Multicriteria coalitional model of decision-making over the set of projects with constant payoff matrix in the noncooperative game Текст научной статьи по специальности «Математика»

Multicriteria Coalitional Model of Decision-making over the Set of Projects with Constant Payoff Matrix in the Noncooperative Game

Xeniya Grigorieva

St.Petersburg State University,

Faculty of Applied Mathematics and Control Processes,

University pr. 35, St.Petersburg, 198504, Russia E-mail: [email protected] WWW home page: http://www.apmath.spbu.ru/ru/staff/grigorieva/

Abstract Let N be the set of players and M the set of projects. The multicriteria coalitional model of decision-making over the set of projects is formalized as family of games with different fixed coalitional partitions for each project that required the adoption of a positive or negative decision by each of the players. The players’ strategies are decisions about each of the project. The vector-function of payoffs for each player is defined on the set situations in the initial noncooperative game. We reduce the multicriteria noncooperative game to a noncooperative game with scalar payoffs by using the minimax method of multicriteria optimization. Players forms coalitions in order to obtain higher income. Thus, for each project a coalitional game is defined. In each coalitional game it is required to find in some sense optimal solution. Solving successively each of the coalitional games, we get the set of optimal n-tuples for all coalitional games. It is required to find a compromise solution for the choice of a project, i. e. it is required to find a compromise coalitional partition. As an optimality principles are accepted generalized PMS-vector (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006) and its modifications, and compromise solution.

Keywords: coalitional game, PMS-vector, compromise solution, multicriteria model.

1. Introduction

The set of agents N and the set of projects M are given. Each agent fixed his participation or not participation in the project by one or zero choice. The participation in the project is connected with incomes or losses by different parametres which the agents wants to maximize or minimize. This gives us an optimization problem which can be modeled as multicriteria noncooperative game. We reduce the multicriteria noncooperative game to a noncooperative game with scalar payoffs by using the minimax method of multicriteria optimization. Agents may form coalitions. This problem we will call as multicriteria coalitional model of decision-making.

Denote the players by i G N and the projects by j G M. The family M of different games are considered. In each game Gj, j G M the player i has two strategies accept or reject the project. The payoff of the player in each game is determined by the strategies chosen by all players in this game Gj. As it was mentioned before the players can form coalitions to increase the payoffs components. In each

game Gj coalitional partition is formed. The problem is to find the optimal strategies for coalitions and the imputation of the coalitional payoff between the members of the coalition. The games Gi, ..., Gm are solved by using the PMS-vector (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006) and its modifications.

Then having the solutions of games Gj, j = 1, to the optimality principle - “the compromise solution" is proposed to select the best projects j* € M. The problem is illustrated by example of the interaction of three players.

2. State of the problem

Consider the following problem. Suppose

— N = {1, ... , n} is the set of players;

— Xi = {0; 1} is the set of pure strategies x* of player i, i = 1, n. The strategy

xi can take the following values: x* = 0 as a negative decision for the some project and x* = 1 as a positive decision;

— = 2 is the number of pure strategies of player i;

— x = (x1;..., xn) is the n-tuple of pure strategies chosen by the players;

— X = n X is the set of n-tuples;

i= 1 , n

— Mi = {£ (xi)}x-eX- , £ (xi) > 0 V xi € Xi, £ (xi) = 1 is the mixed strategy

* * XiEXi

of player i; will be used denotation too = (£0 , £i1), where £0 is the probability of making negative decision by the player i for some project, and £i1 is the probability of making positive decision correspondingly;

— is the set of mixed strategies of the i-th player;

— M is the n-tuple of mixed strategies chosen by players for some project;

— M = n M* is the set of n-tuples in mixed strategies for some project;

i= 1, n

— Ki : X ^ Rr is the vector-function of payoff defined on the set X for each

player i, i = 1, n.

Thus, we have multicriteria noncooperative n-person game G( x):

G(x) = (N, {Xt}l=T{K,(x)}i=r^i,ex) . (1)

Using the minimax method of multicriteria optimization, we reduce the noncooperative n-person game G(x) to a noncooperative game G(x) with scalar payoffs:

G(x) = (N, {Xi}^, {Kt(x)}t=—i xeX) , (2)

where

Ki (x) = max if* (x) , K? (x) eK,(x), x G X , (3)

S = 1 , r

Ei(v)= ^2 ••• \-Ki(x)£(xi)• • -£(x™)]> * = 1 > n- (4)

X1 GX1 Xn^Xn

Now suppose M = {1, ... , m} is the set of projects, which require making positive or negative decision by n players.

A coalitional partitions of the set N is defined for all j = 1, to:

i

sj = {Sj, ..., Sj'} , l < n , n = |N| , Sj n Sj = 0 V k = q, U Sj = N.

fc=i

Then we have m simultaneous /-person coalitional games Gj {xSj), j = 1, to, in normal form associated with the game G (x):

Gj {xEj) = /N, {x^l _____________ . ,\Hsj(xSj)X _______ . \, j = l,m.

J \ I k=l,l, I j fc=l , /, siesi /

_____ (5)

Here for all j = 1, to:

— = {xi\. sj is the /-tuple of strategies of players from coalition Sk , k = 1, Z;

^ k ^ k . _______________________________________________________________

— X* is the set of strategies of coalition Sk , A; = 1, /, i. e. Cartesian product of the sets of players’ strategies, which are included into coalition sl _

— xjj, = ^xsj , ... , xsj j G X, G k = 1, I is the /-tuple of strategies

of all coalitions;

— X = Xsi is the set of /-tuples in the game Gj (xs,);

lSl =

Xsj

= n li is the number of pure strategies of coalition Sj;

— l^j = l Sj is the number of l-tuples in pure strategies in the game Gj (xSj).

--- k

k=l,l

— Msi is the set of mixed strategies fis, of the coalition Sk , k = 1, /;

i ,■

/ 1 A £ ------ £

— Msj = , ..., Msj J , ^ 0, £ = 1 lsj, ^MSj = 1, is the mixed

strategy that is the set of mixed strategies of players from coalition 5j, k =

M; _ _ _

— Mx'j = (Ms3 > • • • > Ms3) e &sj e ’ k = I, I, is the /-tuple of mixed

\ 1 l / k k

strategies;

— M = n MSj is the set of l-tuples in mixed strategies;

--- k

k=1,1

— ¿S. (M) is the payoff function of coalition Sj in mixed strategies and defined as

Esk{p)= ■■■ HSk (x&)i(xs,J ...|(xsi) ,i=T~n. (6)

XSj ^Xei X q3 ^Xoj

S1 S1 Sl Sl

From the definition of strategy xsj of coalition Sj it follows that

xs, = ^xsj , ... , xsj j and x = (x1, ... , xn) are the same n-tuples in the games G(x) and Gj (xSj). However it does not mean that m = Msj.

Payoff function Hs : X ^ R1 of coalition Sj for the fixed projects j, j =

1, m, and for the coalitional partition E3 is defined under condition that:

Hsj (xSj) > H„j (xSj) = 'S'' Ki (x), k = 1, /, j = 1, to, S{ G Ej , (7)

S S k

where K (x) , i G Sj , is the payoff function of player i in the n-tuple xXj.

Definition 1. A set of m coalitional l-person games defined by (5) and associated with noncooperative games defined by (1)-(2) is called multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game.

Definition 2. Solution of the multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game in pure strategies is x— , that is Nash equilibrium (NE) in a pure strategies in l-person game Gj* (x£i*), with the coalitional partition ^j , where coalitional partition ^j is the compromise coalitional partition (see 3.2).

Definition 3. Solution of the multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game in mixed strategies is m^j* , that is Nash equilibrium (NE) in a mixed strategies in l-person game Gj* (m— ), with the coalitional partition , where coalitional partition is the compromise coalitional partition (see 3.2).

Generalized PMS-vector is used as the coalitional imputation (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006).

3. Algorithm for solving the problem

3.1. Algorithm of constructing the generalized PMS-vector in a coalitional game.

Remind the algorithm of constructing the generalized PMS-vector in a coalitional game (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006).

1. Calculate the values of payoff Hs, (xs,) for all coalitions Sk G E3, k= 1, I, for coalitional game Gj (xXj) by using formula (3).

2. Find NE (Nash, 1951) x^- or mX— (one or more) in the game Gj(xXj ). The

payoffs’ vector of coalitions in NE in mixed strategies E = jv (^C) } ___•

Denote a payoff of coalition Sj in NE in mixed strategies by

l £3

v (Sk) = J2PT’3^r,Si (xh)’k = M»',

T = 1

where

— HT sj (x— is the payoff of coalition Sj, when coalitions choose their pure strategies xs3 in NE in mixed strategies Mr3.

~ Pt, j = El A*li > = 1; ¿«3; t = \is probability of the payoff’s realization

--- k

k=l,l k

Ht, sj (x^-) of coalition Sj.

The value HT sj (x^-) is a random variable. There could be many l-tuple of NE in the game, therefore, v ^Sjj , ...., v ^Sjj, are not uniquely defined.

The payoff of each coalition in NE E (mX-) is divided acaording to Shapley’s value (Shapley, 1953) Sh (Sfc) = (Sh (Sj : 1) , ..., Sh (Sj : s)):

sh(s3k:i^= ~—~j~~~“~ lv (S') - v (S’\ {*})] v*=1>s> (8)

s'3i

where s = Sj (s' = |S'|) is the number of elements of sets Sj (S'), and v (S') is

the maximal guaranteed payoff of S' c Sk.

Moreover

Skj = Sh Skj : i .

Then PMS-vector in the NE in mixed strategies mX in the game Gj (xxj ) is

defined as ) a

PMSj (mX) = (PMSj (mXj) ,..., PMsn (mXj)) ,

where ) a

PMSj (^) = sh(s{:^ , i£ s°k, k = Tj.

3.2. Algorithm for finding a set of compromise solutions.

Remind the algorithm for finding a set of compromise solutions (Malafeyev, 2001; p.18).

Cpms (M) = arg minmax < maxPMSj — PMSj i .

j * [ j J

Step 1. Construct the ideal vector R = (R1; ... , R„) , where R* = PMSj =

max PMSj is the maximal value of payoff functions of player i in NE on the set M,

j

and j is the number of project j G M:

PMS1 ... PMSn '

PMS” ... PMS”

I ... I

• * - *

PMSj1 ... PMS^n

Step 2. For each j find deviation of payoff function values for other players from the maximal value, that is Aj = i?* — PMSj , * = 1, n:

R1 — PMS 1 ... Rn — PMSn

R 1 — PMSf ... Rn — PMS”

Step 3. From the found deviations Aj for each j select the maximal deviation

Aj* = max Aj among all players i:

3 i

R1 — PMS1 ... R„ — PMSn \ I A1 ... An \ ^ A1*

R1 — PMS” ... Rn — PMS” / I A” ... A”/ ^ A”

s'Csj

v

Step 4. Choose the minimal deviation for all j from all the maximal deviations

among all players i Aj* = min Aj* = min max Aj.

j* j 3 j i

The project j* G CPMS (M) , on which the minimum is reached is a compromise solution of the game Gj(xXj ) for all players.

3.3. Algorithm for solving the multicriteria coalitional model of

decision-making over the set of projects with constant matrix of payoffs in the noncooperative game.

Thus, we have an algorithm for solving the problem.

1. Reduce the multicriteria noncooperative n-person game G(x) (see (1)) to a noncooperative game G(x) with scalar payoffs (see (2)) using the minimax method of multicriteria optimization.

2. Fix a j, j = 1, to.

3. Construct the coalitional Gj(xXj ) associated with the noncooperative game G(x) for the fixed j.

2. Find the NE mX- in the coalitional game Gj(xXj ) and find imputation in NE, that is PMSj (mX ).

3. Repeat iterations 1-2 for all other j , j = 1, to.

4. Find compromise solution j*, that is j* G CPMS (M).

4. Example

Consider the set M = and the set N = {I\, /2, /3} of three players, each

having 2 strategies in multicriteria noncooperative game G(x): x* = 1 is “yes" and Xj = 0 is “no" for all i = 1, 3. The payoff functions of players in the game G (x) are determined by the table 1.

Table 1: The payoffs of players.

The strategies The payoffs of players

/1 I2 I3 h I2 I3

1 1 1 (4, 3.43, 1.71) (2, 1.71, 0.86) (1, 0.86, 0.43)

1 1 0 (1, 0.86, 0.43) (2, 1.71, 0.86) (2, 1.71, 0.86)

1 0 1 (3, 2.57, 1.29) (1, 0.86, 0.43) (5, 4.29, 2.14)

1 0 0 (5, 4.29, 2.14) (1, 0.86, 0.43) (3, 2.57, 1.29)

0 1 1 (5, 4.29, 2.14) (3, 2.57, 1.29) (1, 0.86, 0.43)

0 1 0 (1, 1, 0.43) (1.14, 2, 0.86) (1.86, 2, 2)

0 0 1 (0, 0, 0) (4, 3.43, 1.71) (3, 3, 1.29)

0 0 0 (0, 0, 0) (4, 3.43, 1.71) (2, 2, 0.86)

Reduce the multicriteria noncooperative n-person game G(x) to a noncooperative game G(x) with scalar payoffs using the minimax method of multicriteria optimization (see (3)). The values of payoff functions of players in the game G (x) are in the table 2.

Table 2: The payoffs of players.

The strategies The payoffs The payoffs of coalition

/1 I2 I3 /1 I2 I3 1—1 to w 1—1 {11, /2, I3}

1 1 1 4 2 1 6 3 5 7

1 1 0 1 2 2 3 4 3 5

1 0 1 3 1 5 4 6 8 9

1 0 0 5 1 3 6 4 8 9

0 1 1 5 3 1 8 4 6 9

0 1 0 1 2 2 3 4 3 5

0 0 1 0 4 3 4 7 3 7

0 0 0 0 4 2 4 6 2 6

1. Compose and solve the coalitional game G2 (1^2), S2 = {{/1, /2}, /3}, i. e. find NE in mixed strategies in the game:

n = 3/7 1 - n =4/7 1 0 0 (1, 1) [6, 1] [3, 2]

0 (0, 0) [4, 3] [4, 2]

£ =1/3 (1, 0) [4, 5] [6, 3]

1 - £ = 2/3 (0, 1) [8, 1] [3, 2].

It’s clear, that first matrix row is dominated by the last one and the second is dominated by third. One can easily calculate NE and we have

y = (3/7 4/7) , x = (0 0 1/3 2/3) .

Then the probabilities of payoffs’s realization of the coalitions S = {/1, /2} and N\S = {/3} in mixed strategies (in NE) are as follows:

'36 7' r 1 11

Y’ 3 — 5-, 27’ 3

n1 n2 £1 00

£2 0 0 .

£3 1/7 4/21

£4 2/7 8/21

The Nash value of the game in mixed strategies is calculated by formula:

1 2 4 8

E (x, y) = - [4, 5] + - [8, 1] + - [6, 3] + - [3, 2] =

In the table 3 pure strategies of coalition N\S and its mixed strategy y are given

horizontally at the right side. Pure strategies of coalition S and its mixed strategy x are given vertically. Inside the table players’ payoffs from the coalition S and players’ payoffs from the coalition N\S are given at the right side.

Divide the game’s Nash value in mixed strategies according to Shapley value

Shi = v (/1) + | [v (/1, /2) - v (/2) - v (/1)] ,

Sh2 = v (/2) + 5 [v (/1, /2) - v (/2) - v (ii)j .

Table 3: The maximal guaranteed payoffs of players 11 and 12.

Math. Expectation

The strategies of N\ S, the payoffs of S and the payoffs of N\ S

min 1 min 2

max

2.286 2.000 4.143 1.000 2.714 2.429 0.000 4.000 v (Ii) v (Ii) 2.286 2.000 0.000 1.000 2.286 2.000

+1

+2

0 - (1 , 1) /(4 , 2) (1 , 2)\

£ = 0.33 +(1 , 2) (3 , 1) (5 , 1)

1 - £ = 0.67 +(2, 1) (5 , 3) (1 , 2)

0 - (2 , 2) V(0 , 4) (0 , 4)/

Find the maximal guaranteed payoffs v (/1) and v (/2) of players /1 and /2. For this purpose fix a NE strategy of a third player as

y= (3/7 4/7) .

Denote mathematical expectations of players’ payoff from coalition S when mixed NE strategies are used by coalition N\S by ES{i^) (y) ,i,j = 1,2. In the table 3 the mathematical expectations are located at the left, and values are obtained by using the following formulas:

ES(i,i)(y) = (M+f -1; f-2+1-2; |-l + f-2) = (2f; 2; if) ES(i,2)(y) = (f '3+f -5; I-l + |.l; f • 5 + f • 3) = (4±; 1; 3f)

!rr,\ - (3 . c , 4 1 . 3 o , 4 o. 3 , 4 _/a5.o3. ,41

E

S(2

, D (y) = (? • 5 + f • 1; f • 3 + f • 2; f • 1 + I • 2) = (2f; 2f ; if)

.. _ /3 . fl I 4 n . 3 a 1 4a. 3 0 1 4 — in- A ■ O3 \

Es(2,2) (v) — (f ' 0 + 7 ' 0 ; | • 4 + | • 4 ; | • 3 + | • 2) — (0; 4 ; 2|) .

Third element here is the mathematical expectation of payoff of the player /3 (see table 2 too).

Then, look at the table 2 or table 3,

miniii (xi = 1, x2, y) = min {2|; 4^} = 2^; minH1 (xi = 0, x2, y) = min {2|; 0} = 0; minii2 (xi, x2 = 1, y) = min {2; 2|} = 2; minH2 (xi, x2 = 0, y) = min{1; 4} = 1;

v (/1) = max {2|; 0} = 2|; v (12) = max {2; 1} = 2.

Thus, maxmin payoff for player I\ is v{I\) = 2^ and for player /2 is v{I2) = 2. Hence,

^1 (y) = v (/1) + \ (5^ — v (/1) — v (/2)) = 2| + ^ (5^ —2^—2) = 2|;

Sfe2 (y) = 2+f = 2f.

Thus, PMS-vector is equal:

5 3 1

PMSi = 2-; PMS2 = 2-; PMS3 = 2- .

Table 4: Shapley’s value in the cooperative game.

The strategies of players The payoffs of players The payoff of coalition Shapley’s value

/1 I2 I3 /1 I2 I3 HN (/1, /2, I3) XiHn X2HN X3HN

1 1 1 4 2 1 7

1 1 2 1 2 2 5

1 2 1 3 1 5 9 2.5 3.5 3

1 2 2 5 1 3 9 2.5 3.5 3

2 1 1 5 3 1 9 2.5 3.5 3

2 1 2 1 2 2 5

2 2 1 0 4 3 7

2 2 2 0 4 2 6

2. Solve the cooperative game G5 (xS5), Z5 = {N = {/1,/2, /3}}, see table 4. Find the maximal payoff HN of coalition N and divide him according to Shapley value (8), (Shapley, 1953):

Sh\ = \[v (h,h) + v (h,I3) - v (/2) - v (/3)] + \ [v (N) - v (h,h) + v (/1)] ;

6 3

Sh2 = \[v (h,h) +v(h,h) - V (/1) - v (/3)] + \ [v (N) - v {h,h) + v (I2)\ ;

6 3

Sh3 = \[v (h,h) + v (h,I2) ~ v (/1) - v (/2)] + ^ [v (N) - v (h,I2) + v (/3)] . 63

Find the guaranteed payoffs:

v (/1, /2) = max {4, 3} =4; v (/1, /3) = max {3, 2} = 3;

v (/2,13) = max{3, 4} = 4;

v (/1) = max {1, 0} = 1; v (/2) = max {2, 1} =2; v (/3) = max {1, 2} = 2 .

Then

Shf' h 1} = Sh^’2’2) = Sh^’2,1) = - + - + - [9 -4] + - = - + - + - + - = 2-,

1 1 1 3 6 3 1 J 3 3 6 3 3 2

Sh$' h 1} = Sh£'2'2) = S,41,2,1) = - + - + - [9-3] + - = - + - + - + - = 3-,

2 2 2 2 3 3 3 2 3 3 3 2

S/42’1’15 = Shi1,2’2) = ^1,2,1) = | + | + |[9-4] + | = | + | + | + | = 3-

3. Solve noncooperative game G1 (x£i), £1 = {S1 = {/1}, S2 = {/2}, S3 = { /3}}. In pure strategies NE not exist.

From p. 3 it follows that the guaranteed payoffs v (/1) = 1; v (/2) = 2; v (/3) =

2. Find the optimal strategies with Nash arbitration scheme, see table 5. Then optimal n-tuple are ((1), (1), (2)) and ((2), (1), (2)), the payoff in NE equals ((1), (2), (2)).

A detailed solution of games for various cases of coalitional partition of players is provided in (Grigorieva, 2009). Present the obtained solution in (Grigorieva, 2009) in the table 6.

Table 5: Solution of noncooperative game.

The strategies of players The payoffs of players Pareto-optimality (P) and Nash arbitration scheme

Ii І2 Із Ii І2 Із Nash arbitration scheme P

1 1 1 4 2 1 (4 - 1) (2 - 2) (1 - 2) < 0 -

1 1 2 1 2 2 (1 - 1) (2 - 2) (2 - 2) = 0 +

1 2 1 3 1 5 (3 - 1) (1 - 2) (5 - 2) < 0 -

1 2 2 5 1 3 (5 - 1) (1 - 2) (3- 2) < 0 -

2 1 1 5 3 1 (5 - 1) (3 - 2) (1 - 2) < 0 -

2 1 2 1 2 2 (1 - 1) (2 - 2) (2 - 2) = 0 +

2 2 1 0 4 3 (0 - 1) (4 - 2) (3 - 2) < 0 -

2 2 2 0 4 2 (0 - 1) (4 - 2) (2 - 2) < 0 -

Table 6: Payoffs of players in NE for various cases of the coalitional partitions of players.

Project Coalitional partitions The те-tuple of NE (II, І2, Із) Probability of realization NE Payoffs of players in NE

1 Si = {{II} {h} {h}} т, (1), (0)) 1 ((1) , (2) , (2))

((0) , (1) , (0))

((1,0), і) 1/7

2 S2 = {{h, І2} {ІЗ}} ((1,0), 0) 4/21 ((2.71, 2.43) , 2.33)

((0,1), 1) 2/7

((0, 1), 0) 8/21

(1, (1), 1) 5/12

3 £з = {{Іі,Із}{І2}} (1, (0), 1) 1/12 (2.59, (2.5) , 2.91)

(0, (1), 1) 5/12

(0, (0), 1) 1/12

4 S, = {{І2, Із} {II}} (1, (0, 1)) 1 (3, (3, 3))

(1, 0, 1) 1

5 Sb = {II, І2, Із} (1, 0, 0) 1 (2.5, 3.5, 3)

(0, 1, 1) 1

Table 7: The set of compromise coalitional partitions.

h І2 Із h І2 Із

1 2 2 Л{{Іі}{І2}{Із}} 2 1.5 1 2

Ї2 = {{h, h} {Із}} 2.71 2.43 2.33 Л{{Іі,І2}{Із}} 0.29 1.07 0.67 1.07

S3 = {{h, Із} {h}} 2.59 2.5 2.91 Л{{Іі,Із}{І2}} 0.41 1 0.09 1

Е4 = {{і2,із}{іі}} 3 3 3 Л{{І2,Із}{Іі}} 0 0.5 0 0.5

S5 = {h, І2, h} 2.5 3.5 3 Л{І1, І2, Із} 0.5 0 0 0.5

R 3 3.5 3

Applying the algorithm for finding a compromise solution, we get the set of compromise coalitional partitions (table 7).

Therefore, compromise imputation are PMS-vector in coalitional game with the coalition partition Z4 in NE (1, (0, 1)) in pure strategies with payoffs (3, (3, 3)) and Shapley value in the cooperative game in NE ((1, 0, 1), (1, 0, 0), (0, 1, 1)

- cooperative strategies) with the payoffs (2.5, 3.5, 3).

Moreover, in situation, for example, (1, (0, 1)) the first and third players give a positive decision for corresponding project. In other words, if the first and third players give a positive decision for corresponding project, and the second does not, then payoff of players will be optimal in terms of corresponding coalitional interaction.

5. Conclusion

A multicriteria coalitional model of decision-making over the set of projects with constant payoff matrix in the noncooperative game and algorithm for finding optimal solution are constructed in this paper, the numerical example is given.

References

Grigorieva, X., Mamkina, S. (2009). Solutions of Bimatrix Coalitional Games. Contributions to game and management. Collected papers printed on the Second International Conference “Game Theory and Management" [GTM’2008]/ Edited by Leon A. Petrosjan, Nikolay A. Zenkevich. - SPb.: Graduate School of Management, SpbSU, 2009, pp. 147-153.

Petrosjan, L., Mamkina, S. (2006). Dynamic Games with Coalitional Structures. Intersectional Game Theory Review, 8(2), 295-307.

Nash, J. (1951). Non-cooperative Games. Ann. Mathematics 54, 286-295.

Shapley, L. S. (1953). A Value for n-Person Games. In: Contributions to the Theory of Games( Kuhn, H. W. and A. W. Tucker, eds.), pp. 307-317. Princeton University Press. Grigorieva, X. V. (2009). Dynamic approach with elements of local optimization in a class of stochastic games of coalition. In: Interuniversity thematic collection of works of St. Petersburg State University of Civil Engineering (Ed. Dr., prof. B. G. Wager). Vol. 16. Pp. 104-138.

Malafeyev, O. A. (2001). Control system of conflict. SPb.: St. Petersburg State University, 2001.