Multicriteria Coalitional Model of Decision-making ... - Semantic Scholar

Applied Mathematical Sciences, Vol. 8, 2014, no. 170, 8473 - 8479 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410894

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 c 2014 Xeniya Grigorieva. This is an open access article distributed under Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1] and its modifications, and compromise solution.

Mathematics Subject Classification: 90Axx

8474

Xeniya Grigorieva

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 ∈ N and the projects by j ∈ M . The family M of different games are considered. In each game Gj , j ∈ 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 G1 , . . . , Gm are solved by using the PMS-vector and its modifications [1]. Then having the solutions of games Gj , j = 1, m the optimality principle - “the compromise solution” is proposed to select the best projects j ∗ ∈ M .

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 xi of player i , i = 1, n. The strategy xi can take the following values: xi = 0 as a negative decision for the some project and xi = 1 as a positive decision; li = 2 is the number of pure strategies of player i; x = (x1 , . . . , xn ) is the n-tuple of pure strategies chosen by the players; Q X= Xi is the set of n-tuples; i=1 , n

µi = {ξ (xi )}xi ∈Xi , ξ (xi ) ≥ 0 ∀ xi ∈ Xi ,

P xi ∈Xi

ξ (xi ) = 1 is the mixed strat-

egy of player i; will be used denotation too µi = (ξi0 , ξi1 ), where ξi0 is the probability of making negative decision by the player i for some project, and ξi1 is the probability of making positive decision correspondingly;

Multicriteria coalitional model of decision-making over the set of projects

8475

Mi is the set of mixed strategies of the i-th player; µ is the n-tuple of mixed strategies chosen by players for some project; Q M= Mi 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 . ˜ ( x): Thus, we have multicriteria noncooperative n-person game G D

E

˜ (x) = N, {Xi } G i=1 , n , {Ki (x)}i=1 , n , x∈X .

(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: E

D

G (x) = N, {Xi }i=1 , n , {Ki (x)}i=1 , n , x∈X ,

(2)

Ki (x) = max Kis (x) , Kis (x) ∈ Ki (x) , x ∈ X ,

(3)

where s=1 , r

X

Ei (µ) =

...

x1 ∈X1

X

[Ki (x) ξ (x1 ) . . . ξ (xn )] , i = 1 , n .

(4)

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 Σj of the set N is defined for all j = 1 , m: o

n

Σj = S1j , . . . , Slj , l ≤ n , n = |N | , Skj ∩ Sqj = ∅ ∀ k 6= q,

l [

Skj = N .

k=1

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

Gj (xΣj ) = N,

n

˜ j X S

k

o

n

o

˜ j (xΣj ) , H S j

k=1 , l , Skj ∈Σ

k

k=1 , l , Skj ∈Σj



, j = 1, m. (5)

Here for all j = 1 , m: x˜S j = {xi }i∈S j is the l-tuple of strategies of players from coalition Skj , k = k

k

1, l; ˜ X

Skj

=

Q i∈Skj

Xi is the set of strategies x˜S j of coalition Skj , k = 1, l, i. e. k

Cartesian product of the sets of players’ strategies, which are included into coalition Skj ;  ˜ x˜ j ∈ X ˜ j , k = 1, l is the l-tuple of strategies xΣj = x˜S j , . . . , x˜S j ∈ X, Sk Sk 1 l of all coalitions;

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Xeniya Grigorieva

˜ j is the set of l-tuples in the game Gj (xΣj ); ˜= Q X X S lS j = k

k

k=1, l

˜ XS j k

lΣj =

Q

Q

=

i∈Skj

li is the number of pure strategies of coalition Skj ;

lS j is the number of l-tuples in pure strategies in the game k

k=1,l

Gj (xΣj ). ˜ j is the set of mixed strategies µ M ˜S j of the coalition Skj , k = 1, l; S k

k

l

µ ˜

Skj

=

µ ˜1S j , k

... , µ ˜

j S k j Sk

l

!

µ ˜ξS j k

,

≥ 0 , ξ = 1, l , Skj

j

S k P

ξ=1

µ ˜ξS j = 1, is the mixed k

strategy, that is the set of mixed strategies of players from coalition Skj , k = 1, l;   ˜ j , k = 1, l, is the l-tuple of mixed ˜ µ ˜S j ∈ M µΣj = µ ˜S j , . . . , µ ˜S j ∈ M, Sk 1 k l strategies; ˜ j is the set of l-tuples in mixed strategies; ˜ = Q M M Sk

k=1, l

E˜Sk (˜ µ) is the payoff function of coalition Skj in mixed strategies and defined as E˜Sk (˜ µ) =

X x ˜

˜ j ∈X

S 1

x ˜

j S 1

h



1

˜ j ∈X

S l





˜ S (xΣj ) ξ˜ x˜ j . . . ξ˜ x˜ j H k S S

X

...

1

i

, i = 1, n.

(6)

j S l

From the definition of strategy x˜S j of coalition Skj it follows that 

k



xΣj = x˜S j , . . . , x˜S j and x = (x1 , . . . , xn ) are the same n-tuples in the 1 l games G(x) and Gj (xΣj ). However it does not mean that µ = µΣj . ˜ → R1 of coalition S j for the fixed projects ˜ j : X Payoff function H k Sk j j, j = 1, m, and for the coalitional partition Σ is defined under condition that: ˜ j (xΣj ) ≥ H j (xΣj ) = H S S k

k

X

Ki (x) , k = 1 , l , j = 1 , m , Skj ∈ Σj ,

(7)

i∈Skj

where Ki (x) , i ∈ Skj , is the payoff function of player i in the n-tuple xΣj . 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 decisionmaking with constant matrix of payoffs in the noncooperative game in pure strategies is x∗Σj∗ , that is Nash equilibrium (NE) in a pure strategies in l∗ person game Gj ∗ (xΣj∗ ), with the coalitional partition Σj , where coalitional ∗ partition Σj is the compromise coalitional partition.

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Multicriteria coalitional model of decision-making over the set of projects

Definition 3. Solution of the multicriteria coalitional model of decisionmaking with constant matrix of payoffs in the noncooperative game in mixed strategies is µ∗Σj∗ , that is Nash equilibrium (NE) in a mixed strategies in l∗ person game Gj ∗ (µΣj∗ ), with the coalitional partition Σj , where coalitional ∗ partition Σj is the compromise coalitional partition. Generalized PMS-vector is used as the coalitional imputation [1].

3

Algorithm for solving the problem

Remind the algorithm of constructing the generalized PMS-vector in a coalitional game [1]. ˜ j (xΣj ) for all coalitions S j ∈ Σj , k = 1. Calculate the values of payoff H k S k

1, l , for coalitional game Gj (xΣj ) by using formula (3). 2. Find NE [2] x∗Σj or µ∗Σj (one or more) in the game Gnj (xΣj ). oThe payoffs’ j ∗ vector of coalitions in NE in mixed strategies E (µΣj ) = v Sk . k=1, l

Denote a payoff of coalition

Skj

in NE in mixed strategies by

l

v



Skj



=

Σj X

˜ j (x∗Σj ), k = 1, lΣj , pτ, j H τ, S k

τ =1

˜ j (x∗ j ) is the payoff of coalition S j , when coalitions choose their pure where H k Σ τ, S k

strategies x˜∗S j in NE in mixed strategies µ∗Σj ; pτ, j = k

Q

µ ˜ξSkj , ξk = 1, lS j , τ =

k=1,l ˜ j Hτ, S (x∗Σj ) k

k

k

1, lΣj , is probability of the payoff’s realization of coalition Skj . ˜ j (x∗ j ) is a random variable. There could be many l-tuple The value H Σ τ, S k









of NE in the game, therefore, v S1j , ...., v Slj , are not uniquely defined. ∗ The payoff of each divided according to Shapley’s  coalition  in NE E (µ  Σj ) is j j value [3] Sh (Sk ) = Sh Sk : 1 , ... , Sh Sk : s :





Sh Skj : i =

X S 0 ⊂Skj S 0 3i



(s0 −1) ! (s−s0 ) ! [v (S 0 ) − v (S 0 \ {i})] ∀ i = 1, s , s!



(8)

where s = Skj (s0 = |S 0 |) is the number of elements of sets Skj (S 0 ), and v (S 0 ) is the maximal guaranteed payoff of S 0 ⊂ Sk . Moreover v



Skj



=

s X i=1





Sh Skj : i .

8478

Xeniya Grigorieva

j Then PMS-vector in theNE in mixed strategies µ∗Σj in  the game Gj (xΣ ) j j j ∗ ∗ ∗ is defined as PMS (µΣj ) = PMS1 (µΣj ) , ..., PMSn (µΣj ) , where





PMSji (µ∗Σj ) = Sh Skj : i , i ∈ Skj , k = 1, l. Also remind that a set of compromise solutions [4], [5] is 

CPMS (M ) = arg min max max PMSji − PMSji j

i

j



.

The project j ∗ ∈ CPMS (M ) , on which the minimum is reached is a compromise solution of the game Gj (xΣj ) for all players. 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 , m. 3. Construct the coalitional Gj (xΣj ) associated with the noncooperative game G(x) for the fixed j. 2. Find the NE µ∗Σj in the coalitional game Gj (xΣj ) and find imputation in NE, that is PMSj (µ∗Σj ). 3. Repeat iterations 1-2 for all other j , j = 1 , m. 4. Find compromise solution j ∗ , that is j ∗ ∈ CPMS (M ).

4

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.

References [1] X. Grigorieva, Solutions of Bimatrix Coalitional Games, Applied Mathematical Sciences, vol. 8, 2014, no. 169, 8435 - 8441. http://dx.doi.org/10.12988/ams.2014.410880 [2] J. Nash, Non-cooperative Games, Ann. Mathematics 54 (1951), 286–295. http://dx.doi.org/10.2307/1969529 [3] L. S. Shapley, A Value for n-Person Games. In: Contributions to the Theory of Games (Kuhn, H. W. and A. W. Tucker, eds.) (1953), 307–317. Princeton University Press.

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[4] X. Grigorieva, Static Model of Decision-making over the Set of Coalitional Partitions, Applied Mathematical Sciences, vol. 8, 2014, no. 170, 8451 8457. http://dx.doi.org/10.12988/ams.2014.410889 [5] O.A. Malafeev, Managed conflict system, SPbSU, SPb., 2001. Received: November 15, 2014; Published: November 27, 2014