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Lecture 1 Mathematical Preliminaries

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Lecture 1 Mathematical Preliminaries

A. Banerji July 26, 2016

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Outline

1

Preliminaries

Sets Logic Sets and Functions Linear Spaces

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Preliminaries

Sets

Basic Concepts

Take as understood the notion of a set. Usually use upper case letters for sets and lower case ones for their elements. Notation.

a ∈ A, b ∈/ A.

A ⊆ B if every element of A also belongs to B. A = B if A ⊆ B and B ⊆ A are both true.

A ∪ B = {x|x ∈ A or x ∈ B}. The ‘or’ is inclusive of ‘both’. A ∩ B = {x|x ∈ A and x ∈ B}. What if A and B have no common elements? To retain meaning, we invent the concept of an empty set, ∅.

A ∪ ∅ = A, A ∩ ∅ = ∅. For the latter, note that there’s no element in common between the sets A and ∅ because the latter does not have any elements.

∅ ⊆ A, for all A. Every element of ∅ belongs to A is vacuously true since ∅ has no elements. This brings us to some logic.

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Preliminaries

Logic

Logic

Statements or propositions must be either true or false. P ⇒ Q. If the statement P is true, then the statement Q is true. But if P is false, Q may be true or false. For example, on real numbers, let P = x > 0 and Q = x2 > 0. If P is true, so is Q, but Q may be true even if P is not, e.g. x = −2. We club these together and say P ⇒ Q.

Let P = x2 < 0 and Q = x = 5. Then P ⇒ Q is vacuously true. ∼ Q ⇒∼ P is the contrapositive of P ⇒ Q.

For example: If x3 ≤ 0, then x ≤ 0 is the contrapositive of “if x > 0, then x3 > 0 ".

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Preliminaries

Logic

Logic

Claim. A statement and its contrapositive are equivalent.

Proof.

Suppose P ⇒ Q is true. Suppose Q is false. Then P must be false, for if not, then via P ⇒ Q, Q would be true, contradicting our assumption that Q is false. On the other hand, suppose P ⇒ Q is false. It follows that P is true and Q is false (for if P is false, then P ⇒ Q is vacuously true). But then, ∼ Q ⇒∼ P cannot be true.

Definition

Q ⇒ P is called the converse of the statement P ⇒ Q. No relationship between a statement and its converse. e.g., if x > 0 then x2 > 0 is true, but its converse is not. On the other hand, if some statement and its converse are both true, we say P if and only if Q or

P ⇔ Q.

5 / 27

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Preliminaries

Logic

Quantifiers and Negation

2 logical quantifiers: ‘for all’ and ‘there exists’. P : For all a ∈ A, property Π(a) holds. The negation of P (i.e. ∼ P) is the statement: There exists at least one a ∈ A s.t. property Π(a) does not hold. e.g. P : For every x ∈ <, x2 > 0. The negation of P : There exists x ∈ < s.t. x2 ≤ 0.

Q : There exists b ∈ B s.t. property Θ(b) holds. ∼ Q : For all b ∈ B, property Θ(b) does not hold.

Note. Order of quantifiers matters. e.g. ‘for all x > 0, there exists

y > 0 s.t. y2 = x’, says that every positive real has a positive square root. This is not the same as ‘there exists y > 0 s.t. for all x > 0, y2 = x’, which says some number y is the common square root of every positive real.

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Preliminaries

Logic

Logic

Let Π(a, b) be a property defined on elements a and b in sets A and B respectively. Let P : For every a ∈ A there exists b ∈ B s.t. Π(a, b) holds. Then, ∼ P : There exists a ∈ A s.t. for all b ∈ B, Π(a, b) does not hold.

Necessary and Sufficient Conditions

Let P ⇒ Q be true. We say Q is necessary for P. Or P is sufficient for Q. So, if P ⇔ Q, P is necessary and sufficient for Q.

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Preliminaries

Sets and Functions

Sets

Set Difference: A − B = {x|x ∈ A and x ∈/ B}. Also called the complement of B relative to A. More familiar is the

notion of the universal set X, and X − B = Bc. Some set-theoretic ‘laws’

Distributive Laws

(i) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Proof.

(i) Suppose x ∈ LHS. So, x ∈ A and x ∈ (B ∪ C). So x ∈ A and either x ∈ B or x ∈ C or both. So, either x ∈ (A ∩ B) or x ∈ (A ∩ C) (or both). Converse?

DeMorgan’s Laws (i) A − (B ∪ C) = (A − B) ∩ (A − C) or more familiarly,

(B ∪ C)c = Bc ∩ Cc

(ii) A − (B ∩ C) = (A − B) ∪ (A − C) or (B ∩ C)c = Bc ∪ Cc.

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Preliminaries

Sets and Functions

Sets

Arbitrary Unions and Intersections

Let A be a collection of sets. Then

S

= {x|x ∈ A for at least one A ∈ A}

A∈A A∈A

T

= {x|x ∈ A for every A ∈ A}

Cartesian Products

We’ll say (a, b) is an ordered pair if the order of writing these 2 objects matters: i.e. if (a, b) and (b, a) are not the same thing. (Think of points on the plane). Alternatively, we can derive the notion of ordered pair from the more primitive notion of a set as follows. Define (a, b) = {{a}, {a, b}}. On the right is a set of 2 sets; the first of these is the singleton that we want to be ‘first’ in the ordered pair. The 2nd is the set of both objects (obviously, {a, b} = {b, a}, so that alone cannot be sufficient to define an ordered pair). Thus (b, a) is defined to be the set {{b}, {a, b}}. Let A and B be sets. We then have

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Preliminaries

Sets and Functions

Functions

Definition

The Cartesian product A × B = {(x, y)|x ∈ A and y ∈ B}. We can formally define a function using the notion of Cartesian product, as follows. Let C, D be 2 sets. A rule of assignment r is a subset of C × D s.t. elements of C appear as first coordinates of ordered pairs belonging to r at most once. The set A of elements of C appearing as first coordinates in r is called the domain of r. The set of elements of D comprising 2nd coordinates of r is called the image set of r. Then

Definition

A function f is a rule of assignment r along with a set B that contains the image set of r.

A is called the domain of f and the image set of r is called the image set or range of f.

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Preliminaries

Sets and Functions

Functions

We write f : A → B and think of f as a rule carrying every element a ∈ A to exactly one element b ∈ B.

Examples. 1. f : < → < defined by f(x) = x2, ∀x ∈ <.

2

2. Using the notation < for the Cartesian product < × < (representing

  • 2
  • 2

the plane), let f : < → < be defined by f(x1, x2) = x1x2, ∀(x1, x2) ∈ < .

  • 2
  • 2
  • 2

3. g : < → < defined by g(x1, x2) = (x1x2, x1 + x2), ∀(x1, x2) ∈ < .

4. Arbitrary Cartesian Products. We first formally define n-tuples in

terms of functions. Let X be a set and define the function x : {1, ..., n} → X. This function is called an n-tuple of elements of X. It’s image at i ∈ {1, ..., n}, x(i), is written as xi. The n-tuple is written as (x1, ..., xn). The order counts. We write Xn for the set of all n-tuples of

n

elements of X. The leading example is < , n-dimensional Euclidean space.

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Preliminaries

Sets and Functions

Functions

Definition

S

Let A1, ...An be a collection of sets, and let X = n Ai. The cartesian

1

product of this collection of sets, written as A1 × ... × An or Πni=1Ai, is the set of all n-tuples (x1, ..., xn) such that xi ∈ Ai, ∀i ∈ {1, ..., n}.

5. Let X be a set. A sequence or infinite sequence of elements of X

is a function x : Z++ → X. (Z++ is the set of positive integers). Sequences are written by collecting images in order. We write x = (x1, x2, ......). This definition generalizes the notion of infinite sequences of real numbers.

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Preliminaries

Sets and Functions

Functions - Images, Preimages

Let f : A → B.

Definition

Let A0 ⊆ A. The image of A0 under f, denoted f(A0), is the set

{b|b = f(a), for some a ∈ A0}.

Let B0 ⊆ B. The preimage of B0 under f, f−1(B0) = {a|f(a) ∈ B0}.

So the image of a set is the collection of images of all its elements, and the preimage or inverse image of a set is the collection of all elements in the domain that map into this set.

Example

For the function f(x) = x2, let A0 = [−2, 2]. Then, f(A0) = [0, 4]. Let B0 = [−2, 9]. Then, f−1(B0) = [−3, 3].

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Preliminaries

Sets and Functions

Functions - Injective, Surjective

Fact. Let f : A → B, A0, A1 ⊆ A, B0, B1 ⊆ B. Then (i)

B0 ⊆ B1 ⇒ f−1(B0) ⊆ f−1(B1).

(ii) f−1(B0 ∗ B1) = f−1(B0) ∗ f−1(B1), where ∗ can be ∪, ∩, −. i.e., f−1 preserves set inclusion, union, intersection and difference. f only preserves the first two of these. The 3rd holds with ⊆ and the 4th with ⊇. To find counterexamples, many-to-one functions (to which we now move) are helpful. Proofs of the above claims are homework.

Definition

f : A → B is injective (one-to-one) if [f(a) = f(a0 )] ⇒ [a = a0 ]. It is surjective (onto) if for every b ∈ B, there exists a ∈ A s.t. b = f(a). A function that is both of these is called bijective.

For example, f : < → < defined by f(x) = x2 is many-to-one and not surjective. If the domain is <+ instead, then f is injective, and further if the codomain is <+, then it is bijective.

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Preliminaries

Sets and Functions

Functions, Inverse

Fact. Let f : A → B, A0 ⊆ A, B0 ⊆ B. Then (i)f−1(f(A0)) ⊇ A0; equality holds if f is injective. (ii) f(f−1(B0)) ⊆ B0; equality holds if f is surjective. You should also show by examples that equality does not in general hold. Now we use the f−1 to mean something different, namely the inverse function. If f is bijective, then define a function f−1 : B → A by f−1(b) = a if a is the unique element of A s.t. f(a) = b. Note that f−1 is also bijective. Indeed, suppose b = b0 and f−1(b) = f−1(b0 ) = a. Then f(a) = b and f(a) = b0 which is not possible. So f−1 is injective. Moreover, for every a ∈ A, there is a b ∈ B s.t. b = f(a), since f is a function. So, there is a b ∈ B s.t. f−1(b) = a. So f−1 is also surjective; hence it is bijective.

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Preliminaries

Sets and Functions

Functions

One way to check whether f is bijective uses the following

Lemma

Let f : A → B. If there are 2 functions g, h from B to A s.t. g(f(a)) = a, ∀a ∈ A and f(h(b)) = b∀b ∈ B, then f is bijective and

  • g = h = f−1
  • .

Proof.

f injective. Suppose a = a0 and f(a) = f(a0 ) = b. Then g(f(a)) = g(b) = g(f(a0 )). So g(b) cannot be equal to both a and a0 . Contradiction. f is also surjective. Indeed, suppose there is a b ∈ B with no preimage under f. However, we require f(h(b)) = b. This implies h(b) is a preimage. Contradiction.

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Preliminaries

Sets and Functions

Simultaneous Equations

  • n
  • m
  • m

Let f : < → < . Fix y ∈ < . Then the equation f(x) = y represents a system of m simultaneous equations in n variables. This is clear since the equation can be rewritten as

f1(x1, ..., xn) = y1 . . . . . . . . . . . . . . . . . . fm(x1, ..., xn) = ym

where y = (y1, ..., ym), x = (x1, ..., xn),

n

f(x) = (f1(x), ..., fm(x)), ∀x ∈ < , where for each i ∈ {1, ..., m}, the

n

component function fi : < → <. An x which satisfies f(x) = y is called

a solution to the equation or system of equations. Note that whether the system of equations has a solution is the same question as

  • 2
  • 2

whether f−1({y}) is a nonempty set. Exercise. f : < → < ,

f(x1, x2) = (x1x2, x1 + x2). For what values of y in the codomain does the equation f(x) = y have a solution?

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Preliminaries

Sets and Functions

Simultaneous Equations - contour surfaces

One way to look at a solution: the intersection of (hyper)-surfaces. For example, in the exercise above, given y = (y1, y2), the equations

2

x1x2 = y1 and x1 + x2 = y2. These are 1-dimensional curves in < , and their intersection is the set of solutions. For the more general n-variable case, fi(x1, ..., xn) = yi describes an (n − 1)-dimensional surface, and the solution set is the intersection of the m such surfaces.

Definition

n

Let g : < → < and let y ∈ <. The contour set of g at y,

n

Cg(y) = {x ∈ < |g(x) = y}. The upper contour set Ug(y) = {x|g(x) ≥ y}. The lower contour set Lg(y) = {x|g(x) ≤ y}.

Observe that Cg(y) = Ug(y) ∩ Lg(y).

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Preliminaries

Sets and Functions

Simultaneous Equations in Economics

First order conditions in optimization problems; general equilibrium; Nash equilibrium in some problems. All these can be cast as solutions

  • n
  • n
  • n

to a system of equations F(x) = 0, where F : < → < and 0 ∈ < . General equilbrium is sometimes described as a fixed point of a

  • n
  • n

function. (i.e., if say f : < → < , x is a fixed point if it satisfies f(x) = x). But x is a fixed point of f if and only if it is a zero of F(x) ≡ f(x) − x, so the the question is really of finding the zeros of F i.e. solving F(x) = 0. Continuity of F is an important player in the existence of a solution. (Just as in 1-dimensional case: if f : < → <, is continuous, and f(x1) > 0 > f(x2), then the intermediate value theorem assures a solution x ∈ (x1, x2).) For the more general higher dimensional case, in computational economics methods like Gauss-Jacobi make use of 1-dimensional solutions to the n different equations in an iterative way to converge to a solution. (see Judd - Numerical Methods in Economics).

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Preliminaries

Sets and Functions

Relations

Recall that we can define a function as a subset of a Cartesian product C × D such that elements of C appear as first coordinates of ordered pairs at most once. Relations are more general in a way.

Definition

A relation ꢀ on a set X is a subset of X2, i.e. ꢀ⊆ X × X. Conventionally, if (x, y) ∈ꢀ, we write x ꢀ y. As in the case of defining functions, the idea of the definition is to not take anything more than the meaning of a set to be understood, and to successively define things in terms of it. (Set -> Cartesian Product -> Relation). But as in the case of a function, we think of relations in specific ways not directly related to the definition. For example, in consumer theory, if X is the consumption set and x, y ∈ X, we think of x ꢀ y directly as “x is preferred to y".

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  • Dominance-Based Solutions for Strategic Form Games

    Dominance-Based Solutions for Strategic Form Games

    Dominance-based Solutions for Strategic Form Games John Duggan Department of Political Science and Department of Economics University of Rochester Rochester, NY 14627 Michel Le Breton Greqam-Leqam Universit´ed’Aix Marseille 2 and Institut Universitaire de France Route des Milles, 13290, Les Milles France Original: November 20, 1996 Revision: April 24, 1998 Abstract We model a player’s decision as a choice set based on an abstract concept of dominance, called a dominance structure, and define a choice-theoretic no- tion of equilibrium. We investigate various properties of dominance struc- tures and provide a general existence result; we give sufficient conditions for uniqueness of “maximal” and “minimal” equilibria; and we explore the logical relationships among several well-known and some new dominance structures. Our results explain many regularities observed in the literature on rationalizability, in which specific dominance structures are used to char- acterize rationalizable strategy profiles under different common knowledge assumptions. Our uniqueness result for “minimal” equilibria extends Shap- ley’s (1964) uniqueness result for the saddle of a two-player zero-sum game. 1 Introduction The focus of game theory, and the source of its richness, is the strategic indetermi- nacy inherent in many social situations. A long tradition, beginning with the work of von Neumann and Morgenstern (1944) and Nash (1951), resolves this indeterminacy by assigning mixed strategies to players and finding equilibrium points, assignments that allow no players to improve their expected payoffs. The equilibrium point ap- proach rests on several assumptions: preferences over uncertain outcomes are given by expected utility; these preferences are common knowledge; rationality of the players is common knowledge; and the mixed strategies of the players are common knowledge.1 Dominance concepts arise in attempts to weaken these assumptions, especially the last.
  • Multi-Type Resource Allocation with Partial Preferences

    Multi-Type Resource Allocation with Partial Preferences

    Multi-Type Resource Allocation with Partial Preferences Haibin Wang,1 Sujoy Sikdar,2 Xiaoxi Guo,1 Lirong Xia,3 Yongzhi Cao,1* Hanpin Wang4,1 1Key Laboratory of High Confidence Software Technologies (MOE), Department of Computer Science and Technology, Peking University, China 2Computer Science & Engineering, Washington University in St. Louis 3Department of Computer Science, Rensselaer Polytechnic Institute 4School of Computer Science and Cyber Engineering, Guangzhou University, China [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], *Corresponding Author Abstract first class: agents consume their favorite remaining item one- We propose multi-type probabilistic serial (MPS) and multi- by-one according to an ordering drawn from the uniform type random priority (MRP) as extensions of the well-known distribution. Each agent is allocated a fraction of each item PS and RP mechanisms to the multi-type resource allocation equal to the probability that they consume the item. It is easy problems (MTRAs) with partial preferences. In our setting, to see that RP satisfies ex-post-efficiency and equal treat- there are multiple types of divisible items, and a group of ment of equals. Additionally, RP satisfies notions of envy- agents who have partial order preferences over bundles con- freeness and strategyproofness through the idea of stochas- sisting of one item of each type. We show that for the unre- tic dominance (sd). Given strict preferences, a fractional al- stricted domain of partial order preferences, no mechanism location p dominates another q, if at every item o, the total satisfies both sd-efficiency and sd-envy-freeness.
  • Matrix-Vector Representation of Various Solution Concepts

    Matrix-Vector Representation of Various Solution Concepts

    MATRIX-VECTOR REPRESENTATION OF VARIOUS SOLUTION CONCEPTS Fuad Aleskerov, Andrey Subochev State University - Higher School of Economics Moscow State University - Higher School of Economics 2009 1 Aleskerov F., Subochev A. Matrix-vector representation of various solution concepts: Working paper WP7/2009/03. – Moscow: State University – Higher School of Economics, 2009. – 32 p. A unified matrix-vector representation is developed of such solution concepts as the core, the uncovered, the uncaptured, the minimal weakly stable, the minimal undominated, the minimal dominant and the untrapped sets. We also propose several new versions of solution sets. The work was partially supported by the Scientific Foundation of the State University – Higher School of Economics (grant № 08-04-0008), by Russian Foundation for Basic Research (joint Russian-Turkish research project, grant N 09-01-91224-CT_a) and by Decision Choice and Analysis Laboratory (DeCAN Lab) of the State University – Higher School of Economics. Fuad Aleskerov – Department of Mathematics for Economics, University – Higher School of Economics (Moscow), [email protected] Andrey Subochev – Department of Mathematics for Economics, University – Higher School of Economics (Moscow), [email protected] 2 1. Introduction In decision making theory solution concepts are of major significance. This stems from the fact that there is no single best solution for different decision making problems - each problem dictates its own reasonable answer. In collective decision making the absence, in general case, of a maximal element in majority relation, i.e. nonexistence of an alternative more preferable for the majority of agents than any other alternative under binary comparisons, is called the Condorcet paradox. This very paradox led to proliferation of solution concepts over last 50 years of research in the area.
  • Set-Valued Solution Concepts in Social Choice and Game Theory Axiomatic and Computational Aspects

    Set-Valued Solution Concepts in Social Choice and Game Theory Axiomatic and Computational Aspects

    TECHNISCHEUNIVERSITÄTMÜNCHEN Lehrstuhl für Wirtschaftsinformatik und Entscheidungstheorie Set-Valued Solution Concepts in Social Choice and Game Theory Axiomatic and Computational Aspects Markus Brill Vollständiger Abdruck der von der Fakultät für Informatik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Tobias Nipkow, Ph.D. Prüfer der Dissertation: 1. Univ.-Prof. Dr. Felix Brandt 2. Prof. Dr. Jérôme Lang, Université Paris-Dauphine/Frankreich Die Dissertation wurde am 5. Juli 2012 bei der Technischen Universität München eingereicht und durch die Fakultät für Informatik am 12. Oktober 2012 angenom- men. SET-VALUED SOLUTION CONCEPTS IN SOCIALCHOICEANDGAMETHEORY markus brill Axiomatic and Computational Aspects Markus Brill: Set-Valued Solution Concepts in Social Choice and Game Theory, Axiomatic and Computational Aspects, c July 2012 Für Papa ABSTRACT This thesis studies axiomatic and computational aspects of set-valued solution concepts in social choice and game theory. It is divided into two parts. The first part focusses on solution concepts for normal-form games that are based on varying notions of dominance. These concepts are in- tuitively appealing and admit unique minimal solutions in important subclasses of games. Examples include Shapley’s saddles, Harsanyi and Selten’s primitive formations, Basu and Weibull’s CURB sets, and Dutta and Laslier’s minimal covering sets. Two generic algorithms for computing these concepts are proposed. For each of these algorithms, properties of the underlying dominance notion are identified that en- sure the soundness and efficiency of the algorithm. Furthermore, it is shown that several solution concepts based on weak and very weak dominance are computationally intractable, even in two-player games.
  • Download (323Kb)

    Download (323Kb)

    Munich Personal RePEc Archive On the equivalence of strategy-proofness and upper contour strategy-proofness for randomized social choice functions Roy, Souvik and Sadhukhan, Soumyarup Economic Research Unit, Indian Statistical Institute, Kolkata, Economic Research Unit, Indian Statistical Institute, Kolkata 21 August 2020 Online at https://mpra.ub.uni-muenchen.de/104405/ MPRA Paper No. 104405, posted 05 Dec 2020 13:34 UTC ON THE EQUIVALENCE OF STRATEGY-PROOFNESS AND UPPER CONTOUR STRATEGY-PROOFNESS FOR RANDOMIZED SOCIAL CHOICE FUNCTIONS Souvik Roy∗1 and Soumyarup Sadhukhan†1 1Economic Research Unit, Indian Statistical Institute, Kolkata August, 2020 Abstract We consider a weaker notion of strategy-proofness called upper contour strategy- proofness (UCSP) and investigate its relation with strategy-proofness (SP) for random social choice functions (RSCFs). Apart from providing a simpler way to check whether a given RSCF is SP or not, UCSP is useful in modeling the incentive structures for certain behavioral agents. We show that SP is equivalent to UCSP and elementary monotonicity on any domain satisfying the upper contour no restoration (UCNR) property. To analyze UCSP on multi-dimensional domains, we consider some block structure over the preferences. We show that SP is equivalent to UCSP and block monotonicity on domains satisfying the block restricted upper contour preservation property. Next, we analyze the relation between SP and UCSP under unanimity and show that SP becomes equivalent to UCSP and multi-swap monotonicity on any domain satisfying the multi-swap UCNR property. Finally, we show that if there are two agents, then under unanimity, UCSP alone becomes equivalent to SP on any domain satisfying the swap UCNR property.
  • Contour Boxplots: a Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles

    Contour Boxplots: a Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles

    6/3/13 8:42 AM Contour Boxplots: A Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles Ross T. Whitaker, Senior Member, IEEE, Mahsa Mirzargar and Robert M. Kirby, Member, IEEE Fig. 1. Contour boxplot for an ensemble of the pressure field of a fluid flow simulation with a LIC background image for context. Abstract— Ensembles of numerical simulations are used in a variety of applications, such as meteorology or computational solid mechanics, in order to quantify the uncertainty or possible error in a model or simulation. Deriving robust statistics and visualizing the variability of an ensemble is a challenging task and is usually accomplished through direct visualization of ensemble members or by providing aggregate representations such as an average or pointwise probabilities. In many cases, the interesting quantities in a sim- ulation are not dense fields, but are sets of features that are often represented as thresholds on physical or derived quantities. In this paper, we introduce a generalization of boxplots, called contour boxplots, for visualization and exploration of ensembles of contours or level sets of functions. Conventional boxplots have been widely used as an exploratory or communicative tool for data analysis, and they typically show the median, mean, confidence intervals, and outliers of a population. The proposed contour boxplots are a generalization of functional boxplots, which build on the notion of data depth. Data depth approximates the extent to which a particular sample is centrally located within its density function. This produces a center-outward ordering that gives rise to the statistical quan- tities that are essential to boxplots.