Games and ContractsLecture 1

Yu (Larry) Chen

School of Economics, Nanjing University

Fall 2015

Introduction

I Game theory studies how the agents (players) strategicallyinteract with each other.Root: each player�s payo¤ depends on not only his own actionbut also on others�actions.

I It is a crucial methodology of analyzing human behaviors.I It has become the core of microeconomic theory.I Contract theory is heavily based on game theory. So one hasto master basic game theory knowledge before thoroughlystudying contract theory.

IntroductionThe core of contract theory is principal-agent games.I The principal needs the agent�s participation in someeconomic activity to realize her own objective in terms ofeconomic interests.

I Normally the agent(s) has some information only observableto himself. � Information AsymmetryHidden action � > Moral Hazard Eg. e¤ort.Hidden information (type) � > Adverse Selection Eg. ability.

I Since the agent has his own objective (interests) andunobservable information, the principal has to design a certainincentive scheme related to allocation of economicinterests, which is based one some available information,to induce the agent to behave in her best possible interests.Such incentive scheme is called contract, or mechanism, orsometimes menu depending on contexts.

I The principal(s) and the agent(s) will thus play a contractinggame.

IntroductionThe core of contract theory is principal-agent games.I The principal needs the agent�s participation in someeconomic activity to realize her own objective in terms ofeconomic interests.

I Normally the agent(s) has some information only observableto himself. � Information AsymmetryHidden action � > Moral Hazard Eg. e¤ort.Hidden information (type) � > Adverse Selection Eg. ability.

I Since the agent has his own objective (interests) andunobservable information, the principal has to design a certainincentive scheme related to allocation of economicinterests, which is based one some available information,to induce the agent to behave in her best possible interests.Such incentive scheme is called contract, or mechanism, orsometimes menu depending on contexts.

I The principal(s) and the agent(s) will thus play a contractinggame.

IntroductionThe core of contract theory is principal-agent games.I The principal needs the agent�s participation in someeconomic activity to realize her own objective in terms ofeconomic interests.

I Normally the agent(s) has some information only observableto himself. � Information AsymmetryHidden action � > Moral Hazard Eg. e¤ort.Hidden information (type) � > Adverse Selection Eg. ability.

I Since the agent has his own objective (interests) andunobservable information, the principal has to design a certainincentive scheme related to allocation of economicinterests, which is based one some available information,to induce the agent to behave in her best possible interests.Such incentive scheme is called contract, or mechanism, orsometimes menu depending on contexts.

I The principal(s) and the agent(s) will thus play a contractinggame.

IntroductionThe core of contract theory is principal-agent games.I The principal needs the agent�s participation in someeconomic activity to realize her own objective in terms ofeconomic interests.

I Normally the agent(s) has some information only observableto himself. � Information AsymmetryHidden action � > Moral Hazard Eg. e¤ort.Hidden information (type) � > Adverse Selection Eg. ability.

I Since the agent has his own objective (interests) andunobservable information, the principal has to design a certainincentive scheme related to allocation of economicinterests, which is based one some available information,to induce the agent to behave in her best possible interests.Such incentive scheme is called contract, or mechanism, orsometimes menu depending on contexts.

I The principal(s) and the agent(s) will thus play a contractinggame.

Key Words

I Incentive: all parties have di¤erent objectives (payo¤s).Especially, the principal(s) have con�icts of interests with theagent(s). So the principal(s) will use contracts as a device toprovide incentives for the agents to perform in her bestinterests.

I Asymmetric information: The agent(s) normally has somehidden actions or information (types). They can takeadvantage of such asymmetric information for self-bene�t. Eg.insurance or loa�ng worker. The principal(s) need to deal withinformation asymmetry in contracting.

I Incentive compatibility: When the principal(s) contractswith the agent(s), she must take into account the interestspursuit of the agent(s). Otherwise, the contracting gamecannot reach a sound equilibrium for both or even just theprincipal.

Key Words

I Incentive: all parties have di¤erent objectives (payo¤s).Especially, the principal(s) have con�icts of interests with theagent(s). So the principal(s) will use contracts as a device toprovide incentives for the agents to perform in her bestinterests.

I Asymmetric information: The agent(s) normally has somehidden actions or information (types). They can takeadvantage of such asymmetric information for self-bene�t. Eg.insurance or loa�ng worker. The principal(s) need to deal withinformation asymmetry in contracting.

I Incentive compatibility: When the principal(s) contractswith the agent(s), she must take into account the interestspursuit of the agent(s). Otherwise, the contracting gamecannot reach a sound equilibrium for both or even just theprincipal.

Key Words

I Incentive: all parties have di¤erent objectives (payo¤s).Especially, the principal(s) have con�icts of interests with theagent(s). So the principal(s) will use contracts as a device toprovide incentives for the agents to perform in her bestinterests.

I Asymmetric information: The agent(s) normally has somehidden actions or information (types). They can takeadvantage of such asymmetric information for self-bene�t. Eg.insurance or loa�ng worker. The principal(s) need to deal withinformation asymmetry in contracting.

I Incentive compatibility: When the principal(s) contractswith the agent(s), she must take into account the interestspursuit of the agent(s). Otherwise, the contracting gamecannot reach a sound equilibrium for both or even just theprincipal.

Key WordsI Individual rationality: When the principal(s) contracts withthe agent(s), she normally needs to take into accountvoluntary participation (and outside options) of the agent(s).Otherwise, the contracting games will end up with "nocontracting."

I Bargaining power: The bargaining power for one party is thepower to determine how much his or her objective will weighin the �nal objective of principal-agent problem. (Generally,contracting is a bargaining problem. cooperative or coalitionalgames)!Normally we assume the principal has full bargaining power,so the �nal objective of principal-agent problem is just theprincipal�s objective. Contracting becomes a noncooperativegames. Bargaining power sometimes can determine the role ofP and A in di¤erence contexts. The party with full bargainingpower will propose contract.*Eg. supplier vs retailer.

Key WordsI Individual rationality: When the principal(s) contracts withthe agent(s), she normally needs to take into accountvoluntary participation (and outside options) of the agent(s).Otherwise, the contracting games will end up with "nocontracting."

I Bargaining power: The bargaining power for one party is thepower to determine how much his or her objective will weighin the �nal objective of principal-agent problem. (Generally,contracting is a bargaining problem. cooperative or coalitionalgames)!Normally we assume the principal has full bargaining power,so the �nal objective of principal-agent problem is just theprincipal�s objective. Contracting becomes a noncooperativegames. Bargaining power sometimes can determine the role ofP and A in di¤erence contexts. The party with full bargainingpower will propose contract.*Eg. supplier vs retailer.

Broad Scopes for Application of Contract Theory

I Employment or Executive CompensationI AuctionI RegulationI Nonlinear PricingI Public good provisionI Supply ChainI International relationsI InsuranceI Voting

Math review

I FunctionsI Metric SpaceI Vector Space and convex setsI Ordered setI Euclidean SpaceI Di¤erential and IntegrationI Probability and Random VariableI Distribution and DensityI ExpectationI Notable Properties of functions under di¤erent structuresI Set-valued functions (maybe)I Mathematical program and KKT conditions.

Functions

I A function f from a set X to a set Y is a rule associatingevery x in X with exactly one element y 2 Y .

I If some rule g from a set X to a set Y associates every x in Xwith more than one element y 2 Y , then g is called set-valuedfunction or multi-function. (Very useful in game theory.)

I A function f with domain X and codomain Y is commonlydenoted by

f : X ! Y .

A multi-function g with domain X and codomain Y iscommonly denoted by

g : X � Y .

Functions

I A function f from a set X to a set Y is a rule associatingevery x in X with exactly one element y 2 Y .

I If some rule g from a set X to a set Y associates every x in Xwith more than one element y 2 Y , then g is called set-valuedfunction or multi-function. (Very useful in game theory.)

I A function f with domain X and codomain Y is commonlydenoted by

f : X ! Y .

A multi-function g with domain X and codomain Y iscommonly denoted by

g : X � Y .

Functions

I A function f from a set X to a set Y is a rule associatingevery x in X with exactly one element y 2 Y .

I If some rule g from a set X to a set Y associates every x in Xwith more than one element y 2 Y , then g is called set-valuedfunction or multi-function. (Very useful in game theory.)

I A function f with domain X and codomain Y is commonlydenoted by

f : X ! Y .

A multi-function g with domain X and codomain Y iscommonly denoted by

g : X � Y .

Functions

I The elements of X are called arguments of f . For eachargument x , the corresponding unique y in the codomain iscalled the function value at x or the image of x under f . It iswritten as f (x). But in some cases, people may also use f (x)to denote a function.

I Intuition: a function is a relation between a set of inputs anda set of permissible outputs with the property that each inputis related to exactly one output.

Functions

I The elements of X are called arguments of f . For eachargument x , the corresponding unique y in the codomain iscalled the function value at x or the image of x under f . It iswritten as f (x). But in some cases, people may also use f (x)to denote a function.

I Intuition: a function is a relation between a set of inputs anda set of permissible outputs with the property that each inputis related to exactly one output.

Metric Space

I A metric d on a set X is a function d : X � X ! R such thatfor all x , y 2 X :(1) d(x , y) � 0 and d(x , y) = 0 if and only if x = y ;(2) d(x , y) = d(y , x) (symmetry);(3) d(x , y) � d(x , z) + d(z , x) (triangle inequality).

I A metric space (X , d) is a set X with a metric d de�ned on X .I Intuition: a metric space is a set where a notion of distance(called a metric) between elements of the set is de�ned. Itcontains the metric (topological) structure concerningmeasuring how close two elements of the set will be.

Metric Space

I A metric d on a set X is a function d : X � X ! R such thatfor all x , y 2 X :(1) d(x , y) � 0 and d(x , y) = 0 if and only if x = y ;(2) d(x , y) = d(y , x) (symmetry);(3) d(x , y) � d(x , z) + d(z , x) (triangle inequality).

I A metric space (X , d) is a set X with a metric d de�ned on X .

I Intuition: a metric space is a set where a notion of distance(called a metric) between elements of the set is de�ned. Itcontains the metric (topological) structure concerningmeasuring how close two elements of the set will be.

Metric Space

I A metric d on a set X is a function d : X � X ! R such thatfor all x , y 2 X :(1) d(x , y) � 0 and d(x , y) = 0 if and only if x = y ;(2) d(x , y) = d(y , x) (symmetry);(3) d(x , y) � d(x , z) + d(z , x) (triangle inequality).

I A metric space (X , d) is a set X with a metric d de�ned on X .I Intuition: a metric space is a set where a notion of distance(called a metric) between elements of the set is de�ned. Itcontains the metric (topological) structure concerningmeasuring how close two elements of the set will be.

Vector Space

A vector space over R is a set V together with the operations ofaddition V � V ! V and scalar multiplication R� V ! Vsatisfying the following properties:(1) Commutativity: u + v = v + u for all u, v 2 V ;(2) Associativity: (u + v) + w = u + (v + w) and (ab)v = a(bv)for all u, v ,w 2 V and a, b 2 R;(3) Additive identity: There exists an element 0 2 V such that0+ v = v for all v 2 V ;(4) Additive inverse: For every v 2 V , there exists an elementw 2 V such that v + w = 0;(5) Multiplicative identity: 1v = v for all v 2 V ;(6) Distributivity: a(u+ v) = au+ av and (a+ b)u = au+ bu forall u, v 2 V and a, b 2 R.

Vector Space

I The elements v 2 V of a vector space are called vectors.

I Intuition: a vector space is a set where linear operations(addition and scalar multiplication) between elements of theset are de�ned. It contains the linear algebra structurecontaining linear operations.

I addition: some element can be accumulated.scalar multiplication: some element can be enlarged by times.e.g. monetary reward

Vector Space

I The elements v 2 V of a vector space are called vectors.I Intuition: a vector space is a set where linear operations(addition and scalar multiplication) between elements of theset are de�ned. It contains the linear algebra structurecontaining linear operations.

I addition: some element can be accumulated.scalar multiplication: some element can be enlarged by times.e.g. monetary reward

Vector Space

I The elements v 2 V of a vector space are called vectors.I Intuition: a vector space is a set where linear operations(addition and scalar multiplication) between elements of theset are de�ned. It contains the linear algebra structurecontaining linear operations.

I addition: some element can be accumulated.scalar multiplication: some element can be enlarged by times.e.g. monetary reward

Normed Vector Space

I Most of the spaces that arise in analysis are vector, orlinear,spaces, and the metrics on them are usually derivedfrom a norm, which gives the �length�of a vector.

I A normed vector space (X , k�k) is a vector space X togetherwith a function k�k : X ! R, called a norm on X , such thatfor all x , y 2 X and k 2 R:(1) 0 � kxk < ∞ and kxk = 0 if and only if x = 0;(2) kkxk = jk j kxk;(3) kx + yk � kxk+ kyk.

Normed Vector Space

I Most of the spaces that arise in analysis are vector, orlinear,spaces, and the metrics on them are usually derivedfrom a norm, which gives the �length�of a vector.

I A normed vector space (X , k�k) is a vector space X togetherwith a function k�k : X ! R, called a norm on X , such thatfor all x , y 2 X and k 2 R:(1) 0 � kxk < ∞ and kxk = 0 if and only if x = 0;(2) kkxk = jk j kxk;(3) kx + yk � kxk+ kyk.

Convex Set

I A convex set is a set of elements from a vector space suchthat all the points on the straight line between any two pointsof the set are also contained in the set.

I Formally, A set S in a vector space V is said to be convex iffor each x1, x2 2 S , the line segment λx1 +(1� λ)x2 2 S , forany λ 2 (0, 1).

I Convex set is important for analysis of optimization.

Convex Set

I A convex set is a set of elements from a vector space suchthat all the points on the straight line between any two pointsof the set are also contained in the set.

I Formally, A set S in a vector space V is said to be convex iffor each x1, x2 2 S , the line segment λx1 +(1� λ)x2 2 S , forany λ 2 (0, 1).

I Convex set is important for analysis of optimization.

Convex Set

I A convex set is a set of elements from a vector space suchthat all the points on the straight line between any two pointsof the set are also contained in the set.

I Formally, A set S in a vector space V is said to be convex iffor each x1, x2 2 S , the line segment λx1 +(1� λ)x2 2 S , forany λ 2 (0, 1).

I Convex set is important for analysis of optimization.

Ordered set

I A binary relation " % " on a nonempty set X is said to be apartial order if(1) x % x for every x 2 X (re�exive)(2) x % y % x implies x = y for every x , y 2 X(antisymmetric)(3) x % y % z implies x % z for every x , y , z 2 X (transitive)

I A partial order on X is called a total (linear) order on X ifeither x % y or y % x for every x , y 2 X (complete)

I A set (X ;%) is called a poset (short for partially ordered set)if % is a partial order on X . Similarly, A set (X ;%) is called aloset (short for linearly ordered set) if % is a linear order on X .

I Intuition: an ordered set contains the order structure betweenelements of the set.

Ordered set

I A binary relation " % " on a nonempty set X is said to be apartial order if(1) x % x for every x 2 X (re�exive)(2) x % y % x implies x = y for every x , y 2 X(antisymmetric)(3) x % y % z implies x % z for every x , y , z 2 X (transitive)

I A partial order on X is called a total (linear) order on X ifeither x % y or y % x for every x , y 2 X (complete)

I A set (X ;%) is called a poset (short for partially ordered set)if % is a partial order on X . Similarly, A set (X ;%) is called aloset (short for linearly ordered set) if % is a linear order on X .

I Intuition: an ordered set contains the order structure betweenelements of the set.

Ordered set

I A binary relation " % " on a nonempty set X is said to be apartial order if(1) x % x for every x 2 X (re�exive)(2) x % y % x implies x = y for every x , y 2 X(antisymmetric)(3) x % y % z implies x % z for every x , y , z 2 X (transitive)

I A partial order on X is called a total (linear) order on X ifeither x % y or y % x for every x , y 2 X (complete)

I A set (X ;%) is called a poset (short for partially ordered set)if % is a partial order on X . Similarly, A set (X ;%) is called aloset (short for linearly ordered set) if % is a linear order on X .

I Intuition: an ordered set contains the order structure betweenelements of the set.

Ordered set

I A binary relation " % " on a nonempty set X is said to be apartial order if(1) x % x for every x 2 X (re�exive)(2) x % y % x implies x = y for every x , y 2 X(antisymmetric)(3) x % y % z implies x % z for every x , y , z 2 X (transitive)

I A partial order on X is called a total (linear) order on X ifeither x % y or y % x for every x , y 2 X (complete)

I A set (X ;%) is called a poset (short for partially ordered set)if % is a partial order on X . Similarly, A set (X ;%) is called aloset (short for linearly ordered set) if % is a linear order on X .

I Intuition: an ordered set contains the order structure betweenelements of the set.

Euclidean Space: a well-behaved space

I Rn is a metric space!In Cartesian coordinates, if p = (p1, ..., pn) andq = (q1, ..., qn) are two points in Euclidean n-space, then theEuclidean distance from p to q, or from q to p is given by

d(p, q) =q(p1 � q1)2 + � � �+ (pn � qn)2

I Rn is a normed vector space!Addition and Scalar multiplication are de�ned. EuclideanNorm is given by

kpk =qp21 + � � �+ p2n

I Rn has a partial order "�"!"�" will become linear order in R.

Euclidean Space: a well-behaved space

I Rn is a metric space!In Cartesian coordinates, if p = (p1, ..., pn) andq = (q1, ..., qn) are two points in Euclidean n-space, then theEuclidean distance from p to q, or from q to p is given by

d(p, q) =q(p1 � q1)2 + � � �+ (pn � qn)2

I Rn is a normed vector space!Addition and Scalar multiplication are de�ned. EuclideanNorm is given by

kpk =qp21 + � � �+ p2n

I Rn has a partial order "�"!"�" will become linear order in R.

Euclidean Space: a well-behaved space

I Rn is a metric space!In Cartesian coordinates, if p = (p1, ..., pn) andq = (q1, ..., qn) are two points in Euclidean n-space, then theEuclidean distance from p to q, or from q to p is given by

d(p, q) =q(p1 � q1)2 + � � �+ (pn � qn)2

I Rn is a normed vector space!Addition and Scalar multiplication are de�ned. EuclideanNorm is given by

kpk =qp21 + � � �+ p2n

I Rn has a partial order "�"!"�" will become linear order in R.