Multistep Bertrand duopoly model with imports Текст научной статьи по специальности «Математика»

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MULTISTEP BERTRAND DUOPOLY MODEL WITH IMPORTS © V. I. Zhukovskiy, M. V. Boldyrev

Lomonosov Moscow State University Faculty of Computational Mathematics and Cybernetics Department of Optimal Control Leninskiye Gory ul., 1, bldg. 52, Moscow, 119234, Russian Federation e-mail: m_boldyrev@list.ru

Introduction

In 1838, French mathematician Antoine Augustin Cournot suggested a mathematical model of interactions between two firms in his book Researches on the Mathematical Principles of the Theory of Wealth [1]. In this model, the firms select their output values, assuming the opponent's policy constant, thereby de facto stating the hypothesis beneath the "Nash equilibrium". Independently from Cournot in 1950, American mathemiatician and economist John Forbes Nash formalized [2, 3] the concept of the equilibrium situation, later named the Nash equilibrium. Forty-four years later (in 1994) he received a Nobel Prize (shared with John Harsanyi and Reinhard Selten) "for their pioneering analysis of equilibria in the theory of non-cooperative games". In his model Cournot assumed that both firms select the amount of the good they can sale while the price is formed as a result of equality of demand and supply. However, it would be more natural for the salesperson to select their price directly. This approach was suggested by French mathematician Joseph Louis François Bertrand in 1883 [4]. The mathematical model considered in this article differs from Bertrand's: first, the problem is formulated in terms of a multistep game; second, it accounts for imports into the market.

A suitable modification of Bellman's method of dynamic programming is suggested for such an extended mathematical model. Thanks to this method, an explicit presentation of a strongly guaranteed equilibrium is found, first suggested by the first author in 1994 in [5, p. 233].

1. Mathematical model

Suppose the market is dominated by two firms, 1 and 2, who produce interchangable goods. Firm 1 declares their price pi, firm 2 declares their price p2. After the prices have been declared, demand for these goods is established (we assume it is linear in respect to the declared prices). The demand for the first firm's good can be represented as

Qi(Pi,P2) = q - liPi + I2(P2 + y), for the second firm's good as

Q2(Pi,P2) = q - I1P2 + I2(Pi + y), where q is the initial demand, l1 > 0 is the coefficient of elasticity that represents the fall of demand after the price of the firm's good has been increased by one currency unit, l2 > 0 is the coefficient of elasticity that represents the rise of demand after the price of the competitor firm's good has been increased by one currency unit, y > 0 is the price of analogous imported good set independently from the seller's actions (y will later be assumed as an uncertainty).

Suppose the cost price of a unit of the good is c. Then the function estimating the profit of the firm i = 1,2 may be represented as

fi(Pi,P2,y) = [q - liPi + l2(P2 + y)](Pi - c)

f2(Pi,P2,y) = [q - liP2 + l2(Pi + y)](P2 - c).

Since /i(Pi,P2,y) is concave downward with respect to Pi (i = 1,2) (since d2fi

-—i = -2li < 0), the sufficient condition of existence of P* that maximizes fi(Pi,P2,y) dPi

with respect to Pi is reduced to fulfillment of the following condition:

d/i(Pi,P2,y)

dpi d/2(Pl,P2,y)

= q - 2/ipl + /2(P2 + y) + lic = 0,

Pi

= q - 2/iP2 + /2(Pi + y) + /ic = 0.

P*2

ÔP2

Then the maximum profit of player 1 is reached when

Pi = q +//iC + ^(P2 + = a + /(P2 + y) VP2 - 0,

analogously for player 2

P2 = a + /(Pi + y) Vpi - 0,

where

q + lic

2li

> 0,

l2

l = 221 > 0

If we account for time lags (which we assume equal to one time period), assume z = ly and ui is control action of the firm i, and if u^k] is the amount of money spent at the moment of time t = k for marketing, modernization of means of production, implementation of new technologies, various measures of stimulation and penalty on the producer, then our controlled system of interactions between the firms and importers may be represented as a "difference scheme":

where p0 : of time t

pi(k + 1) = a + lp2(k) + z[k] + ui[k],

P2(k + 1) = a + lpi(k) + z[k] + U2[k] (k = 0,1, 2,...), (1)

Pi(0)= P0 (i = 1, 2) Pi(0) are initial prices; z[k] = ly[k] is the uncertainty at the moment

0 l

k. With matrix L

l0

and two-dimensional column-vectors

a = (a, a)T ,p = (рьр2)т, e = (1,1)T, u = (ui, )T the system (1) may be represented as

a

+ 1) = a + Lp(k) + ez[k] + u[k], p(0) = p0 = (p?,p0)T (k = 0,1, 2...). (2)

2. Two-step noncoalitional game

From now on, we will suppose the game (1) (or (2)) only lasts for two turns, i.e. k = 0,1. Each firm will be called a player. Each strategy (rule of operation) Uj(k) of the firm i at the moment k = 0,1 will be identified (in accordance with the theory of differential games) as a scalar function uj(k,p) of position (k,pi,p2) = (k,p) at the moment t = k (we will from now on represent this relation as Uj(k) ^ Uj(k,x) ).

The set of strategies Uj(k) will be denoted by the symbol U^(k) (i = 1, 2). Then the strategy of player i in the two-step noncoalitional game defined later will be represented as an ordered set Ui = (Ui(0), Ui(1)) e U = Ui(0) x Ui(1).

We will now consider uncertainties. Supposing informational discrimination of the firms, we will identify the uncertainty Z(k) at the moment t = k as a scalar function z(k,p,u), i.e. Z(k) ^ z(k,p,u) = z(k,p1,p2,u1,u2); we will later use the set Z(k) = {Z(k)}. Then the uncertainty is Z = (Z(0),Z(1)) e Z = 3(0) x 3(1). As time progresses from 0 to 2, the game "unfolds" as follows. Suppose the players

do not form a coalition and each player i (i = 1,2) independently selects their strategy Ui = (Ui(0), Ui(1)) E Ui, i.e. builds two scalar functions ui(0,p1,p2) > 0 and ui(1,Pi,P2) > 0 (where pi > 0, p2 > 0). Each player i selects strategies Ui(0) E Ui(0) and Ui(1) E Ui(1) under their guidance to maximize their outcome (the value of the outcome function Ji(U, Z,p0), p0 = (p0,p0); its composition will be shown below). At the same time, some definite values of strategies u[0] = u(0,p(0)),u[1] = u(1,p(1)) and uncertainties z[0] = z(0,p(0), u[0]), z[1] = z(1,p(1),u[1]) take place. Using (2) under k = 0, z = z[0], and p(0) = p0 (i = 1, 2), we obtain the value of the phase vector p(1) = (pi(1),p2(1)):

p(1) = a + Lp(0) + ez [0] + u[0].

Then, after applying (2) under k = 1, z = z[1], and already selected scalar functions ui(1,pi,p2) (i = 1, 2), we have

p(2) = a + Lp(1) + ez [1] + u[1]. Thereby we have obtained, first, two sequences

Pi(k)k=o (i =1, 2), (3)

that form a discrete trajectory of system (2) that assumes the usage of the mentioned (and selected) specific strategies Ui ^ ui(0,p), ui(1,p), Ui E Ui (i = 1, 2) and implementation of uncertainty Z + {z(0,p,u),z(1,p,u)};

second, two sequences of implementation

ui[k] = ui(k,pi(k),p2(k))k=o (i = 1, 2) (4)

of the selected strategies Ui E Ui (i = 1, 2);

third, a sequence of implementation of uncertainties z[k]k=0.

Using (3), (4), and z[k];i=0, we will build the criterion (outcome function) of player i, whose value (outcome) will estimate the quality of player's operations. While doing this, we will account for three circumstances:

first, each firm i (i = 1, 2) seeks to minimize their price, which, in the end, may be represented as minimization of pf(2) (or, equivalently, maximization of -p2(2)) by player i,

second, each firm seeks to minimize usage of their resources. This may be represented as aspiration to maximize k=0(-2u2[k]),

third, following the principle of guaranteed result formulated by Yu. B. Germeyer and choosing their strategy, player i must assume "maximal antagonism" of the uncertainty,

which can be acounted for by introducing the summand of Y1 k=0 1.5z2[k] to the outcome function. Therefore, we obtain the outcome function of player i:

l

Ji(U, Z,p0) = p2(2) + £(-2u?[k] + 1.5z2[k]) (i = 1, 2). (5)

k=0

Ordered quintuplet

Г = <{1, 2}, E - (2), {Ui}i=i,2,3, {Ji(U,3,p0) — (5)}i=i,2> forms a two-step noncoalitional linear-quadratic game of two persons under uncertainty. Here, E — (2) denotes that the control system E is described by the difference scheme (2), and Ji(U, Z,p0) — (5) is the outcome function of player i that takes the form (5).

Definition 1. Pair (Ue, Je[p0]) is called [6, p.117] a strongly guaranteed equilibrium of the game Г if:

1) there exist uncertainties Z(i) e Z (i — 1, 2) such that

min Ji(U,Z,p0) = Ji(U, Z(i),p0) = Ji[U,p0] (i = 1, 2) ZE3

for VU e U;

2) Ue = (Ue,U2e) e U is the only Nash equilibrium situation in the "game of guaranties"

< {1, 2}, E, Ui=i,2, {Ji[Ui,U2,p0]}i=l,2 >, i.e. Ue is defined by equalitites:

Ji[U e,p0] = Je[p0], J2[U e,p0] = J2e[p0];

3. Method of building of a strongly guaranteed equilibrium

After using the method of dynamic programming, first described by German mathematician Richard Bellman, and results from [6, §3.4], we obtain the following method of finding a strongly guaranteed equilibrium (SGE).

First of all, we need to build two (i = 1, 2) functions of p(k + 1)

Wi(k,p,u,z, v/k+i)(p(k + 1) = a + Lp + ez + u)) =

max Ji[Ui,U2e,p0] = uxeu i 2

max да, U2,p0] = щеи

3) Je[p0] = (Jf[p0],J2e[P0])-

= -2u2 + 1.5z2 + Vi(fc+i)(a + Lp + ez + u) (k = 1, 2). (6)

Step 1. For k = 2 we introduce two scalar functions

V(2)(P) = -P.2 (i =1, 2).

Step 2. For k =1 we need to find two functions z (i)(1,p, u) for Vp E R+ = {p = (pi,P2)|pi > 0},u E R+ in accordance with the equalities

min{-2u2 + 1.5z2 - (a + /p3-i + ui + z)2} =

z

= Idem{z ^ z(i)(1,p, ui, u2)} = Wi[1,p, ui, u2] (i = 1, 2); (7)

then we need to build four functions Vi(1)(p) h uf(1,p) (i = 1, 2) in accordance with

Vl(1)(p) = max{Wi[1,p,ui,u2(1,p)]} = Idem{ui ^ u1(1,p)}, (8)

«1

V2(1)(p) = max{[1,p,ui(1,p),u2]} = Idem{u2 ^ u^(1,p)} (9)

«2

under Vp E R+, and ensure that the pair (uf(1,p),u|(1,p)) is unique; Step 3. For k = 0 we need to find two functions z(i)(0,p, u) (i =1, 2) using equalities min{-2ui + 1.5z2 + V1(1) (p(1) = a + Lp + ez + u)} =

z11

= Wi[0,p,u] = Idem{z ^ z(1)(0,p,u)} Vp E R+,u E R2 (i = 1, 2), (10)

f^t. v/n

analogously to Step 2, find functions uf (0,p), V(0)(p) (i = 1, 2) for Vp E R+ according to

max WL[0,p,uL,u2(0,p)] = /dem{uL ^ u1(0,p)} = VL(0)(p),

«1

max W2[0,p,u1(0,p),u2] = Idem{u2 ^ u2(0,p)} = V2(0)(p) (11

«2

and establish that the pair (ui(0,p), u2(0,p)) is unique. Then the SGE of the game r for all p0 E R+ is formed by the pair (Ue,Je[p0]), where Ue = (Uf,Uf), Uf = (Uf(0),Uf(1)) - (uf(0,p),uf(1,p)) (i = 1,2), and Je[p0] = (V(0)(p0), V(0)(p0)).

4. Explicit presentation of the SGE of the game r

We have applied the scheme suggested in the previous section.

Step 1. (k = 2). Two scalar functions V2(p) = -p2(i = 1, 2) have been built.

Step 2. (k = 1). Equalities (7) only take place under z = z(i)(1,p,u) if

minWi(1)(z) = Wi(1)(z(1)(1,p,u)) Vp,u E R+ (12)

where

W( )(z) = + 1.5z2 - (a + 1p3-i + U + z)2(i = 1, 2). (13)

In turn, (12) takes place if

dWi( 1 )(z)

dz

z(1)( 1 ,p,u)

3z( 1 )(1,p,u) - 2(a + 1p2 + U 1 + z( 1 ) (1, p,u)) = 0,

S2W( 4(z) =1 > 0.

dz2

The first equality leads us to

z( 1 ) (1,p,u) = 2(a + lp2 + u i). (14)

Analogously

z(2) (1,p, u) = 2(a + lp 1 + u2). After substitution of (14) into (13) and taking (7) into account under i =1 we obtain

V} 1 )(p) = max{-2u2 - 3[a + lp2 + u i]2} = /demju i ^ u1 (1,p)}. (15)

This equality takes place for Vp2 > 0 if

dW (1)[u]

du1

«1e)( 1 ,p)

= -4u1 (1,p) - 6[a + 1p2 + u1 (1,p)] = 0, (16)

since

Ô2W1( 1 )[u]

5u2

-10 < 0,

«1e) ( 1 ,p)

where

Wi( i )[u] = -2u2 - 3[a + lp2 + u i]2.

From (16) and (15) we obtain ui(1,p) = —0.6(a + lp2) and V(i )(p) = —1.2(a + lp2)2, analogously u|(1,p) = —0.6(a + lp i) and V2(^(p) = —1.2(a + /pi)2.

Step 3. (k = 0) Finding z(i)(0,p,u) (i = 1, 2) using inequalities

min Wi(0)(z) = W(0)(z(i)(0,p,u)) Vp, u e R2 (i = 1, 2),

z

where (see (10)

Since

=(0) dWi

dz

=(0)

Wl (z) = 1.5z2 - 1.2[a + Ipi + u2 + z)]2 W20)(z) = 1.5z2 - 1.2[a + 1p2 + ui + z)]2

3z(1)(0,p, u) - 2.4/[a + /(a + /pi + u2 + z(1)(0,p,u)] = 0, (17

z(1) (0,p,u)

5 2Wi

(0)

dz2

3 - 2.4/2

z(1) (0,p,u)

and 3 - 2.4/2 > 0 if

/ E [0; 1.11], then, when (18) takes place, (17) yields

18)

z(1)(0,p, u) = Y[a(1 + /) + /2pi + /u2],

where constant

Finally,

2.4/

Y

3- 2.4/2

> 0, V/ E [0; 1.11].

V1(0)(p) = max{-2u2 + —[a(1 + /) + /2pi + /u2(0,p)]} = /dem{ui ^ u1(0,p)}. «1 4/2 - 5

This chain of equalities and its analog for building u1(0,p) takes place under ue (0,p) = u|(0,p) = 0. Then

6

V(0)(P)

6

4/2 5

[a(1 + /) + /2p i ],

analogously

^(pHt^ [a(1 + /) + /2P2].

6

4/2 5

Finally, we obtain the following

Statement 1. If in the game r constant l = 22 € [0; 1.11], then for any initial price P0 = (p i,P2) the strongly guaranteed equilibrium (Ue, Je) is: first, the equilibrium situation

ue = (Ue,Ue), Щ = (U?(0),U?(1)), Ue(0) = Ue(0) - 0,

Ue(1) - -0.6(a + /Р2), U2e(1) - —0.6(a + /p 1 ); second, the equilibrium outcomes

Je = ( Je, je ) , 6

Je = Ji [U e,P°] = i/2^ [a(1 + /) + /2p? ], J2e = J2 [Ue, p°] = 4/^5 [a(1 + /) + /2р2].

Conclusion

Through the use of a suitable variant of dynamic programming, the explicit form of the situation of guaranteed equilibrium in a two-step positional mathematical model of Bertrand duopoly has been found. This research may be extended by examination of N-person games as well as applying the Berge equilibrium instead of the Nash equilibrium.

References

1. COURNOT, A. A. (1838) Recherches sur les principles mathématiques de la théorie de richesses. Paris.

2. NASH, J. F. (1950) Equilibrium points in N-person games. Proc. Nat. Academ. Sci. USA. 36. p. 48-49.

3. NASH, J. F. (1951) Non-cooperative games. Ann. Math.. 54. p. 286-295.

4. BERTRAND, J. (1883) Review of Walras's Theorie mathematique de la richesse sociale and Cournot's Recherches sur les principles mathematiques de la theorie de richesses. Ann. Math.. 68. p. 499-508.

5. Жуковский В. И., Чикрий А. А. Линейно-квадратичные дифференциальные игры. — Киев: Наукова Думка, 1994. — 320 с.

ZHUKOVSKIY, V. I. and CHIKRIY A. A. (1994) Linear-quadratic differential games. Kiev: Naukova Dumka.

6. Жуковский В. И., Кудрявцев К. Н., Смирнова Л. В. Гарантированные решения конфликтов и их приложения. — М.: КРАСАНД, 2013. — 368 с. ZHUKOVSKIY, V. I., KUDRYAVTSEV, K. N., and CHIKRIY A. A. (2013) Guaranteed solutions to conflicts and their applications. Moscow: KRASAND.