Lvy-Prohorov Metric on the Measure Space

QU Li-min,ZHU Ji-yun

(Department of Basic Courses,Qingdao Binhai University,Qingdao 266555,China)

QU Li-min,ZHU Ji-yun

(Department of Basic Courses,Qingdao Binhai University,Qingdao 266555,China)

Under the premise of infinitely many pure strategies,by defining the new LP?metric,striking an equivalence of topology and weak?topology,we prove that the existence of the essential component.

Lvy-Prohorov metric;the essential component;metric;Nash equilibrium

In this paper,we focus on discussing the stability of the set of mixed Nash equilibrium points of N-person non-cooperative games with infinitely many pure strategies.We define LP?metric and in the space M,the topology and the weak?topology which are induced by LP?metric are kept equivalent.We prove the existence of the essential components of the set of Nash equilibrium points.

§1.Introduction

We start with introducing the definition of the measure,weak convergence of measure and so on.

Definition 1.1There is a limited measureμin measurable space(Z,B),μis a nonnegative real-valued countable additive function of space B and the three tuple(Z,B,μ)is a measure space.

Definition 1.2If measureμmeets the conditions ofμ(Z)=1,μis a probability measure on Z.

Definition 1.3There is a proposition p on Z.If there is a collection A∈B,subject to μ(A)=0 and the property p is also established on ZA,then p is established almost everywhere with regard toμ.

Definition 1.4Ifμand ν are respectively two measures on(Z,B)and there are two nonempty disjoint sets A and B in B,subject to the conditions that X=A∪B andμ(A)= ν(B)=0,thenμand ν are regular.

Definition 1.5Let M be a space which is constituted by all of the measures of the compact metric space of Z.If f∈C(Z)and

then the measure sequence{μn∈M}weakly converge toμ∈M.

Definition 1.6For any E∈A×B,we have

and the measureμ×ν is the product ofμand ν.

So M is a nonempty convex subset of C(Z)?which is a local convex weak?topological space and C(Z)?is constituted by all continuous linear functionals of C(Z).Let P be space constituted by all probability measure,so P is a nonempty compact convex subset of M[5]and a nonempty compact convex subset of C(Z)?.

Lemma 1.1Ifμ∈M,thenμis regular.For any A∈B,?ε＞0,there are an open set Gεand a closed set Kεin Z,so that

(1)Kε?A?Gε;

(2)μ(GεKε)＜ε.

Lemma 1.2Ifμ,ν∈M and for any f∈C(Z),R f dμ=R f dν,thenμ=ν.

Lemma 1.3If the measure sequence{μn∈M}weakly converges to two measures ofμ and ν,μand ν∈M,thenμ=ν.

Lemma 1.4[2]If{gn}∞n=1is a countable dense subset of C(Z),for anyμand ν∈M,we define

Then(M,LP)is a separable metric space.We say LP is Lvy-Prohorov metric or Prohorov metric.

Lemma 1.5[3]If Z is a compact metric space,the measure sequence{μn∈M}weakly converges toμ∈M and f is a lower semi continuous real valued function on Z,then

Lemma 1.6If h is a nonnegative measurable function on X×X,then

§2.Lvy-Prohorov Metric on the Measure Space

In order to discuss the minor influence of the change of the set of the best reply correspondences strategy,the existence of essential components of the sets of the Nash equilibrium points for the continuous strategy f∈gc,we will improve the Lvy-Prohorov metric on the measure space.

Suppose Z is a compact metric space and M is a set of all of the measures on Z.Usually LP is called Lvy-Prohorov metric on M,that is to say for anyμand ν∈M,

We introduce the distance function diin every mixed strategy space Siand for anyμiand νi∈Si,we define

We define the distance function of LP?on M,so that for anyμand ν∈M,

§3.Main Results

Theorem 3.1(Si,di)is a metric space and the topology and the weak?topology induced by diin Siare equivalent.

ProofFor anyμi,νiand λi∈Si,by the definition of di,we know di(μi,νi)≥0;ifμi=νi, then di(μi,νi)=0 and di(μi,νi)≤di(μi,λi)+di(λi,νi).

Next,if di(μi,νi)=0,then

This proves that diis the metric of Si.

Theorem 3.2(S,d)is a metric space and the topology and the weak?topology which are induced by d on S are consistent.

Theorem 3.3(M,LP?)is a metric space,and the topology and the weak?topology which are induced by LP?metric on M are equivalent.

Condition(c)[6]For any two nonempty closed sets K1and K2of Y,K1T K2=?and any two points x1and x2of X,if F(x1)T K1=?and F(x2)T K2=?,there is x′∈X,so that d(x′,x1)≤d(x1,x2)and d(x′,x2)≤d(x1,x2),while F(x′)T(K1S K2)=?.

Theorem 3.4For any R∈R,there is at least one essential component.

Let gcbe the set of all such infinite games:the pure strategy set of the players are compact metric space Xi,Siis a metric space of all probability measures on pure strategy set Xi,the expected payofffunction is

For any game ?！蔳,we use N(Γ)to express the set of all Nash equilibrium points of Γ,so N(Γ)/=?and a set valued mapping from g to S is given by Γ→N(Γ).

For anyμ?i∈S?i,the set of the best reply correspondences of the player is

We know that B R(μ?i)is nonempty convex compact set of Siand B Ri:S?i→2Siis upper semi continuous on S?i[1].We define a set valued mapping B R:S→2Sand for any μ=(μ1,μ2,...,μn)∈S,we have

and call it the map of the best reply correspondence of Γ.B R(μ)is a nonempty convex compact set of S,and B R is upper semi continuous on S,so B R∈R.

For any two games of Γ and ?！洹蔳,we define the best reply correspondence metric between them

Let gc={f=(f1,f2,···,fN):for any i,fiis a real continuous function on the compact metric space X}and gcis the set of all N-person continuous games on X.For any two strategies f and f′∈g,we define the metric

Example If there are two-person games:The pure strategy space of the two people are both X=[0,1],the space of the game is g2={f:f is continuous function of X×X},ρ is the metric between the games and l is the best reply correspondence metric between the games.S is the set of all mix strategies on X and we define the Lvy-Prohorov metric LP?after being improved,for any f∈g2,the expected payment is

Take the game of f=(0,0)∈g2and the best reply correspondence is

(1)The game sequence is convergent according to metric ρ,but is not always convergent according to metric l.

For n=1,2,···,take

and then fn∈g2and the corresponding best reply correspondence is

(2)The game sequence converges according to metric l,but does not always converge according to metric ρ.

For n=1,2,···,take fn=(1,1),then fn∈g2and the corresponding best reply correspondence is

Then for any n,l(fn,f)=0;but for any n,ρ(fn,f)=2.

[1]AUBIN J P,EKELAND I.Applied Nonlinear Analysis[M].New York:John Wiley and Sons,1984.

[2]YAN Jia-an.Measure Theory[M].Beijing:Science Press(In Chinese),2004.

[3]SIMON L K.Games with discontinuous payoffs[J].Review of Economic Studies,1987,54:569-597.

[4]YU Jian.Essential equilibria of n-person noncooperative games[J].Journal of Mathematical Economics, 1999,31:361-372.

[5]BILLINGSLEY P.Convergence of Probability Measures[M].New York:Wiley,1968.

[6]YU Jian,TAN Kok Keong,XIANG Shu-wen,et al.The existence and stability of essential components[J]. Acta Mathematicae Applicatae Sinica(In Chinese),2004,27(27):201-209.

[7]YU Jian,XIANG Shu-wen.On essential components of the Nash equilibrium points[J].Nonlinear Analysis: Theory,Methods and Applications,1999,38:259-264.

[8]ZHOU Xuan-wei.Connectedness of cone-efficient solution set for cone-quasiconvex multiobjective programming in hausdorfftopological vector spaces[J].Chin Quart J of Math,2010,25(1):132-139.

tion:28A33,91A44

:A

1002–0462(2017)01–0042–07

date:2015-06-20

Biographies:QU Li-min(1979-),female(Manzu),native of Xinbin,Liaoning,a lecturer of Qingdao Binhai University,engages in nonlinear analysis;ZHU Ji-yun(1980-),female,native of Jiaozhou,Shandong,a lecturer of Qingdao Binhai University,engages in image processing and reconstruction.

CLC number:O225