Network formation in competition model Текст научной статьи по специальности «Компьютерные и информационные науки»

Artem Sedakov

St Petersburg University Faculty of Applied Mathematics and Control Processes 35 Universitetsky prospekt, Petrodvorets,

St Petersburg, 198504, Russia E-mail: a.sedakov@yahoo.com

Abstract A competition model on a market is considered. Each player (firm) attracts customers only by price for a carriage service in a route network between any of its two nodes. It is proposed two types of players behavior: noncooperative and cooperative. In noncooperative scenario each player aims to maximize its profit in a route network, and as a solution concept Nash equilibrium is considered. In cooperative scenario it is supposed that two fixed players can cooperate only on a route which connects their hubs to maximize their total profit on the route, and on other routes their behavior as well as behavior of other players remains noncooperative.

Here we refer to cooperative theory and choose a solution concept (core, the Shapley value). All considered scenarios are illustrated with a numerical example.

Keywords: competition model, network formation, coalition, cooperative solution.

A finite set of players N = {1,...,n}, which provide similar carriage service in route network between its nodes from a given set H = {h\,..., hn}, is considered. Suppose that each player i G N is located in hi G H, i. e. player provides a service in the network via hi, hi = hj for all i,j G N, i = j. One can consider a complete graph which consists of n nodes and a set of links G. For each hi,hj G H a pair hihj G G denotes a route between the nodes. On route hihj both players i G N and j G N provide carriage service using a scheme hi ^ hj, and player s G N \ {i U j} provides carriage service via its hub hs using a scheme hi ^ hs ^ hj.

Let a pair hk,h G H, i. e. route hkhi G G be fixed. Hereafter the following notation kl for route hkhi is used for simplicity. By pki = (pki,... ,plni) denote a price profile that players set up on route M. Define a demand function of player i G N on the route for its service in accordance with pki:

Suppose that parameters aki, bki, cki > 0 are positive, bki > (n — 1)cki and

for each kl G G and for each i, j G N.

A set of prices pi = {pki}k£ea which are assigned by player i can be interpreted as his strategy, and the n-tuple of strategies p = (p\,...,pn) for each i G N gives us a strategy profile.

(1)

j=i

pki G [0, ake/bki]

(2)

Number of potential customers of player i G N on the route ki subject to price profile (pki,... ,p!nf) can be calculated as:

Nki = £Dki (pY,...P), i G N.

i=k

Introduce player costs for service ski > 0, i G N on route k£ (ski = sik for each i G N and k,£ G H):

i ski

nki ____ ) si ,

hi = hk or hi = h£, ski + si, hi G H \ {hk U hi}.

Then one can define profit (payoff function) of player i subject to price profile P = (Pi,...,Pn):

ni(p) = V (pk - ski)Dki(pki,.. .,pkni), i g N.

kieG

(3)

1. Noncooperative scenario

In this section the noncooperative case of competition is considered. It is supposed that players choose their price profiles pi = {plki]kieG, i G N simultaneously and independently from each other from the set (2). Then in accordance with the chosen price profile, both demand functions (1) and profits (3) are calculated. As a solution concept Nash equilibrium is considered.

For each i G N and k£ G G, first order conditions have the form:

aki - 2bkipki + ckiJ2pf + bkiski = 0, i G N.

j=i

In matrix form first-order conditions (4) can be rewritten as:

(4)

2bki -cki cki 2bki

-cki\ (pki\ pki p2

ki

ki ki -c -c

( aki + bkiski\

2b J \pnf)

ki bkiski

\aki +

(5)

Solving system (5) of linear equations, we obtain the following prices in equilibrium:

-M(2bki + cki) + bkicki ski + bki(2bki - (n - 1)cki)

ki ki

n — i)c )s

p-f =

ieN

(2bki + cki)(2bki - (n - 1)cki)

(6)

Equilibrium prices from (6) belong to the corresponding admissible intervals pki G [0, aki/bki] subject to inequalities:

SV < — * — frki

ki

1

(n - 1 )cke

, for each i G N.

Thus, hereafter, it is supposed that problem parameters satisfy inequalities (7).

By substituting prices from (6) to the expression (3), we can calculate players profits in equilibrium for each i G N:

n(p )

Y bke x

k£eG

= X

ake(2bke + cke) + bkicki J2 skk + (bke + cki)(2bki — (n — 1)cke)st ieN

(2bke + cke)(2bke — (n — 1)cke)

(8)

kieo

2. Cooperative scenario

Now consider a case in which two players i and j can cooperate only on direct route hi hj. In this setting both players i and j aim to maximize sum of its profits

on the direct route hihj, i. e. maximize

ni (pij) + nij (pij) = (pij — sij )(aij — bijpj + cij V'pijj) +

/ 1

£=i

+ (pjj — sj )(aij — bij pj + cijy^ p%j),

while other players k G N \ {i,j} want to maximize

nk (pij ) = (pj — sij)(aij — bij pk + c

pijj )■

£=k

(9)

(10)

This problem is reduced to finding equilibrium in an (n — 1)-person game, in which player-coalition {i,j} has a payoff function (9), and players k G N \ {i,j} have a payoff function (10). First-order conditions for this problem can be written as:

aij — 2bijpf + 2cijpj + cij Y pe + (bij — cij)sj = 0,

i=i,i=j

aij — 2bijpj + 2cijpij + cij Y piej + (bij — cij)sj = 0,

e=i,e=j

aij — 2bijpkj + cijYpijj + bijsij =0, k = i,k = j. (11)

e=k

In matrix form first-order conditions (11) have the form:

( 2bij • • • —cij • • —cij • • —cij\ pi1j f aij + bij sij1 \

—cij • • • 2bij • • • —2cij • • —cij ij piij aij + (bij — cij )sij

—cij • • • —2cij • • 2bij • • —cij ij pj aij + (bij — cij )sjj

\ —cij • • —cij • • —cij • • 2bij yi \fj) \ aij + bij snj /

2

Let pij = (pj,... ,p1,j) be a solution of the system (12). Since matrix of the system (5) has inverse, it is possible to find invertible matrix of the system (12) using for example the Woodbury matrix identity, or computational software.

The next problem is to allocate total profit nli3 (p1) + nj (p1) on route hihj among two players i and j. Here we refer to cooperative game theory. For each coalition — empty, {i}, {j}, or grand coalition {i,j} — define a characteristic functions v() as follows:

v({i,j}) = ni1 (pij)+ nj (pij),

(i) = maxmin Wi1 (pj ,piji) =

v(j) = max min ni (pi1, p1„•

(aij - tfisf)2 (aij - bij sj )2

i v'l ’ r-p 4bij ’

v(0) =0?

where p1 = (pj,.. .,pi-1 ,pj+1,... ,pn1), and pl-j = (p\j,.. .,pj_ 1 ,pi+1,.. .,p).

As a solution concept consider the Shapley value (Shi1 (pij),Shij(pij)), which components depend on the characteristic function v() and are calculated as:

shij(pij) = + =

nij(pij) + njj(pij) (sij - si1 )(2aij - bij(sj + si1))

Shi pi) =

2

v({i,j}) - v(i) + v(j)

12

_ II?m + n?№) (s? - sf)(2aV - b+ 4))

2 8 '

Thus, if players i and j cooperate on route ij, players profits are:

Shi1 (pi1) for player i,

Sh1 (pi1) for player j,

nk1 (p1) for player k G N \{i,j}.

Here the Shapley value components Shi (p1) and Sh1 (p1) are calculated using (13), and profit nlj1 (p1) of player k G N \ {i,j} is calculated using (10) subject to price profile p%1, where p%1 = (pi,... ,pl1) is a solution of (12).

3. Network Formation

In this section it is supposed that players are allowed to form a network, i. e. choose those players with whom they want to form mutual links. Following the definition, network is a pair: (N, L). Here N = {1,...,n} is a finite set of players, and a set L is a set of links in the network.

Now consider a network formation mechanism. For each player i G N introduce an n-dimensional vector gi = (gi1,..., gin) G {0,1}n s. t.

_ J 1, iff i wants to cooperate (form a link) with j, gi1 = 0, otherwise or if j = i.

8

The n-dimensional vector gi is called a strategy of player i G N in a network formation game, and a set of all possible strategies of i in this game is denoted by Gi.

In this setting the following network formation mechanism is proposed: players choose simultaneously their strategies gi G Gi,i G N, which constitute a strategy profile g = (g1,. ..,gn). How to construct a set of links L subject to profile g? A link ij G L is formed only if both players i and j want to cooperate with each other, i. e. if gi1 = gji = 1. In other cases link ij is not formed. Thus, strategy profile g uniquely defines the set of links L, which generates a set Ni(L) = {j : ij G L} of neighbours of players i G N. According to L players payoffs are calculated as:

Ui (L) = Y Shk (pik)+ Y nik (Pik) +

keNi(L) keN\(Ni(L)Ui)

+ Y nke(pkt), i G N. (14)

k£^G:k=i,£=i

Here pik = (plk,... ,p1lk) is the solution of system (12), and pik = (p1k,... ,pnk) is taken from (6).

The first term in (14) is the sum of player i profits if he cooperates with his neighbours on routes from hi. The second and the third terms in (14) are the sum of player i profits if he plays individually on other routes.

To formulate a solution concept in such setting, reduce the network formation mechanism to an n-person game in strategic form:

{N, {Gi}i£N, {Ui(')}i£N).

By this transformation, each strategy profile g = (g1,...,gn) in the game in strategic form generates a unique network with the set of links L. Set L helps in turn to calculate price profile for (14).

Nash network. As a solution concept consider the Nash network: a network (N, L*) which is supported by strategy profile g* = (g\,..., gn) such that for each i G N and for each gi G Gi the inequality

Ui(L*) > Ui(Li)

holds. Here Li is a set links in the network which is realized subject to strategy profile (g1 ^.^ g*-^ gi, g*+l, ...,g*n).

Efficient network. Another solution concept is the efficient network: a network (N, L*) is said to be the efficient network if for each L the inequality

Y Ui(L*) > Y Ui(L)

ieN ieN

holds.

Example 1. In the example it is shown that strategy profiles, which generate either the empty or complete network, constitute Nash equilibrium. Therefore, both empty network and complete network are Nash networks. Moreover, it is also shown that the complete network is the efficient network.

Consider a 3-person game with players set N = {1, 2,3}, and set of hubs H = {hi, h2, h3} (see Fig. 1). The set of possible routes contains only 3 elements: {h\h2, h1h3, h2h3}, so carriage service is provided using a scheme:

Route Player 1 Player 2 Player 3

h1h2 hi -O- h-2 h1 O h2 h1 O h3 O h2

h1h3 hi -O h.3 h1 O h2 O h3 h1 O h3

h2h3 h2 O h1 O h3 h2 O h3 h2 O h3

h2

Figurel: Route network

In the example problem parameters (demand parameters and players costs for each route) are taken as

a12 = 150; a13 = 150; a23 = 150;

b12 = 0.13; b13 = 0.15; b23 = 0.17:

c12 = 0.02; c13 = 0.01; c23 = 0.02

s}2 = 134; s}3 = 122; sf = 256;

si2 = 122; s}3 = 216; s}3 = 94;

s}2 = 209; s}3 = 113; sf = 96;

Here superscripts h1h2, h1h3, h2h3 are replaced by 12, 13, 23 only to simplify the notations.

Table1: Players profits on route hihj in noncooperative case.

Route Player 1 Player 2 Player 3 /71*2 51,541.17 52,598.98 45,173.35 hih3 34,637.85 27,996.74 35,310.74 h2h3 24,409.70 36,668.13 36,501.64

Players profits in noncooperative case are calculated using (8) and shown in Table 1. Players profits in cooperative case are calculated using (10), (13) and shown in Table 2. Total players profits in the network are calculated using (14) and shown in Table 3.

From Table 3 it is easy to check that both empty network and complete network are Nash networks. It is also can be seen from the table that the complete network with the total players profits of 348,279.57 is the efficient network.

Table2: Players profits on route hi hj subject to cooperation between i and j.

Route Player 1 Player 2 Player 3 hih2 52,158.05 52,958.21 46,559.67 h1h3 34,723.84 28,146.30 35,319.53 h2h3 24,884.66 36,831.58 36,697.73

Table3: Total players profits in the network (N, L).

Set of links L Player 1 Player 2 Player 3

L = 0 110,558.72 117,263.85 116,985.78

L= {12} 111,205.60 117,623.08 116,985.78

L= {13} 110,674.71 117,263.85 116,994.57

L= {23} 110,588.72 117,427.30 117,181.82

L= {12,13} 111,291.59 117,623.08 116,994.57

L= {12, 23} 111,205.60 117,786.53 117,181.82

L= {13, 23} 110,674.71 117,427.30 117,190.61

L= {12,13, 23} 111,766.55 117,936.09 118,576.93

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