Fields, Ch is; Glazeb ook, James F.
A icle
Nash equilib ia and undecidabili y in gene ic physical
in e ac ions: A ee ene gy pe spec i e
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Fields, Ch is; Glazeb ook, James F. (2024) : Nash equilib ia and undecidabili y in
gene ic physical in e ac ions: A ee ene gy pe spec i e, Games, ISSN 2073-4336, MDPI, Basel, Vol.
15, Iss. 5, pp. 1-22,
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Ci a ion: Fields, C.; Glazeb ook, J.F.
Nash Equilib ia and Undecidabili y in
Gene ic Physical In e ac ions—A F ee
Ene gy Pe spec i e. Games 2024,15,
30. h ps://doi.o g/10.3390/
g15050030
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games
A icle
Nash Equilib ia and Undecidabili y in Gene ic Physical
In e ac ions—A F ee Ene gy Pe spec i e
Ch is Fields 1,* and James F. Glazeb ook 2,3
1Allen Disco e y Cen e , Tu s Uni e si y, Med o d, MA 02155, USA
2Depa men o Ma hema ics and Compu e Science, Eas e n Illinois Uni e si y,
Cha les on, IL 61920, USA; [email p o ec ed]
3Adjunc Facul y, Depa men o Ma hema ics, Uni e si y o Illinois a U bana–Champaign,
U bana, IL 61801, USA
*Co espondence: ields [email p o ec ed]
Abs ac : We s a om he undamen al p emise ha any physical in e ac ion can be in e p e ed as
a game. To demons a e his, we d aw upon he ee ene gy p inciple and he heo y o quan um
e e ence ames. In his way, we place he game- heo e ic Nash Equilib ium in a new ligh in so
a as he incomple eness and undecidabili y o he concep , as well as he na u e o s a egies in
gene al, can be seen as he consequences o ce ain no-go heo ems. We show ha games o he
gene ic imi a ion ype ollow a ci cula i y o idealiza ion ha includes he good egula o heo em,
gene alized synch ony, and undecidabili y o he Tu ing es . We discuss Bayesian games in he ligh
o Bell non-locali y and es ablish he basics o quan um games, which we ela e o local ope a ions and
classical communica ion p o ocols. In his ligh , we also e iew he a ionali y o gaming s a egies
om he playe s’ poin o iew.
Keywo ds: ee ene gy p inciple; Gödel’s heo em; Ma ko blanke ; measu emen ; Nash equilib ium;
quan um e e ence ame; Tu ing es ; undecidabili y
1. In oduc ion
Since i s incep ion by on Neumann and Mo gens e n [
1
], game heo y (GT) has
p o ided a e ile g ound o o mal s udies o algo i hmic decidabili y and undecidabili y.
The de ini ion o equilib ium o non-coope a i e games in [
1
] was es ic ed o wo-pe son,
ze o-sum games. J. F. Nash in [
2
,
3
] in oduced a concep o equilib ium applicable o a much
mo e gene al class o games, based on bes - esponse s a egies, ega dless o he numbe o
playe s and any bounds on he e en ual payo . Speci ically, he Nash equilib ium (NE)
comp ises a se o s a egies, one o each o he
n
game playe s, wi h he p ope y ha
each playe ’s choice o s a egy is hei bes esponse o he choices o he
n−
1 o he
playe s [
4
]. This p inciple, as applied o an a ay o gaming s a egies (mixed and pu e) o
‘bes esponse’ de e mined by p obabili y dis ibu ions, p o oundly in luenced he wo ld
o economic game heo y, and in ecogni ion o his i educibly o iginal idea, Nash was
awa ded he Nobel P ize in 1994 ( o he ounda ions and a ious in e p e a ions and
applica ions o he heo y; see, e.g., [4–6]).
To illus a e he no ion o an NE, conside a simple wo-playe game, he p isone ’s
dilemma (PD). Each ound is de ined by a payo ma ix in which he bes payo is ob ained
by de ec ing (D) when o e ed coope a ion (C), e.g.,
Bob sB
C D
Alice sAC(3,3) (0,5)
D(5,0) (1,1)
Games 2024,15, 30. h ps://doi.o g/10.3390/g15050030 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 30 2 o 22
In such a se ing, D is he dominan s a egy and (D, D) is he Nash equilib ium. The
op imal join s a egy is (C, C), in which bo h playe s coope a e, bu his is clea ly uns able.
While he NE in he PD as de ined abo e is he single poin (D, D), ede ining he
space o possible s a egies can ende he s uc u e o he NE as an a ac o in s a egy
space a bi a ily complex. We could, o example, allow Bob a con inuous dis ibu ion o
s a egies, each labeled wi h a complex numbe , and eplace he ‘C’ and ‘D’ column labels
in he able abo e wi h ‘
{c|c/∈ M}
’ and ‘
{c|c∈ M}
’, espec i ely, wi h
M
he Mandelb o
se , o wi h
M
he pullback a ac o o some andom dynamical sys em
D
on
C
o which
such an a ac o is de ined. While such o mal mo es a e somewha con i ed, he la e
may ealis ically e lec he si ua ion in e olu iona y game- heo e ic se ings [
7
], whe e
he numbe s o dis inc ac ions, and hence, “s a egies” a ailable o bo h an o ganism o
popula ion and i s en i onmen (comp ising o he o ganisms o popula ions) a e e y la ge,
bu only di icul - o-de ine se s o such ac ions ha e disce nibly dis inc sho - e m payo s.
A undamen al ques ion is posed by he exis ence, in all games, o NEs: o a gi en
game
g
and he dynamics
ϕ
on
g
, can i be shown whe he
ϕ
con e ges o an NE o
g
? Va ious esul s ha e been p o ed, many demons a ing noncon e gence o games
wi h pa icula s uc u es. Th ee no able examples om his li e a u e a e p oo s ha
(i) game dynamics ha a e uncoupled, in he sense ha each playe ’s choice o s a egy
depends only on hei own payo unc ion, a e gene ically no Nash-con e gen , e en o
poin a ac o s [
8
]; (ii) games exis o which all dynamics ail o con e ge o an NE [
9
];
and (iii) su icien ly
high-dimensional mul i-playe games can ha e chao ic a ac o s, and
hence, dynamics ha ne e con e ge [
10
], hough in some cases, simple heu is ics can
p edic long- e m ou comes [
11
]. The second o hese esul s was p o ed o degene a e
games, i.e., games in which mul iple mo es ha e he same payo ; howe e , he p oblem
o deciding whe he a game is degene a e is known o be NP-comple e excep in special
cases [
12
]. In ac , o ini e games he compu a ional complexi y o NE has be shown
o be ha o polynomial pa i y a gumen s on di ec ed g aphs (PPAD) p oblems [
13
]. As
in o mally explained in [
14
], PPAD is a class o all sea ch p oblems o which a solu ion is
gua an eed o exis o he same combina o ial eason ha e e y game has a leas one NE.
Resul s such as hese a e gene ally ob ained by ep esen ing games as dynamical
sys ems—hence, he idea o game dynamics—bu as migh be expec ed om Gödel’s he-
o em [
15
], any sel -consis en axioma ic sys em su icien o ep esen game dynamics is
p o ably incomple e, in he speci ic sense ha whe he a gi en combina ion o s a egies
cons i u es an NE is gene ically unp o able, e en o ini e games [
16
]. Gödel undecidabil-
i y ex ends beyond he ques ion o NE: he ques ion o s a egy con e gence in a spa ialized
p isone ’s dilemma (SPD)—e ec i ely, a ques ion o whe he a ini e, he e ogeneous cel-
lula au oma on (CA) will e ol e in o a homogeneous CA gi en a gene ic dis ibu ion o
ules/s a egies—has also been shown explici ly o be undecidable [
17
], as discussed in
de ail in Sec ion 4.3 below.
While he abo e kinds o ques ions a ise when games a e analyzed as abs ac s uc u es,
playe s o games also ace decidabili y ques ions on each ound o play. Is, o example,
one’s cu en ly coope a i e opponen in an i e a ed PD (IPD) playing i - o - a , o is hei
s a egy o de ec a e
n
ounds? This ques ion is clea ly undecidable a ound
k
o any
k<n
.
Indeed i he opponen is modeled as a black box, he undecidabili y o any ques ion abou
u u e s a egy is undecidable by Moo e’s heo em [
18
], which shows ha no ini e sample o
I/O beha io is su icien o cha ac e ize a gene ic black box. Tu ing’s imi a ion game [
19
],
o he wise known as he Tu ing es , p o ides a case in poin : wi h he assump ion ha he
in e oga o is a Tu ing machine (TM), he undecidabili y o whe he he esponden is human
o machine a e any n ounds o ques ions has been p o ed explici ly [20].
He e, we s udy hese ques ions o decidabili y and con e gence o an equilib ium in a
se ing in which he “game” is a gene ic physical in e ac ion. This e ec i ely gene alizes
Milno ’s idea o a “game agains Na u e” [
21
] o si ua ions in which he payo o Na u e
bea s no pa icula ela ionship o he “playe ’s” payo . To e ec his gene aliza ion, we
i s desc ibe gene ic physical in e ac ions in e ms o “s a egies” and he “mo es” ha
Games 2024,15, 30 3 o 22
hey gene a e. We hen ollow Feynman [
22
], and la e F is on and colleagues [
23
–
27
], in
employing he a ia ional ee ene gy (VFE) compu ed by each o he in e ac ing sys ems,
which ea s each sys em’s in e nal dynamics as a model o he o he and measu es i s
p edic ion e o , as an in e se payo measu e. When “compu a ion” is ea ed simply
as a unc ional in e p e a ion o a physical p ocess [
28
], his ep esen a ion is comple ely
gene ic in bo h classical [
25
–
27
] and quan um [
29
,
30
] se ings. Ha ing desc ibed games in
his gene ic se ing, we can appeal o undecidabili y esul s o “no-go” heo ems p o able
o gene ic physical sys ems o unde s and why, gi en ha NEs gene ically exis , ailu e o
con e ge o an NE can be expec ed o gene ic games.
We e iew he o malism needed o ep esen gene ic physical in e ac ions as games
in Sec ion 2, and show how no mal- o m games cons i u e a special case o such gene ic
games. We hen e iew a numbe o undecidabili y esul s p o able in his gene ic se ing
in Sec ion 3, and discuss how pa icula classes o games can be seen as p o iding a p io i
answe s o o he wise undecidable ques ions. We in e p e Nash’s heo em in e ms o
gene ic physical equilib ia in Sec ion 4, showing ha he NE can be iden i ied wi h ins ances
o gene alized p edic i e synch ony in he classical case and wi h en angled s a es in he
quan um case. We conside he di e ences be ween classical and quan um s a egies in
Sec ion 5, and also discuss he case in which he playe s manipula e a sha ed quan um
esou ce as pa o he game. We discuss some emaining issues in Sec ion 6and conclude
wi h some open ques ions in Sec ion 7.
2. Rep esen ing Gene ic In e ac ions as Games
2.1. Physical In e ac ion Is In o ma ion Exchange
Le
U
be an isola ed, ini e, physical sys em. We can, wi hou loss o gene ali y, ega d
U
as comp ising
m
bina y deg ees o eedom. I we conside hese deg ees o eedom o be
classical bi s, he possible s a es o Ua e simply he m-bi s ings wi h, e.g., he Hamming
dis ance as a me ic; i we conside he deg ees o eedom o be quan um bi s (qubi s),
we can ep esen
U
by a complex Hilbe space
HU
wi h dimension
dim(HU) = 2m.
He e,
we employ he quan um o malism o i s g ea e gene ali y. Following he de elopmen
in [
31
–
33
], we selec a Hilbe space decomposi ion
HU=HS⊗HE
in o a “sys em”
S
o
in e es —which we will ega d as he “playe ”—and i s “en i onmen ”
E
which plays
he ole o “Na u e” (
E
can also be in e p e ed as a collec ion o esou ces, e.g., he mo-
dynamic ee ene gy, s igme gic memo y, e c., wi h some deg ee o s ochas ici y). We can
hen w i e he in e ac ion be ween
S
and
E
as a Hamil onian (i.e., o al ene gy) ope a o
HSE =HU−(HS+HE),
whe e
HU
,
HS
, and
HE
a e he in e nal o sel -in e ac ions o
U
,
S
, and
E
, espec i ely. We a e in e es ed in he case in which
HSE
is weak enough ha
he join s a e
|SE⟩
(using Di ac’s no a ion) is sepa able, i.e., no en angled, o e he ime
in e al o in e es . This allows us o w i e HSE as
HSE =βkkBTk
N
∑
i
Mk
i, (1)
whe e
k=S
o
E
,
kB
is Bol zmann’s cons an ,
Tk
is empe a u e,
Mk
i
a e
N
He mi ian
ope a o s wi h eigen alues in
{−
1,1
}
, and
βk≥ln
2 is an in e se measu e o
k
’s he mody-
namic e iciency ha depends on he in e nal dynamics
Hk
. This in e ac ion
HSE
p o ides
a o mal desc ip ion o “playing” he game.
We now assume he holog aphic p inciple (HP), he claim ha no mo e in o ma-
ion can be ob ained abou a physical sys em han can be encoded on ha sys em’s
bounda y [
34
–
36
]; see [
37
] o de ails o how he HP applies in his se ing. The HP
gi es Equa ion
(1)
a s aigh o wa d opological in e p e a ion. Le
B
deno e he decompo-
si ional bounda y gi en implici ly by he Hilbe space ac o iza ion
HU=HS⊗HE
. Gi en
sepa abili y, i.e.,
|SE⟩=|S⟩|E⟩
, he en anglemen en opy
S(|SE⟩)
ac oss
B
is ze o. We
can, he e o e, ega d
B
as a holog aphic sc een, i.e., an ancilla y
N
-qubi a ay, sepa a ing
S om E, and depic HSE as in Figu e 1.
Games 2024,15, 30 4 o 22
Figu e 1. A holog aphic sc een
B
sepa a ing sys ems
S
and
E
wi h an in e ac ion
HSE
gi en by
Equa ion
(1)
can be ealized by an ancilla y a ay o nonin e ac ing qubi s ha a e al e na ely
p epa ed by
S
(
E
), and hen, measu ed by
E
(
S
). Qubi s a e depic ed as Bloch sphe es [
38
]. The e
is no equi emen ha
S
and
E
sha e p epa a ion and measu emen bases, i.e., quan um e e ence
ames, as desc ibed below. Adap ed om [33] Figu e 1, CC-BY license.
Figu e 1makes explici a undamen al obse a ion o Wheele [39]: quan um heo y
allows any in e ac ion be ween sepa able sys ems o be ea ed as communica ion. This
communica ion is bidi ec ional and indeed in o ma ionally symme ic by de ini ion;
S
and
E
in e ac by exchanging
N
-bi s ings. I hus ende s he idea o “passi e obse a ion”
unphysical, as e lec ed in Wheele ’s amous apho ism, “No ques ion? No Answe !”.
2.2. Ac ions Requi e Quan um Re e ence F ames
The bounda y
B
is he “boa d” (o “media”, o “channel”) on o h ough which he
game is played. This channel can be any in o ma ion-encoding space, e.g., he in e ne in
he case o ideo games, o a p i a e channel in a quan um c yp og aphy se ing whe e an
ea esd oppe e ec i ely plays a game wi h a subjec who has assumed o al p i acy [
40
].
Gi en
B
, we can desc ibe he mo es and he s a egies ha d i e hem. Each mo e has
wo componen s: i s
S
(
E
) p epa es each qubi on
B
in some s a e, a e which
E
(
S
)
measu es each qubi . P epa a ion and measu emen , i.e., obse a ion, o qubi
qi
a e dual
p ocesses [
41
] ca ied ou wi h he ope a o
Mk
i
. This ope a o is, e ec i ely, an ins ance
o he
z
-spin ope a o
sz
; i ac s on a qubi o p epa e i in ei he he
↑
(
+
1) o
↓
(
−
1) s a e.
Rende ing his ac ion well-de ined equi es speci ying a physical di ec ion ha coun s as
“up” (o
+z
); he opposi e di ec ion is hen “down” (
−z
). This speci ica ion is achie ed
by employing a quan um e e ence ame (QRF), a physical sys em ha p o ides a ixed,
e-usable s anda d o measu emen s [
42
,
43
]. In a ypical labo a o y, “up” is de ined by
he Ea h’s g a i a ional ield. The QRF is used o “make” he mo e; choosing a QRF o
employ co esponds, in his se ing, o choosing a s a egy.
When
S
encodes a bi s ing on
B
by p epa ing each o he
qi
in some pa icula s a e,
i mus selec a local
+zS
i
QRF o each o he
MS
i
, and hence, o each o he
qi
. When
E
hen eads a bi s ing om
B
, i mus also selec a local
+zE
i
QRF o each o he
qi
. I
S
and
E
a e o emain sepa able, hese choices o local QRFs, which co espond o choices
o basis
|i⟩
in Equa ion
(1)
, mus be made independen ly o “ eely” [
37
]; i
S
’s choice o
Games 2024,15, 30 5 o 22
basis depends on
E
’s o ice- e sa, hey a e en angled. I we iew
B
as a communica ion
channel, independen choice o basis by bo h
S
and
E
can be iewed as in oducing noise
in o he communica ion; in he ex eme case o
S
choosing
+zS
i=−(+zE
i)
o
qi
,
S
will
obse e
E
’s encoded bi as being lipped. As
U
is isola ed, he e is no classical sou ce o
noise in he sys em; he “noise” due o di e ences in QRF/basis choice be ween
S
and
E
is pu ely quan um. The “noise” caused by dis inc QRF choices gene a es s a is ical
su p ise, as discussed below; his is he su p ise induced when one’s opponen chooses a
di e en s a egy.
Mo es in a game can in ol e mo e han one bi ; in such cases, a mul i-bi s a egy is
needed. Single-qubi QRFs can be combined o c ea e QRFs ha ead o w i e pa icula
bi s ings encoded by subse s o qubi s on
B
. We can ep esen hese composi e QRFs by
hie a chies o maps ha o m ca ego y- heo e ic limi s and colimi s o e he ele an single-
qubi QRFs [
33
]; Figu e 2shows such a composi e QRF “a ached” o
B
. These cone–cocone
diag ams (CCCDs) [
44
,
45
] a e logically egula ed, causally and con ex -sensi i e, dis ibu ed
sys ems o in o ma ion low, as based on he heo y o [
46
]. They a e p o ably gene al
ep esen a ions o composi e QRFs, and a e p o ably equi alen o opological quan um
ield heo ies (TQFTs) o e he ele an subse s o qubi s [
47
,
48
]. As wi h single-qubi QRFs,
Sand Eha e independen , ee choice o composi e QRFs o deploy on B.
Figu e 2. “A aching” a CCCD o an in e sys em bounda y
B
depic ed as an ancilla y a ay o qubi s.
The ope a o s
Mk
i
,
k=S
o
E
, a e single-bi componen s o he in e ac ion Hamil onian
HSE
. The
node
C
is bo h he limi and he colimi o he nodes
Ai
; only le wa d-going (cocone-implemen ing)
a ows a e shown o simplici y. See [29–31,47] o de ails. Adap ed om [31], CC-BY license.
We can now ully desc ibe a mo e in a gene ic
S
-
E
game. Assuming
S
has he i s
mo e,
S
deploys some QRF/s a egy
QS
i
o encode a pa icula bi s ing on he subse
dom(
Q
S
i)
o qubi s on
B
, a e which
E
deploys some QRF/s a egy
QE
j
o ead a bi s ing
Games 2024,15, 30 6 o 22
om he subse
dom(
Q
E
j)
o qubi s. The u n hen e e ses, wi h
E
encoding and
S
eading.
No e ha no hing equi es ha
dom(
Q
S
i) = dom(
Q
E
j)
, and no hing equi es ha ei he
S
o
E
deploys he same QRF/s a egy on each mo e. In a gene ic game, bo h playe s can be
expec ed o deploy mul iple QRFs/s a egies, up o some limi imposed by hei a ailable
compu a ional esou ces.
2.3. VFE P o ides a Gene ic Payo Func ion
F om a global pe spec i e, he al e na ing mo es o he
S
-
E
game a e d i en by he
global sel -in e ac ion
HU
; in e posing
B
be ween
S
and
E
does no a ec his global
in e ac ion in any way. F om he pe spec i e o
S
o
E
, he mo es a e d i en by he
da a ha he o he pa y encodes on
B
. We can also say: hey a e d i en by how he
in e nal dynamics
HS
and
HE
espond o he pe u ba ions o he sys em s a es
|S⟩
and
|E⟩
,
espec i ely, by he in e ac ion HSE.
The ee ene gy p inciple (FEP), in oduced by F is on and colleagues [
23
–
27
], p o ides
a s a is ical physics ep esen a ion o he abo e ac s. In o mally, he FEP s a es ha
S
and
E
will emain dis inc only i hey emain su icien ly spa sely o weakly coupled
ha he bounda y be ween hem emains well-de ined[
25
]. I
U
is ea ed as a classical
causal ne wo k, he bounda y becomes a Ma ko blanke (MB), as o iginally de ined by
Pea l [
49
]. In his se ing, he FEP can be o mula ed as he equi emen ha s a es o
S
and
E
each emain in he icini y o some espec i e non-equilib ium s eady s a e (NESS) [
25
],
o ha almos all pa hs h ough he join space ha begin in
S
(
E
) emain in
S
(
E
) [
27
]. We
can, clea ly, e-in e p e hese classical s a emen s simply as equi ing ha
S
and
E
emain
unen angled, i.e., ha bo h ha e “in e nal s a es” ha con ibu e only negligibly o HSE.
The u ili y o he FEP as a guiding p inciple is ha i shi s he ocus om cha ac e iz-
ing he in e nal dynamics
HS
o
HE
, o cha ac e izing he unc ion ha each mus pe o m
o main ain he long- e m in eg i y o i s MB, i.e., o
B
. Again speaking in o mally,
S
’s
abili y o main ain a well-de ined bounda y—i
S
is an o ganism, o s ay ali e—depends on
keeping en i onmen al pe u ba ions o i s s a e ela i ely small. This can be o mula ed
in e ms o p edic ion and su p ise: wha e e
HS
does, i needs o minimize he su p ise
−lnp(b)
, whe e
b
is an MB o bounda y s a e (in he no a ion o Sec ion 2.1, o he holo-
g aphic sc een
B
) ela i e o a p edic ion
η
o
E
’s beha io . The a ia ional ee ene gy (VFE)
measu ed a Bis an uppe bound on su p ise ([25] Equa ion (2.3)):
F=DKL[qµ(η)|p(η)] −Eq[lnp(b|η)],
=DKL[qµ(η)|p(η|b)] −lnp(b),(2)
whe e
qµ(η)
is a a ia ional densi y o e p edic ed ex e nal s a es
η
pa ame e ized by
in e nal s a es
µ
, and
Eq
is an expec a ion alue ope a o pa ame e ized by he a ia ional
densi y
q
. No e ha he Kullback–Leible (KL) di e gence in he second equali y sco es he
(non-nega i e) p edic ion e o as a di e gence be ween he a ia ional p edic ion and he
ue dis ibu ion o e ex e nal s a es, gi en obse able MB/bounda y s a es. Because his
p edic ion e o is non-nega i e, he VFE u nishes a bound on su p ise, which becomes
exac when he p edic ion e o is ze o.
The i s equali y in Equa ion
(2)
exp esses VFE in e ms o complexi y minus accu acy
( i s and second e ms, espec i ely), whe e he e is an in ima e ela ionship be ween he
complexi y (i.e., di e gence be ween he a ia ional pos e io and p io ) and he algo i h-
mic complexi y o he gene a i e model as a desc ip ion o
E
. Heu is ically, his means
ha minimizing VFE p o ides he simples accu a e accoun o an en i onmen ha can
ne e be obse ed di ec ly. In u n, minimizing complexi y speaks o a desc ip ion o
he en i onmen in e ms o minimum message o desc ip ion leng hs, i.e., ha speaks o
uni e sal compu a ion [50–53].
We can now s a e he FEP as he claim ha
S
and
E
will emain dis inc sys ems o he
ex en ha hei espec i e dynamics
HS
and
HE
a e able o each minimize he VFE
F
measu ed
Games 2024,15, 30 7 o 22
a hei side o
B
, i.e., o hei own inpu s and p edic ions. No e ha as hey a e desc ibed by
Equa ion (2), Sand Ea e equally “in he game” o main aining hei mu ual dis inc ion.
In he Bayesian sense,
µ
can be seen as encoding a pos e io o e he ex e nal s a e
η
.
Minimizing he VFE leads o minimizing a p edic ion e o , a p ocess encompassed by a
gene a i e model (GM) implemen ed by he agen ’s in e nal dynamics. He e, i is ap o see
VFE minimiza ion, om an in o ma ional pe spec i e, as a he modynamically d i en p o-
cess, assimila ing he dynamics o an en i onmen whose he modynamic agency becomes
minimized while d i ing in e nal sel -o ganiza ion. Declining VFE is hen sel -e idencing
in he li e al sense o p o iding e idence o he implemen ing sys em’s con inuing exis-
ence [
25
]. F om a dynamical sys ems pe spec i e, his mechanism main ains he in e -
nal s a e
µ
in he neighbo hood o an NESS solu ion o he sys em’s densi y dynamics
as gi en abo e.
No hing in he abo e assumes any hing pa icula abou
HS
o
HE
beyond he unc ion
o main aining hei dis inc ness, o in quan um language, he sepa abili y o
S
and
E
.The
FEP he e o e desc ibes gene ic in e ac ions as a simple game wi h VFE minimiza ion as he payo
unc ion. The objec i e o he game is main aining a dis inc exis ence wi h sel -e idencing.
I is he mos basic game any sys em plays, and all sys ems play i all he ime.
2.4. No mal-Fo m Games A e Special Cases
I is implici in game heo y ha he playe s exis and a e dis inc om one ano he ;
no mal- o m games a e, he e o e, special cases cons uc ed on op o he gene ic “game o
main aining exis ence” desc ibed abo e. Mos games a e no , mo eo e , games agains all
o Na u e, i.e., all o
E
, bu games agains some componen s o
E
, wi h he o he componen s
being neu al, p o iding in as uc u e, o simply being neglec ed al oge he .
The QRF o malism illus a ed in Figu e 2allows us o say p ecisely wha is mean by
S
iden i ying and in e ac ing speci ically wi h some “sys em”
X
embedded in
E
. To emain
iden i iable by
S
o e ime,
X
mus ha e some componen
XR
wi h a s a e
|XR⟩
(o s a e
densi y
ρXR
) ha is in a ian unde he in e ac ion
HSE
. To accomplish he iden i ica ion
o
X
o e ime,
S
mus implemen some QRF
XR
speci ic o
XR
, i.e., ha p oduces an
ou come ‘+1’ when
|XR⟩
is de ec ed and ‘
−
1’ o he wise. To be seen by
S
as making “mo es”
o in e es ,
X
mus ha e some componen
XP
(a “poin e ” componen ) wi h a s a e
|XP⟩
ha a ies unde he in e ac ion
HSE
, and
S
mus implemen a QRF
XP
ha speci ically
de ec s his a iable “poin e s a e” [
29
,
31
,
33
]. The sec o
dom(XR)∪dom(XP)
o
B
is,
e ec i ely, he “image” o , o in he e minology o [54] he “icon” o , X o S.
Playing a mul i-mo e game wi h
X
equi es ha
S
has some memo y o p e ious
mo es. We could conside
S
o ha e a pu ely p ocedu al memo y, e.g., o implemen some
lea ning algo i hm ha upda ed i s decision algo i hms on e e y cycle, as is s anda d o
a i icial neu al ne wo ks [
55
]. Mo e in e es ing om he p esen pe spec i e is he case
in which
S
has decla a i e memo ies o pa icula e en s, and so can ace
X
’s beha io
explici ly h ough ime. W i ing and hen eading a decla a i e memo y
Y
equi es a
dedica ed QRF
Y
, as shown in Figu e 3. The p ocess o i e e sibly w i ing a decla a i e
memo y equi es bo h he modynamic ene gy om he en i onmen and an in e nally
coun ed ime, which we can ep esen by a coun e , o ime QRF, Gij [31,33].
In o de o pa icipa e in a wo-playe game wi h
X
, he e o e,
S
needs a QRF
X
; wo
memo ies
YS
and
YX
o sel - and
X
-ac ions, espec i ely, bo h o some empo al dep h
∆ S≥
1; a VFE measu e
FX
o e he sec o dom(
X
); and a decision unc ion
D
ha compu es
wha o do in he nex imes ep. Gene alizing o
k
playe s is o mally s aigh o wa d. We
can see in his a gene aliza ion o no mal o m, which eplaces he
k2−k
in e playe QRFs
wi h
k
“objec i e” playe s, he decision unc ions
Di
wi h se s o disc e e s a egies, and he
VFE measu es
Fij
wi h maps om selec ed ac ions o
R
. We can, in o he wo ds, see no mal
o m as an assump ion o bo h classical ealism— he playe s a e assumed o pe cei e and
ac wi hin an obse e -independen “game wo ld”—and disc e eness—se s o s a egies
and associa ed payo s a e assumed o be ini e, o e en ac ably small. These condi ions
o malize, as hey we e in ended o, ou in ui i e no ions o wha a “game” is.
Games 2024,15, 30 8 o 22
Figu e 3. Ca oon ep esen a ion o a sys em A ha deploys a QRF
X
( ed iangle) o measu e he
s a e o an ex e nal sys em
X
in i s in o ma ional en i onmen (i.e., a sec o
X
o i s bounda y
B
),
and hen, deploys a second QRF Y(g een iangle) o w i e he ou come o a memo y sec o Y. This
p ocess induces one “ ick” o an in e nal clock
Gij
ha de ines an in e nal elapsed ime
S
. The p ocess
is powe ed by a he modynamic loop om ( he modynamic ee ene gy in) and back o (was e hea
ou ) he physical en i onmen E. Adap ed wi h pe mission om [37], CC-BY license.
2.5. Example: The IPD as a P edic ion Game
The IPD again p o ides a simple example ha illus a es how no mal o m abs ac s
om he physical desc ip ion. The playe s—Alice and Bob—a e embedded o some o e all
sha ed en i onmen ha p o ides hem wi h esou ces, pa icula ly he modynamic ee
ene gy, ha allow hem o play he game i e a i ely. They a e assumed o ha e iden i ied
each o he as playe s, and o each be able o ocus hei a en ion exclusi ely on he o he .
The VFE o each playe , in o he wo ds, is assumed o be a unc ion only o wha he o he
playe does; sou ces o unce ain y in he gene al en i onmen a e iewed as negligible o
i ele an . Clea ly his is an abs ac ion o any eal se ing, e.g., he eal si ua ion o any
pai o o ganisms. The playe s a e also each es ic ed o one o he o he o he same wo
possible mo es on each cycle, again a conside able idealiza ion o mos eal se ings.
Each playe in he IPD has a model o he o he , o mo e speci ically, a mo e- o-mo e
upda ed p obabili y dis ibu ion o e he o he ’s nex mo es. The IPD is a dilemma
because hese p obabili y dis ibu ions a e ypically no unimodal, and e en i hey a e
unimodal—co esponding o a “ce ain” p edic ion— hey may no be p edic i ely accu a e.
Decoupling be ween subjec i e p obabili ies and ac ual ou comes is, o cou se, ypical in
eal si ua ions.
The mo es in he IPD can be iewed as p edic ions: a C mo e p edic s a C on he
opponen ’s pa , while D p edic s D o , mo e hope ully, C. Obse ing D a e C o C a e
D a e bo h su p ising, bu in opposi e di ec ions: D a e C is disappoin ing and dec e-
men s he model p obabili y ha he opponen is a coope a o , while C a e D inc eases
ha p obabili y. P edic ing good esul s om isky mo es is a key componen o ac i e
explo a ion o he en i onmen d i en by he FEP, o en called “epis emic o aging” [
25
].
While in he abs ac ed con ex he IPD i can appea decei ul, in o he si ua ions such
high- isk/high- ewa d beha io is a key indica o —and esul —o in insic mo i a ion [
56
].
I is, o example, a cen al d i e o science [57].
The IPD con e ges o (D,D) when he playe s can, e ec i ely, no longe lea n any mo e
abou each o he . This is an example o gene alized synch ony, he gene ic equilib ium s a e
o sys ems in e ac ing ia he FEP discussed in Sec ion 4below. While his NE exis s, when
i will be eached is unp edic able in eal ime by he playe s, as u he discussed below.
Games 2024,15, 30 15 o 22
E
[
99
,
100
]; see [
101
] o a comp ehensi e e iew. We can ep esen his p ocess as a game
simila o he quan um PD ske ched abo e, in which quan um s a es o he qubi
S
o
in e es a e he “playe s” and i s en i onmen
E
is he “a bi e .” Fo simplici y, we will
jus conside wo playe s, which we can ep esen as he s a es
| ↑⟩
and
| →⟩
, whe e
| →⟩ = (1/√2)(| ↑⟩+| ↓⟩).
When he game begins,
S
is in a cohe en s a e, so
| ↑⟩
and
| →⟩
ha e equal p obabili ies.
We hen in oduce in e ac ion wi h
E
, which we can ep esen by Equa ion
(1)
. Le his
in e ac ion begin by being e y weak and slowly inc ease in s eng h; we can hink o
TE
in Equa ion
(1)
inc easing slowly, o o he equency wi h which
E
in e ac s wi h
S
inc easing slowly. As equi ed by Equa ion
(1)
,
E
chooses a basis o he ope a o
ME
(we
need conside only one); we will assume
E
chooses
(↑
,
↓)
. This amoun s o
E
choosing he
payo ma ix: wi h he choice
(↑
,
↓)
,
| ↑⟩
ecei es one “ou come poin ” (i.e., he ou come
‘1’ is indica ed by a “de ec o click”) while
| →⟩
ecei es ze o (no click). When he game
begins,
| ↑⟩
and
| →⟩
ha e equal sco es, bu as
HSE
gains s eng h,
| ↑⟩
’s sco e slowly
inc eases while
| →⟩
’s sco e does no ; i we ead p obabili y as p opo ional o sco e and
no malize,
| ↑⟩
’s p obabili y inc eases while
| →⟩
’s dec eases. Asymp o ically,
| ↑⟩
has
p obabili y one and | →⟩has p obabili y ze o.
This p ocess o measu emen by
E
“ ewa ding” he s a e o
S
ha is an eigens a e o
E
’s chosen basis is called “einselec ion” [
101
], p ojec i e measu emen , o he “collapse o
he wa e unc ion.” When
E
is aken o be la ge enough o in e ac wi h many independen
obse e s—e.g., when
E
is he ambien pho on ield—
E
can be ega ded as “encoding”
he selec ed s a e o
S
wi h su icien edundancy ha all obse e s can de ec i ; his
compe i i e s a e encoding is called “quan um Da winism” and is p o essed o explain
he eme gence o a “public” quan um- o-classical ansi ion [
102
,
103
]. Like o he quan um
games, quan um Da winism is an LOCC p o ocol: he mul iple obse e s in e ac wi h
E
as a quan um esou ce while ag eeing classically o employ he same measu emen basis,
e.g., o employ ision and a common concep ion o wha coun s as an “objec ” [48].
5.3. Example: The Bell/EPR Game
Bell/EPR expe imen s a e he “gold s anda d” o de ec ing quan um en anglemen ,
and p o ide he concep ual basis o all o he quan um communica ion p o ocols [
104
,
105
].
In he expe imen ’s canonical o m, a cen ally loca ed sou ce dis ibu es an en angled
wo-qubi s a e—e.g., an en angled pho on pai — o wo obse e s, Alice and Bob, who
a e loca ed a equal dis ances om he sou ce, bu in opposi e di ec ions. Each obse e is
equipped wi h a spin-o ien a ion de ec o —e.g., a pola izing il e — ha can be se o any
di ec ion. Alice and Bob know he equency wi h which he sou ce emi s en angled s a es,
and independen ly se hei de ec ion di ec ions du ing he ime equi ed o en angled
s a es o each hei loca ions om he de ec o ; hei mu ual sepa a ion is chosen o bo h
allow his o happen and o p e en collusion be ween hem. They each eco d hei sepa a e
obse a ions o each s a e, and la e exchange hei esul s, ia a classical channel, in o de
o compu e he s a is ics o hei join obse a ions. A iola ion o Bell’s inequali y [
106
,
107
]
indica es de ec ion o en anglemen ; see, e.g., [
108
] o an in o mal discussion o bo h he
expe imen and he s a is ical analysis.
A Bell/EPR expe imen can be iewed as a wo-playe , limi ed-coope a ion game
agains Na u e, which bo h supplies he en angled s a es and ewa ds de ec ion o en an-
glemen [
109
]. The playe s, Alice and Bob, mus ag ee o make spin measu emen s on
e e y ound, bu a e o bidden om sha ing in o ma ion abou hei chosen measu emen
di ec ions. The sco e is he cumula i e alue o a join -measu emen s a is ic, e.g., he
Clause –Ho ne–Shimony–Hol (CHSH) s a is ic [
110
]. Alice and Bob can maximize hei
sco e by making hei measu emen s 45
deg
apa , up o he limi o Tsi elson’s bound,
2
√
2 [
111
]. As shown in [
48
], his Bell/EPR game is also an LOCC p o ocol: Alice and Bob
manipula e a quan um esou ce— he sequence o en angled s a es—and also communica e
classically, bo h o se up he expe imen and o analyze i s esul s.
Games 2024,15, 30 16 o 22
5.4. QRFs, Con ex uali y, and Asymp o ic En anglemen
I he playe s in a Bell/EPR game a e allowed o communica e hei de ec o se ings,
hey can employ hei sha ed en angled s a e as a secu e communica ion esou ce; his
is he basis o quan um communica ion and c yp og aphy p o ocols (see [
38
] o an
ex ensi e e iew). As en anglemen is, e ec i ely, sup aclassical co ela ion, playe s o
games in which en anglemen o o he non-local esou ces can be employed can achie e
payo s la ge han hose possible in classical e sions o hose same games. This leads
o he concep o a quan um (o no-signaling) Nash equilib ium [
109
] o games ha allow
use o quan um esou ces. No e ha as discussed in Sec ion 3abo e, sys ems canno , in
gene al, de e mine by obse a ion whe he hey a e en angled wi h hei en i onmen s [
59
];
hence, playe s canno de e mine by obse a ion whe he he game hey a e playing is
quan um o classical. As he asymp o ic s a e o he quan um FEP is en anglemen [
29
],
his immedia ely implies ha each playe canno de e mine whe he hey ha e eached a
quan um equilib ium wi h hei en i onmen , o ha e no .
While in classical uncoupled, o Bayesian, games he concep o “ u n aking” can o en
be elided, in quan um games his is gene ally no he case. The eason o his is clea in he
example o a Bell/EPR game: i Alice uses he basis
(↑
,
↓)
while Bob uses
(↗
,
↘)
, di e en
o de s o measu emen will esul in di e en p ojec ions o he sha ed en angled s a e. This
is an example o ope a o non-commu a i i y: measu emen s along di e en spin di ec ions
do no commu e, jus as measu emen s o posi ion and momen um do no commu e (bo h a e
examples o Heisenbe g’s unce ain y p inciple). Ope a o non-commu a i i y gene ically
induces quan um con ex uali y [
45
,
59
], de ined as he non-causal dependence o one measu e-
men ou come on wha o he s a e pe o med simul aneously [
107
,
112
,
113
]. Con ex uali y is
gene ally ecognized as a esou ce o bo h quan um in o ma ion and complexi y [114].
In he gene ic language employed in Sec ion 2, con ex uali y can be exp essed as non-
commu a i i y be ween QRFs [
45
,
48
,
59
]. I
Qi
and
Qj
a e non-commu ing QRFs wi h ou -
come p obabili y dis ibu ions
Pi
and
Pj
, espec i ely, when ac ing on some s a e
ψ
, hen
con ex uali y mani es s as he non-exis ence o he join p obabili y dis ibu ion
PiPj
(i.e., non-
commu a i i y implies iola ion o he Kolmogo o axioms; see [
115
] o a comp ehensi e
e iew). In he p esence o con ex uali y, join p obabili ies o e classical s a egies can ail o
be well de ined, hence he shi s in co esponding NEs canno be de ec ed [116].
6. Discussion
6.1. Ra ionali y
In a eas whe e GT is applied ex ensi ely, such as h oughou economics, logis ics, and
he beha io al sciences, i is an o e iding assump ion ha game playe s ac a ionally
owa ds op imizing hei e en ual payo s gi en he in luence o en i onmen al ac o s,
be hese local o global. I is a undamen al p inciple o GT ha when p edic ion and
a ionali y a e ap ly combined, an NE is a ained. Bu he ques ion emains, howe e , in
ou -o -equilib ium condi ions, can a ional playe s success ully p edic hei opponen s’
beha io ? We can easonably hink, and indeed i is hypo hesized in [
117
], ha he e
is inhe en ension be ween a ionali y and p edic ion when playe s a e unce ain o
hei opponen s’ modus ope andi owa ds payo . In ac , Fos e and Young [
117
] p o e
he exis ence o games in which i is impossible o pe ec ly a ional playe s o (e en
app oxima ely) p edic he u u e beha io o hei opponen s, ega dless o any lea ning
ules adop ed. The e a e se e al slan s on his, as in es iga ed in GT and economics (and
likely applicable elsewhe e oo). In [
118
] (Theo em 1), a ional agen s in he sense o
economics a e equi alen o sui ably indexed TMs. This implies ha decision p ocesses,
as implemen ed by such a ional agen s, a e equi alen o he compu ing beha io o a
sui ably indexed TM, and indeed, a ional choice, as unde s ood as maximizing choice, is
undecidable [
118
] (Theo em 2). We could ollow, e.g., Ewe ha [
119
] and conside a ional
playe s as basing hei decisions solely on he p o able implica ions o hei assump ions,
and he exis ence o undecidable s a emen s in GT as causing unde inabili y o a ional
concep s on he basis o dis inc ions in logical beha io , e.g., u h e sus p obabili y. So,
Games 2024,15, 30 17 o 22
his demands a de ini ion o a a ional s a egy: a s a egy is a ional i i is equi alen o a
bes eply o a Bayesian belie , o mo e gene ally, i i is a bes eply o some lexicog aphical
p obabili y sys em ha sa is ies ce ain consis ency condi ions, while no ing ha he e a e
games and s a egies o which i is undecidable i hey can a ise om pe ec ly a ional
beha io o om unique p edic ions, and gi en he i egula i y o belie s and assump ions
ha playe s may ha e abou each o he ’s mo i es [120].
Assump ions o a ionali y can in some cases be ela ed o deep assump ions in ma he-
ma ical logic. The e a e, o example, s a emen s abou wo-playe , ze o-sum games ha a e
undecidable; one such is analogous o he con inuum hypo hesis (CH) in
Ze melo–F aenkel
axiom o choice heo y, es ablished in [
121
]; we e e he eade o [
15
,
122
–
124
] o discus-
sion o his issue om he pe spec i e o Gödel’s heo em. As ano he example, one could
ake wo-pe son games wi h pu e s a egies and o
i=
1,2, conside playe
i
wi h belie
se
∆i(g)
, gi en a game
g
implemen ing an in ini e- eg ess logic, deno ed
EIR2
. Then, o
an unsol able game
g
, he heo y
(EIR2
,
∆i(G))
is incomple e [
125
]. As poin ed ou in [
125
],
we could hink o his as a case o sel - e e en iali y, bu he ac ual sou ce o incomple eness
is a disc epancy a ising om he collec i e independence o payo s, p edic ions, and
decision making.
The e is also he ques ion o o wha ex en (Hamil onian) chaos in luences playe s’
beha io owa ds whe he hey play a ionally o no . In ypes o simple games (such as
ock-pape -scisso ), no s a egy is seen o be dominan , and in pa icula , no pu e s a egy
o NEs can exis [
126
]. The main poin is ha , in such simple games, i is o en he case ha
ques ionable psychological heu is ics on behal o he playe s ine i ably supp ess any a emp
a a ional lea ning o he ex en o noncon e gence o an NE. The iewpoin o [
126
] is simply
his: i can be summa ized by saying ha chaos is a necessa y condi ion o in elligen playe s
o ail a aining o an NE, and he p esence o chaos sugges s ha playing a ionally is no
always a easible assump ion. O e all, ou accoun e lec s upon he p e alence o cogni i e
bias in many shapes and o ms, o he ex en ha , o he bes pa , humans a e ne e eally
close o being (Bayesian) a ional game playe s [
127
]. On he mo e echnical side, classical
economics asse s he exis ence o a leas one NE engaged in s a egies, bu gene ally, mul iple
equilib ia a e mo e likely o be he case. Capping his, i can be compu a ionally in ac able
o playe s o s ic ly con o m o GT p inciples in economics [14].
6.2. En opy o Quan um Games
Since in o ma ion pe mea es h ough his whole ci cle o ideas, le us commen on how
he s a is ical physics o in o ma ion/en opy accoun s o a ional choices o s a egies (o
he shee lack o hem) in quan um games. The amoun o in o ma ion ha a playe can
ob ain abou hei opponen depends on maximum/minimum en opy c i e ia, and he
a ionali y o he playe s in assimila ing his in o ma ion du ing he cou se o he game
is de e mined by he p e ailing en opy. To see his, le us ecall some basic concep s o
s a is ical physics. A (posi i ely alued) densi y ope a o
ρ
speci ies a mixed ensemble in
which each membe has an assigned p obabili y o being in a de e mined s a e. I s on
Neumann en opy,
S(ρ) = −T {ρln ρ}(6)
is a p obabili y dis ibu ion unc ion [
128
]. In [
129
],
S(ρ)
is maximized subjec o
δT (ρ) =
0,
and he in e nal ene gy cons ain δ(E) = 0, leading o
ρii =exp(−βEi)
∑kexp(−βEk)(7)
whe e
β
deno es a he modynamic pa ame e (see below). Wi hou he in e nal ene gy
cons ain
δ(E) =
0, we ha e
ρii =N−1
, whe e
N>
0 is he ‘popula ion size’. E ec i ely
β=T−1
, an in e se empe a u e, and om his he e a e wo cases [
129
]: (i)
β−→
0 is
he high empe a u e limi , in which a canonical ensemble becomes a comple ely andom
ensemble; and (ii)
β−→ ∞
is he low empe a u e limi , in which a canonical ensemble
Games 2024,15, 30 18 o 22
becomes a pu e ensemble whe e only he g ound s a e is popula ed. Seeing
β
as ela ed o
he empe a u e o a s a is ical sys em, i can on his accoun be in e p e ed as a measu e o
he a ionali y o he playe s in ques ion. So, om his s a is ical physics poin o iew, we
ha e he expec ed consequence:
high en opy ⇐⇒ low a ionali y in he playe s’ beha io .
6.3. Al e na i e Equilib ia
We ha e adop ed he NE as a ocal poin o his pape , as he NE has been an idealized
cen al concep o GT ha has p o ided long-s anding analy ic me hods o pa amoun
impo ance. We acknowledge, howe e , ha pa icula ypes o expe imen al da a, as
his pe ains o “noisy” games wi h possibly “i a ional” playe s, can escape his analysis,
and hence, o e he yea s al e na i e o ms o equilib ia ha e been in oduced. One o
hese is he quan al esponse equilib ium (QRE) o [
130
], which is mo e gene al han he
NE in ha i elaxes he assump ion o bes esponse ( o a ‘p obabilis ic’ esponse) and
allows noisy op imizing beha io while main aining consis ency o a ional expec a ions.
A aining a QRE en ails elemen s o Bayesian and s ochas ic choice (e.g., in biological
sys ems and neu oscience) [
131
,
132
]. Howe e , he QRE is seen o con e ge o an NE
as he quan al esponse unc ions s eepen owa d app oxima e bes esponse unc ions,
and in a heo e ical amewo k may no al e p edic ions as de e mined by NEs [
133
].
Fu he , he e a e expe imen s o which QRE solu ions do no ou pe o m hose o NEs
(see, e.g., [
134
]). O he al e na i e GT equilib ium heo ies such as noisy belie and andom
belie a e e iewed in [131].
7. Conclusions
We ha e shown ha he FEP desc ibes gene ic in e ac ions be ween physical sys ems
as games in which VFE minimiza ion is he payo unc ion. Physical in e ac ions a e
Bayesian games in which powe ul no-go heo ems es ic wha playe s can know abou
hei own s a egies as well as hose o hei opponen s in his con ex . We ha e in es i-
ga ed con e gence o gene ic games, and ha e shown ha he classical no ions o good
egula ion, iden ical synch oniza ion, and winning a GIG a e all, in p inciple, idealiza ions.
We ha e e iewed quan um games and shown how hey implemen LOCC p o ocols.
Undecidabili y is pe asi e in GT; indeed he esul s e iewed he e sugges ha all
decidable games in ol e he assump ion o knowledge ha canno , as a ma e o p inciple,
be ob ained by ini e obse a ion. Bo h he undecidabili y o he ame p oblem [
59
,
78
]
and he undecidabili y o whe he wo agen s a e deploying he same QRFs [
48
] s ongly
suppo his conjec u e. A gumen s o he e ec ha undecidabili y is ubiqui ous in physics
ha e p e iously been ad anced by Wol am [135] and Hawking [136].
As all physical sys ems a e, a bo om, quan um sys ems, gene ic physical in e ac ions a e
no jus games, bu quan um games. To he ex en ha hey in ol e classical communica ion—and
hey mus , o be conside ed games a all— hey a e ins ances o LOCC p o ocols. Thei dynamics
depend, he e o e, on he ex en o which quan um esou ces a e manipula ed in a coo dina ed
manne by he playe s. The ex en o which such coo dina ion can be ei he a anged ia classical
communica ion o deduced by obse a ion is undecidable [48].
We can conclude, he e o e, ha GT is much b oade in scope han i is o en ega ded
as being: GT’s scope includes mos , i no all, o physics. Physical sys ems can, he e o e,
be ega ded as game-playing agen s. Tha his should be he case ollows, indeed, om
Conway and Kochen’s amed “ ee will” heo em, which shows ha no physical sys em, a
any scale, can be ully desc ibed by any locally de e minis ic heo y [137].
Au ho Con ibu ions: Concep ualiza ion, C.F. and J.F.G.; o mal analysis, C.F. and J.F.G.; w i ing—o iginal
d a p epa a ion, C.F. and J.F.G.; w i ing— e iew and edi ing, C.F. and J.F.G. All au ho s ha e ead and
ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Games 2024,15, 30 19 o 22
Da a A ailabili y S a emen : All da a a e con ained in he pape .
Acknowledgmen s: The au ho s wish o hank wo anonymous e e ees o hei commen s and
sugges ions, which we e help ul owa ds he o e all p esen a ion o ideas.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
CA Cellula au oma on
CCCD Cone–cocone diag am
CH Con inuum hypo hesis
CHSH Clause –Ho ne–Shimony–Hol
EPR Eins ein–Podolsky–Rosen
FEP F ee ene gy p inciple
GIG Gene alized imi a ion game
GT Game heo y
HP Holog aphic p inciple
I/O Inpu /ou pu
IPD I e a ed p isone ’s dilemma
KL Kullback–Leible
LOCC Local ope a ions and classical communica ion
MB Ma ko blanke
NE Nash equilib ium
NP Nonde e minis ic polynomial
PD P isone ’s dilemma
PPAD Polynomial pa i y a gumen s on di ec ed g aphs
QRE Quan al esponse equilib ium
QRF Quan um e e ence ame
RNN Recu en neu al ne wo k
SPD Spa ialized p isone ’s dilemma
TM Tu ing machine
TQFT Topological quan um ield heo y
VFE Va ia ional ee ene gy
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