Linear Representation of Symmetric Games

0.31 MB·

by  Daizhan Cheng

⋆
LinearRepresentationofSymmetric Games
DaizhanCheng,TingLiu
Academy ofMathematics andSystems Science, Chinese Academy ofSciences, Beijing100190, P.R.China
7
1
0 Abstract
2

r Using semi-tensor product of matrices, the structures of several kinds of symmetric games are investigated via the linear
a representation of symmetric group in the structure vector of games as its representation space. First of all, the symmetry,
M
described as the action of symmetric group on payoff functions, is converted into the product of permutation matrices with
structurevectorsofpayofffunctions.Usingthelinearrepresentationofthesymmetricgroupinstructurevectors,thealgebraic

8 conditions for the ordinary, weighted, renaming and name-irrelevant symmetries are obtained respectively as the invariance
underthecorrespondinglinearrepresentations.Secondly,usingthelinearrepresentationstherelationshipbetweensymmetric

] games and potential games is investigated. This part is mainly focused on Boolean games. An alternative proof is given to
T
show that ordinary, renaming and weighted symmetric Boolean games are also potential ones under our framework. The
G correspondingpotentialfunctionsarealsoobtained.Finally,anexampleisgiventoshowthatsomeotherBooleangamescould
also bepotential games.
.
s
c
[
Keywords: Symmetricgame, linear representation, potential game, Boolean game, semi-tensor productof matrices.

2
v
7
0 1 Introduction nizedclassesofsymmetricgames:(1)ordinarysymmet-
8 ricgame;(2)name-irrelevantsymmetricgame.Thefirst
6 one is very natural, which means all the payoffs are es-
Symmetric game is an important class of games. It has
0 sentiallythesame.Thesecondoneisfirstlyproposedby
drawnaconsiderableattentionfrombothgametheoret-
. Peleget al [23], basedon a simple type of strategy per-
1 icalcommunityandsystemscientists.Thereareseveral
0 reasonsforthis:Firstofall,symmetryrepresents“fair”. mutationsofNash[21].Anewclassofsymmetricgames
7 Inrealworld,afairgameismorerealisticandacceptable. was proposed in [4] as the renaming symmetric game.
1 (See also[6]).
That makes many commonly played games symmetric.
:
v Forinstance,Rock-Paper-Scissors,Prisoner’sDilemma,
i HawkandDove,etc.are allsymmetric.Ifmore general From mathematical point of view, a symmetry means
X
types of symmetric games are considered, many other invariance under the action of certain group. Roughly
r games such as Battle of the sexes, Matching Pennies, speaking, a game is symmetric means its payoff func-
a
etc.arealsosymmetric.Secondly,symmetrycangreatly tionsareinvariantunderapermutationgroup,whichis
simplifytherepresentationandcomputationofgamere- asubgroupofa symmetricgroup[12].
latedissues[22].Finally,symmetrymayberelatedwith
certain other properties. For instance, it is well known Linear representationof a group A in a vector space V
thatafinitesymmetricBooleangame,i.e.,agamewith isanisomorphismA→GL(V)[24].Thisrepresentation
acommonactionsetofsize2,isanexactpotentialgame notonly makesthe groupactioneasily computable but
[15]. alsoprovidesaframeworktoinvestigatecertainproper-
ties ofthe action.In this paper we revealfor eachsym-
The concept of symmetric games was firstly proposed metry of games its linear representation, which is the
by J.Nash inhis famous paper [21]. Lately,it has been actionofsymmetricgrouporitssubgroupsonthestruc-
developed into several classes of interesting games [1], ture vector of games. The structure vector of a game
[18],[4].Accordingto[4],therearetwocommonlyrecog- consistsofthe structurevectorsofits payofffunctions.
⋆ This work is supported partly by NNSF 61333001 and To construct and to investigate the linear representa-
61273013 of China. Corresponding author: Daizhan Cheng. tionsofsymmetricgamesthesemi-tensorproduct(STP)
Tel.: [phone]; fax.: [phone]. of matrices becomes a main tool. STP of matrices is a
Emailaddress: [email protected] (Daizhan Cheng). newlydevelopedmatrixproduct[8].Itisageneralization
Preprintsubmitted toAutomatica 9March 2017
ofconventionalmatrixproduct andkeepsmost proper- (8) A matrix L ∈ M is called a logical matrix if
m×n
tiesofconventionalmatrixproductremainingavailable. the columns of L are of the form of δk. That is,
m
It has been successfully used for studying logical (con- Col(L) ⊂ ∆ . Denote by L the set of m×n
m m×n
trol) systems. We refer to [7], [19], [13], [20], [25], [27], logicalmatrices.
[26], just to name a few. It has also been used to game (9) If L ∈ L , by definition it can be expressed as
n×r
theoreticcontrolproblems[14],[9],[10],[28]. L=[δi1,δi2,··· ,δir].Forthesakeofcompactness,
n n n
itis brieflydenotedasL=δ [i ,i ,··· ,i ].
n 1 2 r
Vector space structure of finite games is a key issue in (10) Mn, the structure matrix of “negation”, that is,
thislinearrepresentationapproachtosymmetricgames. Mn =δ2[2,1].
Aclearpicture ofvectorspacestructureoffinite games (11) Sn: n-thordersymmetric group.
was firstly presented in [3]. This vector space structure (12) Pn: n-thorderBooleanorthogonalgroup.
was then modified and merged into an Euclidean space (13) Let H, G be two groups. H < G means H is a
[11].InthispaperafinitegameGwithplayerset|N|=n subgroupofG.
and strategy set |Si| = ki, i = 1,··· ,n, is simply con- (14) Θ[n;κ]:blockpermutationgroup(Θ[n;κ] <Snκ).
sidered as a point in Rs, where s = n n k , denoted (15) id:identity mapping.
i=1 i
byV ∈Rs.SuchasetisdenotedbyG ,which (16) GL(n,R)(orGL(V)): generallinear group.
hastGhesametopologyandvectorspaceQ[ns;tkr1u,··c·t,kunr]easRs, (17) G :Setoffinitegameswith|N|=n,|S |=
[n;k1,···,kn] i
asdemonstratedin[11]. k .
i
(18) G : |S |=κ, i=1,··· ,n.
[n;κ] i
The first purpose of this paper is to deduce the linear (19) So :the setofordinarysymmetric games.
[n;κ]
representations for each of the aforementioned symme- (20) Sw :the setofweightedsymmetricgames.
tries of finite games, considering as the action of sym- [n;κ]
(21) Sr :the setofrenamingsymmetric games.
metricgrouporitspropersubgroupsinthevectorspace [n;κ]
G ∼ Rs, which consists of the structure vec-
[n;k1,···,kn]
tors of all payoff functions. Then it is proved that a fi-
nite game G is certainkind ofsymmetry, if andonly if,
itsstructurevectorisinvariantunderthecorresponding The rest of this paper is organized as follows: Section
linear representation. This presentation gives an easy 2 considers the action of symmetric group on games.
way to verify whether a finite game G is of this kind Subsection2.1 introduces symmetric groupandits ma-
of symmetry. Moreover, the matrix form of linear rep- trix expression. Subsection 2.2 reviews the semi-tensor
resentations provides a framework for investigating the product(STP)ofmatrices,whichisafundamentaltool
propertiesofsymmetricgamesandmanipulatingthem. in this paper. Using STP, Subsection 2.3 expresses the
groupactiononfinite gameintomatrixform.Section3
Secondly, the relationship between certain symmetric provides the linear representation for ordinary symme-
gamesandpotentialgamesisinvestigated.Wemainlyfo- tryofgames.ContinuedfromSection3,Section4intro-
cusonBooleangames.Theordinaryandrenamingsym- duces the weightedsymmetry andrenamingsymmetry,
metricBooleangamesareprovedtobepotentialgames which are two simple extensions of ordinary symmetry.
andthe weightedsymmetric Booleangames are proved The name-irrelevant symmetry and its linear represen-
tobe weightedpotentialgames. tationarediscussedinSection5.Someexamplesarepre-
sentedtodepictthem.Thepotentialgameisintroduced
inSection6.Theverificationofpotentialgamesandthe
Thirdly, a special kind of Boolean games, called the
formula for calculating potential function are also pre-
negation-symmetricBooleangame,isdefinedandinves-
sented.Section7showstherelationshipbetweenpoten-
tigated.WeprovethatthiskindofBooleangamesisalso
tialgamesandsymmetricgamesforBooleangamesonly.
potential. It shows that (generalized) ordinary symme-
ItwasshownthatanordinarysymmetricBooleangame
try is sufficient for a Booleangame to be potential, but
and a renaming symmetric Boolean game are potential
itis notnecessary.
game.(Theseconclusionsarewellknown,butwegivean
alternative proof.) And a weighted symmetric Boolean
Finally,forstatementease,wegivesomenotations:
game is a weighted potential game. The corresponding
potential functions are also obtained respectively. Sec-
(1) M :the setofm×n realmatrices.
m×n tion8introducesthenegation-symmetricBooleangame
(2) B :thesetofm×nBooleanmatrices,(B :the
m×n n andprovedthatitisalsoapotentialgame.Section9isa
setofndimensionalBooleanvectors.)
concludingremark,whichshowsthatallthesymmetries
(3) D:={0,1}.
discussedinthispapercanbecharacterizedbyaunified
(4) δi:the i-thcolumnofthe identity matrixI .
n n formoflinearrepresentations.Thischaracteristicprop-
(5) ∆ := δi|i=1,··· ,n .
n n ertyrevealsthe relationshipamongthem.Therelation-
(6) 1 =(1,1,··· ,1)T.
ℓ (cid:8) (cid:9) ship between symmetric games and potential games is
also summarized. To streamline the presentation, some
ℓ
(7) 0 :ap×q matrixwith zeroentries. proofsareprovidedinthe Appendix.
p×q
| {z }
2
2 Action ofSymmetricGroup on Games Moreover,itisobviousthatthereisaone-onecorrespon-
dencebetweenP andthesetofn-thorderBooleanor-
n
thogonalmatrices.We, therefore,have P <GL(n,R).
2.1 SymmetricGroupand Boolean Orthogonal Group n
Next,define ψ :S →P as
n n
Let N = {1,2,··· ,n}. Sn is the set of permutations ψ(σ):=Pσ, (3)
of elements of N. With the composed permutation as
the product of two permutations, Sn becomes a group. then ψ is a group isomorphism. That is, ψ is bijective
TheelementsinSncanbeexpressedeitherbyaone-one and
correspondenceor a product of cycles. We use a simple
exampleto depictthem. ψ(µ◦σ)=P =P P =ψ(µ)ψ(σ). (4)
µ◦σ µ σ
Example1 ConsiderS ,σ, µ∈S canbeexpressedas
5 5
Itiseasyto seethat
1 2 3 4 5 1 2 3 4 5
(1) P hasa naturalgroupactionon∆ as
n n
σ : ↓ ↓ ↓ ↓ ↓; µ: ↓ ↓ ↓ ↓ ↓.
P δi =δσ(i) ∈∆ , i=1,··· ,n; (5)
3 5 1 4 2 2 4 5 3 1 σ n n n
(2) if we identify i ∼ δi, then the group action of S
Equivalently, they can beexpressedin cycle form as n n
onN isexactlythesameasthegroupactionofP
n
on∆ .Precisely,
n
σ =(1,3)(2,5); µ=(1,2,4,3,5).
σ(i)∼P δi. (6)
σ n
The product of σ andµ is definedas
2.2 Semi-tensorProduct of Matrices
µ◦σ =(1,5,4,3,2).
This subsection is a brief introduction to semi-tensor
Notethatintheaboveproduct,thepermutationσ isper- product(STP)ofmatrices.Wereferto[7],[8]fordetails.
formed first,and then µ is performed. Say, 1−→σ 3−→µ 5, TheSTP ofmatricesisdefined asfollows:
µ◦σ
which leads to1−−→5,etc.
Definition2 Let M ∈ M , N ∈ M , and t =
m×n p×q
lcm{n,p} be the least common multiple of n and p. The
Let G be a group and X 6= ∅, φ : G×X →X is called
STPof M and N is definedas
agroupactionofGonX,if
M ⋉N := M ⊗I N ⊗I ∈M , (7)
t/n t/p mt/n×qt/p
φ(e,x)=x, e∈Gis the identity, x∈X
where⊗ is th(cid:0)eKroneck(cid:1)er(cid:0)product.(cid:1)
φ(g ,φ(g ,x))=φ(g g ,x), g , g ∈G, x∈X.
2 1 2 1 1 2
TheSTPofmatricesisageneralizationofconventional
matrix product, and all the computational properties
The symmetric group has a natural action on N, that of the conventional matrix product remain available.
is, σ×i7→σ(i). Throughout this paper, the default matrix product is
STP, so the product of two arbitrary matrices is well
defined,andthe symbol⋉ ismostly omitted.
Foreachσ ∈S ,wecanconstructaBooleanorthogonal
n
matrix,calledthe structurematrixofσ, as
First, we give some basic properties of STP, which will
be usedinthe sequel.
P = δσ(1),δσ(2),··· ,δσ(n) . (1)
σ n n n
Proposition3 (1) (Associative Law:)
h i
A⋉(B⋉C)=(A⋉B)⋉C. (8)
Define the set of structure matrices of n-th order per-
mutationsas
(2) (DistributiveLaw:)
Pn :={Pσ |σ ∈Sn}. (2) (A+B)⋉C =A⋉C+B⋉C; (9)
Notethatthe setofn-thorderBooleanorthogonalma-
tricesisasubgroupofthegenerallineargroupGL(n,R). C⋉(A+B)=C⋉A+C⋉B. (10)
3
Proposition4 Let X ∈ Rt be a t dimensional column Definition10 [17,8] Assume A ∈ M , B ∈ M ,
p×s q⊗s
vector,and M a matrix.Then the Khatri-Rao product of A, B, denoted by A∗B, is
definedas
X ⋉M =(I ⊗M)⋉X. (11)
t
A∗B = [Col (A)⋉Col (B),Col (A)⋉Col (B),
1 1 2 2
··· ,Col (A)⋉Col (B)]∈M .
s s pq×s
Next, we consider the matrix expression of logical rela-
(16)
tions.Identifying
Proposition11 Assume x ∈ ∆ , y ∈ ∆ , z ∈ ∆ ,
1∼δ1, 0∼δ2, n p q
2 2 wherey, z arelogicalfunctionsofxandareexpressedin
algebraic form as
thenalogicalvariablex∈D canbeexpressedinvector
formas
y =Mx, M ∈L
p×n
x (17)
x∼ , z =Nx, N ∈L .
q×n
1−x!
whichiscalledthe vectorformexpressionofx Then wehave
A mapping f : Dn → R is called a pseudo-Boolean yz =(M ∗N)x. (18)
function.
Proposition5 Given a pseudo-Boolean function f : 2.3 Matrix Expression of Finite Games under Action
Dn → R, there exists a unique row vector Vf ∈ R2n, of Sn
calledthestructurevectoroff,suchthat(invectorform)
A finite game, denoted by a triple G = (N,S,C), con-
f(x ,··· ,x )=V ⋉n x . (12) sists of three components: (i) the set of players N =
1 n f i=1 i
{1,2,··· ,n}; (ii) the profile set S = n S , where
i=1 i
Remark6 In previous proposition, if D is replaced by Si = {1,2,··· ,ki}, is the set of strategies of player i,
fDukn,ctkio>n2a,ndthtehnetrheesuflutnrcetmioaninfsaisvacaillalebdleawpitsheuadnoo-blovgiiocuasl cii=:S1,→···R,nis;tthheepsaeytooffffpuanycotffiosnCof=pl{acy1e,rQ·i·,·i,=cn1},,·w··h,enre.
modification that xi ∈∆k and Vf ∈Rkn. Denote the set of G with |N| = n and |Si| = ki by
G . If |S | = κ, i = 1,··· ,n, the set of such
[n;k1,···,kn] i
games is briefly denoted by G . This paper concerns
Definition7 A swap matrixW ∈M is de- [n;κ]
[m,n] mn×mn onlyG .
finedas [n;κ]
Assume a game G ∈ G is given. It is obvious that
[n;κ]
W := [δ1δ1 ,δ2δ1 ,··· ,δnδ1 ,δ1δ2 , thepayofffunctionc isapseudo-logicalfunction.When
[m,n] n m n m n m n m (13) i
the strategies x , i = 1,··· ,n, are expressed into their
··· ,δnδ2 ,··· ,δ1δm,··· ,δnδm]. i
n m n m n m vectorform,the payofffunctions canbe expressedas
c (x ,··· ,x )=Vc⋉n x :=Vcx, i=1,··· ,n,
i 1 n i j=1 j i
Thebasicfunctionofaswapmatrixistoswaptwovec- (19)
tors.
where
Proposition8 LetX ∈RmandY ∈Rnbetwocolumn
vectors.Then x=⋉n x ∈∆ (20)
i=1 i κn
W[m,n]XY =YX. (14) is called the STP form of the strategy profile. We can
alsoexpressitina (column)vectorformas
Theswapmatrixhasfollowingdecompositionproperty. ~x:=[xT1,xT2,··· ,xTn]T ∈Bnκ. (21)
Proposition9 Denoting
Φ :=1T ⊗I ⊗1T , i=1,··· ,n, (22)
W[p,qr] = Iq ⊗W[p,r] W[p,q]⊗Ir . (15) i κi−1 κ κn−i
(cid:0) (cid:1)(cid:0) (cid:1)
4
thena matrixΦ canbe constructedas 3 OrdinarySymmetryanditsLinearRepresen-
tation
Φ
1
Define the structurevectorofG∈G as
[n;κ]
Φ2
Φ:= ... ∈Bnκ×κn. (23) VG :=[V1c,V2c,··· ,Vnc]∈Rnκn, (30)
 
Φ 
 n thenthegameGiscompletelydeterminedbyVG.Inthis
  way,we have a vector space structure (precisely, Rnκn)
This Φ can converta profile from its STP form into its forG .Wereferto[3],[11]forthevectorstructureof
[n;κ]
vectorform. finite games.
Proposition12 Let G∈G , x and~x are STP form Hereafter,we adoptthe terminologiesused in [4] to de-
[n;κ]
scribethe symmetry ofgames.
andvector form of its strategyprofilerespectively. Then
Definition14 A game G ∈ G is called ordinary
~x=Φx. (24) [n;κ]
symmetricif for anyσ ∈S
n
Next, assume σ ∈ S . A straightforward computation ci(x1,··· ,xn) = cσ(i)(xσ−1(1),xσ−1(2),··· ,xσ−1(n)),
n
showsthat i=1,··· ,n.
(31)
xσ−1(1) Φσ−1(1)
xσ−1(2) Φσ−1(2) Using(25)-(27),wehavethe followingproposition.
Pσ~x= . = . x. (25)
 ..   .. 
    Proposition15 A game G ∈ G is ordinary sym-
    [n;κ]
xσ−1(n) Φσ−1(n) metric,if and onlyif, for anyσ ∈Sn
   
Vc =Vc T , i=1,··· ,n. (32)
i σ(i) σ
Thefollowingexpressioncanbeprovedbyusingformula
(25)andProposition11.
Denote
Vσ :=[Vc ,··· ,Vc ].
Proposition13 Let σ ∈S .Define G σ(1) σ(n)
n
Thenwehavethe followingresult:
σ(x):=⋉ni=1xσ−1(i). (26) Theorem16 G ∈ G is ordinary symmetric, if and
[n;κ]
onlyif,
Then
V (P ⊗T )=V , ∀σ ∈S . (33)
G σ σ G n
σ(x)=T x, (27)
σ
where Proof.AccordingtoProposition15,itisenoughtoshow
that (33) and (32) are equivalent. From (32), we have
Tσ =Φσ−1(1)∗Φσ−1(2)∗···∗Φσ−1(n). (28) the followingidentity
V =[Vc T ,Vc T ,··· ,Vc T ]=Vσ(I ⊗T )
G σ(1) σ σ(2) σ σ(n) σ G n σ
(34)
Finally,wegiveanexpressionforcalculationΦσ−1(i).In
fact,substituting (24)into(25)andwe have
Onthe other hand, a straightforwardcomputationver-
ifies
Φσ−1(1) Φ1
Vσ =V P . (35)
Φσ−1(2) Φ2 G G σ
. =Pσ . . (29)
 ..   .. 
    Substituting it into (34) gives rise to (33). On the con-
   
Φσ−1(n) Φn trary, splitting (33) to each block component Vic, it is
   
5
easytoobtain(34)whichyields(32).Thiscompletesthe Then wehave
proofofTheorem16. ✷
Definition17 [24] A linear representation of a finite
1 0 0
groupAinafinitevectorspaceV isahomomorphism ψ
from thegroup Aintothegroupof GL(V). Pσ1 =I3 Pσ2 =0 0 1
0 1 0
 
Now consider the vector space G ∼ Rnκn. That is, 0 1 0 0 0 1
[n;κ]
eachG∈G isrepresentedbyitsstructurevectorV .
Thenweha[nv;eκ]alinearrepresentationofσ ∈S inG G Pσ3 =1 0 0 Pσ4 =1 0 0
n [n;κ]
asfollows: 0 0 1 0 1 0
   
0 1 0 0 0 1
Proposition18 Define a mapping of σ ∈Sn in vector Pσ5 =0 0 1 Pσ6 =0 1 0;
spaceG as
[n;κ]
1 0 0 1 0 0
   
   
ϕ(σ):=P ⊗T ∈GL(G ). (36)
σ σ [n;κ]
and
Then ϕ is alinear representation of S in G .
n [n;κ]
Proof. We alreadyhave(4),thatis,
T = id
σ1
T = δ [1,4,7,2,5,8,3,6,9,
P =P P . σ2 27
µ◦σ µ σ
10,13,16,11,14,17,12,15,18,
19,22,25,20,23,26,21,24,27]
Whatdowe needto proveis
T = δ [1,2,3,10,11,12,19,20,21,
σ3 27
T =T T . (37) 4,5,6,13,14,15,22,23,24,
µ◦σ µ σ
7,8,9,16,17,18,25,26,27]
This equationcomesfrom(28)and(29). T = δ [1,10,19,2,11,20,3,12,21,
σ4 27
4,13,22,5,14,23,6,15,24,
✷
7,16,25,8,17,26,9,18,27]
T = δ [1,4,7,10,13,16,19,22,25,
σ5 27
Corollary19 G∈G is ordinary symmetric, if and
[n;κ] 2,5,8,11,14,17,20,23,26,
onlyif,itisinvariantwithrespecttothelinearrepresen-
tation ϕ, definedby(36). 3,6,9,12,15,18,21,24,27]
T = δ [1,10,19,4,13,22,7,16,25,
σ6 27
We giveanexample to depictthis: 2,11,20,5,14,23,8,17,26,
3,12,21,6,15,24,9,18,27].
Example20 Consider G∈G . Denote
[3;3]
S3 ={σi |i=1,2,3,4,5,6}, According to Theorem 16, G is ordinary symmetric, if
andonly if,
where
6
σ1 =id σ2 =(2,3) σ3 =(1,2) VT ∈ PT ⊗TT −I ⊥. (38)
G σi σi 81
σ4 =(1,2,3) σ5 =(1,3,2) σ6 =(1,3). i\=1(cid:0) (cid:1)
6
Asimple computation shows that (38) implies that Then
3c (1,1)=2c (1,1)=6 3c (2,2)=2c (2,2)=12
V = [A,D,E,D,F,G,E,G,H 1 2 1 2
G
3c (1,2)=2c (2,1)=12 3c (2,1)=2c (1,2)=18.
I,J,K,J,B,L,K,L,M 1 2 1 2
N,O,P,O,Q,R,P,R,C
Hence G is a weighted symmetric game with weights:
A,D,E,I,J,K,N,O,P µ1 =3andµ2 =2.
D,F,G,J,B,L,O,Q,R (39)
Similarto Proposition15,wehavethe followingresult.
E,G,H,K,L,M,P,R,C
Proposition24 G∈G is weighted symmetricwith
A,I,N,D,J,O,E,K,P [n;κ]
respect to {µ | i = 1,··· ,n}, if and only if, for each
i
D,J,O,F,B,Q,G,L,R σ ∈S
n
E,K,P,G,L,R,H,M,C],
µ Vc =µ Vc T , (41)
i i σ(i) σ(i) σ
wheretheparametersA,B,··· ,Rarearbitraryrealnum-
whereT is definedin (28).
bers. Denote by So the ordinary symmetric subspace σ
[n;κ]
of G[n;κ]. Then we knowthat Define aweightmatrix Γ as
Γ:=diag(µ ,µ ,··· ,µ ),
dim So =18, 1 2 n
[3;3]
(cid:16) (cid:17) whereµ >0,i=1,··· ,n.UsingΓ,foreachσ ∈S we
i n
anda basis of So can easily befiguredoutfrom (39). define
[3;3]
ϕµ(σ):=Γ(P ⊗T )Γ−1 ∈GL(G ). (42)
σ σ [n;κ]
4 Generalized OrdinarySymmetricGame
Corresponding to Theorem 16, Proposition 18, and
Corollary19,wehave
4.1 Weighted SymmetricGames
Theorem25 (1) Themappingϕµ definedin(42)isa
Definition21 Let G ∈ G[n;κ], µi ∈ R+, i = 1,··· ,n, linear representation of Sn in G[n;κ].
beaset of positive realnumbers,called theweights.
(2) A game G ∈ G is weighted ordinary symmetric
[n;κ]
withrespectto{µ , i=1,··· ,n},ifandonlyif,V
i G
(1) G is said to be weighted symmetric with respect to
is invariant with respect to this linear representa-
σ ∈S and {µ ,i=1,··· ,n},if
n i tion.
µici(x1,··· ,xn)=µσ(i)cσ(i) xσ−1(1),··· , 4.2 RenamingSymmetric Game
(40)
xσ−1(n) , i=1,··· ,n. (cid:0) Definition26 [4] LetG∈G ,then
[n;κ]
(cid:1)
(2) G is said to be weighted symmetric with respect to
(1) A renaming of G is a set of permutations r ∈ S
i κ
{µ ,i = 1,··· ,n} if it is weighted symmetric with
i on S , i=1,··· ,n.
respect toeach σ ∈Sn andµi,i=1,··· ,n. (2) A reinaming game Gr := (N,(Sr) ,(cr) ),
i i∈N i i∈N
where
Remark22 When µ = 1, i = 1,··· ,n, the weighted
i
symmetrybecomes theclassical ordinary symmetry. Sr ={r (1),r (2),··· ,r (κ)},
i i i i (43)
cr(x ,··· ,x )=cr(r−1(x ),··· ,r−1(x )).
Example23 Consider G∈G[2;2].Thepayoffbi-matrix i 1 n i 1 1 n n
is in Table 1.
(3) Gisrenamingsymmetricifthereexistsarenaming
Table1 r = (r ,··· ,r ) such that Gr is ordinary symmet-
Payoff Bi-matrix 1 n
ric.
P1\P2 1 2
1 2, 3 4, 9 Example27 Consider the Battle of the Sexes, where
Player1ishusband,Player2iswife,F isfootballandC
2 6, 6 4, 6
is concert.The payoff bi-matrix is as in Table 2
7
Table2 Theorem29 Given a renaming r = (r ,r ,··· ,r ),
1 2 n
Payoff Bi-matrix of Battle of theSexes
wherer ∈S .
i κ
P1\P2 F C
F 2, 1 0, 0 (1) The mapping ϕr :Sn →GL(G[n;κ])definedas
C 0, 0 1, 2
ϕr(σ):=P ⊗Tr, σ ∈S (49)
σ σ n
It is obvious that this game is not ordinary symmetric.
is alinear representation.
ButifwerenamethestrategiesinS asF ∼2andC ∼1,
2 (2) G ∈ G is renaming symmetric, if and only if,
i.e., choose r = id and r = M , then it becomes ordi- [n;κ]
1 2 n there exists a renaming such that V is invariant
nary symmetric. Hence Battle of the Sexes is renaming G
withrespect tothelinear representation ϕr.
symmetric.
Remark30 Both weighted symmetry and renaming
SimilartoProposition15,wecanhavethefollowingre-
symmetry are theoretically almost trivial. In fact, any
sult.
property of ordinary symmetry has its corresponding
property for weighted or renaming symmetry with some
Proposition28 G∈G[n;κ]isrenamingsymmetricwith obvious modifications. But from application point of
renamer =(r1,··· ,rn), if andonly if, for anyσ ∈Sn view,thetwogeneralizations enlargedtheregionofsym-
metric games a lot, which tremendously expands the
Vc =Vc Tr, i=1,··· ,n, (44) application of symmetricgame theory.
i σ(i) σ
where 4.3 Verification of Symmetry
Tσr =ΓTrTσΓr (45) LetSbeafinitegroupandV beafinitedimensionalvec-
torspace,andϕ:S→GL(V)bealinearrepresentation.
and ThenW ⊂V iscalledarow(column)invariantsubspace
of ϕ, if Wϕ(σ) ⊂ W, (correspondingly, ϕ(σ)W ⊂ W),
∀σ ∈S.
Γ =P ⊗P ⊗···⊗P . (46)
r r1 r2 rn
In this paper, taking the structure of V into consider-
G
ation, the row invariance is considered as a default in-
Proof.Denotethestructurevectorsoftherenamingsym-
variance.
metric gameasVr, i=1,··· ,n.
i
The following proposition comes from the definition of
Bydefinition, wehave linearrepresentationimmediately.
Vr⋉n x = Vc⋉n r−1(x ) Proposition31 Let ϕ : S → GL(V) be a linear repre-
i j=1 j i j=1 j j
sentation,and
= Vc⋉n P−1x
i j=1 rj j
= VicΓ−r1⋉(cid:16)nj=1xj (cid:17) Ω:={ω1,ω2,··· ,ωs}⊂S
= VcΓT ⋉n x ,
i r j=1 j be a set of generators of S, i.e., S = hΩi. Then W ⊂ V
is invariant withrespect toϕ, ifand onlyif,
whereΓ isdefined in(46).Hence we have
r
Wϕ(ω)⊂W, ∀ω ∈Ω. (50)
Vr =VcΓT. (47)
i i r
On the other hand, Using the fact that Gr is ordinary This proposition makes the verification of ordinary (or
symmetric,we have,inlightofProposition15, weighted, or renaming ) symmetry easier. Because ac-
cording to Corollary 19, Theorem 25 or Theorem 29,
instead verifying the invariance with respect to all ele-
Vr = Vr T =Vc ΓTT . (48)
i σ(i) σ σ(i) r σ mentsofSn,weneedonlytoverifyitforasetofgener-
ators.
Comparing(47)with(48)yields(44)-(45). ✷
Next,werecallsomestandardsets ofgeneratorsofS .
n
The following result is parallel to Theorem 16 with a
mimic proof. Proposition32 [16]
8
(1) S is generated by transpositions (switches). That Next, we give an alternative expression of θ ∈ Θ .
n [n;κ]
is, Denote
Sn =h(i,j)|1≤i<j ≤ni. (51) Mθ := Pπθ;D1θ,··· ,Dnθ , (55)
where (cid:0) (cid:1)
(2) S is generatedbytranspositions with1.That is,
n Dθ :=P , i=1,··· ,n.
i θ|π−1(i)
θ
Sn =h(1,r)|1<j ≤ni. (52) Denote the setofsuchMθ byM[n;κ].Thatis
Example33 RecallExample 20. Since M := M |θ ∈Θ . (56)
[n;κ] θ [n;κ]
(cid:8) (cid:9)
S =hσ =(1,2),σ =(1,3)i,
3 3 6
Let
to check whether a game G is ordinary symmetric, it is Mα =(Pπα;D1α,··· ,Dnα)∈M[n;κ],
veenroiufigchattioonchshecokws(3t8h)e,sfaomreσr3esaunldt,σa6ssohnolyw.nAinn(u3m9)e,rcicaanl Mβ = Pπβ;D1β,··· ,Dnβ ∈M[n;κ].
beobtained. ✷ Define aproduct(cid:16)onM[n;κ] as (cid:17)
M := P ;Dβ◦α,··· ,Dβ◦α , (57)
5 Name-irrelevant Symmetry and Its Linear β◦α πβ◦α 1 n
Representation (cid:16) (cid:17)
where
This section considers the strategy permutations. We P =P P
needsomepreparations. πβ◦α πβ πα (58)
Dβ◦α =DβDα , i=1,··· ,n.
i i π−1(i)
β
Definition34 Consider S , which can be considered
nκ
as a set of permutations on grouped (or double-labeled) Thenitis easyto verifythat M ,◦ is agroup.
[n;κ]
set
Letθ ∈Θ andset (cid:0) (cid:1)
[n;κ]
O := {(1,1),(1,2),·,(1,κ);··· ;(n,1),(n,2),·,(n,κ)}
P := P |θ ∈Θ . (59)
:= {O ,O ,··· ,O }. [n;κ] θ [n;κ]
1 2 n
Thenitis readyto verif(cid:8)ythat (cid:9)
θ ∈ S is called a block permutation if there exists an
nκ
object(i,α)suchthatθ((i,α))=(j,β),thenthenθ(i,·): P <GL(nκ,R).
[n;κ]
O →O is a bijection. The set of block permutations is
i j
denotedas Θ . Moreover,astraightforwardcomputationshowsthefol-
[n;κ]
lowingresult:
According to above definition, it is easy to verify the
following Proposition36 ϕ:P[n;κ] →M[n;κ] as
Proposition35 (1) θ ∈ Θ[n;κ], if and only if, there ϕ(P ):= P ;P ,··· ,P , θ ∈Θ .
exist π ∈S , which is a permutation on {O | i = θ πθ θ|π−1(1) θ|π−1(n) [n;κ]
θ n i (cid:18) θ θ (cid:19)
1,··· ,n},anddθ,··· ,dθ ∈S ,where (60)
1 n κ
is agroupisomorphism.
dθi =θ|π−1(i) :Oπ−1(i) →Oi, (53)
θ θ
We useanexampleto depictthis.
suchthat
Example37 Consider α = (1,6,3,4,2,5) ∈ S , and
9
Pθ =DθPπθ, (54) β =(1,2,3)(4,9,6,8)(5,7)∈S9.Assumetheyareacting
on
where
(O1,O2,O3):=(O1,O1,O1;O2,O2,O2;O3,O3,O3),
Dθ =diag(Dθ,Dθ,··· ,Dθ), 1 2 3 1 2 3 1 2 3
1 2 n
andDθ is thestructurematrix ofdθ,i=1,··· ,n. which is aset ofblocked elementswith n=3and κ=3.
i i
(2) Θ <S and Θ =n!(κ!)n. Then wediscuss thematrix expression
[n;κ] nκ [n;κ]
(cid:12) (cid:12)
(cid:12) (cid:12)
9
• It is easy to verify that both α, β ∈ Θ . Moreover, • SinceP , P ∈P , wehave
[3;3] α β [3;3]
π =(1,2)and π =(2,3).
α β
• Their structurematrices are
0 0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
     
Pα =0 1 0 0 0 0 0 0 0; Pβ =0 0 0 0 0 0 1 0 0 Pγ =PβPα =0 0 0 0 0 0 1 0 0∈P[3;3].
    0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  
     
0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
     
    0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0  
     
   
•
α1 =α|πα−1(1) =α|O2 =(1,2) It follows that
0 1 0
Pα1 =1 0 0
0 0 1 γ1 =γ|πγ−1(1) =γ|O2 =(1,3)
  0 0 1
α2 =α|π0α−10(21)=α|O1 =(1,3) Pγ1 =0 1 0
1 0 0
Pα2 =01 10 00 γ2 =γ|πγ−1(2)=γ|O3 =(1,2)
  0 1 0
αP3 ==αI|πα−1(3)=α|O3 =id Pγ2 =1 0 0 (61)
α3 3
0 0 1
 
β1 =β|πβ−1(1) =β|O1 =(1,2,3) γ3 =γ|πγ−1(3)=γ|O1 =(1,2)
0 0 1 0 1 0
Pβ1 =1 0 0 Pγ3 =1 0 0
0 1 0 0 0 1
 
β2 =β|π−1(2)=β|O3 =(1,2)  
β
0 1 0
• We already know that Θ is isomorphic to P .
Pβ2 =1 0 0 To check that M[n;κ] is a[nls;κo]isomorphic to them[,n;wκ]e
0 0 1 useformula (59) tocalculateD1γ,D2γ,and D3γ:
 
β3 =β|π−1(3)=β|O2 =(1,3,2)
β
0 1 0
Dγ = P =P P
Pβ3 =0 0 1 1 γ1 β|πβ−1(1) α|πα−1(πβ−1(1))
= P P
1 0 0 β|O1 α|πα−1(1)
• Consider γ =β◦α. Itiseasytocalculate that = Pβ|O1Pα|O1
0 0 1 0 1 0 0 0 1
γ =(1,8,4,3,9,6)(2,7,5).
= 1 0 01 0 0=0 1 0;
π =π ◦π =(1,3,2). 0 1 00 0 1 1 0 0
γ β α     
    
10