CC BY
3Xeniya Grigorieva1 and Svetlana Mamkina2
1 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/
2 E-mail: [email protected]
Abstract The PMS-vector is defined and computed in (Petrosjan and Mamkina, 2006) for coalitional games with perfect information. Generalization of the PMS-vector for the case of Nash equilibrium (NE) in mixed strategies is proposed in this paper.
Keywords: bimatrix games, coalitional partition, Nash equilibrium, Shap-ley value, PMS-vector, games with perfect information.
1. Introduction
The new approach to the solution of bimatrix coalitional games is proposed. Suppose N-person game r with finite sets of strategies is given. The set of players N is divided on two subsets (coalitions) S, N\S each acting as one player. The payoff of player S (N\S) is equal to the sum of payoffs of players from S (N\S). The Nash equilibrium (NE) in mixed strategies is calculated (in the case of multiple NE (Nash, 1951) the solution of the correspondly coalitional game will be not unique). Mathematical expectation of the payoffs coalition S (N\S) in the NE in mixed strategies is allocated according to the Shapley value (Shapley, 1953). The resulting payoffs vector will be a generalization of the PMS-vector defined and computed in (Petrosjan and Mamkina, 2006) for coalitional games with perfect information. Then the payoff of coalitions S, N\S which appears with positive probability in the NE is allocated proportionally to the PMS-vector.
2. Statement of the Problem.
Suppose N-person game
r = {N,Xi,... ,XN,Hi,...,Hn}
is given. The set of players N is divided into two separated coalitions S and N\S each acting as one player. Rename the players of coalition S from 1 to s = |S|, and the players of coalition N\S from s + 1 to N. For each of the player the strategy sets are defined Xi = j, where ji = 1, m*, i = 1, N.
Denote by
x = (xj1 xj2
1 2 N
the n-tuple of strategies in the game. Let Hi (x) , i <G N, be the payoff of player i, x <G n Xi. The payoff of coalition S (N\S) is equal to the sum of payoffs of
ieN
players from S (N\S):
HS (N\S) (x) = (x)'
ieS(N\S)
Then we get the coalitional bimatrix game
r(A, B) = {N,Xs,Xn\s,Hs,Hn\s} ,
where
A4 = S ¡3 ., xs, xs+i,''' ,xN) '.' Hs (x^.. 1 ms+1 ' ,xs,xs+i , ' ' ' , xmN , xN
\hs (x^1 ," x'ms x1 • , xs , xs + i, ' 3 ,x ... Hs (xm1 ,.. xms xms+1 ' , xs ,xs+i , ' xmN ' ' , xN
B= I / HN\S (x1, ' ' ' ,xs,xs+1, ' ' 1; ,x '' HN\S (x1, ' 1 ms+1 '' ,xs,xs+i ,' ' xmN A ' , xN
1 \Hn\s (xm1,. x'ms x1 .'., xs , xs+i , -,xN)' ..Hn\s (xm1,. xms xms+1 ' ' , xs , xs+i , ''', xmN ''', xN
dim ^4 = dim B = mi x mi.
ieS ieN\S
It’s required to find the optimal imputation rule for each coalition and in some sense optimal strategies for the coalitions.
3. Solution of the Problem
1. We shall consider the case when the matrixes A and B can be reduced to square matrixes A and B: det A = 0, det B = 0. In other cases the iterational methods can be used.
Solve the bimatrix game r (A, B), i. e. find a Nash equilibrium (NE) in the mixed strategies by formulas
x = V2uB-i ; y = viA-iu,
where vi = 1/(uA-i u) , v2 = 1/(uB-i u) , u = (1, , 1) , using the theorem
about a completely mixed equilibrium (Petrosjan et al., 1998, p. 135).
In the case of multiple NE (Nash, 1951) the solution of the corresponding coali-tional game will be not unique.
2. Calculate the NE value in mixed strategies:
E (x, y) = [v (S) ,v (N\S)],
where
v (S)= aij&nj, v (N\s) = bij&nj,
ieXs jeXN\s ieXs jeXN\s
x = liGXs ,y = {nj}jeXN\s '
One can show that v (S) >5^ vi, where vi — maximal guaranteed payoff of i-th
ieS
player, i £ S, under condition that the players from coalition N\S use mixed strategy from NE. This follows from the superadditivity of the characteristic function defined as maximal guaranteed payoff of the coalition S.
3. In (Petrosjan and Mamkina, 2006) the definition of PMS-vector in pure strategies for coalitional games with perfect information has been given. Find PMS-vector (imputation) in mixed strategies as the mathematical expectation over the 2-tuples of strategies generated by NE.
In the beginning define PMS-vector in mixed strategies. Let the game
r = {N,Xi ,'.' ,Xn ,Hi, ... ,Hn }
in normal form with the coalitional partition
S = {Si, ..., S;} , l < n, Si n Sj = 0, V i = j,
be given.
Consider the game in normal form
= {N,Xi, ' ' ' , X;, Hi, ' ' ' ,H;}
between l players, where the players are coalitions from partition S. Consider coalition Si, consisting of si players.
Notions:
— is the number of strategies of player j;
— Xj = {
xj J’fc-TTrT se^ strategies of player j;
— Xs, = II Xj is the set of the strategies of coalition Si, i. e. Cartesian product
jeSi
of the sets of players’ strategies, which are included into coalition Si;
— vector xi £ XSi of dimension si is the strategy of player i in the game ;
— HSi = ^ Hj, i. e. payoff of player Si is a sum of payoffs of the players from
jeSi
coalition Si.
— lsi = |Xsi | = n lj is the number of pure strategies of coalition Si;
jeSi
— l^ = lSi is the number of l-tuples in pure strategies in the game .
¿=17
Let p = (pi, '.', p;) be l-tuple NE in mixed strategies in the game , where each mixed strategy of coalition Si is a vector
lsi
pi = (p\, ..., /x-Si) , >0, j = 1, ¿Mi = 1 •
j = 1
Denote a payoff of coalition Si in NE by
Is
v(Si)=J2PkHk(Si),i = lJ,
k=i
where
= n MiS k = 1,^,
¿=17
— probability of the payoff’s realization Hk (Si) of coalition Si , when players choose their pure strategies xji in l-tuple NE in mixed strategies p, i. e.
Hk (Si )= ^ Hj (xi, ''',x;)'
jeSi
The value Hk (Si) is random variable. There could be many l-tuple NE in the game, therefore, v (Si) , ''.', v (S;), are not uniquely defined.
Consider for each coalition Sj <G S, i = 1,1, a cooperative game Gst supposing that the players which are not included into the coalition Si, use NE strategies from l-tuple p.
Definition 1. Let w (Si : K) = v (K) be characteristic function in the cooperative game GSi, where K C Si. Divide payoff w (Si) = v (Si) between the players of coalition Si, according to Shapley value (Shapley, 1953) Sh = (Shi, '- , ShSi):
Ei's'-l'l T (s-sr)! ___
^---------------MS') -w(S'\{i})] V* = l, s, (1)
s!
S'cS
where s = |S| (s' = |S'|) is the number of elements of set S (S') and w (S') is a maximal guaranteed payoff of the subcoalition S' С S. Denote
Sh (Sk) = (Sh (Sk : 1) , ...,Sh (Sk : sfc)) ,
Si
where sk is the number of elements of set Sk. Moreover w (Sj) = ^ Sh (Sj : j).
j=i
Then PMS-vector for the NE in mixed strategies in the game Ге is defined as
PMS (Ге) = (PMSi (Ге), ..., PMSn (Ге)),
where
PMSj (Ге) = Sh (Si : j), j Є Si, і = ЇД
4. Divide payoffs Hk (Sj) of coalition Sj for every г = 1,/ occurring with pos-
itive probability when l-NEtuple in the game Ге is played, proportionally to the components of Shapley value:
л Sh(Si:j) ,^a —
£ Sh(S%:j)’ JG ’
jGSi
ls
One can show that is Sh (Sj : j) = £ pkHjk (Sj), where Hjk (Sj) = AjHk (Sj),
___ fc=i
j € Sj V* = 1,L Then in bimatrix coalitional game matrixes of players’ payoffs j = 1, N, are defind as follows
A = Aj A, j Є S; = AjB, j є N\S.
4. Examples
Example 1. Let there be 3 players in the game. Each of them has two strategies (see table 1). The payoffs of each player are defined for all three-tuples.
1. Compose and solve the coalitional game, i. e. find NE in mixed strategies in the game:
П = 3/7 1 - n =4/7 1 2 0 (1, 1) [6, 1] [3, 2]
0 (2, 2) [4, 3] [4, 2]
£ =1/3 (1, 2) [4, 5] [6, 3]
1 - £ = 2/3 (2, 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) .
Table 1.
The strategies The payoffs The coalitions The NE strategies TheXE payoffs
I II m I II ni I II in I E m
1 1 1 4 2 1 Mi) 1 1 7 = 'r, 1 ■ 1 2--- 2 i-7 7
1 1 2 1 2 2 (iU)=2)
] 2 1 3 1 5 (M.1) 1 2 7 = ''3' 7 4 ■,7, 4I 1 3^ 7 7
1 2 2 5 1 3 (M):2)
2 1 1 5 3 1 mm 2 1 (V Hi \JJ 2- 2- l-:- 7 7 7
2 1 2 1 2 2 (№)
2 2 I 0 4 3 ((2=2)a) 2 2 Y = ■'"3"' 7 4 \1. 3 0 4 27
2 2 2 0 4 2 ({2=2)=2)
Then the probabilities of payoffs’s realization of the coalitions S and N\S in mixed strategies (in NE) are as follows:
ni n2 £i 0 0 £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]
Rewrite table 1 in table 2.
Table 2 comments. In 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 the total payoff of the coalition S are given at the right side.
'36 7' 11
Y’ 3 = 5-, 27 3
Table 2.
The strategies of MS, the payoffs of S y 0,43 0,57
Math. Expectation jf 1 S 2 S
2,286 2,000 0,00 1, 1 4 2 6 1 2 3
4,143 1,000 0,33 1,2 3 1 4 5 1 6
2,714 2,429 0,67 2, 1 5 3 8 1 2 3
0,000 4,000 0,00 2,2 0 4 4 0 4 4
vl v2 vl v2 vl v2
2,286 2,000 min 1 3 2 1 2
0,000 1,000 min 2 0 1 0 1
2,286 2 max 3 2 1 2
2. Divide the game’s Nash value in mixed strategies according to Shapley’s value
(1):
Sin = v {1} + \ [v {I, II} - v {II} - v {I}] ,
Sh2 = v {II} + | [v {I, II} - v {II} - v {I}] .
Find the maximal guaranteed payoffs v {I} and v {II} of players I and II. For this purpose fix a NE strategy of a third player as
y= (3/7 4/7) .
Table 2 comments (continuation). Denote mathematical expectations of the players’ payoffs from coalition S when mixed NE strategies are used by coalition
N\S by Es(itj) (y) , i,j = 1, 2. In table 2 the mathematical expectations are located
at the left, and values are obtained by using the following formulas:
ES(i,i)(y)= (f -4+f -1; f-2+|-2; f-l + f -2) = (2f; 2; if)
Es(1,2) (y) = (f ■ 3 + f ■ 5; y • 1 + | • 1; y • 5 + | • 3) = (4^; 1; 3y)
Es(2,i)(y) = (f '5+f -1; f • 3+|.2; f- -1 + f -2) = (2f; 2f ; l|)
Es(-2;2) (y) = (7 ' 0 + 7 ' 0 ; f • 4 + | • 4 ; y ■ 3 + | • 2) = (0; 4 ; 2y) .
Third element here is mathematical expectation of payoffs of the player III (see table 1 too).
Then, look at the table 1 or table 2,
miniii (xi = 1, x-2, y) = min {2|; 4y} = 2y ;
miniii (xi = 2, X2, y) = min {2|; 0} = 0; minH-2 (xi, X2 = 1, y) = min {2; 2|} = 2; minH2 (xi, x2 = 2, y) = min{1; 4} = 1;
vi = max {‘2'y; 0} = 2|;
v2 = max {2; 1} = 2.
Thus, maxmin payoff for player 1 is v {1} = 2y and for player 2 is v {II} = 2. Hence,
Shi (¿0 = ^1 + j (5y — vi — V2) = 2y + i (5y — 2y — 2) = 2y;
Sh-2 (y)= 2 + f = 2f .
Thus, PMS-vector is equal:
5 3 1
PMSi = 2-; PMS2 = 2-; PMS3 = 2- .
Now dividing the payoffs of coalition S in pure strategies proportional to the Shapley vector we get:
PMSi 2 f 19 ^ PMS2 17
PM Si + PMS2 ~ 5Ï “ 36’ 2 “ PM Si + PMS2 ~ 5Ï “ 36'
>7 * —-2 Hence, the newly defind payoffs of players I and II from coalition S are:
Au = MA = ^ , _ , , _ _
'9 -l12
hi46)- H 2f
36 I 83 V3|lA
98 12
and the matrix of the game became equal to
n = 3/7 1 - n = 4/7
£=1/3 +(1,2) [(2i, If), 5] [(3i,2§),3]
_ . . ' ' 7_ ' "
c9’ ^9/ > -*-J LV12 ’ J’12>
l-£ = 2/3 +(2,1) [(4§,3§) ,1] [(l£, 1^,2] .
5. Conclusion
In this paper the algorithm of getting imputation proportional to the PMS-value in the NE in mixed strategies, and the example which show realization of the proposed approach are given.
References
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.
Petrosjan, L., Zenkevich N., Semina E. (1998). The Game Theory. - M.: High School.
