Quantum Nash-equilibrium and linear representations of ortholattices Текст научной статьи по специальности «Физика»

Andrei A. Grib1 and Georgy N. Parfionov2

1 Department of Theoretical Physics and Astronomy, A.I.Herzen State Pedagogical University, Russia. E-mail: Andrei-Grib@mail.ru 2 St.Petersburg State University of Economics and Finances, St. Petersburg, Russia..

E-mail: gogaparf@gmail.com

Abstract A natural connection between antagonistic matrix games and ortholattices (quantum logics) is established. It is shown that the equilibrium in the corresponding quantum game defines the operator representation of the quantum logic. The conditions of the quantum equilibrium are formulated.

Keywords: quantum equilibrium, ortholattices, linear representations.

1. Introduction

Theorem of existence of equilibrium as a main result of the matrix game theory has an interesting story. The creator of the game theory von Neumann proposed in case of absence of the equilibrium in terms of usual acts to expand the strategic possibilities of the participants of the conflict si,...sn by including their probability combinations

p = pi ■ Si + ...+ pn ■ Sn, pi + ...+ pn = 1, pi,...pn > 0 (1)

- so called mixed strategies. It occurred that the mixed expansion of the game was sufficient for solution of any game in the dynamics-repeating the game many times. This idea was taken by von Neumann from the quantum mechanics in construction of which he actively participated at that time. In quantum theory besides pure states their convex combinations mixed states are also used. So the probability concept came to the game theory. Mixed strategies in game theory and mixed states in quantum mechanics have the same statistical interpretation. However in quantum theory pure states can be combined in a more general way - one can take not only convex, but any any nonzero complex linear combinations of pure states

0 = Xi ■ Si + ... + Xn ■ Sn (2)

- so called wave functions. In spite of the formal similarity of combinations (1) and (2) there is a serious distinction between them: no wave function is a mixed state. This leads to the idea about the secondary expansion of the strategic possibilities of the player-expansion of the set of initial pure strategies by new si_,...sn pure strategies - wave functions (2). Can it occur that quantum strategies - wave functions will be more successful than the classical mixed strategies? This idea leading to discovery of the quantum game theory was first realized by D.A.Meyer (Meyer, 1999), who demonstrated the very possibility of use of quantum strategies. At last in (L.Marinatto and T.Weber, 2000) and in (A.Grib, G.Parfionov, 2005) it was found that sometimes there exists the advantage of quantum strategies over classical ones

- their use can increase the profit.

2. Quantum strategies: the first steps

Introducing of quantum strategies into a game theory is based on the simple mathematical construction generalizing the idea of von Neumann. The set of initial pure strategies of the player si,...sn is taken as the orthonormal basis of the complex Hilbert space. The expanded set of pure strategies is formed by linear combinations of the form

0 = Xi ■ si + ... + Xn ■ sn |Xi |2 + ... + \Xn\2 = 1 (3)

It is important to note that such strategies are basically different from the strategies (1), used in game theories: they are not mixed. Really if the linear combination

0 would be a mixed strategy then Xi + ... + Xn = 1, Xi,...Xn ^ 0. But then A2 + ... + X2n < 1. So the idea of the game quantization is in expanding of the set of pure strategies. As to mixed strategies they initially don’t play any role in this scheme. At the first glance such expansion looks unreasonable because to any quantum strategy (3) automatically corresponds mixed strategy of the form

p^> = |Xi|2 ■ si + ...+ |Xn|2 ■ sn, (4)

where the square of moduli of the coefficients could be considered as probabilities of use of corresponding pure strategies. In paper (L.Marinatto and T.Weber, 2000)i this interpretation of quantum strategies was analyzed and it was shown that it gives nothing new in comparison with classical game theory. However one can understand it from general considerations. Any probability vector can be formally represented as a set of squares of moduli of the components of normalized complex vector. However in such formalism important properties of the probability amplitudes - phases are not used. That is why one must not expect any quantum effects. The advantages of quantum strategies and new effects are manifested only if one takes into account not squares of moduli of the coefficients but original coefficients. L.Marinatto and T.Weber found the advantages of quantum strategies for entangled states:

^ cjk sf 0 sf, ^ = 1 (5)

j,k

complex linear combinations of factorised states sA <g> sf.

The entangled state looks similar to the special kind of the mixed strategy used in classical game theory joint mixed strategy:

J^pjk sf0 sf, J2pjk = ^ pjk ^ 0 (6)

j,k j,k

expressing the correlated behavior of the players. So the possibility of getting the increase of the profit due to use of entangled states is not a surprise because such states as well as joint mixed strategies express coordinated behaviour. In the majority of papers on quantum game theory it is entangled state which is used for getting some new results. However can one get something new due to quantum effects in terms of independent choice of quantum strategies i.e. using only factorizable states?

1 See L.Marinatto and T.Weber ”A quantum approach to static games of complete information”

3. From quantum games to the quantum logic

Theory of quantum games is still at the initial stage and there is no unique system of notions and interpretations. This can be seen from the fact that each group of researchers write its own "Introduction to quantum games theory”. General feature of all concepts is the following.

A quantum game is a strategic use of a quantum system by participating parties who are identified as players. The players have the necessary means to perform actions on the quantum system and knowledge is shared among them about what constitutes a strategy. Often the strategy space is the set of possible actions that players can take on the quantum system. The players’ payoff functions, or utilities, are associated with their strategies. Payoffs are obtained from the results of measurements performed on the quantum system. A two-player quantum game, for example, is a set:

r = (H, p, Sa, Sb , Ha, Hb)

consisting of an underlying Hilbert space H of the physical system, the initial state p, the sets Sa and Sb of allowed quantum operations for two players, and the payoff functions Ha and Hb . In most of the existing set-ups to play quantum games the initial state p is the state of one or more qubits, or qutrits, or possibly qudits.

The explicit form of the initial state and the definition of payoff function is defined in different papers differently. In the majority of papers the procedures of quantization of classical games, i.e. the means of introducing quantum strategies into a classical game are proposed. We do not have the aim to invent the quantum version of the classical game to use the quantum strategies in the context of microscopic phenomena as the majority of researchers are doing. microscopic phenomena. In the basic paper (A.A.Grib, G.N.Parfionov, 2005)2 it was shown by us that quantum strategies naturally arise in description of macroscopic interactions, when the laws of classical Boolean logic are broken. The quantization of the antagonistic quantum

game of two persons is made due to the scheme similar to the procedure of canonical

quantization in mechanics when one takes instead of the classical Hamiltonian some self adjoint Hamiltonian operator. The rules of quantization are the following. If h = ||hjk|| the payoff matrix of the antagonistic game, then the payoff function of the first player can be written as:

h(a, 3) = ^2 hjk a3k (7)

j, k

where a, 3 - vectors, with components aj, 3k equal to zero or one and

E aj = E 3k = 1 (8)

jk

In case when the players act independently, the average profit is defined by the expression

(h) = E hjk pj qk (9)

j, k

which arises under the consideration of the mixed expansion of the game. Note that in this consideration there is no place for quantum strategies. The necessity

2 See A.A.Grib, G.N.Parfionov - Can a game be quantum?

of quantum strategies is arising when the components of vectors a, 3 satisfy other, different from the (8), relations. Then putting these or those equations for the components of the mentioned vectors lead to this or that quantum game. The form of these relations depends on the logical conditions of the participants of the conflict.

In paper (L.K.Franeva, A.A.Grib, G.N.Parfionov, 2007)3 examples of macroscopic game interactions with breaking of relations of classical logic are given. In this paper the simple version of these relations was studied:

ai + a3 = 1, a2 + a4 = 1, 3i + 33 = 1, 32 + 34 = 1 (10)

leading to breaking of the Kolmogorovian axiomatic of the probability:

Pi + P2 + P3 + P4 =2, qi + q2 + q3 + q4 = 2

One can notice that same relations are present in quantum mechanics of the electron. This occurs due to the fact that the measurement procedures are described in the physics of the microworld in terms of quantum logic.

4. Probability on ortholattices

In quantum theory the notion of probability is modified and it does not satisfy Kolmogorov’s axioms and it must be modified. The main axiom - axiom of additivity

a fl b = 0 =^ Pr(a U b) = Pr(a) +Pr(b)

is changed on the more weak axiom

a ± b =^ Pr(a V b) = Pr(a)+Pr(b)

based on the notion of orthogonality of events. Orthogonality is some specification of the notion of disjointness coinciding with the latter in case of Boolean logic. The notion of orthogonality implies consideration of new structures - ortholattices. An ortholattice is a mathematical structure L = {L, ^, ^, 1, 0} where

I. {L, ^, 1, 0} is a bounded lattice, where 1 is the maximum and 0 is the minimum. In other words:

(i) ” ^ ”isa partial order relation on L (reflexive, antisymmetric and transitive);

(ii) any pair of elements a, b has an infimum a A b and a supremum a V b such

that:

a A b ^ a,b and Vc: c ^ a,b ^ c ^ a A b;

a,b ^ a V b and Vc: a,b ^ c ^ a V b ^ c;

(iii) Va : 0 ^ a; a ^ 1.

II. the operation ”±” (called orthocomplement) satisfies the following conditions:

(i) (a^)^ = a (double negation);

(ii) a ^ b ^ b^ ^ a^ (contraposition);

(iii) a A a^ = 0 (non contradiction).

The notion of the ortholattice makes possible to formulate the definition of orthogonal events:

a _L b -<=> a < b ^

3 See L.K.Franeva, A.A.Grib, G.N.Parfionov - Quantum games of macroscopic partners

Boolean algebras are ortholattices and their orthocomplements are identical to complements. But, differently from Boolean algebras, ortholattices do not generally satisfy the distributive laws of A and V. There holds only

(a A b) V (a A c) ^ a A (b V c)

and the dual form

a V (b A c) ^ (a V b) A (a V c).

The lattice C(H) of all closed subspaces in a Hilbert space is a characteristic example of a non distributive ortholattice. Another example of the non distributive ortholattice - is the set P(H) of selfadjoint projectors in a Hilbert space with partial order relation: a ^ b ab = ba = a. The morphism a ^ ima, putting to projector its image realizes the isomorphism of these two ortholattices.

5. The specific of the quantum probability

The set of all probability measures on the ortholattice is described by relations:

Pi + P3 = 1, P2 + P4 = 1, P1,P2,P3,P4 > 0

These distributions can be parameterized by points (p1,p2) of the unit square of the coordinate plane 0 ^ p1 ^ 1, 0 ^ p2 ^ 1. However in quantum theory not all of these distributions have sense. It is due to the fact that probabilities arise to describe the result of measurement procedure, which supposes the act of preparation of the system in definite state. Quantum probability measures are constructed on the basis of the representations of the elements of an ortholattice a eL by means of projectors E(a) in a Hilbert space H such that

E(a A b) = E(a) ■ E(b), E(a±)= I - E(a), E(1) = I

for any elements a,b eL, that commute. In this case the probabilities of the elements a e L are calculated according to Prp(a) = tr(pE(a)) where p is a density matrix. If 0 e H is pure states, then p = \0){0\ and probabilities are calculated according to Prp(a) = (0\ E(a) \0), (0\0) = 1. It occurs that in case of the nondistributive ortholattice (10) the set of quantum probability measures, i.e. those obtained by the procedure of measurement does not full all the square.

Let us take aj ^ «j - some representation of the ortholattice (10) by projector operators. Then due to orthogonality relations a1± = a3, a2^ = a4, pairs of operators a1, a3 and a2, a4 commute, which is not true about operators a1, a2 and a3,a4. Really, if [a1 a2] = 0 and [a3 a4] = 0, then one has the relations a1± = a2, a3^ = a4 and the ortholattice (10) occurred to be Boolean, but this is not correct because of non Boolean relations (10). From this one sees that nontrivial commutation relations

[«1 S2] = 712 = 0, [S3 S4] = 734 =0 (11)

introduce auxiliary constraints for probabilities p1,p2 and p3,p4. In other words besides natural relations for probabilities, there are some hidden relations for probabilities induced by the very measurement procedure. To find these relations one must analyze the representations of the ortholattice.

6. The representations of the ortholattice ♦ + ♦

Let us fix some Hilbert space H as the space of representation. Taking self adjoint projection operators «j : H^H so that

«1 + «3 = I, «2 + «4 = I (12)

then for the normalized vector 0 e H the probability is pj = (0\aj\0). From this immediately follows

P1 + P3 = 1, P2 + P4 = 1

Other relations are defined by the choice of these or those commutation relations for pairs of operators a1, a2 a3, a4. From symmetry considerations let us agree that operators a1, a2, a3, a4, representing the elements of the lattice have the same rnnge rk aj = dimim aj = r.

Then due to a3 = I — a1 and [a1 a3] = 0, one obtains r = n — r, where n = dim H and so the space H - is even dimensional on C. Then by using the property that projectors of the same range are unitary equivalent one can find such operator u e SU(n), that the relation a2 = uta1u takes place. It is easy to see that then a4 = Ua3u. Depending on the choice of the corresponding unitary operator one gets different models. It is natural to consider among different possible representations of the ortholattice irreducible representations, when the family of projecting operators {aj } does not have nontrivial subspaces. It is easy to check that such representations are two dimensional on C. More easily are formed the representations on R. To find them introduce on TL some complex structure * - the morphism of TL on itself such that (x + y)* = x* + yk, (Ax)* = Ax*, x** = x.

The operator of the complex structure is involution * 2 = I, and its eigen spaces are +1 and —1. The eigen subspace L = {x eH\x* = x} is the two dimensional real linear space. The real part Re(x, y) of the Hermitian scalar product (x, y) in Hilbert space H transforms L into two dimensional Euclidean space. Consider the special case when the operators of the representation are invariant relative to some complex structure *, i.e each of the projective operators commutes with the operator of the complex structure: aj * = * aj. Then L is the invariant subspace for each operator aj and all these orth-projectors will have in space L range equal to one. Due to rk aj = 1 for all j. The element u e SO(2), exists that a2 = u-1 a1u. Calculation shows that in this case a4 = u-1 a3u.

The operator u is rotation on the plane. This rotation transforms the line £1 = im a1 into a line l2 = im a2 and correspondingly the line £3 = im a3 into a line^4 = ima4. Let 0 is the angle between the lines ^1, ^2, then u is the operator of rotation on the angle 0. Note that 0° ^ 0 < 180° - because by rotation on 180° every line is transformed into itself. The angle can be calculated (13), from the operators of the representation. Let us find the commutation relations for the operators of the representation. To do this consider s1 = 2a1 — I, s2 = 2a2 — I - the reflection operators relative to the lines £1,£2. One can see from elementary geometric considerations that the composition S1S2 is the rotation on the angle 20. So s1s2 = u2 Reflections are involutions, so s2 = s2 = I one obtains s2s1 = s-1s-1 = (s1s2)-1 = u-2. Calculating the commutator one obtains [s1s2] = 4[a1 a2] and so

22 u2 — u-2 =-----------

Figurel. The realization of the space of strategies as some real space of strategies

Let x € L some unit operator on the plane. Due to u2 being the rotation on the angle 20, one obtains from elementary geometric considerations that vector u2x-u~‘2x is perpendicular to the vector x and has the length 2 sin 20. So u2-u-2 = 2isin 20 where i - is the operator of rotation on the direct angle: i2 = —I. One can write with the help of this operator all nontrivial commutation relations:

[SiS2] = [S2S3] = [S3S4] = [S4S1] = — (,sin20 (13)

Construct matrix representation of the ortholattice (10). Let us take in the real space of states L the normalized eigen basis of the projector Si. Matrices of operators Si, S2 in this basis have the form

^ /1 0 \ ^ f cos2 0 sin 0 cos 0

Si \0 0/ , S2 \sin 0 cos 0 sin2 0

where 0 - is the angle between l1,l2. Calculation shows that

[Si, S2] = ^tsin20,

Now consider the general complex case. Take f €H- the normalized eigen vector of the projection operator S2, with the eigen value 1. Then the projector S2 = \f )(f |. Expanding the vector f in eigen basis of the operator S1, one obtains the coordinate representation f \ = eiLp • (cos 0, e-iX sin 0) where p and A are some real parameters. From this one gets

cos2 0 e lX sin 0 cos 0 \ ^ ^ , _ 1

e*Asin0cos0 sin2 0) ’ ^aiQ!2J 2

where

(A) '0 e -

i(A) =

In the result one obtains commutation relations

[SiS2] = [S2S3] = [S3S4] = [S4S1] = —t(A) sin 20 (14)

similar to those obtained for the real version (13). From i(A)J(A) = I, the operator i(A) - is unitary. Due to i(A)2 = —I, One sees,that i(A) is th "rotation” in two dimensional complex space on the direct angle. There is a set of such rotations.

To classify them note that operators k(A) = —i(A)i(0) form the one parametric subgroup SU(2): K(a + 3) = K(a) • k(3) In terms of the double list enveloping SU(2) ^ SO(3) the operators i(A) are classified as rotations on the direct angle around the axices in R3.

7. The quantum strategies on the ortholattice ♦ + ♦

In case of the irreducible representation of the ortholattice the coordinate representation of the normalised vectors in the eigen basis of the operator S1 has the form (0\ = (cos£, sin£). Calculating the probabilitiesp1 = (0\S1\0), p2 = (0\S2\0), and using the matrix representation of operators S1, S2, one finds that p1 = cos2 £,

0 0 o o o

p2 = cos2 £ cos2 0 + 2 cos £ cos 0 sin £ sin 0 + sin2 £ sin2 0 = cos2(£ — 0)

where 0° ^ 0 < 180° - the parameter of the representationof the ortholattice, corresponding to the given commutation relation. Excluding from these equations the parameter of the state £ one obtains the equation of the constraint for p1,p2:

p2 — 2p1p2 cos 20 + p2 — 2p1 sin2 0 — 2p2 sin2 0 + sin4 0 = 0 (15)

The corresponding quadratic form has the positive discriminant S = sin2 20, so this equation defines the ellipse.

This ellipse is included into the square 0 ^ p1, p2 ^ 1. For 0 = 45° it is a circle. For 0 close to 0° and to, 90° it is dealated in the direction of diagonals. Write the ellipse equation in the canonical form using the affine transformation

p1 + p2 = cos2£ + cos2(£ — 0) = 1 + cos(2£ — 0)cos0

—p1 + p2 = cos2£ — cos2(£ — 0) = sin(2£ — 0)sin0 The change of variables

— 1+p1 + p2 —p1 + p2

XI = --------------, x2 = ---------------

cosO sinO

gives the parametric equations x1 = cos(2£ — 0), x2 = sin(2£ — 0) and in the new coordinate system the equation of ellipse becomes the equation of the circle

x1 + x2 = 1.

Pay attention to the important property that the transformation £ ^ £ + 180° means going from the wave function 0 to the wave function —0 and so the quantum state is not changed(only the phase factor is changed). This agrees with the equation of the circle, invariant under the shift £ on 180°. This makes possible to represent the quantum strategy by the unit vector on the plane or by a point on the unit circle. It is easy to see that the correspondence 0 ^ (x1,x2) is the double list envelope of the space of quantum strategies by the set of wave functions.

8. Quantization of classical games

Let us apply to the expression of the payoff function of the matrix game

h(a, 3) = ^ hjk aj3k j, k

the quantization procedure, changing Boolean values a.j, 3k on the projection operators aj ^ Sj, 3k ^ Sk in Hilbert spaces Ha , Hb, describing the players strategies Sj : Ha —► Ha, Sk : Hb —► HB In the result one obtains the self adjoint payoff operator

h = ^ hjk Sj ® 3k (16)

j, k

The players strategies are described by the wave functions and the game situations are represented by the resolved vectors of the tensor product Ha ® Hb .

Let p € Ha, 0 € Hb - are quantum strategies. Calculation gives the following expression for the average profit in the factorized state p <g> 0:

(h) = E hjk {p ® 0\Sj <g> 3k\p ® 0) = E hjk (p\Sj\p){0\3k\0) j, k j, k

Denoting pj = (p\Sj\p), qk = (0\Sk\0) one comes to the same as in the classical game theory expression for the average profit

(h) = E hjk pj qk

j, k

with the difference that pj, qk, are in general not classical probabilities. The operator representation makes possible to consider as games with classical Boolean probabilities as their generalisations on ortholattices. It needs only to choose properly the projection operators. If one takes the operators Sj, Sk so that

S1 + S2 + S3 + S4 = I, [SjSk] = 0, j = k (17)

S1 + 32 + 33 + S4 = ^ \3jSk] = 0 j = k (18)

then one obtains the classical game with usual mixed strategies constrained by the conditions

p1 + p3 + p2 + p4 = 1, q1 + q3 + q2 + q4 = 1

If one takes the projection operators Sj, Sk so that

S1 + S3 = I, [<21(23] =0, a.2 + S4 = I, \a2<24 ] = 0 (19)

^1 + S3 = ^ \/?1S3] = 0 S2 + ^4 = ^ \^2^4] = 0 (20)

then one comes to limitations:

p1 + p3 = 1, p2 + p4 = 1, q1 + q3 = 1, q2 + q4 = 1 (21)

At last one of the players can be classical but the other can follow the quantum

logic. In other words the quantization of the game is defined by the choice of these or those commutation relations.

9. Search of equilibria

Results of this part generalize the previous results obtained for special cases. By the linear transformation of variables

2p = xMg + e, 2q = yMT + e (22)

where p = (P1,P2), q = (q1,q2), x = (x1,x2^ y = (y1,y2)

MY

cos y cos y sin y sin y

e =(1, 1)

the equations of constraint can be written as x1 + x2 = 1, y2 + y^ = 1. So each player chooses some point on the unit circle and the quantum game occurs to be the classical game on torus.

Let h = \ \hjk \\ - the matrix of the antagonistic game of two persons each having four strategies and p = (p1,p2,p3,p4), q = (q1,q2,q3,q4) are quantum strategies of players in the probability representation. Then the average quantum profit is (h) = p h q1. Due to relations (21), the four dimensional vectors p, q can be linearly expressed through two dimensional vectors p, q:

p = pZ + k, q = qZ + k

where

Z =

The average profit in new variables is

10 —1 0

0 1 0-1

k = (0,0,1,1)

(h) = (pZ + k) h (Z1q1 + k1) = pZhZ1q1 + pZhk1 + khZ1 q1 + khk1

Changing the variables (22) and putting away the scale factors and additive constants one gets

(h) - (xMg + e)ZhZt(MTtyt + et) + 2(xMe + e)Zhkt + 2khZt(MTtyt + et)

Putting additive constants once more one has

(h) - xMgZhZtMTtyt + xMgZh(Ztet + 2kt) + (eZ + 2k)hZtMTtyt

Then calculate eZ = (1,1, —1, —1), w = eZ + 2k = (1,1,1,1). Use the notations:

C = ZhZ1', A = MeCMTt, a = whtZt, b = whZt, u = aMgt, v = bMTt

Then (h) — xAy1 + xu1 + vy1. To find Nash equilibria use the following criterium of the equilibrium: (G.Parfionov, 2008)4 Pair (x, y) is the Nash equilibrium if and only if for some A, p ^ 0 one has the equalities :

yA1 + u = Ax, xA + v = py (23)

Search of eigen equilibria corresponds to finding such values of parameters of the representation d,r, for which for some values s,t one has the relations:

vA1 = su, uA = tv (24)

See G.Parfionov - Multiple Nash-equilibrium in Quantum Game

4

Supposing u, v = 0 consider vectors x system (23) one gets

u/\u\, y = v/\v\. Putting them into the

yA^ + u = —-(s + \v\)x, xA + v= -—-At + \u\)y

So if the system of equations (24) for d,r,s,t has the solution with limitations s + \v\ ^ 0, t + \u\ ^ 0, then the pair (x, y) is the Nash equilibrium. Let us analyze the system of equations (24). Using evident manipulations one gets

tbMT 1 = aMe 1Me CMT 1,

From this one has

However

aMe 1Me C = tb, bMT 1MT C = sa

(25)

Ny

1 cos 2y

cos 2y 1

It is important that the matrix C and vectors a, b from angle parameters 0, t are not dependent.

Supposing matrix C to be non-degenerate take f = bC-1, g = aC-1 The the equations (25) can be written as aNe = tf, bNT = sg. From this one obtains the relations a1 + a2 cos 20 = tf1, a1 cos 20 + a2 = tf2. Excluding the parameter t, one obtains the equation for defining the parameter of the representation 0 and t

a1 + a2 cos 20 a1 cos 20 + a2

/1 bt + b2 cos 2r _ gi

f2 ’ 6i cos 2r + b2 g2

But the values a1,a2,b1,b2,f1,f2,g1,2 depend only on the elements of the payoff matrix h = \\hjk\\, so we can formulate the final result:

In case of the equilibrium the linear representations of the lattices of the players of the game are totally defined by the elements of the payoff matrix.

So one comes to a conclusion that to define the quantum game one needs only the payoff matrix and logical relations forming the corresponding ortholattice.

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