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2 Bertrand duopoly problem

Let us recall the classical problem of Bertrand duopoly. There are two firms (players) who compete in the price of a homogenous product. The demand q of the product is a function of the price, q(p) = max a p, 0 for every p > 0. The firm with a lower price captures the entire market. If both firms charge the same{ − price,} they split the market equally. We assume that each firm has the same marginal cost c such that 0 6 c < a. If player 1 sets the price as p1 and player 2 sets the price as p2 the payoff function of player 1 is

(p1 c)(a p1) if p1 < p2 and p1 6 a, 1 − − u1(p1, p2) =  (p1 c)(a p1) if p1 = p2 and p1 6 a, (1)  2 − − 0 ifotherwise.  Similarly, the payoff function of player 2 is (p c)(a p ) if p < p and p 6 a, 2 − − 2 2 1 2 = 1 u2(p1, p2)  (p2 c)(a p2) if p1 = p2 and p2 6 a, (2)  2 − − 0 ifotherwise.  The Bertrand model [29] was proposed as an alternative to the Cournot model [30] in which the players compete in quantities (see also [31] for more details about these two models). While it seems that the ratio- nal players would obtain similar payoffs in both games, comparison of the Cournot and Bertrand duopoly examples with respect to Nash equilibria exhibits a paradox. In the Cournot duopoly, the payoff is (a c)2/9. On the other hand, the game defined by (1)–(2) has the unique Nash equilibrium = − (p1∗, p2∗) (c, c) that arises from intersection of best reply functions β1(p2) and β2(p1), p p > p if p < c, { 1| 1 2} 2  > = , =  p1 p1 c if p2 c β1(p2) { | } + (3) ∅ if c < p 6 a c ,  2 2  a+c a+c  if p2 > ,  2 2 n o  p2 p2 > p1 if p1 < c, { | }  p p > c if p = c, β (p ) = { 2| 2 } 1 (4) 2 1 ∅ 6 a+c  if c < p1 2 ,  + +  a c if p > a c .  2 1 2 n o The equilibrium implies the payoff of 0 for both players.

3 Quantum Bertrand duopoly

In [32] we discussed two well-known quantum duopoly schemes [20, 33]. We pointed out that under some condition the Li–Du–Massar scheme [20] appears to be more reasonable. In what follows, we apply that scheme to Bertrand duopoly problem and study the resulting game with respect to Nash equilibria. Next, we investigate the duopoly problem with the use of a simpler two-qubit scheme.

2 3.1 The Li–Du–Massar approach to Bertrand duopoly Let us recall the key elements of the Li–Du–Massar scheme that are needed to consider the Bertrand duopoly. In the original paper [20], the quantities q1 and q2 in the quantum Cournot duopoly are deter- mined by the measurements Xˆ and Xˆ on the final state Ψ . Formally, the final state is of the form: 1 2 | f i

Ψ = Jˆ(γ)†(Dˆ (x ) Dˆ (x ))Jˆ(γ) 0 0 , (5) | f i 1 1 ⊗ 2 2 | i1| i2 where Jˆ(γ) is the entangling operator, • γ(ˆa†aˆ† aˆ1aˆ2) Jˆ(γ) = e− 1 2− , (6)

Dˆ (x ) for x [0, ) and j = 1, 2 are unitary operators • j j j ∈ ∞

x j(ˆa† aˆ j)/ √2 Dˆ j(x j) = e j − (7)

that correspond to player j’s strategies,

operatorsa ˆ anda ˆ† satisfy the following commutation relations: • j j = = = [ˆai, aˆ †j] δi j, [ˆai†, aˆ †j] [ˆai, aˆ j] 0. (8)

Then the quantities q1 and q2 are obtained by formula

q Ψ Xˆ Ψ = x cosh γ + x sinh γ, (9) 1 ≡ h f | 1| f i 1 2 q Ψ Xˆ Ψ = x cosh γ + x sinh γ. (10) 2 ≡ h f | 2| f i 2 1 In what follows, we provide the reader with detailed calculation needed to obtain (9). The same reasoning applies to the case (10). First, we recall the following operator relation that involves the function eA (see also [34]):

λA λA [A, B] e Be− = B cosh γ β + sinh γ β (11) √β p p for operators A and B that satisfy [A, [A, B]] = βB, (β: constant). In a special case [A, B] = µ1 (µ: constant), formula (11) leads to λA λA e Be− = B + λµ1. (12) From (8) and (11) we have ˆ ˆ = + J(γ)ˆa1 J†(γ) aˆ 1 cosh γ aˆ 2† sinh γ. (13) Thus, we have Oˆ Jˆ(γ)Xˆ Jˆ†(γ) = Xˆ cosh γ + Xˆ sinh γ. (14) 1 ≡ 1 1 2 Applying (12), we obtain + xi = aˆ i √ if i j, Dˆ †(x )ˆa Dˆ (x ) = 2 (15) i i j i i  aˆ j if i , j,  for i, j = 1, 2. Therefore 

Oˆ Dˆ †(x )Dˆ †(x )Oˆ Dˆ (x )Dˆ (x ) = (Xˆ + x )cosh γ + (Xˆ + x )sinh γ. (16) 2 ≡ 2 2 1 1 1 1 1 2 2 1 1 2 2 Since, Jˆ†(γ)ˆa Jˆ(γ) = aˆ cosh γ aˆ† sinh γ, (17) i i − j 3 for i, j = 1, 2 and i , j, we thus get

Oˆ Jˆ†(γ)Oˆ Jˆ(γ) = Xˆ + x cosh γ + x sinh γ. (18) 3 ≡ 2 1 1 2

According to the theory of quantization of the electromagnetic field, operatorsa ˆi anda ˆ i† satisfy relations

aˆ n = √n n 1 , aˆ† n = √n + 1 n + 1 . (19) i| i | − i i | i | i Hence 0 Oˆ 0 = x cosh γ + x sinh γ. (20) h |1 3| i1 1 2 We now apply the Li–Du–Massar scheme to the Bertrand duopoly example. From a game-theoretical point of view, the players 1 and 2 are to choose x1, x2 [0, ), respectively. Then, the players’ prices 3 ∈ ∞ p1 and p2 are determined as functions pi : [0, ) [0, ) of x1, x2 and a fixed entanglement parameter γ [0, ), ∞ → ∞ ∈ ∞ p1(x1, x2,γ) = x1 cosh γ + x2 sinh γ,  (21) p2(x1, x2,γ) = x2 cosh γ + x1 sinh γ,  (see [20] and the papers [27, 28, 35] directly related to Bertrand duopoly-type problems for justifying formula (21) in terms of quantum theory). Substituting (21) into (1) and (2) and noting that the sign of p1(x1, x2,γ) p2(x1, x2,γ) depends on the sign of x2 x1 we obtain the following quantum counterpart of (1) and (2): − −

(p (x , x ,γ) c)(a p (x , x ,γ)) if x < x , p (x , x ,γ) 6 a, 1 1 2 − − 1 1 2 1 2 1 1 2 Q = 1 γ u (x1, x2)  (p1(x1, x2,γ) c)(a p1(x1, x2,γ)) if x1 = x2, x1e 6 a, (22) 1  2 − − 0 ifotherwise,   (p2(x1, x2,γ) c)(a p2(x1, x2,γ)) if x2 < x1, p2(x1, x2,γ) 6 a, Q 1 − − γ u (x1, x2) =  (p2(x1, x2,γ) c)(a p2(x1, x2,γ)) if x1 = x2, x2e 6 a, (23) 2  2 − − 0 ifotherwise.   Nash equilibrium analysis

In order to find all the Nash equilibria, we determine the best reply functions β1(x2) and β2(x1) and find the points of intersection of the graphs of these functions. For γ = 0 we have p1(x1, x2, 0) = x1 and p2(x1, x2, 0) = x2. Then β1(x2) and β2(x1) coincide with the classical best reply functions (3) and (4). We thus assume that γ > 0. Let us consider several cases to settle β1(x2).

γ 1. If x2 < c/e , player 1 obtains a negative payoff by choosing x1 6 x2. Indeed, c c x cosh γ + x sinh γ c < cosh γ + sinh γ c = 0. (24) 1 2 − eγ eγ −

and a (x1 cosh γ + x2 sinh γ) > a c > 0. Hence, according to (22), it is optimal for player 1 to take − − γ x1 > x2. By a similar argument, if x2 = c/e , then any x1 < x2 yields player 1 a negative payoff. For γ this reason, player 1’s best reply is x1 > c/e . In that case, player 1 obtains the payoff of 0.

γ γ 2. Let us now consider the case c/e < x2 6 (a + c)/(2e ). Note that

p1(x1, x2,γ) = x1 cosh γ + x2 sinh γ = (a + c)/2 (25)

4 maximizes expression

(p (x , x ,γ) c)(a p (x , x ,γ)) 1 1 2 − − 1 1 2 = (x cosh γ + x sinh γ c)(a (x cosh γ + x sinh γ)). (26) 1 2 − − 1 2 γ γ Hence, if x2 = (a + c)/(2e ), term (26) as a function of variable x1 is maximized at x1 = (a + c)/(2e ). However, equality x1 = x2 implies that the players split the payoff given by (26). Thus, player 1 would

benefit from choosing x1 slightly below x2. But then any x1′ in between x1 and x2 would yield a better γ γ payoff. As a result, there is no best reply in this case. If c/e < x2 < (a + c)/(2e ) then it follows γ from (25) that expression (26) is maximized at point x1 > (a + c)/(2e ) > x2. But by taking into account payoff function (22), it would result in player 1’s payoff of 0. Thus, player 1 again obtains γ more by choosing x1 slightly below x2. In the same manner as in case x2 = (a + c)/(2e ) we can see γ that the set of best responses of player 1 is empty when x2 < (a + c)/(2e ).

γ 3. If (a + c)/(2e ) < x2 6 (a + c)/(2 sinh γ), then from the fact that x1 cosh γ + x2 sinh γ = (a + c)/2 maximizes (26) the player 1’s best reply is x = ((a + c)/2 x sinh γ)/ cosh γ 1 − 2

4. If (a + c)/(2 sinh γ) < x2 < a/ sinh γ, function (26) of variable x1 is monotonically decreasing in interval [0, ). Hence, player 1 would obtain the highest payoff if x = 0. ∞ 1 > > Q = 5. For the case x2 a/ sinh γ we have p1(x1, x2,γ) a for any x1 [0, ). It follows that u1 (x1, x2) 0, and then the set of best replies is [0, ). ∈ ∞ ∞

Summarizing, we obtain the following best reply function β1(x2):

x : x > x if x < c , { 1 1 2} 2 eγ  c = c x1 : x1 > γ ifx2 γ ,  e e ∅n o if c < x 6 a+c , β (x ) =  eγ 2 2eγ (27) 1 2  a+c a+c 6 a+c  x2 sinh γ sech γ if γ < x2 ,  2 − 2e 2 sinh γ   a+c a 0 if 2 sinh γ < x2 < sinh γ ,  [0, ) if x > a .  2 sinh γ  ∞  Similar arguments to those above show that player 2’s best reply function β2(x1) is

x : x > x if x < c , { 2 2 1} 1 eγ  c = c x2 : x2 > γ if x1 γ ,  e e ∅n o if c < x 6 a+c , β (x ) =  eγ 1 2eγ (28) 2 1  a+c a+c 6 a+c  x1 sinh γ sech γ if γ < x1 ,  2 − 2e 2 sinh γ   a+c a 0 if 2 sinh γ < x1 < sinh γ  [0, ) if x > a .  1 sinh γ  ∞  It is clear now that the players best reply functions β1(x2), β2(x1) for γ , 0 are more complex compared with , + γ a+c (3). If γ 0, the best reply functions on interval ((a c)/(2e ), ) (being the counterpart of case xi > 2 if γ = 0) are more specified, and take into account different intervals∞ ((a + c)/(2eγ), (a + c)/(2 sinh γ)], ((a + c)/(2 sinh γ), a/ sinh γ) and [a/ sinh γ, ). This implies that new equilibria arise. Since a Nash equilibrium in a two-person game is a profile∞ in which the strategies are mutually best replies, we can easily determine the Nash equilibria by studying the points of intersection of β1(x2) and β2(x1) (see Fig 1 for the γ γ graphs of β1(x2) and β2(x1)). An example of such a point is profile (c/e , c/e ) that coincides with the unique classical Nash equilibrium (c, c) in case γ = 0. Another and more interesting example is a profile

5 Figure 1: Graphs of best reply functions (27) and (28).

(0, (a + c)/(2 sinh γ)) that implies the payoff profile ((a c)2/4, 0) and has no counterpart in the classical Cournot duopoly. It is a particular case of a general equilib− rium profile

a + b c x1, x1 cosh γ cschγ for x1 0, . (29) 2 − ! ! ∈  eγ 

2 γ To see that this type of Nash equilibrium yields the payoff profile ((a c) /4, 0), note first that x1 6 c/e γ − implies e x1 < a + c. From this we conclude a + c x1 < x1 cosh γ cschγ = x2. (30)  2 −  ff ff Q As a result, player 2’s payo is equal to zero and player 1’s payo function u1 (x1, x2) comes down to

uQ(x , x ) = (p (x , x ,γ) c)(a p (x , x ,γ)). (31) 1 1 2 1 1 2 − − 1 1 2 Since

p1(x1, x2,γ) = x1 cosh γ + x2 sinh γ a + c = x1 cosh γ + x1 cosh γ cschγ sinh γ  2 −  a + c = (32) 2 it follows that 1 uQ(x , x ) = (a c)2. (33) 1 1 2 4 − Similarly, the set of Nash equilibrium profiles

a + c a (0, x2), x2 , (34) ( ∈ 2 sinh γ sinh γ!)

6 Table 1: Nash equilibria in the game determined by (22) and (23) for γ , 0. They correspond to the points of intersection of graphs of (27) and (28). Nashequilibrium Payoff profile

c c eγ , eγ (0, 0)  x , a+c x cosh γ cschγ , x 0, c 1 (a c)2, 0 1 2 − 1 1 ∈ eγ 4 − n a+c x cosh γ csch γ, x  , x h0, c io 0, 1 (a c)2 2 − 2 2 2 ∈ eγ 4 − n(0, x ), x a+c , a  h io ((x sinh γ c) (a x sinh γ) , 0) 2 2 ∈ 2 sinh γ sinh γ 2 − − 2 n(x , 0), x  a+c , a o (0, (x sinh γ c) (a x sinh γ)) 1 1 ∈ 2 sinh γ sinh γ 1 − − 1 n(x , x ), x  0, c , x o a , (0, 0) 1 2 1 ∈ eγ 2 ∈ sinh γ ∞ n(x , x ), x h a i, , x h 0, c o (0, 0) 1 2 1 ∈ sinh γ ∞ 2 ∈ eγ n(x , x ), x h a ,  , x h a io, (0, 0) 1 2 1 ∈ sinh γ ∞ 2 ∈ sinh γ ∞ n h  h o is more profitable for player 1. Since p1(x1, x2,γ) = x2 sinh γ and x2 < a/ sinh γ, we have p1(x1, x2,γ) < a. Hence, uQ(x , x ) = (x sinh γ c)(a x sinh γ). (35) 1 1 2 2 − − 2 Thus, there are continuum many nonclassical equilibria that favor player 1. Another part of equilibrium profiles favors player 2 (see Table 1). Moreover, the great part of the equilibria implies payoff of 0 for each player.

3.2 Bertrand duopoly with fully correlated quantities We have known from the previous subsection that the Li–Du–Massar approach to the Bertrand duopoly does not bring us closer to the unique and paretooptimal outcome. It results from similar structure of payoff functions (22), (23) and (1), (2). In both cases, a unilateral (slight) deviation from profile (x, x) may yield the player almost twice as high payoff. In this way, we cannot obtain a symmetric equilibrium that would be profitable for both players. However, the correlation between prices can be defined in many different ways. In paper [32], we introduced a simplified model that correlate the players choices x1, x2 [0, ) in the following way: ∈ ∞ π = 2 + 2 = 2 + 2 p1′ (x1, x2,γ) x1 cos γ x2 sin γ, p2′ (x1, x2,γ) x2 cos γ x1 sin γ, γ 0, . (36) ∈  4 = = The value pi′ for i 1, 2 is obtained by formula pi′ tr(Mi(x1, x2)ρi) where

x 0 0 + x 1 1 if i = 1, , = 1 2 1. Mi(x1 x2)  | ih | | ih | x2 0 0 + x1 1 1 if i = 2,  | ih | | ih |  2. ρ1 and ρ2 are the reduced density operators tr2( Ψ Ψ ) and tr1( Ψ Ψ ), respectively, and Ψ = cos γ 00 + i sin γ 11 . | ih | | ih | | i | i | i

It is clear from (36) that the correlation between x1 and x2 increases with increasing parameter γ. If γ = 0, = = = we obtain the classical Betrand duopoly. In the maximally entangled case, γ π/4, we have p1′ p2′ p′ and p′ equally depends on x1 and x2. We will show below that this case is crucial in obtaining paretooptimal equilibria.

7 Nash equilibrium analysis Let us first consider the case γ (0, π/4). Substituting (36) into (1) and (2) gives ∈ p′ (x , x ,γ) c a p′ (x , x ,γ) ifx < x , p (x , x ,γ) 6 a, 1 1 2 − − 1 1 2 1 2 1 1 2 Q 1    u (x1, x2) =  p′ (x1, x2,γ) c a p′ (x1, x2,γ) if x1 = x2, x1 6 a, (37) 1  2 1 − − 1 0    if otherwise,    p′ (x1, x2,γ) c a p′ (x1, x2,γ) if x2 < x1, p2(x1, x2,γ) 6 a, 2 − − 2 Q 1    u (x1, x2) =  p′ (x1, x2,γ) c a p′ (x1, x2,γ) if x1 = x2andx2 6 a, (38) 2  2 2 − − 2 0    if otherwise,   where we use the fact that inequalities p′ < p′ and x < x are equivalent for γ (0, π/4). We see that 1 2 1 2 ∈ functions (37) and (38) have the same form as (22) and (23) up to the values pi. Hence, the method to find β1(x2) and β2(x1) and then the Nash equilibria is similar to that used for (27) and (28). We obtain x x > x if x < c, { 1| 1 2} 2  x x > c if x = c, { 1| 1 } 2  a+c ∅ if c < x2 6 , =  2 β1(x2)  a+c 2 2 a+c a+c (39)  x2 sin γ cos− γ if < x2 6 2 ,  2 − 2 2 sin γ   a+c a 0 if 2 < x2 < 2 ,  2 sin γ sin γ  a [0, ) if x2 > 2 .  ∞ sin γ  Symmetric arguments apply to β2(x1), x x > x if x < c, { 2| 2 1} 1  x x > c if x = c, { 2| 2 } 1  a+c ∅ if c < x1 6 , =  2 β2(x1)  a+c 2 2 a+c a+c (40)  x1 sin γ cos− γ if < x1 6 2 ,  2 − 2 2 sin γ   a+c a 0 if 2 < x1 < 2 ,  2 sin γ sin γ  a [0, ) if x1 > 2 .  ∞ sin γ  The resulting Nash equilibria in the game are given in table 2. Comparison of Tables 1 and 2 shows that there is no new type of Nash equilibria in the game defined by (37) and (38). The set of Nash equilibria changes if γ = π/4. For any x1 and x2 we have = = + p1′ (x1, x2, π/4) p2′ (x1, x2, π/4) (x1 x2)/2. (41) = Substituting pi′(x1, x2, π/4) for i 1, 2 into (1) and (2), respectively, we can rewrite (37) and (38) as + + 1 x1 x2 c a x1 x2 if x + x 6 2a uQ(x , x ) = uQ(x , x ) = 2 2 − − 2 1 2 (42) 1 1 2 2 1 2      0 if x1 + x2 > 2a.  Q  = + Note that function u1 (x1, x2) attains its maximum at x1 a c x2 for fixed x2 [0, 2a). Player 1’s best reply is therefore x = a + c x if 0 6 x 6 a + c and x = 0− for case a + c < x∈ < 2a. If x > 2a, then 1 − 2 2 1 2 2 for any x1 [0, ) player 1’s payoff is zero. As a result, player 1’s best reply function β1(x2) is given by formula ∈ ∞ a + c x if x 6 a + c, − 2 2 =  + β1(x2) 0 ifa c < x2 < 2a, (43)  [0, ) if x2 > 2a. ∞   8 Table 2: Nash equilibria in the game determined by (37) and (38) for γ (0, π/4). They correspond to the points of intersection of (39) and (40). ∈ Nashequilibrium Payoff profile (c, c) (0, 0) a+c 2 2 1 2 x1, 2 x1 cos γ sin− γ , x1 [0, c] 4 (a c) , 0 a+c − 2 2 ∈ 1 − 2 n  x cos γ sin− γ, x  , x [0, c]o 0, (a c)  2 − 2 2 2 ∈ 4 − n a+c a  o  2  2 (0, x2), x2 2 , 2 x2 sin γ c a x2 sin γ , 0  ∈  2 sin γ sin γ  − − a+c a  2   2  (x1, 0), x1 2 , 2 0, x1 sin γ c a x1 sin γ  ∈  2 sin γ sin γ  − − a      (x1, x2), x1 [0, c] , x2 2 , (0, 0)  ∈ ∈  sin γ ∞ a (x1, x2), x1 2 , , x2 [0, c] (0, 0)  ∈  sin γ ∞ ∈  a a (x1, x2), x1 2 , , x2 2 , (0, 0)  ∈  sin γ ∞ ∈  sin γ ∞

Table 3: Nash equilibria in the game determined by (42). Nashequilibrium Payoff profile

(x1, x2), x1, x2 [2a, ) (0, 0) {(x , a + c x )∈, x ∞[0,}a + c] ((a c)2/8, (a c)2/8) { 1 − 1 1 ∈ } − −

We conclude similarly that player 2’s best reply function β2(x1) is of the form

a + c x if x 6 a + c, − 1 1 =  + β2(x1) 0 if a c < x1 < 2a, (44)  [0, ) if x1 > 2a.  ∞  From intersection of (43) and (44) (see Fig 2), we conclude that there are two types of Nash equilibria. One of the types is characterized by profiles (x , x ) such that x , x [2a, ). In this case, each player’s payoff 1 2 1 2 ∈ ∞ is zero. The other one is what is desired. Each strategy profile (x1, x2), where x1 + x2 = a + c, constitutes a Nash equilibrium and implies payoff (a c)2/8 for both players. The payoff profile ((a c)2/8, (a c)2/8) is paretooptimal since − − − (a c)2 max (u1(p1, p2) + u2(p1, p2)) = − , (45) p1,p2 [0, ) 4 ∈ ∞ where ui(p1, p2) for i = 1, 2 are the payoff functions (1) and (2).

4 Conclusions

Our research has shown that the quantum approach to the Bertrand duopoly exhibits Nash equilibria that are not available in the classical game. It has been proved that the Li–Du–Massar scheme does not imply the unique and paretooptimal equilibrium outcome as is shown in the Cournot duopoly example. Instead, we have the two equivalent types of Nash equilibria where, depending on the type, one of the players obtains payoff (a c)2/4 and the other one obtains zero payoff. On the other hand, we have shown that it is possible to attain− the symmetric paretooptimal equilibrium if the correlation between the players’ prices is defined in another way. If the players share the maximally entangled two-qubit state then with the appropriately defined quantum scheme we obtain the symmetric and paretooptimal equilibrium outcome

9 Figure 2: Graphs of best reply functions (43) and (44).

((a c)2/8, (a c)2/8). Bertand doupoly, even though it is regarded as unrealistic, attracted a lot of attention in the− economic− literature. We have shown that ”quantization” of the model substantially extends the class of possible player behaviors and Nash equilibria. The above-mentioned paretooptimal equilibrium outcome is exceptionally interesting. We do not claim that there are any quantum correlations among the agents. But, from the phenomenological point of view, agent behaves as if some ”coordination” really occurs. This probably means that there are big incentives to cooperate (). We refrain from speculation on the causes of such behavior [6, 7]. Our discussion is yet another argument for using the formalism of quantum games in the analysis of the oligopoly modeling.

Acknowledgments

We would like to thank the Reviewers for discussion which undoubtedly improved the quality of our work. Work by Piotr Fra¸ckiewicz was supported by the Ministry of Science and Higher Education under the project Iuventus Plus IP2014 010973.

References

[1] Meyer, D. A.: Quantum strategies. Phys. Rev. Lett. 82 10521055 (1999) [2] Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83 3077 (1999) [3] Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272 291 (2000) [4] Baaquie, B.E.: Interest Rates and Coupon Bonds in Quantum Finance. Cambridge University Press, Cambridge (2010) [5] Khrennikov, A.: Ubiquitous Quantum Structure. Springer, Berlin (2010)

10 [6] Busemeyer, J.R., Bruza, P. D.: Quantum Models of Cognition and Decision. Cambridge University Press, Cambridge (2012)

[7] Haven, E., Khrennikov, A.: Cambridge University Press, Cambridge (2013)

[8] Baaquie, B.E.: Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, Cambridge (2004)

[9] Fra¸ckiewicz, P.: Application of the Eisert–Wilkens–Lewenstein quantum game scheme to decision problems with imperfect recall. J. Phys. A: Math. Theor. 44 325304 (2011)

[10] Piotrowski, E.W., Sładkowski, J.: Trading by quantum rules - quantum anthropic principle. Int. J. Theor. Phys. 42 1101 (2003)

[11] Piotrowski, E.W., Sładkowski, J.: Quantum games in finance. Quant. Finance 4 C61 (2004)

[12] Piotrowski, E.W., Sładkowski, J.: Quantum diffusion of prices and profits. Phys. A 345 185 (2005)

[13] Piotrowski, E.W., Sładkowski, J.: Quantum English auctions. Phys. A 318 505 (2003)

[14] Piotrowski, E.W., Sładkowski, J.: Quantum auctions: facts and myths. Phys. A 387 3949 (2008)

[15] Sładkowski, J.: Giffen paradoxes in quantum market games. Phys. A 324 234 (2003)

[16] Aerts, D.: Quantum structure in cognition. J. Math. Psychol. 53 314 (2009)

[17] Yukalov, Y.I., Sornette, D.: Quantum probability and quantum decision making. Philos. Trans. R. Soc. A 374 20150100 2016

[18] Makowski, M.,Piotrowski, E.W., Sładkowski, J.: Do transitive preferences always result in indifferent divisions? Entropy 17 968 (2015)

[19] Puu, T.: Oligopoly: Old Ends-New Means. Springer, Berlin (2011)

[20] Li, H., Du, J., Massar, S.: Continuous-variable quantum games. Phys. Lett. A 306 73 (2002)

[21] Chen, X., Qin, G., Zhou, X., Du, J.: Quantum games of continuous distributed incomplete informa- tion. Chin. Phys. Lett. 22 1033 (2005)

[22] Li, Y., Qin, G., Zhou, X., Du, J.: The application of asymmetric entangled states in quantum games. Phys. Lett. A 335 447 (2006)

[23] Sekiguchi, Y., Sakahara, K., Sato, T.: Uniqueness of Nash equilibria in a quantum Cournot duopoly game. J. Phys. A: Math. Theor. 43 145303 (2010)

[24] Wang, X., Yang, X., Miao, L., Zhou, X., Hu, C.: Quantum Stackelberg duopoly of continuous dis- tributed asymmetric information. Chin. Phys. Lett. 24 3040 (2007)

[25] Yu, R., Xiao, R.: Quantum Stackelberg duopoly with isoelastic demand function. J. Comput. Inf. Syst. 8 3643 (2012)

[26] Wang, X., Hu, C.: Quantum Stackelberg duopoly with continuous distributed incomplete information. Chin. Phys. Lett. 29 120303 (2012)

[27] Lo, C.F., Kiang, D. Quantum Bertrand duopoly with differentiated products. Phys. Lett. A 321 94 (2004)

11 [28] Qin, G., Chen, X., Sun, M., Du, J.: Quantum Bertrand duopoly of incomplete information. J. Phys. A Math. Gen. 38 4247 (2005)

[29] Bertrand, J.: Th´eorie math´ematique de la richesse sociale. Journal des Savants 67 499 (1883)

[30] Cournot, A.: Recherches sur les Principes Math´ematiques de la Th´eorie des Richesses. Paris: Hachette (1838)

[31] Peters, H.: Game Theory: A Multi-Leveled Approach. Springer, Berlin (2008)

[32] Fra¸ckiewicz, P.: Remarks on quantum duopoly schemes. Quantum Inf. Process. 15 121 (2015)

[33] Iqbal, A., Toor, A.H. Backwards-induction outcome in a quantum game. Phys. Rev. A 65 052328 (2002)

[34] Merzbacher, E.: , 3rd edn. Wiley, Hoboken (1998)

[35] Sekiquchi, Y., Sakahara, K., Sato, T.: Existence of equilibria in quantum Bertrand-Edgeworth duopoly game. Quantum Inf. Process. 11 1371 (2012)

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