Three Game-Theoretical Consequences of the Strategic-Equilibrium for n-person Cooperative Games

Transcription

1 Three Game-Theoretical Consequences of the Strategic-Equilibrium for n-person Cooperative Games By Gabriel Turbay Keywords: Strategic-equilibrium, Strategic-core, Stronger player paradox, Linear Programming characterization. Abstract A summary presentation of the theorems that define the strategic-equilibrium for n-person games with transferable utility is given in terms of balanced collections, utility transfers, and theorems on the alternative for matrices. The following consequences are demonstrated (1) An s-equivalent definition of the fundamental strategic-equilibrium, that allows to address in a simple manner the issues of existence, invariance and uniqueness; is given in terms of Linear Programming. (2) Based on strategic-equilibrium originated outcomes, the stronger player-paradox and the importance of the order in which binding agreements occur in von Neumann and Morgenstern solutions is presented by means of an example for triangular three-person cooperative games. (3) A new strategic-equilibrium based solution concept: The Strategic-core (or the core of the Core) is introduced as an emergent property of the fundamental equilibrium and a self-similarity relation to the imputations simplex is exhibited. 1

2 1. Introduction The strategic- equilibrium here presented may be considered a generalization of the von Neumann and Morgenstern (vn-m) equilibrium found for the 3-person zero-sum game, as expressed in the symmetric solution [26] p. 261 and in parametric form of the solution for the 3- person zero-sum game with coalitions of different strength, pp In von Neumann and Morgenstern analysis of the zero-sum 3-person cooperative game, the non-discriminatory solution is identified as a system of imputations in a relative bargaining equilibrium. The heuristic process, by which this equilibrium is found, is based on establishing whether player s claims in excess of certain payoff level could be reasonably maintained under all possible circumstances. In the attempt to determining all solutions to the general-sum n-person cooperative game, vn-m introduced the concepts of composition and decomposition of imputations and those of extended-imputations [26], p.360. They pointed out a simple intuitive meaning of composition of imputations as corresponding to the same operation of viewing as one two separate occurrences. In a retrospective view of their theoretical development, von Neumann and Morgenstern symmetric solution (equilibrium) system may be thought of as a detached extended-imputation [26] p. 370 which is the composition of the three mutually exclusive and collectively exhaustive occurrences represented by three payoff vectors of maximum sustainable claims (extended imputations) each one having as efficient set one of the two-person coalitions that may form. Thus, the non-discriminatory solution to the three person zero-sum game, as a system, may be described: Either as a pair ( x, C ) where x = (½, ½, ½ ) and C = {{1,2}, {1,3},{2,3}} is a cover of N that supports the extended imputation x. Or equivalently, as a conditional system consisting of matrix of payoffs where each row describes the payoffs that may occur if the corresponding coalition forms: Here, for the characterization of the fundamental strategic-equilibrium of the cooperative game we will follow a heuristic line of thinking similar to the one used by vn-m in the definition of the objective solution, non-discriminatory solution. Properties of existence, invariance and uniqueness are shown to characterize the fundamental strategic-equilibrium of any n-person cooperative game with transferable utility. Also, based on 2

3 equilibrium payoffs taken as disagreement points for syndicate bargaining in the three person general sum game, we are able to construct strategic-equilibrium based outcomes from where vn- M solutions emerge. In particular, we exhibit for the three-person triangular game the Stronger player paradox and show the importance for outcome stability of the order in which binding agreements occur along the bargaining processes. The strategic-equilibrium based outcomes introduced in this paper allow us to understand why some Core outcomes have stronger stability characteristics than others. Here we show that the Core of a cooperative games, by no means represents a scenario equivalent to one of pure bargaining as it has been implicitly assumed in conventional game theory and economic theory traditions. A Strategic-core is define in terms of the fundamental strategic-equilibrium a may be thought of as the core of the Core 1 2. Basic concepts and definitions Let N = {1, 2,, n} denote de set of n players and R n denotes the n-dimensional linear vector space. A game in characteristic function form is a pair = (N,v), where v is a real-valued set function defined on the power set of N denoted by 2 N so that v : 2 N R and satisfies the following properties: (i) v( ) = 0 (ii), for all C in some cover C of N A game = (N,v) is said to be essential (in the extended sense) if it satisfies condition (ii). A covering collection of N or simply a cover of N is a set C = {C 1,..., C k } of subsets of N such that C with j C i. C = k is the cardinality of C. Clearly,. A payoff vector for coalition S N is any distribution vector x R n where x = (x 1,,...,x n ) t, x j 0 if j S, and x j =0 if j S, j=1,, n. Similarly, a payoff vector x for a cover C of N, in the game =(N,v) is a column n-vector defined by x = (x 1, x 2,...,x n ) t such that x j 0 for all j N. A payoff vector for the players in N satisfies: group rationality if, coalitional rationality if for all S N, individual rationality if for all j N,. An imputation for the game v is a payoff vector for N that satisfies both 1 The term of core of the Core is borrowed from an observation to this regard made by Professor Guillermo Owen. 2 The concept of extended- imputation and detached extended- imputation was introduced in vn-m p.364for analytical purposes. 3 Here p-balanced corresponds to balanced and w-balanced to weakly-balanced in Shapley[20], 4 A fundamental strategic-equilibrium for the game is said to be strong when the collection is p-balanced and it consists 3

4 individual and group rationality conditions above. That is :. A coalition N is efficient for a payoff vector x if The core of a cooperative games = (N,v) denoted by C ( ) is defined as the set of imputations that satisfy coalitional rationality. An extended imputation 2 is a payoff vector x for N that satisfies individual rationality. In a 0-normalized game a payoff vector is an extended imputation if and only if it is a non-negative vector. Here, the set of all extended imputations consist of the non-negative orthant of R n. In what follows we assume the games under consideration are in 0-normalized form so that extended imputations are simply nonnegative n-payoff vectors. For any extended-imputation x, its value-level is the quantity. An extended imputation is said to be detached [26] p.364(vn-m) if and only if Definition 1 A cover C = {C 1,..., C k } of N is said to be efficient or simply a cover support for an extended imputation x, or x if C, Definition 2 An extended imputation x is said to be attainable or realizable if and only if there exist a cover C of N that is efficient for x. An extended-imputation x attainable through C may be thought of as the composition of k-different payoff vectors: x (1),..., x (k), for coalitions C 1,..., C k respectively; so that C j is an efficient set for x (j), j =1,..., k. Remark 1 Attainable extended-imputations through a cover C of N, are payoff vectors for the covering collection C of N in consideration. The vectors x (i),i=1,...,k are related to each other by the fact that if C r and C s are two different coalitions and if their intersection is not empty, that is C r C s, then for any j C r C s, the payoff to player j in x (r) equals the one in x (s). That is, (r) x j = x (s) j, whenever j C r C s. Definition 3 If x and y are extended imputations, the vector = y- x is referred to here as a utility transfer vector. 2 The concept of extended- imputation and detached extended- imputation was introduced in vn-m p.364for analytical purposes. 4

5 The transfer vector summarizes the transfers of utility that take place whenever players are bargaining in relation to a given payoff vector x and as a result, the payoff vector y is obtained, so that y = x+. Here our analytical uinit is the cover of N as a self contained structure. Definition 4 If x and y with are two different extended imputations and have the same cover support C, then = y- x is said to be a non-null utility transfer admissible by C. Let : 2 N R n be a vector valued set function that assigns to every subset S of N the corresponding characteristic row-vector so that (S) = w t = (w 1,, w n ) and So that (N) = J t =(1,,1 n times) and ( ) = (0,,0 n-times). Now we extend the function to be a matrix-valued set function that maps every cover of N in its corresponding characteristic matrix so that (C ) = W(k x n) is the, 0-1 characteristic matrix of the cover of N given by C ={C 1,,C k }, where C = k and the i th row of W denoted by W i. is the characteristic vector of the coalition C i in C for i= 1,, k. The element of W in row i and column j is W ij = 1 if j C i, W ij = 0 otherwise. The characteristic vector of the set N of all players as noted above is denoted here by the unit row n-vector J t = (1,,1 n-times). Definition 5 Relative to a cover C of N, an admissible utility transfer is said to be Pareto-optimal if and only if J t = 0. Whenever the admissible utility transfer under consideration decreases the value level of attainable extednded imputations, ie.. J t < 0, then it is said to be Paretosuboptimal. Remark 2 Clearly, if is a non-null utility transfer admissible by a cover C so is. We will observe that in a given cover that admits non-null utility transfers, only one of the two possible 5

6 directions is enforceable from the point of view of interpersonal bargaining based on available alternatives within the cover. Example 1 (a) Consider the a 3-person game and the cover C ={1, 2}, {2,3}} with characteristic matrix W and admissible transfer =( -e, e, -e) such that W = 0 t = [ -e e -e ] = 0 W = = 0 Player 2 may ask player 1 for a utility transfer by using as leverage his alternative to go with player 3. Similarly he may ask player 3 by arguing his alternative with player 1. On the other hand, it is clear that the utility transfer = (e, -e, e) is not enforceable. Players 1 and 3 have no bargaining alternatives and hence cannot enforce the utility transfer e in an interpersonal bargaining context. To enforce such demands would require for players 1 and 3 to act as one ie: as a syndicate in Harsanyi s sense. (b) Consider the cover C ={{1, 2, 3, 4}, {2, 3, 4, 5}, {1, 4, 5}, {1, 3, 5}} W = The characteristic matrix is given by : The cover in consideration admits a utility transfers of the form = (-,-3,2,2,- ). Clearly - is also admissible. However, - is not enforceable for it would imply player 2 demanding from player 3.Clearly, the former has no bargaining alternative against but strictly depends on the later. In both cases above the common characteristic of the transfers with enforceable direction is the Pareto-suboptimal nature of the transfer. The observations above suggests, from a constructive point of view, that we could separate the analysis of possible outcomes of a cooperative game in different stages being the first stage the 6

7 one of interpersonal bargaining considered above. And we expect our equilibrium to be one that doesn t admit enforceable utility transfers. Definition 6 A non-null utility transfer admissible by a cover C of N is said to be bargainingenforceable (from an interpersonal bargaining point of view) if and only if it is Pareto-suboptimal, ie: If and only if Jt e < 0. Clearly, bargaining enforceable utility transfers decrease the value level of supported extended imputations. Such transfers can only be stopped when the players affected adversely can protect by means of realizable bargaining alternatives the actual claims under consideration (ie. a possible alternative coalitions that support the defensible set of realizable claims). The concept of bargaining alternative within a cover of N will be fully developed later when introducing the concept of strategic independence. Remark 3 A utility transfer is admissible by a cover C of N if and only if W = 0. So that the existence of an admissible Pareto-optimal transfer for a cover indicates that the augmented system:, has a non trivial solution. This happens whenever the rank of W is less that n and also J t, the characteristic vector of the coalition N, is linearly dependent on the characteristic vectors of the coalitions in the cover C of N. The following concept summarizes the above two conditions: Definition 7 A utility transfer 0 is said to be a zero-sum for C (i) W = 0 (admissibility) (ii) J t = 0 ( pareto-optimality) if and only if Remark 4 If x is an extended imputation attainable through a cover C of N then x satisfies the sytem of equations Wx = v/c where v/c is the restriction of the characteristic vector v to the collection of subset of N in C. The vextor x is said to be a vector of sustainable claims in all eventualities considered in C. 7

8 3. Stability in Bargaining Scenarios Let X(C, v) be the set of extended imputations that may be supported by a given cover C ={C 1,,C k } of N. That is, X(C, v) ={x R n, x i 0 Wx=v/C }. Here we will simply write Wx = v, assuming that the vector v always denotes the corresponding restriction of the characteristic function to the coalitions in C. Any solution to the system W x = v is an n-vector of claims realizable through C since these are distributions of the value of the coalitions in the cover C. Conditions (i) and (ii) in definition 6 characterize two types of bargaining instability encountered in relation to the structure of covering collections of N. Condition (i) indicates that different extended imputations, say x and y, attainable through the cover C,, are claims that can be maintained by the players under all possible circumstances, with no bargaining argument in favor of either the claims in x or those in y. It characterizes a core-like condition of bargaining symmetry (this is so while negotiations are confined to the bargaining scenario defined by the cover C ). The type of bargaining instability that occurs when admissible transfers are allowed by the structure of the coalitions under consideration is termed here as structural-instability. The specific instability that occurs when the admissible utility transfers are Pareto-optimal characterizes the type of transfers that may occur in pure bargaining scenarios and will be referred as instability of the Pareto-optimal type and will be considered as neutral or symmetric structural instability. The sub- Pareto optimal structural instability is the type of instability that we want to characterize in order to determine the conditions under which such instability cannot occur. Definition 8 A pair [X(C, v),c ] is said to be a bargaining scenario for the game = (N,v) if and only if C is a cover of N and X(C, v) is the set of extended-imputations attainable through C. Clearly, X(C, v) = { x W x = v, x Theorem 1 (structural instability) Let C be a cover of N with 0 < C = k < n. Then, there exist always a utility transfer 0 admissible by C.. Corollary 1 The set of all admissible utility transfers, together with the null transfer form a linear subspace and corresponds to the nullity of W. Thus, the dimension of the utility transfer subspace is Dim [ (C )] = n-k 8

9 Remark 5 The rank of the characteristic matrix W of a cover C of N depends on the number of linearly independent rows of W. If the characteristic vectors of a collection of subsets of N are linearly independent we say that the corresponding coalitions are independent subsets of N. Theorem 2 (complete structural stability) Let C. = {C 1,,C n } be a covering collection consisting of n linearly independent subsets of N. Then there exist no utility transfer 0 admissible by the cover C. Corollary 2 Since C is a set of n linearly independent coalitions and it has maximal dimension then the rank of W is n, and the system W x = v has a unique extended-imputation solution given by x = W -1 v. The above corollary tells us that linearly independent balanced collections of maximal dimension may support only one admissible extended imputation. Now, we proceed to obtain necessary and sufficient conditions for the structural admissibility of transfers as an application of the fundamental theorem of linear algebra in the Fredholm alternative for matrices form. In our particular case it gives us the following: Theorem 3 (Utility transfer admissibility theorem) Given a cover C of N with characteristic matrix W, then either the system W t = J has a solution, or the system W = 0, J t 0 has a solution. But not both can have a solution. Corollary 3 If the system W t = J has a solution and there exist 0 such that W = 0, then it must be that J t = 0. Definition 9 Whenever the system W t = J has a solution, the corresponding cover C of N is said to be a linearly-balanced collection of subsets of N. That is, if a linearly balanced collection admits a utility transfer, it has to be zero-sum. In other words, balanced collections admit, if any, only Pareto-optimal transfers. The following definitions of balanced collections discriminate more deeply than those given in Bondareva [6 ] and Shapley[20] : 9

10 Definition 10 A covering collection C = {C 1,..., C k } of subsets on N with characteristic matrix W, is said to be: l-balanced if W t = J, unrestricted (linearly-balanced) n-balanced if W t = J, and i i < 0, i=1,...,k (negatively balanced) nn-balanced if 0 W t = J, (non-negatively-balanced) Non-negatively- balanced 3 coverings may be partitioned into the following two kinds, of covering collections: p-balanced if > 0 W t = J, (positively-balanced) w-balanced if 0 W t = J, and i = 0, for at least one element of. The vector R m is referred to as the balancing vector of the collection C. The elements i of, i=1,,m are the balancing coefficients of the corresponding sets in C. A p-balanced collection C is said to be minimal if it has no p-balanced proper sub-collection. Theorem 4 Any l-balanced collection C of subsets of N is a covering collection for N Remark 4 Partitions of N are a special kind of p-balanced collections. The coefficients of the balancing vector of any partition are 1 s. Any partition with cardinality less than n, by theorem 1, admits always non-null utility transfers, and by corollary 3 these must be Pareto-optimal. Corollary 4 If C = {C 1,,C k } is a p-balanced collection of subsets of N, with balancing vector such that i K1 for all i =1,, k, then the collection C =C C i is a cover of N. Theorem 5 (Pareto-optimal instability of partitions) Any non-null utility transfer admissible by a partition is zero-sum. Remark 5 Any partition is a cover collection of N. It may have at most n non-empty sets. And at least one non-empty set. The only partition that doesn t admit any non-null utility transfers is the partition of N in to n mutually exclusive and collectively exhaustive singletons. The corresponding characteristic matrix W is the n by n identity matrix, so that W = I and the system I = 0 has no solution 0. The partition of N into a collection with 1 set is {N} and the characteristic matrix is ({N })= W (1 x n) =J t. 3 Here p-balanced corresponds to balanced and w-balanced to weakly-balanced in Shapley[20], 10

11 Example 3-Person General-Sum Game v({i})=0, i=1,2,3, v({1,2}) = 80, v({1,3}) = 70, v({2,3}) = 50, v({1,2,3}) = 100 Extended imputations: Value level x(n) = x 1 + x 2 + x 3 Utility Transfer = ( 1, 2, 3 ) Admissible utility transfers (S) = 0, Pareto Optimal utility transfer (N) = 0 (x, A) = x -A = (50, -50,-50) (x, y ) = x -H = (25, -25,-25) (1+) = ( e, - e, -e ) = = 0 J <0 A(N) = 150 B(N) = 120 C(N) = 130 D (N) = 100 A (0, 80, 70) G (0, 80, 100) x 2 H (25, 55, 45) E (0, 100, 0) E(N) = 100 F(N) = 100 H(N) = (x, B) = (-20, -20, 20) = x -B (3+) = ( - e, - e, e ) = = 0 J < 0 B (70, 50, 0) X o (50, 30, 20) x 2 + x 3 = 50 F(100, 0, 0) x 1 + x 2 = 80 x 1 + x 2 +x 3 = 100 x 1 x 3 D (0, 0, 100) C(80, 0, 50) Zero-sum utility transfer W = 0, Admissible J =0, Pareto Optimal (x, C) = (-4, 4,-4) = x -c (2+) = ( -e, e, -e ) = = 0 J < 0 The cover C = {{1, 2}, {1,3}} admits strategically-viable utility transfers = (, -,- ), here W = so that W = 0 notice in particular the admissible utility transfer from point A(0, 80, 70) to point H(25, 55, 45), both supported by the cover C, is given by H-A = (25, -25, - 25) = with = 25 with J t < 0. Note that for the cover C = {{1, 2, 3}} the transfer from D (0, 0, 100) to X (50, 30, 20) given by X -D = (-50, -30, 80) is zero-sum sicnce J t = 0. The cover {{1} {1,2}, {1,3}} is n-balanced with balancing vector (-1, 1, 1) = t However, the cover { {1,2}, {1,3}, {2, 3}} is p-balanced with balancing vector (1/2, 1/2, 1/2) = t Note that neither cover admits utility transfers. 11

12 4. Structural and Strategic Equilibriums In a given game = (N,v), whenever a bargaining scenario [X(C,,v), C ] is analyzed we have determined that two types of instability relative to a vector of claims may occur. In both cases the instability consists on the existence of non-null admissible transfers by the cover C relative to an extended imputation payoff vector x supported by C. The instability of the Pareto-optimal type is the one that occurs whenever de cover is either a partition with cardinality different from n or whenever the collection is linearly balanced but has dimension less than n. Thus, we expect to have stability in a bargaining scenario whenever no non-null admissible utility transfers may take place unless these are of the Pareto-optimal type. Definition 11 Given an extended-imputation x with cover efficient cover support C, the pair [ x,c ] is said to be a structural-equilibrium for the game = (N,v) if and only if every utility transfer, if any, admissible by the cover C relative to x is zero-sum. A bargaining scenario [X(C, v), C] is said to be structurally-stable if and only if every x in X(C, v), the pair [x, C] is a structuralequilibrium. With the above definition, we may combine theorems 2 and 3 and obtain the following theorem structural equilibrium theorem: Theorem 6 A bargaining scenario [X(C,v),C ] is structurally-stable for the game = (N,v) if and only if the cover support C, of every x in X(C,v) is l-balanced. Corollary 4 (Complete structural equilibrium) If the cover C above has cardinality C = n, and the coalitions in C are linearly independent, then the set X(C ) = {x } is a singleton. The particular bargaining scenario [x,c ] is referred to, here, as a complete structural-equilibrium for the game = (N,v). Clearly, no non-null admissible utility transfer may take place among players. Remark 7 The set of all complete structural equilibriums for a cooperative game are in one-to-one correspondence with the possible basis that define the basic solutions of the system of equations [V t J] = J where V is the characteristic matrix of the collection of non-empty proper subsets of N (Here we have the matrix V augmented with the characteristic vector corresponding to the grand 12

13 coalition). Any basic solution gives a 2 n 1 vector whose n basic components components are the balancing coefficients of the linearly independent coalitions that constitute the l-balanced collections of n linearly independent coalitions that constitute any of basis for the given system. Clearly some of the balancing coefficients of a basis may be zero. The non basic values are always zero. Thus we may restrict our attention to the balancing vector of the coalitions in the basis which is always of dimension n. Theorem 7 Given any cover C ={ C 1,, C n } of N, consisting of n linearly independent subsets of N with balancing vector. For any C h C the collection C = C - { C h } has a characteristic matrix (C ) = W n-1x n. and admits only utility transfers with basic structure given by the h th column of W -1 denoted by [W -1 ] (h). Corollary 5 For such transfers: W 0, W = 0 and W h > 0, with J t > 0 if h > 0, J t < 0 if h < 0 and J t = 0 if h = 0. Remark 8 A comparative analysis between complete structural equilibriums with n-balance coefficients versus those with p-balanced coefficients suggest that if players adhere to claims supported by a coalition with negative coefficients they constraint themselves unnecessarily to a condition that prevent them from obtaining better payoffs. This condition is clearly strategically unstable. So, by excluding the n-balanced collections from our complete structural equilibriums we may hopefully be left with those strategically stable. The fact that W h >0 and and J t <0 indicates that some or all players in the complement of C h to N have no alternative within C to protect the current claims. Theorem 8 Let C = {C 1,, C k } be an l-balanced collection of subsets of N with corresponding balancing vector, and let E be a proper sub-collection of C such that C ˆ = (C - E ) E * is also l-balanced with balancing vector where 13

14 Corollary 6 Whenever one of the coalitions in an l-balanced collection of maximal dimension with nonzero balancing coefficient is replaced by its complement to N, the resulting collection is also linearly balanced, The sign of the balancing coefficients remain the same except for the replaced coalition which takes the opposite sign. Remark Our observations above on interpersonal bargaining-enforceable utility transfers and theorems 7 and 8 motivate the following definitions. Definition 12 An admissible utility transfer 0 by some sub-cover of a structural equilibrium [X,C ] is said to be strategically-viable for C if and only if (i) W >= 0 (admissibility and special interest) (ii) J t < 0 ( Pareto-sub-optimality) That is, strategically-viable utility transfers are those bargaining-enforceable transfers that may benefit some or all of the members of some of the supporting coalitions so that there may take place a special interest on part of some or all of its members not to adhere to the current claims supported by the cover in consideration. Now we proceed to define the concept of strategicequilibrium. Definition 11 A bargaining scenario [X (C,v),C ] is said to be a strategic-equilibrium for the game = (N,v) if and only if the cover C of N, admits no strategically-viable transfers. The following theorem can be shown to be a particular case of the Farkas lemma. Theorem 9 (Strategic stability theorem) A bargaining scenario [X (C,v),C ] is a strategicequilibrium for the game = (N, v) if and only if the collection C is nn-balanced Clearly, every strategic-equilibrium is a structural equilibrium but the converse doesn t hold. Also, complete strategic equilibrium s al nn-balanced collections with maximal dimension 14

15 5. LP and the Fundamental Strategic Equilibrium If one could find a minimal p-balanced collection that supports one or more vn-m detached extended imputations, the bargaining scenario in consideration would clearly be a dominance related equilibrium. This would be so, because detached extended imputations are un-dominated sets, see vn-m [28] p Since by definition an extended imputation satisfies individual rationality, it follows then, that the corresponding decomposed conditional payoff vectors in the given systemic equilibrium would all be coalitionally rational. Thus if we can associate detached extended imputations with a p-balanced cover support, the resulting equilibrium would have corelike stability properties as a fundamental differentiating element from all the many strategic equilibriums that we may encounter in a cooperative game with transferable utility. We would like to consider the analysis of the strategic-equilibriums in relation to the rationality conditions in two phases: (1) previous to considering the formation of the grand coalition and (2) when considering the formation of the grand coalition. In the phase we restrict our attention to the set of extended imputations that satisfy coalitional rationality but may or may not violate group rationality. In the second phase we restrict our attention to those that satisfy coalitional rationality and do not violate group rationality. Definition 18 A strategic-equilibrium [ X, C ] is a fundamental-equilibrium for the cooperative game = (N, v) if and only if the extended imputations in the stable-set X satisfy coalitional rationality. The core-like properties of the equilibriums associated with balanced collections brings us back to the dual pair of linear programming problems and their corresponding feasible sets that appear recurrently in relation to the Bondareva [ 6 ], Shapley[ ] theorem on the existence of the core, the theorems of Charnes and Kortanek [ ], Zakarov and Kwon [ ] and in Owen [16 ]. Also in related concepts introduced by vn-m[28] where it is shown the existence of a numerical value bound for the set of detached imputations (un-dominated) and in Turbay [24] where the a linear programming problem is introduced to compute the fundamental strategic-equilibrium of a 3-person general sum cooperative game. Let V denote the characteristic matrix of the collection all proper subsets of N denoted here by V = 2 N -2, so that (V ) = V, is a 0-1 matrix with dimensions (2 n -2) by n. The set of n- vectors Z = { x Vx v } consist of all coalitionally rational extended imputations for the game 15

16 = (N,v) and Y= { V t = J t 0, R V } is the set of balancing coefficients of the nn-balanced collections of subsets of N Clearly both sets X and Y are non empty convex polyhedral sets. The following dual pair of linear programming problems is commonly associated with the study of balanced collections and their applications to the theorems on the existence of the core, see Bondareva[ ] Shapley [ ] Charnes and Kortaneck [ ] and Owen[ ]: (i) Minimize J t x Subject to x Z, and (ii) Maximize v t Subject to Y The feasible set for the primal problem is simple the set of all coalitionally rational extended imputations of which von-neumann and Morgenstern detached extended imputations are a subset. These two sets become the same set whenever the core is empty. Theorem 13 For any cooperative game = (N,v), there exist such that ith Theorem 14 The core of a cooperative game = (N,v) is non-empty, C ( ) K, if and only if v(n) x, where x minimizes J t x for all x Z. Theorem 15 (Existence) Every cooperative = (N,v) has a fundamental-equilibrium. Theorem 16 The core of a cooperative game = (N,v) is non-empty if and only if there exist at least one detached extended-imputation in the strategic stable set of the fundamentalequilibrium. Theorem 17 (Uniqueness) The fundamental-equilibrium of a cooperative game = (N,v)has a unique strategically-stable set X defined by the solution set to the primal LP problem (i) above. Further, the value level of any x in X is unique and given by u* = where x X Z and is an optimal solution to the dual LP problem (ii). Let ={x x X(C,, v), where [X(C,, v),c ] is a strategic-equilibrium for the game = (N,v) } then 16

17 Theorem 18 If x is a fundamentally-stable extended-imputation for the game = (N,v), then That is, the fundamental equilibrium maximizes the value level overall Strategic-stable extended imputations in a game. Theorem 19 (invariance) The fundamental-equilibrium [X (C, v),c ] of a cooperative game = (N, v), is a relative invariant under strategic equivalence. The above theorem follows from the linearity characteristic of the fundamental-strategicequilibrium. 6. The Stronger Player Paradox Emergence of a paradox Here we will use the strategic-equilibrium generic form for a triangular 3-person cooperative to construct stable outcomes in a process that exhibits the emergence of stages that require strategic interaction, player awareness and learning. Based on bargaining alternatives, players demand utility transfers from each other until strategic-equilibrium is reached. A generic description of the strategic-equilibrium shows a conditional stable system, where different occurrences can be viewed as one. It represents interrelated subsystems of game outcomes presented as an extended-imputation of maximum claims associated with a cover of N or as an equivalent conditional matrix of payoffs. The, players accept their maximum sustainable claims as payoffs of a cooperative division agreement; and then take these as disagreement payoffs or dividends if they are to form a syndicate. A syndicate may dispute with the excluded player his incremental contribution to the grand coalition. Binding agreements on splitting rates must take place: (1) an internal division rate is required for a syndicate to form, (2) an external rate must be agreed on if the grand coalition is to form. Remarkable results emerge: The order of binding agreements on rates determines whether or not the outcomes constitute a non-discriminatory solution. Equally remarkable is the fact that a dominating branch in these objective solutions emerges: The weaker players, at the coalition formation level, syndicate against the stronger one which becomes the weaker one at this emerging syndicate bargaining stage. This inherent condition to all non-discriminatory von Neumann and Morgenstern solutions of general-sum cooperative game here is referred to as THE STRONGER PLAYER PARADOX. 17

18 The tiangular3-person game strategic-equilibrium Consider the generic characteristic function for the 3-person cooperative game in zeronormalization : v(n) = v 123, v({1,2}) = v 12, v({1,3}) = v 13, v({2, 3}) = v 23, and v({j}) = 0 j=1,2,3 We will assume: (1) v 123 v 12 v 13 v 23 (with no loss of generality ) (2) v 13 + v 23 v 12 (triangular inequlity). Games that satisfy the triangular inequality are said to be triangular games. The fundamental strategic-equilibrium for triangular 3-person cooperative games is given, in generic form, by the pair [x, C ], here x = (x 1, x 2, x 3 ) t and C = with The collection C has a p-balanced support given by t = J t W -1 = (½, ½, ½ ) Here W is the characteristic full rank, nxn characteristic matrix of the cover C of N that supports the unique extended imputation x. where x i + x i = v ij i j, i, j = 1, 2, 3. The fundamental strategic-equilibrium for triangular games is a strong-strategicequilibrium 4 [x,c ]. It represents the conditional system of the maximum sustainable claims for the players under all possible bargaining alternatives: 4 A fundamental strategic-equilibrium for the game is said to be strong when the collection is p-balanced and it consists on n-linearly independent subsets of N. the set of attainable extended imputations through the cover is X is a singleton. 18

19 Syndicate formation process When players realize the strategic-equilibrium that imbeds them, as members of a 2-person coalition they may realize also, in considering the formation of the grand coalition, that they have the possibility of forming a syndicate. That is a coalition that behaves as a single player. By doing so, the characteristic function value of the game to each coalition may be used as a disagreement base and then a pure bargaining game emerges between the 2-player coalition that forms a syndicate v.s. the excluded one. We denote here by [i, j] the syndicate that forms versus player [k]. For each possible syndicate the new game that emerges is a two-player pure bargaining game that has the following characteristic function: û ([i, j]) = v ij, û([k]) = v k = 0, û ([i, j], [k]) = v(n) i j k, i, j, k = 1, 2, 3, The corresponding 0-normalized form of the emerging bargaining games is: u([i, j]) = 0, u([k]) = 0, u([i, j], [k]) = v(n)-v ij = e i j k, i j k = 1, 2, 3 For a syndicate to form several binding agreements among its members must take place : (1) The members of the syndicate must give up his bargaining possibilities and personal initiatives. The syndicate presents a unique bargaining front and negotiations with outside players may only take place through the official representative or leader of the syndicate. (2) Since syndicates are borne from formed coalitions, the members of the syndicate must have agreed on how the proceeds of the coalition are to be divided. So a coalitional splitting rate must be decided on so that each player secures for himself the amount z k = k v ij, i =, 0 < k < 1, k {i, j}, i, j = 1,2,3, i j. (3) A syndicate external division rate must eventually be agreed on between the syndicate [i, j] and the exclude player [k] syndicate in bargaining for player k s incremental contribution if the grand coalition is to be formed: The syndicate obtained the amount e k and the excluded player the amount (1-. (4) Finally, The members of the syndicate [ i j ] vs. [k] generally agree, before forming a syndicate, on how they will divide the proceeds, if any, are to be obtained in the negotiations with the excluded player k. The bargaining that takes place is for the amount e k = v N - v ij. Namely: the incremental contribution of player k to the grand coalition provided it forms. The syndicate internal division rate is denoted here by so that the syndicate s gain e k is divided between players i and j so that player i receives the amount e k and plyer j receives (1- ) e k where 0< <1. 19

20 That is, if a syndicate [i, j ] forms they divide the proceeds of the coalition {i, j}, namely v i j, obtaining each its maximum sustainable claim x i and x j respectively. Then, players i ad j may use the amount v i j as disagreement value for their syndicate in the negotiations with the excluded player k for the amount e k. An agreement on how to internally divide the syndicate proceeds obtained may preferably be made before the negotiations with the excluded player take place. Otherwise, subsequent conflict may result if syndicate members happen to disagree on the splitting rate after the bargained amount has been secured. The description of possible outcomes to the derived game can be summarized as follows: y [i, j] = (1- k ) e k, y [k] = k e k, 0 k 1, i, j, k=1, 2, 3, i j k So that a general description of strategic-equilibrium based outcomes in the triangular 3- person cooperative game would be given by the following interrelated conditional system of imputations: 1 e 1 x β 1 (1 1 ) x β 1 (1 1 )e 1 x β e 2 2 e 2 x (1 β 2 ) 1 2 e 2 x β e 3 x (1 β 3 )(1 3 ) 3 e 3 0 < i <1, 0< i <1, i=1, 2, 3. This general description of stable outcomes may be viewed as an emergent process composed of two phases. And it may be obtained as the sum of the two matrices that correspond to two interdependent conditional systems: (1) A conditional system that summarizes the maximum sustainable claims that result from the negotiations among players as potential members of coalitions; coalitions that are exhibited as bargaining alternatives to less desirable outcomes, and (2) A conditional system that summarizes the internal and external agreements of the syndicates. 20

21 Phase I Phase I 1 e 1 β 1 e 1 β e 1 + β 2 e 2 2 e 2 β 2 e 2 β 3 e 3 β 3 e 3 3 e 3 0 < I <1, 0< I <1, i=1, 2, 3 Our strategic-equilibrium based system when analyzing a generic game may give us a considerable multiplicity of stable mutually exclusive interrelated sets of outcomes. The mayor determinants of variation are conditions inherent to (1) the characteristic function and (2) the binding agreements on splitting syndicate rates. By discriminating and managing some aspect of these determinants the following scenarios unfold: Case I ( =x (N)). In this case the core of the game exists and consists of a singleton that is C( ) = {x 0 }, also it can be readily show that = x (N) implies x i = e i, x 0 = ( e 1, e 2, e 3 ). Our conditional system above becomes: 1 e 1 β 1 e 1 β e 1 β 2 e 2 2 e 2 β 2 e 2 β 3 e 3 β 3 e 3 3 e 3 0 < I <1, 0< I <1, i=1, 2 This solution system, graphed as interrelated separate occurrences in the imputation simplex viewed as a two dimensional object gives the following figure: x (1) ( 0, e e 1, e 3 +(1-1)e1 e 1 (1-1 )e 1 e 3 x ( e 1, e 2, e 3 ) 1 e 1 e 1 e 2 e 3 Figure-A Strategic-equilibrium general description of possible outcomes when v N = x (N) and syndicates negotiate with excluded player after its members have agreed on. 21

22 The graph reflects the order in which the parameters must be agreed upon in the syndicate inner and outer negotiations. First the syndicate internal agreement of its members must be ratified as a rational prerequisite to its formation. This appears as an obvious rational step, to avoid possible future conflict between the members. Then the external rate is negotiated in a pure bargaining game. However, if the members of the syndicate agree to settle the complementary division rate, say a posteriori, it becomes clear that in case of disagreement, either a third party must enter to arbitrate the division of the syndicate s gain or the members of the syndicate may find themselves in what we may term a prelude to serious conflict that may end up in players taking measures that are not contractually self enforcing. In such case the graph of the solutions would be as shown in the following figure: e 1 (1-1 )e 1 e 3 x ( e 1, e 2, e 3 ) 1 e 1 1 e 1 e 2 e 3 Figure-B Strategic-equilibrium solution when v N = x (N), is reach then is negotiated internally We may readily verify that the first of the two solutions obtained is a vn-m solution while the second one is not. Remark: It is remarkable that of the two particular strategic-stability descriptions of possible outcomes that emerge as consequences of having a different sequential order for the same agreement decisions; the one that leads to possible uncontrolled conflict is precisely the one that is not a vn-m non-discriminatory solution For the following two cases only general observations will made. The specifics of the analysis are left to the interested reader. 22

23 Case II ( < x (N)). In this case C( ) = The corresponding graph when the standard is agreed on previous to is: Figure-C Strategic-equilibrium possible outcomes when <0. Also a vn-m solution All the observations made in Case I regarding the sequential order in which the parameters and are agreed, determine equally whether the solutions obtained constitute or not vn-m solutions. If the syndicates form without defining (the way they will split the proceeds of the bargaining for the incremental contribution of the excluded player) and they settle for an with the excluded player, the outcomes with fixed and variable would look as in the following graph: Figure-D Strategic-equilibrium solution when <0. Not a vn-m solution To explain the core as strategic-equilibrium based possible outcomes requires working with weak-strategic-equilibrium [11]. In general the core as a solution concept has little explanation on how core-outcomes may come to occur. The formation of syndicates gives a good example on why core outcomes may not be enforceable so that actual solutions may well violate coalitional 23

24 rationality requirements. Clearly the branches in vn-m solutions are not coalitional-rational. These results are developed in detail in a forthcoming paper where the strategic-core defined as the convex combination of strategic-equilibrium based outcomes or equivalently as imputation payoffs to the players that are greater or equal to the equilibrium claims, are always contained in the core of the game so that it becomes a stronger solution concept not previously identified in the theory of games. The following case is the one where the core is not empty and is not either a singleton. In our constructive framework, the possible outcomes of syndicate bargaining, in order to conform a vn-m non-discriminatory solution require a unique value for the splitting rate. Case III ( < x (N)). In this case C( ) {x }, The figures bellow shows graphs of strategic-equilibrium based outcomes that include core outcomes: Figure-E Strategic-equilibrium solution with syndicate internal agreement rate = ½, In both cases the strategic-equilibrium solution is a non-discriminatory vn-m solution. When the game has a non-singleton core and the value of the internal syndicate splitting rate is other than = ½, the system interrelated sets do not conform a vn-m solution That is, whenever the game has a non-empty core, a strategically- stable solution is not vn- M stable unless the syndicate internal bargaining rate, agreed previous to the formation of the 24

25 corresponding syndicate internal standard of behavior parameter value, is the one corresponding to the equity standard 5. The particular scenario for possible outcomes where the internal splitting rate is settled after the external rate is agreed on as in the preceding cases do not constitute a vn-m objective solution. Syndicate power and the stronger player paradox One may think that in the mathematical construction that describes the strategic-equilibrium based possible outcomes of the game all strategic considerations have been examined. This may be so, for the two emergent bargaining stages encountered: The coalitional bargaining and the syndicate bargaining stages. However, we may observe that of the three possible ways that 2- players coalitions may form, the one with the largest excess is the one that rational players would choose form. The reasoning may go as follows: We players 2 and 3 can secure the amounts x 2 and x 3 in two of the three cases. By getting together and forming a syndicate versus player 1, not only we secure our maximum sustainable claims but also as a syndicate we may dispute an additional amount, higher than in any other syndicate that under similar rate conditions, the share we obtain of the syndicates gain would be higher. In bargaining with player 1 for the amount e 1 suppose the syndicate internal agreement rate follows an egalitarian standard, so that =1/2 and the syndicates external bargaining rate is determined by a standard based on number of participants so that = 2/3 then under similar conditions the player will receive conditioned tom the formation of the respective coalitions: x 2 + (1/3) e 3 if [1, 2] forms x* 2 ([i,j]) = 0 + (1/3) e 2 if [1, 3] forms x 2 + (1/3) e 1 if [2, 3] forms Similarly, 0 + (1/3) e 2 if [1, 2] forms x* 3 ([i,j] ) = x 3 + (1/3) e 2 if [1, 3] forms x 3 + (1/3) e 1 if [2, 3] forms 5 If is defines as a piecewise continuous function or as a special non-linear function of, it is possible to obtain outcomes that conform a vn-m solution however such cases are not easily explainable in terms of our simple construction of rational behavior responses to the conditions that emerge in the different bargaining stages of the game. 25

26 Since, v12 > v13 >v23, this implies both x 1 > x 2 > x 3 and e 1 > e 2 > e 3, then x* 2 and x* 3 are both maximized in [2, 3]. It follows that the player that can sustain the higher claim is the one who will end up ganged up against. Paradoxically, the strongest player at the coalitional bargaining level becomes the weakest one at the syndicate bargaining level. This paradox appears here as an emergent characteristic that may be present in all games of cooperation and that can be readily recognized as a socio-economic behavioral archetype. It shows the how vulnerable are strong players and the effective power of syndicates when they are allowed to be formed. 7. The Strategic-Core Consider the class of cooperative games with transferable utility S ={ = (N,v)} with a strong-fundamental- strategic- equilibrium [x, C]. For these games, the cover support of x is p- balanced consisting of n linearly independent coalitions. Definition Given game = (N,v) in S, the set of all imputations x, s-c( ) ={ x x j x j } is referred to as the Strategic-core of the game Remark Since the extended imputation any x in a fundamental-strategic-equilibrium is coalitionally rational, the existence of imputations x for which x x is a necessary and sufficient condition for the existence of the Core. Hence the strategic-core is always a subset of the core. The definition of the strategic core as set of vectors x such that x(n) = v(n) and x j x j makes s-c( ) for all games an n-1 simplex similar to the imputations simplex. Example 1) For the triangular (0.100) - normalized game v 12 = 70, v 13 = 60, v 13 = 50, v 123 = 100, the fundamental-strategic-equilibrium is given by [x, C ] = [ { x = (40, 30, 20) }, C ={{1, 2}, {1, 3}, {2, 3} ] Or equivalently is given by the conditional system: 26

27 The Strategic-Core is given by the set s-c ( ) = { x x j x j, j =1, 2, 3} That is, the 2-simplex given by the Imputation set self-similar set s-c ( ) = {x = (x 1, x 2, x 3 ) x 1 40, x 2 30, x 3 20} as indicated by the dark region inside the core below THE STRATEGIC CORE & VN-M SOLUTIONS x 1 + x 2 = 70 x 2 + x 3 = 50 x 1 + x 3 = 60 s-c( ) The extreme points of the core minimize, among all core points the amount x j imputed to player j and maximizes the sum x k + x l if syndicate [k, l] forms versus player j. Compared to the strategic-equilibrium based outcomes, we may observe that only the equity standard of behavior k = l = ½ to split the proceeds of the syndicate when bargaining with player j for the excess e j = v kl (x k + x l ), j, k, l = 1, 2, 3; j k l, may take the bargaining to an internal agreement on a hypothetical departing bargaining point for players k and l to obtain further possible gains from player j as indicated by the irregular curved-lines o the vn-m solution above. ( Clearly the standard of behavior in these branches of the vn-m solution wouldn t longer be the equity standard. 27

28 2) Consider the general-sum 4-person cooperative game = (N, v) where N={1, 2,3}; and v ({1, 2, 3, 4}) = 1, v({i, j, k}) = ½, i K j Kk i =1, 2, 3, 4, otherwise v(s) = 0, S N Here the imputations n-1 simplex of imputations I = { x R 4 x 1 + x 2 + x 3 + x 4 } may also be described as the convex hull of the 4-dimensional unit vectors as shown in the graph below: IMPUTATIONS SIMPLEX (0, 1, 0, 0) (0, 0, 0, 1) (0, 0, 1, 0) (1, 0, 0, 0) The core is given by C= { x R 4 x i + x j + x k ½, i K jkk, i, j, k =1, 2, 3, 4 }, with corresponding graph: 28