### prj

代写project 这个项目是project代写的代写题目

### Game Theory

#### Supermodular Games

#### Ruitian Lang

#### ANU

### General idea

#### Finding all the Nash equilibria is usually a very challenging exercise.

#### Can we say something about the set of Nash equilibria without

#### actually computing all of them?

#### Can we do comparative statics for games: are Nash equilibrium

#### strategies increasing or decreasing in some parameters?

#### We can for a special class of strategic form games: supermodular

#### games.

### Table of contents

#### 1 Topkiss Theorem

#### 2 Rationalizability

#### 3 Supermodular games

### Increasing differences

#### Can we do comparative statics without invoking the Implicit Function

#### Theorem, for example when our Objective function is not differentiable?

#### Definition

#### Let I 1 and I 2 be two intervals. A function f : I 1 I 2 R is said to have

#### increasing differencesif for all x , x

#### I 1 , y , y

#### I 2 with x

#### > x and y

#### > y,

#### f ( x

#### , y

#### ) f ( x , y

#### ) f ( x

#### , y ) f ( x , y ).

#### If the strict inequality always holds, the function f is said to havestrictly

#### increasing differences.

#### Assume that f ( x ,)has increasing differences and for everythere is a

#### unique maximum x

#### ()of f (,). Then x

#### ()is non-decreasing in.

### Necessary generalizations

#### In a two-player game, if S 1 and S 2 are intervals and u 1 ( s 1 , s 2 )has

#### increasing differences, then BR 1 ( s 2 )is non-decreasing in s 2.

#### What if there are more than two players? Can we still say something

#### useful?

#### What if BR 1 is not a function (ie there are more than one best

#### responses)?

### Partial order of vectors

#### Definition

#### For two n-dimensional vectors x =( x 1 ,…, xn ) and y =( y 1 ,…, yn ) , we write

#### x yif xi yifor i = 1 ,…, n andx > yif x y and x = y.

#### There are vectors x , y such that x y and y x are both false;

#### therefore, this defined above is not an order in that not every

#### pair of vectors are comparable.

#### That x > y does not mean that xi > yi for i = 1 ,…, n.

#### Thisis transitive: if x y and y z , then x z , with strict

#### inequality if x > y or y > z.

### Examples from microeconomics

#### Consumer theoryA consumers utility function u is called monotonic if

#### u ( x

#### ) u ( x )whenever x

#### x. Being monotonic means that

#### the consumer prefers more to less.

#### General equilibriumLet N be the set of consumers in an economy with ui

#### being the utility function of consumer i N. An allocation x

#### is said to be Pareto dominated by x

#### if

#### ( u 1 ( x

#### ),…, un ( x

#### ))>( u 1 ( x ),…, un ( x )).

#### The definition of increasing differences remain the same with I 1 and I 2

#### replaced with subsets ofR

#### n

#### , as long asis understood to be the partial

#### order.

### Comparative statics

#### Theorem

#### (Topkis) Let I be an ordered space, T be a subset of R

#### n

#### and f : I T R

#### be a function with increasing differences. Assume that for some t 1 , t 2 T

#### with t 1 t 2 , x 1 is a (global) maximum of f (, t 1 ) and x 2 is a (global)

#### maximum of f (, t 2 ). If x 1 x 2 , then x 2 is also a maximum of f (, t 1 ) and

#### x 1 is also a maximum of f (, t 2 ).

#### It is easier to visualize the theorem in the special case where the set of

#### maxima for each t T is an interval; the theorem says that x 1 and x 2

#### must be in the overlap part (intersection) of the intervals corresponding to

#### t 1 and t 2.

### Compact ordered space I (optional)

#### Definition

#### A compact ordered space is an ordered set equipped with its order

#### topology such that the set is a compact space.

#### In the topology defined by the order, a set is open if and only if it is a

#### union of open intervals.

### Compact ordered space II (required)

#### Two examples:

(^1) A bounded and closed interval. (^2) A finite set equipped with any order. Every function on a finite set is

#### continuous.

#### Two properties:

(^1) Every closed subset of a compact ordered space has a greatest

#### element and a smallest element.

(^2) Every monotone sequence in a compact ordered space has a limit.

### Topkiss Theorem for compact ordered spaces

#### Theorem

#### Let I be a compact ordered space, T be a subset of R

#### n

#### , and f : I T R

#### be a function with increasing differences. In addition, assume that f ( x ,)

#### is continuous in x for every T. Then the following are true.

(^1) *For every* *T, f* (,) *has a maximum; in fact, the set of maxima of*

#### f (,) has a largest element x () and a smallest element x ().

(^2) *For* , *T with* *,* *x* () *x* () *and x* () *x* ()*.*

### Increasing differences and derivatives

#### Proposition

#### Let I be an interval T be a convex open subset of R

#### n

#### , and f : I T R

#### be a continuous function that is differentiable in the interior of the domain.

#### Denote by f 1 jthe mixed second order partial derivative of f with respect to

#### its first variable and its jth variable, for j = 2 ,…, n + 1. If f 1 j ( x , t ) 0 for

#### every x in the interior of I and t T, then f has increasing differences.

#### Interior means not at the boundary of I in case I is not an open interval.

### Example: the producers problem

#### Assume that the cost for producing q units is c ( q )for q 0.

#### A competitive producer solves the problemmax q 0 pq c ( q ), where p

#### is the market price of its product.

#### The objective function has increasing differences in( q , p )without any

#### assumption on c. What can we say about the optimal quantity?

### Table of contents

#### 1 Topkiss Theorem

#### 2 Rationalizability

#### 3 Supermodular games

### Rational players

#### A player of a strategic form game is calledrationalif he plays a best

#### response to some probability distribution over the opponents strategy

#### profiles.

#### Some probability distribution is not the same as a mixed strategy

#### profile, as correlations among different players are allowed.

#### What can we say about a rational players behavior without knowing

#### the probability distribution he plays for?

### Strictly dominated strategies I

#### Definition

#### Fix a strategic form game and a player i. A strategy si Siis called

#### strictly dominatedif there exists a mixed strategy iof Player i such that

#### ui ( si , s i )< ui ( i , s i ) , for every s i S i.

#### No matter what the opponents play, si is strictly worse than i.

#### In searching for a strategy that strictly dominates si , we include mixed

#### strategies.

### Strictly dominated strategies II

#### Theorem

#### Fix a strategic form game and a player i. If a strategy si Siis strictly

#### dominated, then it is not a best response to any probability distribution

#### over the opponents strategies.

#### This means that a rational player never plays a strictly dominated

#### strategy.

#### The converse of the theorem is also true for finite games (and some

#### generalizations): if a strategy is never a best response, it is strictly

#### dominated. The proof of this assertion is much more diicult.

### Knowledge of rationality

#### Assume that Player i is rational himself and knows that the other

#### players are rational. What can we say about Player i s play?

#### Let S

#### ( 1 )

#### j be the set of Player j s strategies that are not strictly

#### dominated. Player i knows that Player j will only choose from S

#### ( 1 )

#### j.

#### Let S

#### ( 1 )

#### i be the set of strategy profiles of Player i s opponents such

#### that each j plays a strategy from S

#### ( 1 )

#### j.

#### Player i plays a best response to some probability distribution over

#### S

#### ( 1 )

#### i

#### . The proof of the theorem implies that Player i will not play a

#### strictly dominated strategy with the opponents strategies restricted to

#### S

#### ( 1 )

#### i.

### Common knowledge

#### Informally, something is a common knowledge if

(^1) Everybody knows it. (^2) Everybody knows that everybody knows it. (^3) Everybody knows that everybody knows that everybody knows it. (^4) …

#### If it is common knowledge that every player is rational, then the

#### elimination procedure can continue indefinitely.

### Rationalizable strategies

#### Definition

#### Let S

#### ( 0 )

#### i = Sifor every Player i. Recursively, call a strategy si Si

#### eliminated by Round n + 1 if there exist a mixed strategy of Player i such

#### that

#### ui ( si , s i )< ui ( i , s i ) , for every s i S

#### ( n )

#### i ,

#### and define S

#### ( n + 1 )

#### i as the set of Player is strategies that are not eliminated

#### by Round n + 1. Put S

#### ()

#### i =

#### T

#### n = 1 S

#### ( n )

#### i. A strategy in S

#### ()

#### i is called

#### rationalizable.

### Nash equilibrium strategies are rationalizable

#### Theorem

#### Assume that is a Nash equilibrium of a strategic form game in mixed

#### strategy. Then for every Player i and strategy siin the support of i, siis

#### rationalizable.

#### Definition

#### If in a strategic form game, every player has only one rationalizable

#### strategy, such a game is calledsolvable by rationalizability.

#### Remark. A game that is solvable by rationalizability needs not have a

#### Nash equilibrium.

### Guessing 2/3 of the average

#### There are n players, each choosing a real number si [ 0 , 100 ].

#### Player i s payoff is

(^) *s*

#### i

#### 2

#### 3

#### s

(^) , where *s* = *n* ^1

#### P n

#### i = 1 si.

#### This game is solvable by rationalizability.

### Table of contents

#### 1 Topkiss Theorem

#### 2 Rationalizability

#### 3 Supermodular games

### Definition

#### Definition

#### A strategic form game with finitely many players is called supermodular if

#### for every player i, Siis a compact ordered space, uiis continuous and

#### ui ( si , s i ) has increasing differences in siand s i.

#### The statement on compactness and continuity encompasses the

#### following two cases:

(^1) *Si* is a bounded and closed interval and *ui* is continuous. (^2) *Si* is a finite set.

#### A compact ordered space always has a largest element and a smallest

#### element.

#### To obtain nice result in the end, the definition also requires that

#### ui ( si , s i )is continuous in s i.

### Examples

#### The investment game.

#### Battle of sexes.

#### Bank run.

#### Two-player Cournot competition after relabeling one firms strategies.

### The main theorem

#### Theorem

#### The following is true for a supermodular game.

(^1) *For every player i* *N, there exists a smallest rationalizable strategy s*

#### i

#### and a greatest rationalizable strategy si.

(^2) *sand* *s are Nash equilibria.*