Static model of decision-making over the set of coalitional partitions Текст научной статьи по специальности «Математика»

Static Model of Decision-Making over the Set of Coalitional Partitions*

Xeniya Grigorieva

St.Petersburg University,

Faculty of Applied Mathematics and Control Processes, University pr. 35, St.Petersburg, 198504, Russia E-mail: kseniya196247@mail.ru WWW home page: http://www.apmath.spbu.ru/ru/staff/grigorieva/

Abstract Let be N the set of players and M the set of projects. The coali-tional 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.

Players can form 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.

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 which the agents wants to maximize or minimize. Agents may form coalitions. This gives us an optimization problem which can be modeled as game. This problem we call as static 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. In each game Gj coalitional partition is formed and 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 new optimality principle

- “the compromise solution” is proposed to select the best projects j* G M. The problem is illustrated by example of the interaction of three players.

* This work was supported by the Russian Foundation for Fundamental Researches under grants No.12-01-00752-a.

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: 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 is the n-tuple of pure strategies chosen by the players;

— X = n Xi is the set of n-tuples;

i= 1 , n

— ii = (£i , €i) is the mixed strategy of player i, where is the probability of making negative decision by the player i for some project, and is the probability of making positive decision correspondingly;

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

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

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

i= 1, n

— Ki (x) : X ^ R1 is the payoff function defined on the set X for each player

i, i = 1, n, and for some project.

Thus, for some project we have noncooperative n-person game G (x):

G(x) = (N, {Xt}l=T{Kt(x)}l=T^,xeX) . (1)

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, m:

i

£j = { Sj ,...,Sj} , l < n, n = \N\ , SI n Sj = 0 V k = q, \JSi = N.

k = 1

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

Gj (xSj) = /N, l^si) ___________ . ,\Hsj(xSj)X ___ . \, j = l,m.

J \ I k = l,l, I j k=l,l, S3kezi /

_________ (2)

Here for all j = 1, m:

— xsj = {xi}ieSi is the /-tuple of strategies of players from coalition S3k , k = 1, I;

— Xs, = Xi is the set of strategies xsj of coalition S3k , k = 1, I, i. e. Carte-

i'^Sk

sian product of the sets of players’ strategies, which are included into coalition

_

— xSj = (xsj , ... , ^ *s3 ^ ’ A: = 1, / is the /-tuple of strategies

of all coalitions;

— X = J} Xsj is the set of l-tuples in the game Gj (xs,);

k

k= 1,1

lsj —

Xsj

— n li is the number of pure strategies of coalition S3k;

lEj — n ls j is the number of l-tuples in pure strategies in the game Gj (x^j ). k=~l k

Msi is the set of mixed strategies flsj of the coalition S3k , k = 1, I;

( lsj \ , _________________________________________ si ,

— flsj — yj'Sj , J , #Sj - 0 , ^ — 1 lsj > — 1, is the mixed

strategy, that is the set of mixed strategies of players from coalition S3k , k —

177; _

— = \tlsi > ■ ■ ■ ’ (“s3) ^ (“s3 ^ ’ k = 1,1, is the /-tuple of mixed

\ 1 l / k k

strategies;

— MI — n MIsi is the set of l-tuples in mixed strategies.

k

k= 1,1

From the definition of strategy Xsi of coalition S3k it follows that

xSi — [xsi , ••• , Xsj^ and x — (xi, ••• , xn) are the same n-tuples in the games

G(x) and Gj (xSj). However it does not mean that i — .

Payoff function Hsi : X ^ R1 of coalition S3k for the fixed projects j, j —

1, to, and for the coalitional partition Sk is defined under condition that:

Hsj (xsi) > Hsj (xsi) = V' Ki (x), k = 1,1, j = 1, m, S3k e S3 , (3)

k k ‘ ■

where Ki (x) , i € S3k , is the payoff function of player i in the n-tuple x^i.

Definition 1. A set of m coalitional l-person games defined by (2) is called static coalitional model of decision-making.

Definition 2. Solution of the static coalitional model of decision-making in pure strategies is x*sj*, that is Nash equilibrium (NE) in a pure strategies in l-person game Gj* (xSj*), with the coalitional partition £k*, where coalitional partition Sk* is the compromise coalitional partition (see 2.2).

Definition 3. Solution of the static coalitional model of decision-making in mixed strategies is i*si*, that is Nash equilibrium (NE) in a mixed strategies in l-person game Gj* (n ), with the coalitional partition Sk , where coalitional partition Sk is the compromise coalitional partition (see 2.2).

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, (xjj, ) for all coalitions S3k G Sk , k =1, I,

k

for coalitional game Gj (x^i ) by using formula (3).

2. Find NE (Nash, 1951) x*si or i*si (one or more) in the game Gj(xs, ). The payoffs’ vector of coalitions in NE in mixed strategies E } ___•

Denote a payoff of coalition S3k in NE in mixed strategies by

lEj

v (Sk) = J2PT’i^T,Si (Xh)> k=l>

where

HT sj (x*sj) is the payoff of coalition S°k, when coalitions choose their pure strategies x*i in NE in mixed strategies i*si.

Pt, j = n i £,k = 1, lsj, t = 1, Ijjj, is probability of the payoff’s realization k=l,l k HT si {x*Sj) of coalition S3k.

The value H si (xU) is random variable. There could be many l-tuple of NE

’ k

in the game, therefore, v ^S{ j , ••••, v (Sk^j, are not uniquely defined.

The payoff of each coalition in NE E {p*si) is divided according to Shapley’s value (Shapley, 1953) Sh (Sk) — (Sh (S°k : ^ , •••,Sh(S0k : s)):

k

s' 3i

Sh (sl ■ *) = E (s' 1}^S s°! b (S') ~ v (S'\ {*})] V * = 177, (4)

where s

(s' — |S'|) is the number of elements of sets Sk (S'), and v (S') are the total maximal guaranteed payoffs all over the S' C Sk.

Moreover

Sk) — J2 Sh(Sl : ^ • i=1

Then PMS-vector in the NE in mixed strategies i*si in the game Gj (xSi ) is defined as

PMSk (p*s,) — (PMSl (p*s,) , •••, PMSJn (p*s, )) ,

where

PMSf ) = Sh (,Si : i) , * G Si, k = ITT.

3.2. Algorithm for finding a set of compromise solutions.

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

Cpms (M) — arg minmax < maxPMSk — PMSk i •

k i k i i

v

Step 1. Construct the ideal vector R — (R1, ••• , Rn) , where fR — PMSj —

maxPMSj is the maximal value of payoff’s function of player i in NE on the set M,

ki

and j is the number of project j G M:

PMSi ••• PMSn

PMS™ ••• PMS

n

PMSi ••• pms^

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

I R1 — PMSi ••• Rn — PMSn * — ( ......................

\Ri — PMSf ••• Rn — PMS^

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

Aj* — max Aj among all players i:

iiii

Ri — PMSi ••• Rn — PMSn \ / A1 ••• An \ ^ Al1^

R1 — PMSf ••• Rn — PMS^/ \Af ••• A^J ^ Am

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

among all players i Aj, — min Aj, — min max Aj.

i* k i k i

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

3.3. Algorithm for solving the static coalitional model of decisionmaking.

Thus, we have an algorithm for solving the problem.

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

2. Find the NE /j,*si in the coalitional game Gj (xSi ) and find imputation in NE, that is PMSk (j*sj).

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

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

m

4. Example

Consider the set M = {j}^=Y~g and the set N = {I\, /2 , /3} of three players, each having 2 strategies in noncooperative game G (x): xi — 1 is “yes” and xi — 0 is “no” for all i = 1,3. The payoff’s functions of players in the game G (x) are determined by the table 1.

Tablel: The payoffs of players.

The strategies The payoffs The payoffs of coalition

h h h h h h {h, h} {h, h} {h, h} {h, h, h}

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 (xS2), S2 = {{Ii, I2} , I3}, 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]

e = 1/3 (1, 0) [4, 5] [6, 3]

1 - e =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 = {I1, I2} and N\S = {I3} in mixed strategies (in NE) are as follows:

ni n2 ei 00 e2 00 .

e3 1/7 4/21 e4 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] =

'36 7' 11

T’ 3 = 5-, 27’ 3

In the table 2 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’s value

Table2: The maximal guaranteed payoffs of players I1 and I2.

Math. Expectation

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

min 1 min 2

max

2.286 4.143 2.714 0.000 v (I1) 2.286 0.000

2.000

1.000

2.429

4.000 v (Ii)

2.000 1.000

0

e = 0.33

n = 0.43 1 - n = 0.57

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

(1, 2)

1 - e = 0.67 +(2, 1) 0 - (2 , 2)

(3 , 1) (5 , 1)

(5,

V(°,

3) (1 ,

4) (0,

2)

4)

2.286 2.000

Shi = v (/i) + \[v (ii, h) ~ v (I2) - v (h)} ,

Sh2 = v (/2) + 5 [v (/i, h) ~ v (h) - v (ii)] .

Find the maximal guaranteed payoffs v (I1) and v (I2) of players I1 and I2. For this purpose fix a NE strategy of a third player as

y= (3/7 4/7) .

Denote mathematical expectations of the players’ payoffs from coalition S when mixed NE strategies are used by coalition N\S by (y) ,i,j = 1, 2. In the table

2 the mathematical expectations are located at the left, and values are obtained by using the following formulas:

ES(i,i)(y) = (!'4+f -1; f-2+1-2; |-l + f-2) =(2f;2; if)

Es( l, 2) (y) = (r • 3 + y • 5 ; ^-l + |- l;y-5+y-3) = (4^; 1; 3y)

Es(2, i) (y) = (I • 5 + f • 1; | • 3 + | • 2; | • 1 + | • 2) = (2|; 2:;: I 1

ES(2,2) (y) = (f ■ 0+|-0; 3 .4+4 ,4; 3.3+4 ,2) = (0; 4; 2f) .

Third element here is mathematical expectation of payoffs of the player I3 (see table

1 too).

Then, look at the table 1 or table 2,

min Hi (xi = 1, x2, y) = min {2|; 4^} = 2|; min Hi (xi = 0, x2, y) = min {2|; 0} = 0; min H2 (xi, x2 = 1, y) = min {2; 2|} = 2; min H2 (x1, x2 = 0, y) = min {1; 4} = 1;

Thus, maxmin payoff for player I1 is v (I1) Hence,

v (/i) = max {2|; 0} v (I2) = max{2; 1} = 2.

2-7 J

21 and for player /2 is w (/2)

(y) — v (h) + \ (5^ — v (Ii) — 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- .

2. Solve the cooperative game G5 (xs&), S5 = {N = {I1,I2, I3}}, see table 3.

Table3: Shapley’s value in the cooperative game.

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

h h /3 h h /3 Hn (h, h, h) 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

Find the maximal payoff HN of coalition N and divide him according to Shapley’s value (4), (Shapley, 1953):

Shi = \ [v {h,h) + v (h, h) - v (/2) - v (/3)] + \ [v (N) - v (h,h) + v (/i)] ; 63

Sh2 = \[v (h,h) + v (/2, /3) - v (/1) - v (/3)] + \ [v (N) - v (/1, /3) + v (/2)] ; 63

Sh3 = \[v (/3, h) + v (/3,12) ~ v (/1) - v (/2)] + ^ [v (N) - v (h,h) + v (/3)] . 63

Find the guaranteed payoffs:

v (I1,I2) = max{4, 3} =4; v (I1,I3) = max {3, 2} = 3;

v (I2, I3) = max {3, 4} = 4; v (I1) = max{1, 0} = 1; v (I2) = max{2, 1} =2; v (I3) = max {1, 2} =2 .

Then

Shf’hl) = Sh(i'2'2) = Sh(i'2'1) = - + - + - [9 -4] + - = - + - + - + - =2-,

1 1 1 3 6 3L J 3 3 6 3 3 2’

Sh{2'1'1} = Sh£'2’2) = Sh£'2,1) = - + - + - [9-3] + - = - + - + - + - = 3-,

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

= Sh£'2'2) = Shi1’2’1) = | + | + |[9-4] + | = | + | + | + | = 3.

3. Solve noncooperative game G1 (x^i), S1 = {S1 = {I1} , S2 = {I2} ,

S3 = {I3}} . In pure strategies NE not exist.

Static Model of Decision-making over the Set of Coalitional Partitions 105

Table4: Solution of noncooperative game.

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

h h h h h h 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 -

From p. 2 it follows that the guaranteed payoffs v (/i) = 1; v (I2) = 2; v (/3) =

2. Find the optimal strategies with Nash arbitration scheme, see table 4. 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 the coalitional partition of players is provided in (Grigorieva, 2009). Present the obtained solution in (Grigorieva, 2009) in the table 5.

Table5: Payoffs of players in NE for various cases of the coalitional partition of players.

Project Coalitional The n-tuple of Probability Payoffs

partitions NE (Iu /2, /3) of realization NE of players in NE

1 Zi = {{h}{h}{h}} ((1), (1), (0)) 1 ((1), (2), (2))

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

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

2 £2 = {№, h} Us}} ((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 £3 = {№, h} {h}} (1, (0), 1) 1/12 (2.59, (2.5), 2.91)

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

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

4 £4 = {{h, h}{h}} (1, (0, 1)) 1 (3, (3, 3))

(1,0, 1) 1

5 £5 = {h, I2, h} (1, 0, 0) 1 (2.5, 3.5, 3)

(0, 1, 1) 1

Applying the algorithm for finding a compromise solution, we get the set of compromise coalitional partitions (table 6).

Table6: The set of compromise coalitional partitions.

h І2 h h І2 h

2i = {{h}{h}{l3}} 1 2 2 A{{h}{I2}{I3}} 2 1.5 1 2

^2 = {{h, h} {-ґз}} 2.71 2.43 2.33 A{{h,I2}{I3}} 0.29 1.07 0.67 1.07

£з = {{Iи h} {h}} 2.59 2.5 2.91 A{{h,I3}{I2}} 0.41 1 0.09 1

E* = {{h, h} {h}} 3 3 3 A{{I2,I3}{h}} 0 0.5 0 0.5

-£5 = {її, I2, Із} 2.5 3.5 3 A{IUI2,I3} 0.5 0 0 0.5

R 3 3.5 3

Therefore, compromise imputation are PMS-vector in coalitional game with the coalition partition S4 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 static coalitional model of decision-making and algorithm for finding optimal solution are constructed in this paper, and 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.

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