Matematicas Aplicadas. c1998 Universidad de Chile A CONVERGENT TRANSFER SCHEME TO THE. Av. Ejercito de Los Andes 950, 5700 San Luis, Argentina.

Transcription

1 Rev. Mat. Apl. 19:23-35 Revista de Matematicas Aplicadas c1998 Universidad de Chile Departamento de Ingeniera Matematica A CONVERGENT TRANSFER SCHEME TO THE CORE OF A TU-GAME J.C. CESCO Instituto de Matematica Aplicada San Luis (UNSL-CONICET) Av. Ejercito de Los Andes 950, 5700 San Luis, Argentina jcesco@linux0.unsl.edu.ar and Centro Regional de Estudios Avanzados, Gobierno de San Luis, Argentina Abstract We present an algorithm for computing points in the core of a game with transferable utility. We prove that it converges if and only if the core of the game is nonempty. The algorithm resembles the transfer scheme model introduced in Stearns [13]. AMS Classications: D90. Key Words: TU-game, core, transfer scheme, algorithm. 1.- Introduction The object of this paper is to describe a model of dynamic bargaining that converges to the core [6] of a TU-game (game with transferable utility) provided this set is nonempty. Because this set is a polyhedron determined by linear inequalities, linear programming methods are usually employed for computational purposes. However, our algorithm is not based upon a linear programming approach; rather, it is more closely related to the transfer scheme model suggested by Stearns [13] to describe processes of dynamic negotiations. There, two transfer schemes, one converging to the kernel [5] and other to the bargaining set [1] of a TU-game, were presented. Later, Justman [7] elaborated an extension of Stearns' model allowing for a wider class of transfers than those considered by Stearns. In this work, a transfer scheme which converges to the nucleolus [11] of a TU-game was developed. Because the nucleolus is always included in the core (provided this set is 23

2 24 J.C. Cesco nonempty) Justman's scheme achieves the same goal we intend in this paper. However, his scheme and ours rest upon dierent intuitive notions and the bargaining processes that they describe are, generally, not related. Besides, since Justman's scheme is designed to reach a very specic point, namely, the nucleolus, the analytic side of his model is more complex than ours. This is an important point in discussing simple practical computational procedures. All the models mentioned above are basically discrete in nature. Continuous transfer schemes have been studied in [2], [3], [8] and [10]. The paper is organized as follows. The background, basic denitions and algorithm are set forth in the next section. In section 3 we prove our main convergence theorem (Theorem 5) and other related convergence results. Section 4 includes alternative convergence proofs that can be used to obtain improvements in the performance of the algorithm. We close with some concluding remarks. 2.- Background For the purpose of this paper, a TU-game is given by an ordered pair G = (N; v) where N = f1; :::; ng is the set of players and v, the characteristic function, is a real function on the family of subsets (coalitions) P(N) of N satisfying v(n) = 1; v() = 0 and v(fig) = 0 for all i 2 N (0-1 normalization). The value v(s) represents the utility the coalition S is able to assure for itself if all of its members agree to participate together. As usual, the set of pre-imputations is dened by E = ( x = (x 1 ; :::; x n ) 2 R n : X i2n x i = 1 ) and the set of imputations or payo vectors by A = fx 2 E : x i 0; i 2 Ng Here R n denotes the n-fold cartesian product of the set of real numbers R. A pre-imputation is a way to split the value of the grand coalition N between its members. An imputation is a pre-imputation where each member gets as much as he would obtain participating alone (individual rationality). For S 2 P(N); S 6= and x 2 E we dene the excess of S in x by e(s; x) = v(s)? X i2s x i We set e(; x) = 0:

3 A Convergent Transfer Scheme to the Core of a TU-Game 25 e(s; x) represents the gain to coalition S if its members depart from an agreement that yields x in order to form their own coalition. A widely used concept of solution for a TU-game G is the core. It is dened by C = fx 2 E : e(s; x) 0; S 2 P(N)g It represents the set of imputations which cannot be improved on by any coalition. Next we provide an alternative denition in terms of demands. Given x 2 E, a demand against x is a pair (S; z) where S 2 P(N); S 6= and z = (z i ) i2s is a real vector with components indexed by the members of S. Moreover, it satises that X i2s z i 0 for all i 2 S (2:1) z i = e(s; x) (2:2) P We call w(s; z) = z i the worth of the demand. Conditions (2.1) and i2s (2.2) are a way of rule out `exorbitant' demands. We say that a demand is essential if its worth is positive. There is no essential demand (N; z) against any x 2 E: The following simple result that we present without proof characterizes the core of a TU-game. Proposition 2.1. Let G(N; v) be a TU-game. Then, a pre-imputation x belongs to the core if and only if there is no essential demand against it. A satisfaction to a demand (S; z) against x is a pre-imputation y such that y i = x i + z i for all i 2 S: From now on, we shall restrict ourselves to a particular class of demands and demand satisfactions. We say that (S; z) is a U-demand against x 2 E if z i = e(s;x) for all i 2 S: Here jsj denotes the jsj cardinality of S. It follows that for each pre-imputation x and nonempty coalition S; there exists a unique U-demand (S; z) against x: We call a U- demand (S; z) against x maximal if w(s; z) w(t; ^z) for any other demand (T; ^z) against x: Because of (2.2) above, this implies that e(s; x) e(t; x) for all nonempty T 2 P(N): Finally, we say that y is a U-satisfaction to a U-demand (S; z) against x; S 6= N; if y = x + e(s; x) S jsj? Sc js c j Here S c denotes the complement of S in N. S the vector of R n dened by ( S ) i = 1 if i 2 S and 0 otherwise. For S = N; x itself is the U- satisfaction to the unique U-demand z = 0: Given S; x and an U-demand (S; z) against x; each member of S will be always better o under y than under x: Indeed, a U-demand species

4 26 J.C. Cesco each member of S requests the same increase in his utility. On the other side, a U-satisfaction species that those increases are equally supported by transfers of utility from the remaining players. In contrast, the transfers considered in [13] were of the form y = x + ( i? j ) where i; j 2 N; 0: A U-transfer scheme (or U-sequence) is a sequence of pre-imputations fx r g r1 such that x r+1 is a U-satisfaction to a U-demand against x r for all r: We call a U-scheme maximal if for all r; x r+1 is a maximal satisfaction. If for all r ^r; ^r 1 is x r = x^r we say that the scheme is nite. Remark 1. The elements x r+1 ; r 1 of a U-transfer scheme are of the form x r + e(s r ; x r ) S r. Here S = S? Sc jsj js c j N jn j if S 6= ; N if S = N for all nonempty coalitions S: We point out that e(s; x r ) 0 for all r 1: Lemma 2.2. The vector S is orthogonal to the ane submanifold for each nonempty coalition S: E(S) = fx 2 E : e(s; x) = 0g The orthogonality is considered with respect to the natural scalar product of R n denoted by h:; :i Proof. Let y, x 2 E(S), S 6= ; N. Then Since x; y 2 E(S); and S ; x? y = D S = X E jsj? Sc js P c j ; x? y x i? P i2s jsj i2s x i = X i2s X X x i = i2s c y i i2s? y i = v(s) i2s c y i = 1? v(s) P i2s c x i? P js c j i2s c y i we conclude that h S ; x? yi = 0: The case S = N is proved similarly. Remark 2. It is easy to see that a U-satisfaction y to a U-demand (S; z) against x is always in E(S): There, the members of S get v(s): At each stage of a U-transfer sequence, the members of demanding coalition obtain

5 A Convergent Transfer Scheme to the Core of a TU-Game 27 as much as they could get if they would decide to form a separate coalition S: Lemma 2.3. Let fx r g r1 be a nite U-sequence. Then e(s r ; x r ) = 0 for all r ^r for some ^r 1: Conversely, if a U-sequence is maximal and e(s ^r ; x^r ) = 0 for some ^r 1; then it is nite. Proof. If the sequence is nite then there exists ^r 1 such that x^r = x^r + e(s r ; x r ) S r for all r ^r: This implies that e (S r ; x r ) S r = 0: Since S r 6= 0; we conclude that e(s r ; x r ) = 0 for all r ^r: To see the second part we note that if e(s ^r ; x^r ) = 0 then x^r+1 = x^r Besides, since the sequence is maximal, 0 = e(s ^r ; x^r ) e(t; x^r ) = e(t; x^r+1 ) for each nonempty T: This implies that every maximal demand (S; z) against x^r+1 will have an S with e(s; x^r+1 ) = 0: Consequently, x^r+2 = x^r+1 + e? S r+1 ; x r+1 S ^r +1 = x^r The proof is completed by induction. Remark 3. Since the set of coalitions is nite, it follows that, for each n 2 IN, the set of natural number, there are positive constants k(n) and K(n) such that k(n) k S k 2 K(n) for all S 6= ; N: U-sequences can be interpreted as the result of a process of bargaining carried out as follows: the group N of individuals seeks an agreement on the division of the value v(n) which they can reach by acting together. A starting proposal x is made and announced. After a period during which communication among the players is allowed, all coalitions make their demands which express the total increase in utility requested in order to obtain as much as they could get by playing alone. The rules of the process establish the formation of a new proposal by giving satisfaction to a high of such demands. Moreover, the rules determine that such a new proposal is obtained by having the members outside the coalition supporting the demand, transfer equal amounts of utility to the members inside it, in such way that each player receives an equal increase in his utility. The result of these transfers is another potential division of the value v(n) which is used as a starting point for a new stage in the process of negotiation. The surprising fact is that this very simple method of bargaining (when carried

6 28 J.C. Cesco out in a TU-game framework) converges to one of the most widely accepted solution concept, namely, the core. We remark here that in some of the pre-imputations occurring during the above process, individual rationality (x i 0) may be absent. However, we note that none of the pre-imputations obtained at each intermediate stage of the bargaining process is actually implemented; their only role is to furnish a base for a further negotiation. In fact, the only pre-imputation to be implemented will be the nal result (if any) of this dynamic procedure. This will be an imputation. 3.- Convergence Results The main goal of this section is to prove that every maximal U-sequence converges to a point in the core of a TU-game so long as the game has nonempty core. Besides, we prove other related convergence results. All convergence properties to be shown are with respect to the topology related to the euclidean norm of R n : h:; :i will denote, as before, the natural scalar product on that space. From here now, we are going to avoid the use of the prexes TU and U. Proposition 3.1. Let G = (N; v) be a game. Let fx r g r1 be a maximal transfer scheme converging to a point x 2 R n : Then, x belongs to the core C of G: Proof. Since x r 2 E for all r; and E is a closed set, x 2 E: On the other hand, because x r+1 = x r + e(s r ; x) S r for all r and taking into account remark 1, given > 0 there exists ^r 1 such that for all r ^r: Therefore e(s r ; x r )k S r k 2 = kx r+1? x r k 2 e(s r ; x r ) k(n) (cfr. remark 3). Besides, because the sequence is maximal, e(s; x r ) e(s r ; x r ) " k(n) for all coalition S and r ^r: Due to the continuity (in x) of the excess functions, we conclude that e(s; x) " k(n) for all coalition S: Since " can be made arbitrarily small we get that e(s; x) 0

7 A Convergent Transfer Scheme to the Core of a TU-Game 29 for all S and this proves that x 2 C: This result is similar to Proposition 4.9 in [7]. An obvious corollary is the following. Corollary 3.2. If a maximal transfer scheme converges, the core is nonempty. The remarkable fact is that the converse of this result holds. This is our next theorem. Before proving it, we state the following result which has useful consequences. It was suggested by L. Shapley. Lemma 3.3. Let G = (N; v) a game with nonempty core C. Let x 2 E and S be a coalition with e(s; x) > 0: If y = x + e(s; x) S then, ky? zk 2 < kx? zk 2 for all z 2 C: Proof. Let z 2 C. Then, e(s; z) 0. Since e(s; x) > 0 and e(s; z) 0, E(S) (cfr. Lemma 2.2) is a separating linear submanifold for x and z. Besides, y 2 E(S) (Remark 2) and y? x is orthogonal to the submanifold E(S) (Lemma 2.2). Then, hx? y; z? yi 0. To see this, let p 2 E(S) be the orthogonal projection of z on E(S). Therefore hx? y; z? yi = hx? y; z? p + p? yi = hx? y; p? yi + hx? y; z? pi = hx? y; z? pi since x? y and p? y are orthogonal vectors. Now, the claim follows by noting that x? y and z? p are colinear vectors (both are ortogonal to E(S)) but with opposite directions. Then, vectors x; y; z determine a plane triangle with an obtuse (or perhaps a right) angle at y. In any case, since kx? yk 2 > 0; ky? zk 2 < kx? zk 2 : This result has an important consequence that shows that a transfer scheme is always bounded. We present it without proof. Corollary 3.4. Let G be a game with nonempty core. Then, for each transfer scheme fx r g r1 (not necessarily maximal) there exists a positive number K such that kx r k 2 K for all r: Next we prove our main convergence result. Theorem 3.5. Let G = (N; v) a game with nonempty core C. Then, every maximal transfer scheme converges to a point in the core of G. Proof. Let fx r g r1 be a transfer scheme. If it is nite, then it obviously converges. Otherwise, e(s r ; x r ) > 0 for all r. Let us denote by

8 30 J.C. Cesco d r = inffkx r? zk 2 : z 2 Cg; the distance from x r to C. Let z r 2 C be a pre-imputation such that kx r? z r k 2 = d r : We point out that this point always exists because C is a nonempty closed set. Due to lemma 3.3, the sequence fd r g r1 is strictly decreasing and bounded from below by zero. In fact d r+1 = kx r+1? z r+1 k 2 kx r+1? z r k 2 < kx r? z r k 2 = d r Hence, it converges to a number d 0. Therefore, given 0 < " < 1 there 2 exists ^r 1 such that d r?d " for all r ^r: Like in lemma 3.3, we consider the `triangle' determined by the pre-imputations x r ; x r+1 and z r which also has an obtuse angle at x r+1 : An application of the cosine theorem from elementary geometry yields kx r+1? x r k 2 2 kx r? z r k 2 2? kx r+1? z r k 2 2 = (d r? kx r+1? z r k 2 )(d r + kx r+1? z r k 2 ) "(2d + 1) (3:1) for all r ^r: Because of corollary 3.4, the sequence fx r g r1 has a subsequence fx r m g m1 converging to a point z 2 E: We claim that this point belongs to the core C. By (3.1), e(s r ; x r ) "(2d+1) for all r ~r; thus k(n) x 2 C. Finally, we apply lemma 3.3 to obtain kx r? zk kx r m? zk " for all m ~r and r r m : These inequalities imply that the whole sequence fx r g r1 converges to z 2 C: Remark 4 (Modied U-sequences). U-transfer sequences are a particular case of a more general class of transfer schemes studied in [9]. Let x; y 2 E. We say that y arises from x by a modied U-transfer, if there is a nonempty coalition S N such that y = x + S for some 0 max(e(s; x); 0). A modied U-transfer scheme is a sequence of pre-imputations fx r g r1 such that x r+1 = x r + r S r is a modied U-transfer from x r for all r. Since for each U-transfer schemes we always have that e(s r ; x r ) 0 for all r, it is also a modied U-transfer scheme. And so is every maximal U-transfer scheme. The result proved in the previous section shows that some modied U-transfer scheme converge to a point in the core. Besides, a modied U-transfer scheme with r = 0 for all r, converges to the initial pre-imputation x 1 although this vector may not belong This fact was pointed out by a referee.

9 A Convergent Transfer Scheme to the Core of a TU-Game 31 to the core of the game. These are particular cases of the following general result. Theorem 3.6. let G = (N; v) a game with nonempty core C. Then, every modied U-transfer fx r g r1 scheme converges. Proof. The proof is very similar to that of theorem 3.5 with minor changes. We get that the sequence fx r g r1 is bounded by proving an equivalent of lemma 3.3. There, the only dierence is to consider the ane manifold E(S; x; ) = fy 2 E : y(s) = v(s)?(e(s; x)?)g as a separating hyperplane instead of E(S). The convergence result is proved exactly in the same way as in theorem 3.5. The only part that cannot be repreduced, is the proof that the limit point belongs to the core. It depends heavily on the fact that, for maximal U-transfer sequences, r = max S6= (e(s; x r )) for all r. 4.- An Algorithm Based upon the model of maximal transfer scheme, we can now provide a pseudo code of an algorithm for computing a point in the core of a game. Step 1 Choose a starting point x 2 E Step 2 Compute all the quantities e(s; x); S 2 P(N); S 6= ; N if m = maxfe(s; x); S 2 P(N); S 6= ; Ng 0 go to step 5 Step 3 Select S 2 P(N); S 2 P(N); S 6= ; N such that e(s; x) = m Step 4 Compute S and update x as x = x + m S ; go to step 2 Step 5 End An implementation of the algorithm will be then, a practical device for the generation of maximal sequences. Hence, in games with nonempty cores, they will converge, sometimes in a nite number of steps (depending on the starting point), to a core point. This is illustrated in g.4.1. The triangle represents the set of imputations of a 3-person game. The lines denoted with v(s) represent the set of pre-imputations in E(S). x denotes the starting point and y the nal one in the core. One of the main problems associated with the computation of solution points in cooperative games is related to the fact that almost all of them are dened in terms of coalitions. Since the number of coalitions growth exponentially with n; the number of players, algorithms designed with computational purposes tend to be very expensive even for moderately large n. In many situations there are, however, `superuous' coalitions which do not participate actively in the determination of the solution set to be computed. Their recognition is a key fact in order to reduce running times. For the algorithm presented in this section, great reduction can be achieved for games having (in 0-1 normalization) several coalition with non positive values.

10 32 J.C. Cesco Figure 4.1. Given a game G = (N; v) we consider the reduced game G + = (N; v + ) where v + : P(N; v)! R is dened by and where v + (S) = v(s) P(N; v) = fs 2 P(N) : v(s) > 0 or jsj = 1g A U + -sequence in G + is a U-sequence in which the transfers are supported by coalitions in P(N; v). Maximality is also dened with respect to P(N; v). Theorem 4.1. Let G = (N; v) be a game with nonempty core C. Then every maximal U + -sequence dened w.r.t. G + converges to a point in C: Proof. The same proof of that given for theorem 3.5 also holds to prove that all maximal sequences converge to elements of C + = fx 2 E : e(s; x) 0 for all S 2 P(N; v)g which is obviously nonempty. But is also easy to see that C + = C: This result allows us to eliminate in step 2 all computations due to coalitions outside P(N; v): In some cases this entails a great computational reduction. However, it is well known that there are games in which all the 2 n? 2 coalitions dierent from ; N participate actively in the denition of the boundary of the core. For such cases, the result is useless. For further geometric properties of the core see [10].

11 A Convergent Transfer Scheme to the Core of a TU-Game 33 Next we describe a dierent kind of reduction that can also be used. It is based on the observation that the denition of maximality given in section 2 is unnecessarily strong in order to guarantee the convergence. In fact, like in [13, theorem 2 ] or in [7, Proposition 7.9], it suces to consider only sequences having an innite subsequence of maximal U-transfers. Let us call these sequences, weakly maximal U-transfer schemes. Proposition 4.2. The result in theorem 3.5 still holds if we replace maximal U-sequences by weakly maximal U-transfer schemes. The proof of theorem 3.5 still works if we choose x as an accumulation point of the innite sequence of maximal U-transfers instead of an arbitrary accumulation point of the sequence fx r g r1 : Some computational experience based upon an implementation of the algorithm described in this section shows that the numerical sequences generated, after an `accommodation' period, start to include U-transfers supported by coalitions belonging to a subset, usually small, of P(N): This suggests that after a rst `heavy' part, once that subset of coalitions is identied, the amount of computation could be drastically reduced. The strategy is, then, to use the result of proposition 4.2 in order to justify some actions to be taken during the execution of the rst part of the algorithm. We propose the following rule: we start computing the excesses indicated in step 2 until the rst positive number is found. Then, we compute the associated U-transfer. This rule applied indiscriminately could lead to a point outside the core. Therefore, we should include the computation of maximal U-transfers at some points of the sequence. Proposition 4.2 guarantees that the insertion of maximal U-transfers will preserve convergence to a corepoint. In order to balance the computational saving achieved because of the fewer number of excesses to be computed and the increase resulting from an increase in the number of iterations to be performed, some numerical experience is required. 5.- Concluding Remarks We present an alternative method for computing points in the core of a TU-game which exhibits some encouraging features. Besides its convergence properties, it can be used as a practical tool to gather empirical information about the existence of nonempty core (because of proposition 4.1) The Shapley-Bondareva theorem [12] gives a characterization of TU-games with nonempty core but, because it is expressed in terms of balanced families of coalitions, practical methods of verication based on it become very expensive. Linear programing provides another practical method to achieve this goal. Also, from dierent starting points, dierent points in the core

12 34 J.C. Cesco can be reached. This could be used to collect some information about the geometrical shape of the core itself. From the computational point of view we point out that the simplicity and the independence of the computations to be carried out due to step 2, allow for a good interaction between the dierent types of memories in any serial implementation for large scale problems. These computations are also suitable to be handled by parallel computation. We close by mentioning some related problems worthy to be studied. Because of theorem 3.5 we know how to design rules to stop the algorithm when the core of the game is nonempty. But when the core is empty, any sequence generated by the algorithm does not converge and therefore it is not clear what rule has to be implemented. Some computational experience within this class of TU-games (with empty core) has shown that all numerical sequences generated by the algorithm always `converged' to a cycle of pre-imputation; this regularity can be used to design a stop-rule. These cycles can be used to accelerate the speed of convergence of the algorithm and to collect more information about the game itself. Some partial results are reported in [4]. Finally, a comparative study with linear programming is necessary. We know that in some cases our algorithm is faster but an extensive study about their relative performances would be interesting, especially in games with large number of players. Acknowledgments The author wishes to thank T. Quint and two anonymous referees. Their comments and suggestions helped to improve the style of the paper. Remaining errors are responsibility of the author only. References [1] Aumann, R.J. and Maschler, M., The Bargaining Set for Cooperative Games, Annals of Mathematical Studies 52 (1964) [2] Billera, L.J., Some Recent Results in N-Person Game Theory, Mathematical Programming 1 (1871) [3] Billera, L.J. and Wu, L., On a Dynamic Theory of the Kernel of an N-Person Game, International Journal of Game Theory 6 (1977) [4] Brown, K.J.; Cosner, C. and Fleckinger, J., Principle Eigenvalues for Problems with Indenite Weight Function on R n, Proc. A.M.S. 109 (1990) [5] Cesco, Juan C., U-Cycles in TU-Games with Empty Cores, working paper, IMASL, San Luis (1998).

13 A Convergent Transfer Scheme to the Core of a TU-Game 35 [6] Davis, M. and Maschler, M., The Kernel of a Cooperative Game, Naval Research Logistic Quarterly 12 (1965) [7] Fleckinger, J. and Serag, H., On Maximum Principle and Existence of Solution for Elliptic System Dened on R n, J. Egypt. Math. Soc. 2 (1994) [8] Gillies, D.B., Some Theorems on N-Person Games, Ph.D. Thesis, Princeton University (1953). [9] Justman, M., Iterative Processes with Nuclear Restrictions, International Journal of Game Theory 6 (1977) [10] Kalai, E.; Peleg, B. and Owen, G., Asymptotic Stability and other Properties of Trajectories and Transfer Sequences Leading to the Bargaining Set, TR 73-3 Department of Operation Research, Stanford University (1973). [11] Maschler, M.; Peleg, B. and Shapley, L.S., Geometrical Properties of the Kernel, Nucleolus and Related Solution Concepts, Mathematics of Operation Research 4 (1979) [12] Schmeidler, D., The Nucleolus of a Characteristic Function Game, SIAM Journal on Applied Mathematics 17 (1969) [13] Shapley, L.S., On Balanced Sets and Cores, Naval Research Logistic Quarterly 14 (1967) [14] Stearns, R.E., Convergent Transfer Schemes for N-Person Games, Transactions of the American Mathematical Society 134 (1969) [Paper received March 1998]