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4Quantum Games of Macroscopic Partners
Ludmila Franeva1, Andrei Grib2, George Parfionov3
1St. Petersburg State University of Economics and Finances, e-mail address: [email protected] 2 Russian State Herzen Pedagogical University e-mail address: Andrei- [email protected] 3 St.Petersburg State University of Economics and Finances, e-mail address: [email protected]
Abstract. Examples where quantum Nash Equilibrium has been found are considered. It turned out that under certain conditions opportunist behavior leads to a greater gain than non-opportunist one. Besides, the analysis of the structure of the quantum game has shown that it is isomorphic to a sum of two inter-connected classical games. This link named “quantum cooperation” is expressed by means of a non-linear relation between the probabilities of the choice of corresponding pure strategies. The difference between the quantum cooperation and the usual correlation has been demonstrated. Mixed strategies for participants of these games are calculated using probability amplitudes according to the rules of quantum mechanics in spite of the macroscopic nature of the game and absence of the Planck’s constant. Possible role of quantum logical lattices for existence of macroscopic quantum equilibria is discussed. The games modelling opportunist behavior have the structure that is characteristic of the description of microparticles with a spin equal to i and 1 when additional variables are measured.
Introduction
It often happens that mathematical structures find natural applications somewhere out of their origination. The formalism of quantum mechanics is not an exception of this rule. Applied initially to the microworld, this formalism can be used for modeling some macroscopic interactions with an element of indeterminacy. Quantum game theory based on quantum theory of micro-particles described by quantum physics is a well-developed theory today. The concept of quantum strategy proposed by J.Eisert, M.Wilkens, M.Lewenstein [Eiser, 1999] and D. Deutsch [Deutsh, 1999] has been effectively applied to the game theory [Barnum, 1999], [Piotrowsky, 1999],
[Polley, 1999], [Flitney, 2002]. This approach has enriched our understanding of the Nash equilibrium and made it possible to solve the Prisoner’s Dilemma [Dilemma, 2004], as well as the problem of the choice of equilibrium in some games with multiple Nash Equilibrium [Marinatto, 2000].
However, one can say that the class of phenomena finding their natural explanation in terms of principles of quantum mechanics is much wider. In the physics of the micro-world, non-distributivity has an objective status and must be present in principle. For macroscopic systems, the non-distributivity of random events expresses some specific case of the observer’s ‘ignorance’ different from the standard probabilistic interpretation. Then it is possible due to the Luders rule to make calculations of the probabilities of definite answers for physical questions concerning properties of the quantum system. These questions are described by self-conjugate operators or in more simple cases by ‘yes-no’ questions, being elements of the Boolean (distributive) sublattices of the general non-Boolean lattice of properties.
As to the application of our results, we look for them not in physics but in economics and social sciences where some similarities with quantum physics can occur. In our examples, we look for macroscopic imitation of only some quantum properties arising due to non-distributivity of the lattice or non-commutativity of operators leading to complementarity. By now, a relatively complete formalism of quantization of classical games has been elaborated. It is based mainly on the approach adopted in theoretical physics which relies on the notion of a microparticle which behavior is described by a complex probability amplitude.
However, the notion of probability amplitude is also applicable in the macroworld, in the context that extends far beyond purely physical phenomena, including such fields as economics, sociology, psychology and law. In this paper we consider the examples of interactions governed not by classical Boolean logic, but by “quantum logic” (non-distributive ortholattices). Although, because of distributivity breaking, Kolmogorovian probabilities can no longer be used, the behavior of the partners are adequately described in terms of the probability amplitude.
Opportunistic use of information asymmetry in economic interactions is a typical example of breaking of distributivity. For example, one of the participants in a quantum game (the observer) tries to obtain information on the intentions of his partner by asking questions. If it is to advantage of his partner to change his current state and if such possibility is available, he gives a positive answer, thus demonstrating opportunist behavior. If such possibility is unavailable a negative answer is given. The observer is interested only in negative answers. However, it is not a Boolean algebra that is formed, but a non-distributive ortholattice which excludes application of the probability theory in general and the notion of mixed strategy in particular. In this case the behavior of both partners can be described in terms of probability amplitudes. Payoff functions are calculated according to the rules of quantum mechanics as the mean value of a sum of non-commuting operators.
1. The Stern—Gerlach Quantum Game
It is very easy to organize the quantum game using some well-known experiments with quantum microparticles. To do this one must write some payoff matrix showing what sums of money one partner must pay to the other depending on the results of the experiments. The advantage of the quantum games in comparison with the classical ones is the ‘objective’ nature of chance in it. In classical games chance occurs due to some ignorance, and that is why it is always possible for one of the partners with more exact information to have the privilege over the other one. In the quantum game based on measurements of some complementary observables, the result of the individual measurement is unpredictable in principle and only some average values can be predicted if the wavefunction is known to the participants of the game. Any game supposes the possibility of participants making some choices dependent on their abilities. So in quantum games participants have the freedom of preparation of the wavefunctions or density matrices for microparticles, and their profit will depend on their skill. In quantum games one has the combination of two different choices: the first choice is the manifestation of the free will of the human participant in the preparation procedure, and the second is the free choice of nature manifested in the result of measurements.
Let us consider the example of what can be called the Stern-Gerlach quantum game based on the well-known Stern-Gerlach experiment. There are two different beams of silver ions in different experimental set-ups which can be located at different places. The participants called Alice and Bob prepare their atoms in the state with some wavefunctions, so that every particle in one beam has the same wavefunction. In different beams the wavefunctions are different. Call them ^a and . Then Alice and Bob measure using Stern-Gerlach magnets at first one projection of the spin and then the other one. Spin projections can be different for different participants but they are fixed. The only freedom for the participants is in change of the wave functions. The payoff matrix can be such that if Alice obtained some definite result for one projection and Bob for the other one fixed by this matrix then Bob pays to Alice some money. However, these results can be obtained only with some probabilities. The average profit of Alice is calculated then by the rule of the quantum physics as the expectation value of some combination of spin operators for the two particle system. If the beams are different as they are supposed in our game then there is no symmetrization of wave functions.
The average profit is different for different choice of wave functions. The aim of Alice is to get the maximal average profit. She can control her wave function but not that of Bob. For some choice of both partners it leads to Nash equilibrium, i.e. the choice optimal for both partners. This means that for the antagonistic game if one partner gets the maximal profit the other one has the minimal loss. An interesting feature of the quantum game is that in spite of the fact that the wave function is the description of the pure state the expectation value defining the average profit from the point of view of the game theory corresponds to the mixed strategy used by Nature. So, at first, we consider the quantum game when only two spin projections
are measured. Let the payoff matrix be given by the table 1
Table 1: Payoff matrix
Bob
Alice 1 2 3 4
1 0 0 C3 0
2 0 0 0 C4
3 Cl 0 0 0
4 0 C2 0 0
values of the spin projections Sz and Sg so that to definite eigenvalues of one spin projection operator correspond two orthogonal projector operators, call them A1 and A3 for one projection and A2, A4 for the other. The same is valid for Bob. But his projectors will be called B1, B2, B3, B4:
Ai + A3 = I, A2 + A4 = I, Bi + B3 = I, B2 + B4 = I.
The meaning of the payoff matrix for our quantum game is that if Alice gets the result of her measurement 1 and Bob gets 3 then Bob pays to Alice the sum. If Alice gets 2 and Bob 4 then he pays, and so on. Alice in the result of the game gets the average profit calculated by the rule of quantum mechanics as the expectation value of the “profit operator”
H = C3A1 <8> B3 + C1A3 <8> Bi + C4A2 <8> B4 + C2A4 ® B2.
So, the average profit occurs to be
(H) = (^a\(^B\H\^B )\^A) = C3P1q3 + C1P3q1 + C4P2q4 + C2P4q2,
where ^a , ^B normalized vectors of states expressing the mixed strategies of Alice and Bob, pi = (^a\Ai\^a), qj = (^B\Bj\^B) the squares of the probability amplitudes given by the Luders rule
P1 + P3 = 1, P2 + P4 = 1, q1 + q3 = 1, q2 + q4 = 1.
In this game Alice gets money and Bob is paying her. So Alice is interested to get the maximal profit and Bob to pay the minimal sum. This leads to the idea of the Nash equilibrium, i.e. to the choice of such wave functions that the expectation value is maximal for one variable depending on the choice of Alice and minimal for the other one depending on Bob’s choice.
Our quantum game can be compared with the “classical” game with the same payoff matrix but with mixed strategies. The payoff function for this game is written as
. Alice measures some
h — C3«1^3 + C1«3^1 + C4«2^4 + C2«4^2,
where ai, (3j - some Boolean variables with values 0 or 1 depending on the use of the corresponding strategy. So,
a1 + a2 + a3 + a4 = 1, @1 + @2 + @3 + @4 = 1.
The average profit for the unlimited repeating of the game is calculated by the classical von Neumann expression:
(h) = C3P1q3 + C1P3q1 + C4P2q4 + C2P4q2,
P1 + P2 + P3 + P4 = 1, q1 + q2 + q3 + q4 = 1.
The comparison of the expression for the quantum game average profit with the classical one shows that the quantum game can be obtained from the classical one by the “quantization” procedure. The Boolean variables ai, @j are transformed into projector operators so that some of them don’t commute. These projectors form the structure of the non-distributive ortholattice (Fig. 1) called the quantum logical lattice. After writing the average profit in terms of probabilities one can see that the
I
0
Fig. 1 : Ortholattice of projectors A-i,A2,A3,A4
main difference between the quantum game and the classical one is in normalization of probabilities. For the classical case the probabilities are normalized on 1, for the quantum case due to Luders rule they are normalized on 2 if two non-commuting observables are measured. This corresponds to existence of two Boolean sublattices of the non-Boolean quantum logical lattice. Kolmogorovian probability measure can exist only on these sublattices, on non-Boolean lattice only the so, called quantum measure defined by the wave function can be defined. Stern-Gerlach quantum game can be generalized for the cases when three and more non-commuting spin projections are measured. Then the average profit of Alice will be constructed as the sum of three or more expectation values of the corresponding observables which can be written as the expectation value of the profit operator being the sum of non-commuting operators. For the spin one half Stern-Gerlach quantum game if two non-commuting spin operators are measured one can consider for simplicity real two dimensional space and take two-dimensional vectors in it as wave functions. Then our projectors
can be defined as projectors on two vectors on the plane with some angle between them. So, in this simple case there are two different angles: one parameterizing the wave function, the other one the spin projections. In our game the angle between spin projections is considered to be fixed while the angle defining the wave function can be varied expressing, thus, the freedom of participants of the game to prepare their wave functions in different ways.
2. Macroscopic quantum games
To look for macroscopic examples of games described by the mathematical formalism of quantum physics here we consider the simple case based on the Luders rule understood as some dependence of probability measures for different experiments.
If some macroscopic player Alice is playing two games at once using for her strategies probabilities different for different games where the difference is described just by the quantum Luders rule then this will be our quantum game. The average profit is calculated as the sum of profits in two games and it is calculated as the quantum expectation value. Nash equilibrium for this combination of two games considered as one game can be found as in microscopic quantum game by varying the angle defining the wave function. However, in our macroscopic case there is no necessity to use the notion of the wave function. In macroscopic situations quantum games occur due to special form of dependence of strategies in different classical games. This dependence can be due to some asymmetry in acts of the player simultaneously playing different classical games. For example, he (she) cannot have the same frequency for acts done by the right or left hand etc. For the quantum game when three non-commuting spin observables are measured this dependence can be manifested in Heisenberg uncertainty relations for spin written in the form of some relations for frequencies in three classical games. Luders rule gives for the probabilities of getting definite answers for spin projections expressions:
P1 = cos2 a, p2 = cos2 (a — 0), q1 = cos2 @, q2 = cos2(@ — t). (1)
Let Alice is playing the games on two desks: one called “even”, the other one “odd”. The same is for Bob. The average profits for Alice in each of the parallel games are
(H)odd = C3P1q3 + C1P3q1, (H)
even C4P2q4 + C2P4q2.
So, for the average profit in two games one obtains
(H) = C3P1q3 + C1P3q1 + C4P2q4 + C2P4q2.
The important feature of these classical games making them different from well known situations is the existence of “quantum cooperation” given by formulas (4) (see Fig 2.) with fixed 0 < 0 < 90°, 0 < t < 90°. This cooperation can be written in more symmetric form as some equation for p1, p2. To do this one can introduce new variables
£ = —1+P1 + P2, n = —P1 + P2.
So, that by use of (4) after simple trigonometric operations one obtains
£2 n2
cos2 0 sin2 0
i.e. equation of the ellipse with axes defined by cos 0, sin 0. The same equation with angle t one obtains for Bob. So Luders rule in our case means the existence of specific “quantum correlation”. The existence of this correlation is the new feature of our games, making possible to consider it as one macroscopic quantum game.
Let us note that the “quantum correlation” arising due to the existence of the wave function, and Luders rule is not the same as classical correlation. Really if one considers two games as one antagonistic classical game the possible strategies can be considered as “1 * 2”, “3 * 2”, “1 * 4”, “3 * 4” The same is for Bob. Then introducing
Table 2: Payoff-matrix of Alice in binary game
A B 1*2 1*4 3*2 3*4
1*2 0 C4 C3 C3 + C4
1*4 C2 0 C2 + c3 C3
3*2 Cl Cl + C4 0 C4
3*4 Cl + Co Cl C2 0
mixed strategies of Alice and Bob in this classical matrix game as pik, qik one has
P1 = P12 + P14, P3 = P32 + P34, P2 = P12 + P32, P4 = P14 + P34-
It is evident that p1 + p3 = 1, p2 + p4 = 1. Acts of Alice in two games can be independent, i.e.
P12 = P1P2, P32 = P3P2, P14 = P1P4, P34 = P3P4
(2)
But equation for correlation can be still valid. On the contrary classical correlation means breaking of (8). One can see the other sense of “quantum cooperation”. Quantum cooperation means that if a = 0 then the “even” game is deterministic but the “odd” game for 0 = 0 cannot be deterministic. For a = 0 the “odd” game is deterministic but then the “even” is random. This is manifestation of complementarity due to non-commutativity of the corresponding operators. Can one look for such situations in economics, politics? It seems that the answer is positive.
3. Quantum Nash equilibrium
The definition of the Nash equilibrium 0A <g> 0°b for the quantum case is not much different from the classical case
where 0a, 0B, normalized vectors of states expressing the mixed strategies of Alice and Bob: p^ = {0a\Ai\0a), qj = (0B\Bj\0b) - the squares of the probability amplitudes. Introduce the variables x1, x2 for Alice and y1, y2 for Bob as
So, the strategy of the participant of the game is defined by a point on the unit circle and to the game situation corresponds the point on the two dimensional torus. Back transformations from vectors to probability distributions written as 2p = Mqx +
e, 2q = Mty + e, where
1+P1 + P2 -P1 + P2
1 + q1 + q2
q 1 + qi sin t
cos 0 sin 0
cos t
, y2
Then the equations for “quantum cooperation” become
2 2 2 2 x1 + x2 = 1, y1 + y2 = 1
Introduce notations:
Then the average payoff function can be written in matrix form as
1
4
[g(x,y) + {e,Ce)],
where
g(x, y) = -{x, Ay) + (x, u) - (v, y)
with
A = MjCMt , u = MJw, v = Mjw.
The vector variables x, y satisfy the limitations \x\ = 1, \y\ = 1. So, one has the problem of Nash equilibrium of g on torus T2 = S'1 x S1.
It is interesting that for the classical game Nash equilibrium always exists in mixed strategies while for quantum game it is not always so. Solving of equilibrium problem based on some general properties: xmin, xmax are the points of minimum and maximum of a function f (x) = (k, x) + b on the circle \x\ = 1 if and only if for some non-negative X the equalities k = Xxmax, k = -Xxmin are valid.
Using this properties one obtains [Grib et.al., 2002]:
Proposition 1. The point (xo,yo) is the Nash equilibrium for the function g(x,y) if and only if nonnegative numbers X and i exist satisfying equalities
-Ayo + u = Xxo, A'xo + v = iyo. (3)
Corollary 1. If the point (xo ,yo) is the Nash equilibrium for the game on torus with the payoff function g(x, y) then for some nonnegative X, fi the following equations are valid
(AAJ + X^I)xo = iu - Av, (A^A + X^I)yo = Xv + A^u.
Corollary 2. If a = b then the Nash equilibrium is impossible.
Examine some special cases of existence of Nash equilibrium for the quantum game. Let w is not equal to zero. Then u = MJw, v = MT.w are also not zero.
Proposition 2. If m = n, 0 = t = 45° and n2 ^ w2 + w2 then there exists one point of Nash equilibrium (x,y), such that
= = 1 ( W2-W1 \
X V v ^2 + Wi J '
The probabilities are equal to
= i fc3-ci + with i,| = ,&
V P2 J \<12 ) 2|w| V c4 - C2 + kl J
Then the optimal value of the profit is (H) = n/4.
Examples (m = n, 0 = t = 45°)
► For c1 = 2, c2 = 1, c3 = 8, c4 = 9 the optimal strategies of Alice and Bob are
P1 = q1 = 0.8, p2 = q2 = 0.9, p3 = q3 = 0.2, p4 = q4 = 0.1.
► For c1 = 8, c2 = 9, c3 = 2, c4 = 1 the optimal strategies of Alice and Bob are
P1 = q1 = 0.2, p2 = q2 = 0.1, p3 = q3 = 0.8, p4 = q4 = 0.9.
► For c1 = 1, c2 = 2, c3 = 9, c4 = 8 the optimal strategies of Alice and Bob are P1 = q1 = 0.9, p2 = q2 = 0.8, p3 = q3 = 0.1, p4 = q4 = 0.2.
These results are in agreement with the properties of quantum logic (see Fig. 1). Really the second set of payoffs is obtained from the first by permutation
' 1 2 3 4'
3 4 12
expressing the automorphism of the lattice changing the elements on their orthocomplements. The third set of payoffs is obtained from the first one by the automorphism of the lattice
12 3 4
2 14 3
The average profit for all three cases will be the same and is equal to (H) = 2.5. This can be compared with the average profit for the Nash equilibrium for the classical game (without quantum cooperation):
h = (c-1 + c-1)-1 + (c-1 + c-1)-1 = 2.5
So, in this case it has the same value for the quantum case and the classical one. However ,such a coincidence is not necessary. This can be shown by the following example:
For c1 = 1, c2 = 9, c3 = 10, c4 = 2 the optimal strategies of Alice and Bob are
130 + 9a/T30 _ 130-7a/T30
»i = </i =--------------« 0.895, p2 = </2 =-----------------~ 0.193
1 H 260 260
The optimal profit in the classical game is smaller than in the quantum one: (h) = 28/11, (H) = 11/4. Now let us consider another special case which can be called the case of “eigenequilibrium”.
Proposition 3. Let w be the general eigenvector of operators CMqMJ and CMtMj, s is the eigenvalue of the operator CMqMJ. Pair of strategies x = u/\u\, y = v/\v\ defines the Nash equilibrium if and only if s < \u\.
Corollary 3. Let both components of w be different from zero, and there are inequalities
2 4 2 4 2 2 2 2 2 2 2 2 2 2
m W1 — n W2 > m n (W1 — W2), (W1 — W2)(m W1 — n W2) > 0.
Then the condition
(n - m)w1 w2
cos It =------s-----= cos 10
mw2 - nw^
is necessary and sufficient for the existence of equilibrium. Thus, the equilibrium probabilities are equal to
Pi \ _ ( qi \ _ 1 ( . I ^1-^2 ( mu 1 \ ( 1
Example. Let 6 = t = 45° and c1 =0, c2 = 1, c3 = 1, c4 = 1 then the optimal strategies of Alice and Bob are
P1 = q1 = 1, P2 = q2 = 0.5, p3 = q3 = 0, p4 = q4 = 0.5.
In this case the profit has the same value for the quantum case and the classical one: (H) = (h) = 0.5.
4. Macroscopic quantum games and quantum logics
In the previous part we discussed the idea of the macroscopic quantum game as the system of classical games with special condition on the strategies. However, we did not consider the origin of this condition, i.e. in what situations such conditions necessarily arise.
Here we give some examples when this is so. These examples are based on the connection first mentioned by D. Finkelstein and then developed in the works of A.A.Grib and R.R. Zapatrin [Grib, Zapatrin,1990] between quantum logical lattices and graphs. These examples are taken from publications [Parfionov, 2005]. Considering the idea of existence of macroscopic situations described by the formalism of quantum physics one must also mention the publications of D. Aerts [Aerts, 1995].
It was J. von Neumann who in his paper with G. Birkhoff [Birkhoff, 1936] was the first man to see that the structure of properties of the quantum system for simple spin one half system is the structure of the orthocomplemented non distributive lattice. Non-distributivity leads to non-commutativity of projector operators representing the abstract lattice. These lattices were called quantum logical lattices, or simply “quantum logics”. Later the ideas of von Neumann were developed by Jauch and Piron [Piron, 1976] for more general cases and now form the basis of the axiomatic of the quantum physics. Non-distributivity means that if there are properties A, B, C then using notation A for “and”, notation V for “or”, then
(A V B) A C = (A A C) V (B A C).
Breaking of the distributivity means that operations A, V cannot be understood as usual conjunctions and disjunctions of the set theory. The structure of the nondistributive lattice is not Boolean, and one cannot define on such structures the standard Kolmogorovian probability measure. Besides quantum mechanics non-Boolean lattices arise for topologies (see [Zapatrin, 1992]) so that if topologies are considered as random one also cannot define for them the standard probability measure. However, for quantum mechanical examples one can define the probability amplitude or “quantum probability measure” represented by some vector in Hilbert space.
In [Parfionov, 2005] the game called “wise Alice” was considered. Let Alice and Bob play the following game. Alice and Bob have two quadrangles, one for Alice and one for Bob. Let Bob puts some ball to the vertex of the quadrangle and Alice must guess to what vertex he did that. She asks Bob the question: “Did you put it
Fig, 3: Binary game of Alice and Bob
into 1?”. The rule of the game is such that Bob always answers “yes” if he is in 1, but he gives the same answer if the ball was in 2 or 4, i.e. in the vertices connected with 1 by one arc. However, it is prohibited for Bob to move by two steps from 3 to 1, and so, if he is in 3 then he always answers to her question “no”. The same rule is valid for any vertex. Alice, however, knows this property of “accommodation” of Bob to her questions. This leads to specific logic of Alice - she pays no attention to affirmative answers of Bob and notices only his negative answers.
Then it is easy to see that different positions of the ball of Bob will be described due to negative logic as disjunctive, i.e. 1A2 = 1A3 = 1A4 = 3A4 = 2A3 = 2A4 = 0 but the disjunction is now not unique 1 V 2 = I. Here I means “always true”, 0 -means “false”. From the structure of the graph and the rules of the game it is easy to see that due to Bob’s “accommodation” there is no difference for Alice between the situation 1 V 2 and 1 V 2 V 3 V 4. On the Fig. 4 we show the connection of the graph and the quantum logical lattice. This lattice is a well known lattice for
0
Fig. 4: Graph and Lattice of Alice’s questions and Bob’s answers
Stern-Gerlach experiment when two different spin projections are measured. Lines going “up” intersect at V (“or”), lines going down intersect at A (“and”). Lower drawings is called the Hasse diagram [Birkhoff, 1993].
However, to simulate the Stern-Gerlach quantum game considered in the first part of this paper one must do the game symmetric for both partners. This means that the same rule is valid for Bob guessing to what vertex of her quadrangle Alice put her ball. So, Bob also comes to the same quantum logical lattice. Asking questions
one to another Alice and Bob obtain some numbers of truly guessed due to negative answers positions of the balls and neglecting all “yes” answers.
Let these numbers be for Alice N1, N3 and N2, N4 for opposite vertices of the graph. Similar numbers are obtained by Bob. To transform these numbers into probabilities
Ni N3 N2 N4
N1 + N3 ’ N1 + N3 ’ N2 + N4 ’ N2 + N4
as it is in the quantum Stern-Gerlach game one can do the following.
Let the game consists of two parts as it was proposed in [Piron, 1976] preparation and measurement. Defining the numbers N means preparation. The second part -measurement - corresponds to the changed situation: Alice and Bob cannot accommodate one to another and now in N1 cases Bob will be in 1, in N3 cases in 3 etc., but Alice every time does not know exactly if he is in 1 or 3.
Instead of one game with quadrangle there are two games with two diagonals of the quadrangle. The strategies of Alice are defined by the probabilities obtained from the first stage due to her knowledge of the numbers N. These probabilities due to the properties of the quantum logical lattice satisfy limitations defined by the wave function for spin one half system. The same rule is valid for Bob, and he also plays two games with strategies defined by numbers obtained in the first part. The payoff is made according to the payoff matrix (Tab. 1) defined in section 1 and the average profit of one of the partners is calculated according to the quantum rule.
The necessity of going to the second part-measurement is motivated by the fact that it is only for Boolean sublattices of the non Boolean lattice that one can define probabilities. The difference of our macroscopic non-Boolean game from the microscopic Stern-Gerlach game is due to the fact that in macroscopic case Alice and Bob necessarily put their ball into position defined by the question of the partner while in microscopic case there is indeterminism, so that there is no such necessity. However, this freedom of choice is simulated in macroscopic game by the freedom of choice of the players in the second part to put their balls to any vertices with prescribed probabilities. The other difference is that the abstract quantum logical lattice which is the same for the microscopic and macroscopic cases has many different representations in terms of projectors. This means that the angle between projectors is not fixed by the lattice and can be any. In microscopic Stern Gerlach experiment this angle is chosen by the will of the experimentalist choosing the direction of the magnetic field in his magnet. In macroscopic case the angle is fixed by the ratio of probabilities for different games on the second stage. Connection between quantum logical lattices and graphs leads to the possibility of certain classification of macroscopic quantum games. For example, one can consider three classical games with strategies defined by the probabilities satisfying limitations due to the existence of the wave function, i.e. Stern-Gerlach experiment with three different spin projections for spin one half system being measured. The graph and the quantum logical lattice are shown on Fig. 5. The payoff matrices and the average profit for cases on Fig. 6. were considered in [Grib, 2003].
Fig. 5: Graph and ortholattice for spin i with three spin projections measured
0
Fig. 6: Graph and ortholattice for spin 1 with two spin projections measured
What is the sense of Nash equilibrium for macroscopic quantum games based on quantum logics? The angle between two “projections” is defined by the ratio of probabilities in two classical games corresponding to Boolean sublattices of the non-Boolean lattice. Nash equilibrium for fixed angle for projections corresponds to some “patterns” of stability for players. One of the partners receives the maximal profit and the other one has the minimal loss in this situation, so the partners can come to mutual agreement on their behavior after experiencing many games of this type. This can have meaning for some economical situations.
One can construct some generalization of the macroscopic quantum game based on the use of quantum logic. It is possible to change the second part of the game so that only the angles between observables are defined in the first part for Alice and Bob. They have the possibility to choose any wave function which means the angles defining the probability amplitudes. This will correspond exactly to the problem for Nash equilibrium considered in section 3 of this paper. Some “quantum casino” can be organized following this rule.
5. Opportunism as a Quantum Phenomena
Social interactions are often accompanied by information asymmetry, in which case one of the participants fully controls the situation while the others do not at all. The simplest model of this kind is the well-known “Leader-Follower” model proposed
by von Stackelberg [Moulin, 1981] which analyses the behavior of duopolists. One of them, the Leader, is fully informed on the intentions of the other one, while his partner, the Follower, is fully ignorant of the preferences of his competitor. Taking advantage of his information superiority, the Leader relies in his choice on all possible responses of the Follow. He openly proclaims the strategy, that he finds most profitable for himself. The Follower has to search the optimal strategy for himself within the scope of the opportunities left available for him. The outcome of such a game is known as Stackelberg equilibrium.
The Stackelberg model is a limit case of information asymmetry, in which each party is actually devoid of choice. Real interactions, in most cases, do not comply with such a rigid scheme. Information asymmetry, as a rule, does not exclude the initiative of the participants, thus leading to the opportunistic behavior. This results, typically, in “moral hard” and “adverse selection. G. Akerlof [Akerlof, 1970] was the first to investigate this sort of phenomena on the “lemon market” . However, his model is based entirely on the statistical characteristics of the ensemble of the players, without taking into account the reciprocal action (the “feedback”, counter-reaction), when the control on the part of one of the players pushes the partner towards opportunistic behavior. The effect of reciprocal action can by illustrated by an example of the interaction of a police interrogator and a suspect. Being questioned by the former, the latter tries to do his best to hide undesirable information.
This phenomenon is well known in quantum mechanical measurements. In section 4, to imitate this effect we will consider the game of Alice and Bob with a ball located on a square. In this case the effect of adaptation is a consequence of the information asymmetry of the freedom of choice, which the Stackelberg model lacks: having received the answer of Alice Bob has an opportunity to move the ball to any of the adjacent vertices. Due to the fact that negative answers are not profitable for him he, in all possible cases, moves the ball to a convenient adjacent vertex. So being in vertices 2 or 4 and getting from Alice the question “Are you in the vertex 1?” Bob quickly puts his ball in the asked vertex and honestly answers “yes”. However, if the Bob’s ball was initially in the vertex 3, to whatever vertex he moves his ball, he cannot escape the negative answer and, consequently, fails. It should be noted that in this case Alice not only gets the profit but also obtains the exact information on the initial position of the ball: Bob’s honest answer immediately reveals its initial position. This example illustrates the mechanisms of the creation of opportunism in the presence of information asymmetry. The payoff matrix of Alice (Tab. 1) shows that the model of the antagonistic game considered is based on a simplistic model of opportunism:
- In spite of the difference in outcomes (1 : 1), (1 : 2), (1 : 4), the payoff of Alice is the same in all these situations;
- outcomes (1:2) and (1:4) also differ, although, actu- In real interac ally, they correspond to different opportunistic trajectories
of Bob.
tions different opportunistic trajectories may result in different payoff. A more realistic model is described by a bimatrix game
Table. 3: Payoff bymatrix
Bob
Alice 1 2 3 4
1 an : 0 a 12 : 0 <2i3 : —63 ai4 : 0
2 a2 i : 0 <222 : 0 <223 : 0 <224 : 0
3 a3i : -bi a32 : 0 <233 : 0 <234 : —64
4 a4i : 0 <242 : - &2 <243 : 0 (244 : 0
in which Bob’s payoff is non-positive. However, even a very simple antagonistic model leads to an unexpected problem. It can be easily shown that in this case (see Tab. 1) there is no Nash equilibrium in pure strategies. However, any attempt to seek the solution of the game in mixed strategies encounters an unexpected problem of calculation of the average payoff of a player in the framework of information asymmetry. According to the game theory, the average payoff of Alice is given by the following expression:
H = c1p1q3 + c3p3qi + C2P2q4 + C4P4q2,
where pj, qk are corresponding probabilities of application of pure strategies. In this case the latter expression is not valid. The point is that the logic underlying the present game interaction is not Boolean and a conventional probability concept leads a contradiction.
In contrast with the Boolean logic, where “not 1” is equivalent to “or 2 or 3 or 4”, in this case proposition “not 1” is equivalent to “3”. Similarly, “not 2” is equivalent to “4”. As a result, a set of true propositions
Pr(1 V 3) = Pr(1) + Pr(3), Pr(2 V 4) = Pr(2) + Pr(4)
leads to an absurd conclusion
Pr(1 V 2 V 3 V 4) = 2
It should be noted that although, contrary to classical logic, 2 A 4 = false, 2 V 4 =
true, “2” is not “not 4”. Thus, the principal axiom of the probability theory
a A b = 0 =^ Pr(a V b) = Pr(a)+Pr(b) (4)
is no longer valid.
It is the mathematical formalism developed in quantum mechanics that provides a means to overcome this difficulty. This technique, elaborated long ago, allows to calculate the average in the case when random events are no longer governed by the Boolean logic. In quantum mechanics the notion of probability is transferred from
the Boolean algebra to more general structures called ortholattices [Kalmbach, 1983]. An ortholattice is a set L with three operations
V : Lx L —>L, A : LxL —>L, — : L —>L
a) a A b = b A a, a V b = b V a - commutativity
b) (a A b) A c = a A (b A c), (a V b) V c = a V (b V c) -
associativity
c) (a V b) A a = a, (a A b) V a = a - absorption
sUc a : d) -(a V b) = —a A —b, -(a A b) = —a V — - de Morgan'
laws
e) 3 0,1 : a V 0 = a, a A I = a - zero and unit exist
f) a V —a = 1, a A —a = 0, ——a = a - invertibility
In the general case distributivity
(a V b) A c = (a A c) V (b A c), (a A b) V c = (a V c) A (b V c)
is not observed for ortholattices. However, if distributivity laws are satisfied, an or-
tholattice amounts to a common Boolean algebra. The relation “c” for ortholattices:
a c b a A b = a
is an analogue of the inclusion “c” in Boolean algebra. Elements a and b are called
compatible, if a c b or a c b. We say that a commutes with b if
a = (a A b) V (a A —b)
In ortholattices beside disjoint (a A b = 0) a stronger relation, the orthogonality:
a .L b a c —b
is considered. It is instrumental in the formulation of the quantum version
a L b = 0 =^ Pr(a V b) = Pr(a) + Pr(b).
of the Kolmogorov’s axiom (4).
According to the Gleason’s theorem [Gleason, 1957], quantum probability measures, “quantum states”, are constructed on the basis of the representations of the elements of an ortholattice a gL 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(I) = I
for any elements a,b gL, that commute. In this case the probabilities of the elements a g L are calculated according to:
Prw (a) =tr(W ■ E(a))
where W is a density matrix. In present paper only pure states: w gH, {w\w) = 1 are considered. Their probabilities are calculated according to
Prw(a) = (E(a)w \w)
where w gH - “probability amplitudes”.
Let us consider a general game model of an interaction, when one participant acts in an opportunistic way. We suppose that each player has a finite number of strategies. Alice is given the right of control over the situation via finding out Bob’s strategy. However, her means of obtaining information are limited, which makes opportunistic behavior possible for Bob. On a set of strategies 0 of an opportunist there exists a structure of a non-oriented graph. While answering a corresponding
question Bob can replace his initial strategy by an adjacent one and claim it to be his
initial strategy without the risk that this will be discovered. However, if he chooses a non-adjacent strategy Bob’s opportunism may be uncovered and punished. This is reflected in the structure of his payoff matrix:
< 0, if i, k are not adjacent ^ 0, in the opposite case.
As for the payoff matrix of Alice \\aik\\, no specific requirements are put forward. Thus, the model is described by the following characteristics:
- the payoff matrix of Alice;
- the payoff matrix of Bob;
- the vertex of adjacency of the graph.
The interaction between the players takes place according to their payoff matrices, while the logic of Bob’s behavior is formalized by the following mathematical structure. Each strategy s G 0 of Bob is associated with an opportunistic neighborhood O(s), i.e. a set of graph vertices comprised of the vertex s and adjacent vertices. Thus, an opportunist can avoid his previous obligation s by shifting to any point O(s). The complement of the opportunistic neighborhood: C(s) = 0 \ O(s) is called zone of control of the vertex s. It is comprised of the vertices non-adjacent to s. The definition of the opportunistic structure, comprised of all possible intersections of the zones of control, is based on the set comprised of the zones of control:
L = {C(si) n ... n C(sn) \ s, G 0}.
The natural order A c B existing on the set L allows to introduce the following operations:
A V B = min{X G L\ X D A, B} = Q {X gL}
XdA,B
A A B = max{X G L\ X c A, B} = A n B, —A = f] C(s).
s£A
It turns out that thus defined set L is an ortholattice. This ortholattice expresses the logic of the actions of an opportunist. For example, the following Boolean logic corresponds to a disjoint graph, consisting of three isolated points: While opportunistic translations along a four-link linear chain result in a ortholattice of 10 elements. The
I
0
Fig. 8. Linear chain and their logic
interconnection between vertex-connection of graph and the logic of the behavior of an opportunist is revealed in the following facts.
Proposition 4. To each ortholattice there corresponds a non-oriented graph, for which the former represents an opportunistic structure. Thus, different logics of interaction correspond to different types of opportunism. Besides, the specificity of opportunism is entirely expressed in the logic of interaction.
Proposition 5. An orthola,ttice is Boolean if the corresponding graph is without ribs. Thus, the presence of opportunism inevitably leads to a failure of the traditional logic.
The analysis of the correspondence between the graphs and the logics engendered by these graphs (see Fig. 9, 10, 11) reveals an interesting feature: more numerous the possibilities for manifesting opportunism are simpler the logic of the interaction of the players is organized. The information asymmetry in the considered game manifests itself in the fact that Alice does not change her strategies, while her partner demonstrates opportunism. That is why in a repetitive game Alice uses classical mixed strategies, while Bob’s strategies are described by state vectors. As a result, a quantum game amounts to a classical game with the payoff function
Fig. 9. Graph of opportunism type 1 and their lattice
Fig. 10: Graph of opportunism type 2 and their lattice
A(p, x) = ^2 aikPi\\Skx||2, B(p, x) = - ^2 bikPi\\Skx\\2,
i,k = i i,k=l
where Sk are the projectors representing the ortholattice L and
n n
p = (pi, . . . Pn), Pi > 0, "^jPi = 1, x = (xi, . . Xn), ^2 \xi\2 =
Fig. 11.: Graph without opportunism Boolean algebra
1
Acknowledgement
One of the authors (A.A.G.) is indebted to the Ministry of Science and Education of Russia for the financial support,grant RNP.2.1.1.6826,of this work.
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