discovered the game,

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES Contents 1. Probabilistic experiments 2. Sample space 3. Discrete probability

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### . Introduction to Game Theory Lecture Note 3: Mixed Strategies. HUANG Haifeng University of California, Merced

.. Introduction to Game Theory Lecture Note 3: Mixed Strategies HUANG Haifeng University of California, Merced Mixed strategies and von Neumann-Morgenstern preferences So far what we have considered are

### 10.1 Integer Programming and LP relaxation

CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely

### On Stability Properties of Economic Solution Concepts

On Stability Properties of Economic Solution Concepts Richard J. Lipton Vangelis Markakis Aranyak Mehta Abstract In this note we investigate the stability of game theoretic and economic solution concepts

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Working Paper Series

RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos Hervés-Beloso, Emma Moreno- García and

### CHAPTER III - MARKOV CHAINS

CHAPTER III - MARKOV CHAINS JOSEPH G. CONLON 1. General Theory of Markov Chains We have already discussed the standard random walk on the integers Z. A Markov Chain can be viewed as a generalization of

### Non-Cooperative Game Theory

Non-Cooperative Having Fun with Strategic Games Ph.D. Candidate, Political Economy and Government TEDy Outline 1 What Is A Non-Cooperative Game? Motivating Examples Formal and Informal Definitions of Non-Cooperative

### Midsummer Examinations 2013

[EC 7089] Midsummer Examinations 2013 No. of Pages: 5 No. of Questions: 5 Subject [Economics] Title of Paper [EC 7089: GAME THEORY] Time Allowed [2 hours (TWO HOURS)] Instructions to candidates Answer

### NON-COOPERATIVE GAMES. Department of Economics, MIT, 1. Normal-Form Games

NON-COOPERATIVE GAMES MIHAI MANEA Department of Economics, MIT, 1. Normal-Form Games A normal (or strategic) form game is a triplet (N, S, U) with the following properties N = {1, 2,..., n} is a finite

### THE CARLO ALBERTO NOTEBOOKS

THE CARLO ALBERTO NOTEBOOKS Complete Monotone Quasiconcave Duality Simone Cerreia-Vioglio Fabio Maccheroni Massimo Marinacci Luigi Montrucchio Working Paper No. 80 November 2008 www.carloalberto.org Complete

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### General Equilibrium Theory: Examples

General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer - 1 consumer several producers and an example illustrating the limits of the partial equilibrium approach

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### The Real Numbers. Here we show one way to explicitly construct the real numbers R. First we need a definition.

The Real Numbers Here we show one way to explicitly construct the real numbers R. First we need a definition. Definitions/Notation: A sequence of rational numbers is a funtion f : N Q. Rather than write

### 4.9 Markov matrices. DEFINITION 4.3 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if. (i) a ij 0 for 1 i, j n;

49 Markov matrices DEFINITION 43 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if (i) a ij 0 for 1 i, j n; (ii) a ij = 1 for 1 i n Remark: (ii) is equivalent to AJ n

### Topology and Convergence by: Daniel Glasscock, May 2012

Topology and Convergence by: Daniel Glasscock, May 2012 These notes grew out of a talk I gave at The Ohio State University. The primary reference is [1]. A possible error in the proof of Theorem 1 in [1]

### Oligopoly: Cournot/Bertrand/Stackelberg

Outline Alternative Market Models Wirtschaftswissenschaften Humboldt Universität zu Berlin March 5, 2006 Outline 1 Introduction Introduction Alternative Market Models 2 Game, Reaction Functions, Solution

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### The Isolation Game: A Game of Distances

The Isolation Game: A Game of Distances Yingchao Zhao a,1 Wei Chen b Shang-hua Teng c,2 a Department of Computer Science, Tsinghua University b Microsoft Research Asia c Department of Computer Science,

### 6.207/14.15: Networks Lectures 19-21: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning

6.207/14.15: Networks Lectures 19-21: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning Daron Acemoglu and Asu Ozdaglar MIT November 23, 25 and 30, 2009 1 Introduction

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Random Union, Intersection and Membership

Random Union, Intersection and Membership Douglas Cenzer Department of Mathematics University of Florida June 2009 Logic, Computability and Randomness Luminy, Marseille, France Random Reals Introduction

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### Monotonic transformations: Cardinal Versus Ordinal Utility

Natalia Lazzati Mathematics for Economics (Part I) Note 1: Quasiconcave and Pseudoconcave Functions Note 1 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21). Monotonic transformations:

### SOLVING LINEAR SYSTEM OF INEQUALITIES WITH APPLICATION TO LINEAR PROGRAMS

SOLVING LINEAR SYSTEM OF INEQUALITIES WITH APPLICATION TO LINEAR PROGRAMS Hossein Arsham, University of Baltimore, (410) 837-5268, harsham@ubalt.edu Veena Adlakha, University of Baltimore, (410) 837-4969,

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative