Strong isomorphism in Marinatto-Weber type quantum games

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by  Piotr Frąckiewicz

Strong isomorphism in Marinatto-Weber type quantum games
PiotrFra˛ckiewicz
InstituteofMathematics
PomeranianUniversity,Poland
January24, 2017
Abstract
Ourpurposeistofocusattentiononanewcriterionforquantumschemesbybringingtogetherthenotionsof
7
quantumgameandgameisomorphism. Aquantumgameschemeisrequiredtogeneratetheclassicalgameasa
1
0 specialcase. Now,givenaquantumgameschemeandtwoisomorphicclassicalgames,weadditionallyrequire
2 theresultingquantumgamestobeisomorphicaswell. Weshowhowthisisomorphismconditioninfluencesthe
players’strategysets. WeareconcernedwiththeMarinatto-Webertypequantumgameschemeandthestrong
n
isomorphismbetweengamesinstrategicform.
a
J

1 1 Introduction
2

] The Marinatto-Weber (MW) scheme introduced in [1] is a straightforward way to apply the power of quantum
T
mechanicstoclassicalgametheory. Inthesimplestcaseof2 2games,theplayersmanipulatetheirownqubits
G ofatwo-qubitstateeitherwiththeidentity1orthePauliopera×torσ . Therefore,ithasfoundapplicationinmany
x
. otherbranchesofgametheory: fromevolutionarygametheory[2],[3]toextensive-formgames[4]andduopoly
s
c examples[5], [6]. Inpaper[7]wepointedoutafewundesirablepropertiesoftheMWschemeandintroduceda
[ refinedquantumgamemodel.

1 ThoughitispossibletoextendboththeMWschemeandourrefinementtoconsidermorecomplexgamesthan
v 2 2, possible generalizationscan be defined in many differentways. A result concerning3 3 games can be
0 fo×undin[2]and[8].Theauthorsproposedsuitablethree-elementsetsofplayers’strategiestoob×tainageneralized
0
3 3 game. On the other hand, our work [9] providesanotherway to define players’strategy sets thatremains
1 ×
validforanyfiniten mgames.
6
×
0 Certainly,onecanfindyetotherwaystogeneralizetheMWscheme. Henceitwouldbeinterestingto place
. additionalrestrictionsonaquantumgameschemeandexaminehowtheyrefinethequantummodel. Inthispaper
1
weformulateacriterionintermsofisomorphicgames.Giventwoisomorphicgameswerequirethecorresponding
0
7 quantumgamestobeisomorphicaswell. If,forexample,twobimatrixgamesdifferonlyintheorderofplayers’
1 strategies, they describe the same problem from a game-theoretical point of view. Given a quantum scheme,
v: it appears reasonable to assume that the resulting quantum game will not depend on the numbering of players’
i strategiesintheclassicalgame.
X
r
a 2 Preliminaries
2.1 Marinatto-weber type quantum gamescheme
Inpaper[7]and[10]wepresentedarefinementoftheMarinatto-Weberscheme[1].Themotivationofconstructing
ourschemewastwofold. Ourmodelenablestheplayerstochoosebetweenplayingafixedquantumstrategyand
classicalstrategies.Thesecondaimwastoconstructtheschemethatgeneratestheclassicalgamebymanipulating
theplayers’strategiesratherthantheinitialquantumstate. Inwhatfollows,werecalltheschemeforthecaseof
2 2bimatrixgame,
×
l r
t (a ,b ) (a ,b )
00 00 01 01 , where(a ,b ) R. (1)
b (a10,b10) (a11,b11)! ij ij ∈
Definition1 Thequantumschemeforgame(1)isdefinedonaninnerproductspace(C2) 4bythetriple
⊗
Γ =(H,(S ,S ),(M ,M )), (2)
Q 1 2 1 2
where
1
H isapositiveoperator,
•
H =(1 1 11 11) 00 00 + 11 11 Ψ Ψ, (3)
⊗ −| ih | ⊗| ih | | ih |⊗| ih |
and
Ψ =α00 +β01 +γ10 +δ11 C2 C2 (4)
| i | i | i | i | i∈ ⊗
suchthat Ψ =1,
k| ik
S = P(1) U(3),i, j=0,1 , S = P(2) U(4),k,l=0,1 are the players’ strategy sets, and the upper
• 1 i ⊗ j 2 k ⊗ l
indicesnidentifythesubspaceoC2of(Cn2) 4onwhichtheoperoators
⊗
P = 0 0, P = 1 1, U =1, U =σ , (5)
0 1 0 1 x
| ih | | ih |
aredefined,
M andM arethemeasurementoperators
1 2
•
M =1 1 a (b )xy xy (6)
1(2) ⊗ ⊗ xy xy | ih |
x,Xy=0,1 
thatdependonthepayoffsa andb from(1).
xy xy
TheschemeproceedsinthesimilarwayastheMWscheme–theplayersdeterminethefinalstatebychoosingtheir
strategiesandactingonoperatorH. Asaresult,theydeterminethefollowingdensityoperator:
ρ = P(1) P(2) U(3) U(4) H P(1) P(2) U(3) U(4)
f i ⊗ k ⊗ j ⊗ l i ⊗ k ⊗ j ⊗ l
(cid:16) (cid:17) (cid:16) (cid:17)
11 11 U(3) U(4)Ψ ΨU(3) U(4) ifi= j=1,
= | ih |⊗ j ⊗ l | ih | j ⊗ l (7)
ij ij U(cid:16)(3) U(4)00 00U(3) U(4)(cid:17) ifotherwise.
| ih |⊗ j ⊗ l | ih | j ⊗ l
Next,thepayoffsforplayer1and2are (cid:16) (cid:17)
tr(ρ M ) and tr(ρ M ). (8)
f 1 f 2
Asitwasshownin[10],scheme(2)canbesummarizedbythefollowingmatrixgame
P(2) 1(4) P(2) σ(4) P(2) 1(4) P(2) σ(4)
0 ⊗ 0 ⊗ x 1 ⊗ 1 ⊗ x
P(1) 1(3) X X X X
0 ⊗ 00 01 00 01
P(1) σ(3)  X X X X 
PP(1(1011))⊗⊗⊗σ1((xx44))  XX101000 XX101111 ∆∆101000 ∆∆101111 , (9)
where
X =(a ,b ), for i, j=0,1
ij ij ij
∆ = α2X + β2X + γ2X + δ2X ,
00 00 01 10 11
| | | | | | | |
∆ = α2X + β2X + γ2X + δ2X , (10)
01 01 00 11 10
| | | | | | | |
∆ = α2X + β2X + γ2X + δ2X ,
10 10 11 00 01
| | | | | | | |
∆ = α2X + β2X + γ2X + δ2X .
11 11 10 01 00
| | | | | | | |
2.2 Strong isomorphism
Thenotionofstrongisomorphismdefinesclassesofgamesthatarethesameuptothenumberingoftheplayers
andthe orderofplayers’strategies. Thefollowingdefinitionsaretakenfrom[11](see also [12], [13] and[14]).
Thefirstonedefinesamappingthatassociatesplayersandtheiractionsinonegamewithplayersandtheiractions
intheothergame.
Definition2 LetΓ=(N,(S ) ,(u) )andΓ =(N,(S ) ,(u) )begamesinstrategicform.Agamemapping
i i N i i N ′ i′ i N ′i i N
f fromΓtoΓ isatuple f =(η∈,(ϕ) ∈),whereηisabijecti∈onfrom∈NtoNandforanyi N,ϕ isabijectionfrom
′ i i N i
∈ ∈
S toS .
i η′(i)
2
Ingeneralcase,themapping f from(N,(S ) ,(u) )to(N,(S ) ,(u) )identifiesplayeri N withplayer
i i∈N i i∈N i′ i∈N ′i i∈N ∈
η(i) and maps S to S . This means that a strategy profile (s ,...,s ) S S is mapped into profile
i η(i) 1 n 1 n
(s ,...,s )thatsatisfiesequations =ϕ(s)fori N. ∈ ×···×
′1 ′n ′η(i) i i ∈
The notion of game mapping is a basis for the definition of game isomorphism. Depending on how rich
structureofthegameistobepreservedwecandistinguishvarioustypesofgameisomorphism.Onethatpreserves
theplayers’payofffunctionsiscalledastrongisomorphism.Theformaldefinitionisasfollows:
Definition3 Given two strategic games Γ = (N,(S ) ,(u) ) and Γ = (N,(S ) ,(u) ), a game mapping
i i N i i N ′ i′ i N ′i i N
f =(η,(ϕ) )iscalledastrongisomorphismifrelatio∈nu(s)∈=u (f(s))holdsfor∈eachi ∈N andeachstrategy
i i∈N i ′η(i) ∈
profiles S S .
1 n
∈ ×···×
FromtheabovedefinitionitmaybeconcludedthatifthereisastrongisomorphismbetweengamesΓandΓ,they
′
maydiffermerelybythenumberingofplayersandtheorderoftheirstrategies.
Thefollowinglemmashowsthatrelabelingplayersandtheirstrategiesdonotaffectthegamewith regardto
Nashequilibria. If f isastrongisomorphismbetweengamesΓandΓ,onemayexpectthattheNashequilibriain
′
ΓmaptoonesinΓ under f.
′
Lemma1 Let f beastrongisomorphismbetweengamesΓandΓ. Strategyprofiles =(s ,...,s ) S S
isaNashequilibriumingameΓifandonlyif f(s ) S S′ isaNashequilib∗rium∗1inΓ. ∗n ∈ 1×···× n
∗ ∈ 1′ ×···× n′ ′
3 Application of game isomorphism to Marinatto-Weber type quantum
game schemes
ItisnothardtoseethatwecandefineawidevarietyofschemesbasedontheMWapproach.Wecanmodifyoper-
ator(3)andtheplayers’strategiestoconstructanotherschemestillsatisfyingtherequirementaboutgeneralization
oftheinputgame.Thefollowingexampleofsuchaschemeisparticularlyinteresting.
Letusconsideratriple
Γ =(H ,(S ,S ),(M ,M )) (11)
′Q ′ 1′ 2′ 1 2
withthecomponentsdefinedasfollows:
H isapositiveoperator,
′
•
H = 00 00 00 00 + 01 01 0 0 ρ + 10 10 ρ 0 0 + 11 11 Ψ Ψ, (12)
′ 2 1
| ih |⊗| ih | | ih |⊗| ih |⊗ | ih |⊗ ⊗| ih | | ih |⊗| ih |
where Ψ C2 C2 such that Ψ = 1, ρ and ρ are the reduced density operators of Ψ Ψ, i.e.,
1 2
ρ =tr|(Ψi ∈Ψ)an⊗dρ =tr (Ψ Ψk|),ik | ih |
1 2 2 1
| ih | | ih |
S = P(1) 1(3),P(1) σ(3),P(1) 1(3) andS = P(2) 1(4),P(2) σ(4),P(2) 1(4) aretheplayers’strat-
• 1′ 0 ⊗ 0 ⊗ x 1 ⊗ 2′ 0 ⊗ 0 ⊗ x 1 ⊗
egysents, o n o
M andM arethemeasurementoperatorsdefinedbyequation(6).
1 2
•
Itisimmediatethattheresultingfinalstateρ isadensityoperatorforeach(pureormixed)strategyprofile. For
′f
example,player1’sstrategyP(1) σ(3) andplayer2’sstrategyP(2) 1(4)imply
0 ⊗ x 1 ⊗
ρ = P(1) P(2) σ(3) 1(4) H P(1) P(2) σ(3) 1(4) = 01 01 1 1 ρ . (13)
′f 0 ⊗ 1 ⊗ x ⊗ ′ 0 ⊗ 1 ⊗ x ⊗ | ih |⊗| ih |⊗ 2
(cid:16) (cid:17) (cid:16) (cid:17)
Asaresult,theplayers’payofffunctifonsu andu givenbytr(ρ M )andtr(ρ M ),respectively,arewell-defined.
′1 ′2 ′f 1 ′f 2
It is also clear that scheme (11) produces the classical game in a similar way to scheme (2). The players play
the classical game as long as they choose the strategies P 1 and P σ . This can be seen by determin-
0 0 x
⊗ ⊗
ing tr(ρ M ) for each strategy profile and arranging the obtained values into a matrix. As an example, let us
′f 1(2)
determinetr(ρ M )forthefinalstateρ givenby(13). Let Ψ representageneraltwoqubitstate,
′f 1(2) ′f | i
Ψ =α00 +β01 +γ10 +δ11 . (14)
| i | i | i | i | i
Since
ρ =(α2+ β2)0 0 +(αγ +βδ )0 1 +(γα +δβ )1 0 +(γ2+ δ2)1 1,
1 ∗ ∗ ∗ ∗
| | | | | ih | | ih | | ih | | | | | | ih | (15)
ρ =(α2+ γ2)0 0 +(αβ +γδ )0 1 +(βα +δγ )1 0 +(β2+ δ2)1 1,
2 ∗ ∗ ∗ ∗
| | | | | ih | | ih | | ih | | | | | | ih |
theplayers’strategiesP(1) σ(3) andP(2) 1(4)generatethefollowingformofthefinalstate:
0 ⊗ x 1 ⊗
ρ = 01 01 (α2+ γ2)10 10 +(αβ +γδ )10 11 +(βα +δγ )11 10 +(β2+ δ2)11 11 . (16)
′f | ih |⊗ | | | | | ih | ∗ ∗ | ih | ∗ ∗ | ih | | | | | | ih |
(cid:16) (cid:17)
3
Hence
(tr(ρ M ),tr(ρ M ))=(α2+ γ2)(a ,b )+(β2+ δ2)(a ,b ). (17)
′f 1 ′f 2 | | | | 10 10 | | | | 11 11
Thevalues(tr(ρ M ),tr(ρ M ))forallstrategycombinationsaregivenbythefollowingmatrix:
′f 1 ′f 2
P(2) 1(4) P(2) σ(4) P(2) 1(4)
0 ⊗ 0 ⊗ x 1 ⊗
P(1) 1(3) X X ∆
0 ⊗ 00 01 02
P(1) σ(3)  X X ∆  (18)
P0(11)⊗⊗1(x3)  ∆1200 ∆1211 ∆1222 
where
X =(a ,b ), for i, j=0,1
ij ij ij
∆ =(α2+ γ2)X +(β2+ δ2)X ;
02 00 01
| | | | | | | |
∆ =(α2+ γ2)X +(β2+ δ2)X ;
12 10 11
| | | | | | | |
∆ =(α2+ β2)X +(γ2+ δ2)X ;
20 00 10
| | | | | | | |
∆ =(α2+ β2)X +(γ2+ δ2)X ;
21 01 11
| | | | | | | |
∆ = α2X + β2X + γ2X + δ2X . (19)
22 00 01 10 11
| | | | | | | |
Itfollowseasilythatmatrixgame(18)isagenuineextensionof(1). Althoughpayoffprofiles∆ , ∆ arealso
ij 22
achievablein(1),theplayers,ingeneral,arenotabletoobtain∆ whenchoosingtheir(mixed)strategies.
22
Tosumup, scheme(11)mightseemto beacceptableaslongasscheme(2)isacceptable. Matrixgame(18)
includes(1)anddependingontheinitialstate Ψ itmaygiveextraordinaryNashequilibria. Itisworthpointing
| i
outthattheNashequilibriain(18)correspondtocorrelatedequilibriain(1),(see[15]).Howeverscheme(11)fails
toimplytheisomorphicgameswhentheinputgamesareisomorphic. Wecanmakethisclearwiththefollowing
example.
Example1 Letusconsiderthegameof“Chicken"Γ andits(strongly)isomorphiccounterpartΓ ,
1 2
l r l r
′ ′
t (6,6) (2,7) t (2,7) (6,6)
Γ : , Γ : ′ . (20)
1 b (7,2) (0,0)! 2 b′ (0,0) (7,2)!
Thecorrespondingisomorphism f =(π,ϕ ,ϕ )isdefinedbycomponents
1 2
π(i)=i for i=1,2, ϕ =(t t ,b b),ϕ =(l r ,r l). (21)
1 ′ ′ 2 ′ ′
→ → → →
Set Ψ =(00 + 01 + 10 )/√3. Using(9)wecanwritequantumapproach(2)togames(20)as
| i | i | i | i
P(2) 1(4) P(2) σ(4) P(2) 1(4) P(2) σ(4)
0 ⊗ 0 ⊗ x 1 ⊗ 1 ⊗ x
P(1) 1(3) (6,6) (2,7) (6,6) (2,7)
0 ⊗  
ΓQ1: PPP(1(0(1111)))⊗⊗⊗σσ1(((xx443)))  (((776,,,226))) (((002,,,007))) (4((7531,,,225))32) (2((3032,,,430))31)  (22)
and
P(2) 1(4) P(2) σ(4) P(2) 1(4) P(2) σ(4)
0 ⊗ 0 ⊗ x 1 ⊗ 1 ⊗ x
P(1) 1(3) (2,7) (6,6) (2,7) (6,6)
0 ⊗  
It is fairly easy to seeΓthQa2t:gPPPa(1(0(1m111)))e⊗⊗⊗sσσ1(2(((xx4243)))) and((((0022,,,3007))))differ in(((776th,,,226e)))order o(2f((3032th,,,430e))31fi)rst tw(4o((5731s,,,t252ra))32t)egies and the second(t2w3o)
strategiesofplayer2.Thus,thegamesarestronglyisomorphic.Moreformally,onecancheckthatagamemapping
f˜=(η,ϕ˜ ,ϕ˜ ),where
1 2
ϕ˜ = P(1) 1(3) P(1) 1(3),P(1) σ(3) P(1) σ(3) ,
1 (cid:16) i ⊗ → i ⊗ i ⊗ x → i ⊗ x (cid:17) (24)
ϕ˜ = P(2) 1(4) P(2) σ(4),P(2) σ(4) P(2) 1(4) ,
1 k ⊗ → k ⊗ x k ⊗ x → k ⊗
(cid:16) (cid:17)
4
fori,k=0,1isastrongisomorphism.
Inthenextsectionweproveamoregeneralresultaboutscheme(2)).
Letusnowconsiderscheme(11). Matrix(18)intermsofinputgames(20)implies
P(2) 1(4) P(2) σ(4) P(2) 1(4)
0 ⊗ 0 ⊗ x 1 ⊗
P(1) 1(3) (6,6) (2,7) (42,61)
Γ′Q1: PP0(0(111))⊗⊗⊗σ1((x33))  (6(731,,42)32) (1(031,,40)32) (4(5332,,15)331)  (25)
and
P(2) 1(4) P(2) σ(4) P(2) 1(4)
0 ⊗ 0 ⊗ x 1 ⊗
P(1) 1(3) (2,7) (6,6) (31,62)
Γ′Q2: PP0(1(011))⊗⊗⊗1σ((x33))  (1(031,,40)32) (6(731,,42)32) ((2233231,,432331)) . (26)
With Lemma 1 we can show that games (25) and (26) are not isomorphic. Comparing the sets of pure Nash
equilibriainbothgameswefindtheequilibriumprofiles
(P(1) 1(3),P(2) σ(4)),(P(1) σ(3),P(2) 1(4)),(P(1) 1(3),P(2) 1(4)) (27)
{ 0 ⊗ 0 ⊗ x 0 ⊗ x 0 ⊗ 1 ⊗ 1 ⊗ }
inthefirstgameand
(P(1) 1(3),P(2) 1(4)),(P(1) σ(3),P(2) σ(4)) (28)
{ 0 ⊗ 0 ⊗ 0 ⊗ x 0 ⊗ x }
inthesecondone.
4 ApplicationofgameisomorphismtogeneralizedMarinatto-Weberquan-
tum game scheme
Additionalcriteriaforaquantumgameschememayhaveasignificantimpactonthewayhowwegeneralizethese
schemes. It can be easily seen in the case of the MW scheme [1] (or the refined scheme (2)), where the sets of
unitarystrategiesarefinite. TheMWschemeprovidesuswithaquantummodel,wherethestrategysetsconsistof
theidentityoperator1andthePaulioperatorσ . Underthisdescription,whatsubsetsofunitaryoperatorswould
x
besuitableforgeneraln mgames? Thecaseofa3-elementstrategysetcanbeidentifiedwithunitaryoperators
1 ,CandDactingonα0× +β1 +γ2 C3,where
3
| i | i | i∈
1 0 = 0 , C0 = 2 , D0 = 1 ,
3
1 |1i=|1i, C|1i=|1i, D|1i=|0i, (29)
3
1 |2i=|2i, C|2i=|0i, D|2i=|2i.
3
| i | i | i | i | i | i
This construction can be found in [2] and [8]. Another way to generalize the MW scheme was presented in
[9]. Having given a strategic-form game, we identify the players’ n strategies with n unitary operators V for
k
k=0,1,...,n 1. Theyactonstatesofthecomputationalbasis 0 , 1 ,..., n 1 asfollows:
− {| i | i | − i}
V i = i ,
0
V |ii=|ii+1modn ,
1
.|i | i (30)
.
.
V i = i+n 1modn .
n 1
− |i | − i
BothwaystogeneralizetheMWschemeenableustoobtaintheclassicalgame. Soatthislevel,neither(29)nor
(30) is questionable. If we seek otherproperties, we see that the MW scheme outputsthe classical game (or its
isomorphiccounterpart)whentheinitialstateisoneofthecomputationalbasisstates. Given(29)and(30),only
thelattercasesatisfiesthiscondition. FurtheranalysiswouldshowthattheMWschemeisinvariantwithrespect
tostronglyisomorphicinputgames.Itturnsoutthatneither(29)or(30)satisfiestheisomorphismproperty.
Example2 Letustakealookatthefollowing2 3bimatrixgames:
×
l m r l m r
′ ′ ′
t (4,8) (0,0) (8,8) t (0,0) (4,8) (8,8)
Γ: , Γ : ′ . (31)
b (0,4) (4,0) (8,0)! ′ b′ (4,0) (0,4) (8,0)!
5
ConsidertheMW-typeapproachesΓ andΓ togames(31)accordingtothefollowingassignement:
Q ′Q
l m r
t P P P
00 01 02 , whereP = j j j j . (32)
b P10 P11 P12! j1j2 | 1 2ih 1 2|
Then
Γ =(Ψ ,(D ,D ),(M ,M )), Γ =(Ψ ,(D ,D ),(M ,M )), (33)
Q | i 1 2 1 2 ′Q | i ′1 ′2 1′ 2′
where
M =4P +8P +4P +8P , M =4P +8P +4P +8P ,
1 00 02 11 12 1′ 01 02 10 12 (34)
M =8P +8P +4P , M =8P +8P +4P .
2 00 02 10 2′ 01 02 11
Settheinitialstate Ψ = (1/2)00 +(√3/2)12 C2 C3 andassumefirstthat D = D = 1 ,σ andD =
| i | i | i ∈ ⊗ 1 ′1 { 2 x} 2
aDn′2d=i={113,,C2,,aDn}dadsoiinn(g2s9i)m.iDlaertecramlciunliantgiotnrs(cid:16)(iUn1th⊗eUca2s)e|ΨoifhΨM|(Uw1†e⊗obUta2†i)nMi(cid:17)foreveryU1⊗U2 ∈{1,σx}⊗{13,C,D}
i′
1 C D 1 C D
3 3
1 (7,2) (2,5) (6,0) 1 (6,0) (5,2) (7,2)
Γ : 2 , Γ : 2 . (35)
Q σx (6,7) (5,6) (7,6)! ′Q σx (7,6) (2,0) (6,7)!
Ontheotherhand,replacing(29)by(30)givesD = D = 1 ,V ,V ,where
2 ′2 { 3 1 2}
1 0 0 0 0 1 0 1 0
13 = 0 1 0 , V1 = 1 0 0 , V2 = 0 0 1 . (36)
 0 0 1   0 1 0   1 0 0 
Then,wehave
1 V V 1 V V
3 1 2 3 1 2
1 (7,2) (0,3) (5,2) 1 (6,0) (4,2) (2,5)
Γ : 2 , Γ : 2 . (37)
Q σx (6,7) (4,6) (2,0)! ′Q σx (7,6) (0,1) (5,6)!
ThereisnopureNashequilibriuminthefirstgameof(35)and(37),whereastherearetwoNashequilibriainthe
secondgames.Asaresult,eachpairofthegamesdonotdetermineastrongisomorphism.
Example2showsthatplayers’strategysetsdefinedby(29)and(36)needtoberevisedinordertohaveageneral-
izedMWschemeinvariantwithrespecttotheisomorphism. Weshallstickforthemomenttoconsideringgames
(31). Let A ,A ,A ,A ,A ,A beplayer2’sstrategysetdefinedtobe
012 102 021 120 201 210
{ }
A 0 = 0 A 0 = 1 A 0 = 0 A 0 = 1 A 0 = 2 A 0 = 2 ,
012 102 021 120 201 210
A |1i=|1i A |1i=|0i A |1i=|2i A |1i=|2i A |1i=|0i A |1i=|1i, (38)
012 102 021 120 201 210
A |2i=|2i A |2i=|2i A |2i=|1i A |2i=|0i A |2i=|1i A |2i=|0i.
012 102 021 120 201 210
| i | i | i | i | i | i | i | i | i | i | i | i
Each A isapermutationmatrixthatcorrespondstoa specificpermutationπ = (0 j ,1 j ,2 j )of
j1j2j3 → 1 → 2 → 3
the set 0,1,2 . Note also that operators(29) and (36) are included in (38). Hence, the MW scheme with (38)
{ }
implies,inparticular,theclassicalgame. Wenowcheckifitoutputstheisomorphicgames. Sincetherearenow
sixoperatorsavailableforplayer2,theresultinggamemaybewrittenasa2 6bimatrixgamewithentries
×
tr (U U )Ψ Ψ(UT UT)M (39)
1⊗ 2 | ih | 1 ⊗ 2 i
(cid:16) (cid:17)
forU 1 ,σ andU A : π permutationsof 0,1,2 .Asaresult,weobtain
1 2 x 2 π
∈{ } ∈{ − { }}
A A A A A A
012 102 021 120 201 210
1 (7,2) (6,0) (4,2) (0,3) (5,2) (2,5)
Γ : 2 (40)
Q σ (6,7) (7,6) (0,1) (4,6) (2,0) (5,6)
x  
and
A A A A A A
012 102 021 120 201 210
1 (6,0) (7,2) (0,3) (4,2) (2,5) (5,2)
Γ : 2 . (41)
′Q σ (7,6) (6,7) (4,6) (0,1) (5,6) (2,0)
x  
6
Thegamesdeterminetheisomorphism f˜=(id ,ϕ˜ ,ϕ˜ ),where
N 1 2
ϕ˜ =(1 1 ,σ σ ),
1 2 2 x x
→ → (42)
ϕ˜ =(A A ,A A ,A A ,A A ,A A ,A A ).
2 012 102 102 012 021 120 120 021 201 210 210 201
→ → → → → →
UsingpermutationmatricesleadsustoformulateanothergeneralizedMWscheme.Forsimplicity,weconfine
attentionto(n+1) (m+1)bimatrixgames.
×
LetS bethesetofallpermutationsπof 0,1,...,n . WitheachπthereisassociatedapermutationmatrixA ,
n π
{ }
A i = π(i) fori=0,1,...,n. (43)
π
|i | i
WeletB denotethepermutationmatrixassociatedwithapermutationσ S . Given(n+1) (m+1)bimatrix
σ m
gameΓwedefine ∈ ×
Γ =(Ψ ,(D ,D ),(M ,M )), (44)
Q 1 2 1 2
| i
where
n m
Ψ = α j j Cn+1 Cm+1, D = A : π S , D = B : σ S ,
| i j1j2| 1 2i∈ ⊗ 1 { π ∈ n} 2 { σ ∈ m}
Xj1=0Xj2=0 (45)
n m
(M ,M )= (a ,b )P .
1 2 j1j2 j1j2 j1j2
Xj1=0Xj2=0
BeforestatingthemainresultofthissectionwestartwiththeobservationthattheMWschemeremainsinvariant
tonumberingoftheplayers.Considertwoisomorphicbimatrixgames:
t t t
0 1 m
···
s (a ,b ) (a ,b ) (a ,b )
0 00 00 01 01 0m 0m
···
s (a ,b ) (a ,b ) (a ,b )
Γ: s...n1 (an100,...bn100) (an111,...bn111) ··.··.··. (an1mm,...bn1mm) (46)
and
s s s
′0 ′1 ··· ′n
t (b ,a ) (b ,a ) (b ,a )
0′ 00 00 10 10 ··· n0 n0
t  (b ,a ) (b ,a ) (b ,a ) 
Γ′: tm...1′′ (b00m1,...a00m1) (b11m1,...a11m1) ··.··.··. (bnnm1,...annm1). (47)
Clearly,theisomorphismisdefinedbyagamemapping f = π,ϕ ,ϕ ,where
1 2
{ }
π=(1 2,2 1), ϕ (s )= s , ϕ (t )=t (48)
→ → 1 j1 ′j1 2 j2 ′j2
for j = 0,1,...,n, j = 0,1,...,m. The generalMW schemefor(46)is simplygivenby(44). ForΓ, we can
1 2 ′
write
Γ =(Ψ ,(D ,D ),(M ,M )), (49)
′Q | ′i ′1 ′2 1′ 2′
where
n m
Ψ = α j j Cm+1 Cn+1,
| ′i j1j2| 2 1i∈ ⊗
Xj1=0Xj2=0
D = B ,σ S ,D = A ,π S , (50)
′1 { σ ∈ m} ′2 { π ∈ n}
n m n m
M = b j j j j ,M = a j j j j .
1′ j1j2| 2 1ih 2 1| 2′ j1j2| 2 1ih 2 1|
Xj1=0Xj2=0 Xj1=0Xj2=0
Gamesdeterminedby(44)and(49)arethenisomorphic.Toprovethis,let f˜=(π,ϕ˜ ,ϕ˜ )beagamemappingsuch
1 2
that
π=(1 2,2 1), ϕ˜ : D D , ϕ˜ (A )= A , ϕ˜ : D D , ϕ˜ (B )= B . (51)
→ → 1 1 → ′2 1 π π 2 2 → ′1 2 σ σ
7
OnaccountofDefinition3wehave
f˜(A B )=ϕ (B ) ϕ (A )= B A . (52)
π σ 2 σ 1 π σ π
⊗ ⊗ ⊗
Asaresult,
u (f˜(A B ))=u (f˜(A B ))
′π(1) π⊗ σ ′2 π⊗ σ
=tr f˜(A B )Ψ Ψ f˜(A B )TM
π⊗ σ | ′ih ′| π⊗ σ 2′
(cid:16) (cid:17)
=tr (ϕ (B ) ϕ (A ))Ψ Ψ (ϕ (B )T ϕ (A )T)M
2 σ ⊗ 1 π | ′ih ′| 2 σ ⊗ 1 π 2′
(cid:16) (cid:17)
n m
=tr (B A )Ψ Ψ (BT AT) a j j j j
 σ⊗ π | ′ih ′| σ⊗ π j1j2| 2 1ih 2 1|
 nXj1=0mXj2=0 
=tr (A B )Ψ Ψ(AT BT) a j j j j
 π⊗ σ | ih | π ⊗ σ j1j2| 1 2ih 1 2|
=u(A B ). Xj1=0Xj2=0  (53)
1 π σ
⊗
Byasimilarargument,wecanshowthatu (f˜(A B )) = u (A B ). Wecannowformulatethefollowing
′π(2) π⊗ σ 2 π⊗ σ
proposition:
Proposition1 AssumethatΓandΓ arestronglyisomorphicbimatrixgamesandΓ andΓ arethecorresponding
′ Q ′Q
quantumgamesdefinedby(44). ThenΓ andΓ arestronglyisomorphic.
Q ′Q
Proof Let Γ and Γ be bimatrix games of dimension n m and let f: Γ Γ, f = (η,ϕ ,ϕ ) be the strong
′ ′ 1 2
× →
isomorphism. Since the MW scheme is invariant to numbering of the players, there is no loss of generality in
assumingη=id . Now,itfollowsfromDefinition3thatgamesΓandΓ differintheorderofplayers’strategies.
N ′
Let us identify players’ strategies (i.e., rows and columns) in Γ with sequences (0,1,...,n) and (0,1,...,m),
respectively. Then,we denotebyπ andσ thepermutationsofthesets 0,1,...,n and 0,1,...,m associated
∗ ∗
with the order of strategies in game Γ. A trivial verification shows tha{t the payoff} oper{ator M in}Γ may be
′ i′ ′Q
writtenas
M =(A B )M (A B )T, (54)
i′ π∗ ⊗ σ∗ i π∗ ⊗ σ∗
where A and B are the permutation matrices corresponding to π and σ . Define a game mapping f˜ =
π σ ∗ ∗
∗ ∗
(id ,ϕ˜ ,ϕ˜ ),where
N 1 2
ϕ˜ : A : π S A : π S , ϕ (A )= A A ;
1 π n π n 1 π π π
{ ∈ }→{ ∈ } ∗ (55)
ϕ˜ : B : σ S B : σ S , ϕ (B )= B B .
2 σ m σ m 2 σ σ σ
{ ∈ }→{ ∈ } ∗
Hence, f˜mapsA B toA A B B . Thus,weobtain
π σ π π σ σ
⊗ ∗ ⊗ ∗
u(f˜(A B ))=tr f˜(A B )Ψ Ψ f˜(A B ) T M
′i π⊗ σ (cid:18) π⊗ σ | ih |(cid:16) π⊗ σ (cid:17) i′(cid:19)
=tr (A A B B )Ψ Ψ(A A B B )T(A B )M (A B )T
π π σ σ π π σ σ π σ i π σ
∗ ⊗ ∗ | ih | ∗ ⊗ ∗ ∗ ⊗ ∗ ∗ ⊗ ∗
(cid:16) (cid:17)
=tr (A B )T(A A B B )Ψ Ψ(A A B B )T(A B )M
π σ π π σ σ π π σ σ π σ i
∗ ⊗ ∗ ∗ ⊗ ∗ | ih | ∗ ⊗ ∗ ∗ ⊗ ∗
(cid:16) (cid:17)
=tr AT A A BT B B Ψ Ψ ATAT A BTBT B M
π∗ π∗ π⊗ σ∗ σ∗ σ | ih | π π∗ π∗ ⊗ σ σ∗ σ∗ i
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
=tr (A B ) Ψ Ψ (A B )T M =u(A B ). (56)
π σ π σ i i π σ
⊗ | ih | ⊗ ⊗
(cid:16) (cid:17)
Thisfinishestheproof. (cid:4)
Note that operators (43) come down to 1 and σ for n = 1. Therefore, the original MW scheme preserves the
x
isomorphism.ThesameconclusioncanbedrawnfortherefinedMWscheme(2).
Corollary1 If Γ and Γ are strongly isomorphic bimatrix games and Γ and Γ are the corresponding games
′ Q ′Q
definedby(2). ThenΓ andΓ arestronglyisomorphic.
Q ′Q
Proof Let Γ and Γ be strongly isomorphic 2 2 bimatrix games. By Proposition 1 there exists a strong iso-
′
morphism f˜ = (id ,ϕ˜ ,ϕ˜ ) between the game×sΓ and Γ playedaccordingto (44)-(45). Given the quantum
1,2 1 2 Q ′Q
approach(2)toΓa{nd}Γ wedefineg˜ =(id ,ξ˜ ,ξ˜ ),where
′ 1,2 1 2
{ }
ξ˜ : S S , ξ˜ P(1) U(3) = P(1) ϕ˜ U(3) ,
1 1 → 1 1(cid:16) i ⊗ j (cid:17) i ⊗ 1(cid:16) j (cid:17) (57)
ξ˜ : S S , ξ˜ P(2) U(4) = P(2) ϕ˜ U(4) .
2 2 → 2 2 k ⊗ l k ⊗ 2 l
(cid:16) (cid:17) (cid:16) (cid:17)
8
Now,wehave
u g˜ P(1) P(2) U(3) U(4)
′i i ⊗ k ⊗ j ⊗ l
(cid:16) (cid:16) (cid:17)(cid:17)
=tr g˜ P(1) P(2) U(3) U(4) Hg˜ P(1) P(2) U(3) U(4) T M
(cid:18) (cid:16) i ⊗ k ⊗ j ⊗ l (cid:17) (cid:16) i ⊗ k ⊗ j ⊗ l (cid:17) 1′(cid:19)
=tr P(1) P(2) ϕ˜ U(3) ϕ˜ U(4) HP(1) P(2) ϕ˜ U(3) T ϕ˜ U(4) T M
(cid:18) i ⊗ k ⊗ 1(cid:16) j (cid:17)⊗ 2(cid:16) l (cid:17) i ⊗ k ⊗ 1(cid:16) j (cid:17) ⊗ 2(cid:16) l (cid:17) i′(cid:19)
=tr P(1) P(2) f˜ U(3) U(4) HP(1) P(2) f˜ U(3) U(4) T M . (58)
(cid:18) i ⊗ k ⊗ (cid:16) j ⊗ l (cid:17) i ⊗ k ⊗ (cid:16) j ⊗ l (cid:17) i′(cid:19)
ForfixedP P ,wecanwritetherightsideof(58)intheform
i k
⊗
tr(cid:18)|ikihik|⊗ f˜(cid:16)U(j3)⊗Ul(4)(cid:17)ρf˜(cid:16)U(j3)⊗Ul(4)(cid:17)T Mi′(cid:19), ρ=||Ψ00iihhΨ0|0,|, ((ii,, jj))=,((11,,11));. (59)
Byreasoningsimilarto(56)weconcludethat 
u g˜ P(1) P(2) U(3) U(4)
′i i ⊗ k ⊗ j ⊗ l
(cid:16) (cid:16) (cid:17)(cid:17)
=tr ik ik f˜ U(3) U(4) ρf˜ U(3) U(4) T M
(cid:18)| ih |⊗ (cid:16) j ⊗ l (cid:17) (cid:16) j ⊗ l (cid:17) i′(cid:19)
=tr ik ik U(3) U(4)ρ U(3) U(4) T M
(cid:18)| ih |⊗ j ⊗ l (cid:16) j ⊗ l (cid:17) i′(cid:19)
=u P(1) P(2) U(3) U(4) . (60)
i i ⊗ k ⊗ j ⊗ l
(cid:16) (cid:17)
WehavethusprovedthatΓ andΓ areisomorphic. (cid:4)
Q ′Q
Itisworthnotingthattheconversemaynotbetrue.GivenisomorphicgamesΓ andΓ ,theinputgamesΓandΓ
Q ′Q ′
maynotdeterminethestrongisomorphism.Indeed,bimatrixgames
l r l r
′
t (3,1) (0,0) t (4,0) (0,0)
and ′ (61)
b (0,0) (1,3)! b′ (0,0) (0,4)!
are notstronglyisomorphic. Howeverthe MW approach(with the initialstate (00 + 11 )/√2) to each one of
| i | i
(61)impliesthesameoutputgamegivenby
1 σ
x
1 (3,1) (0,0)
(62)
σx (0,0) (1,3)!
5 Conclusions
The theory of quantum games does not provide us with clear definitions of how a quantum game should look
like. In fact, only one condition is taken into consideration. A quantum game scheme is merely required to
generalize the classical game. As a result, this allows us to define a quantum game scheme in many different
ways. However, a wide variety of techniques to describe a game in the quantum domain can imply different
quantumgameresults. Therefore,itwouldbeconvenienttospecifythatsomequantumschemesworkundersome
furtherrestrictions. Wehavebeenworkingundertheassumptionthataquantumschemeisinvariantwithrespect
to isomorphic transformationsof an input game. We have shown that this requirementmay be essential tool in
definingaquantumscheme.Theprotocolthatreplicatesclassicalcorrelatedequilibriaisanexamplethatdoesnot
satisfyourcriterion.Therefineddefinitionforaquantumgameschememayalsobeusefultogeneralizeprotocols.
OurworkhasshownthatdependenceoflocalunitaryoperatorsintheMWschemeonthenumberofstrategiesin
a classical gameis notlinear. Infact, the generalizedapproachto n m bimatrixgamecanbe identifiedwith a
×
gameofdimensionn! m!.
×
Acknowledges
ThisworkwassupportedbytheMinistryofScienceandHigherEducationinPolandundertheProjectIuventus
PlusIP2014010973intheyears2015–2017.
9
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