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The Relative Interior of the Base Polyhedron and the Core*

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The Relative Interior of the Base Polyhedron and the Core* Forthcoming in Economic Theory The Relative Interior of the Base Polyhedron and the Core* Jingang Zhao Department of Economics Iowa State University 260 Heady Hall Ames, Iowa 50011-1070 USA Fax: (515) 294-0221 Tel: (515) 294-5245 (Email: [email protected]) Summary: This note provides two necessary and sufficient conditions for the relative interior of the core (and the base polyhedron) to be non-empty: (i) the second largest excess of the prenucleolus is negative; (ii) the grand coalition's payoff is greater than the minimum no-blocking payoff. Such conditions imply an intuitive method in proving core existence, they also imply results on the sensitivity of the base polyhedron and the core. Keywords: Base polyhedron, core, minimum no-blocking payoff, nucleolus, sensitivity analysis. JEL Classification Number: C62, C71, D71 AMS#: 05B35, 52B40, 90C31, 90D12. Running Head: The Interior of Base Polyhedron and Core * The author would like to thank Rick Percy, David Schmeidler, Lin Zhou, an anonymous referee, and the editor for valuable comments and suggestions. All errors, of course, are my own. An earlier version of the paper was circulated with the title "The comparative statics of the core and new conditions for core existence." 1. Introduction The core of a convex game and the base polyhedron of a submodular set function are dual to each other. This note provides two necessary and sufficient conditions for their relative interior (R-interior henceforth) to be non-empty. The first condition relies on a new property of the prenucleolus: the core has a non-empty R-interior if and only if the second component in the excess vector of the prenucleolus is negative. The second condition uses the new concept of the minimum no-blocking payoff (MNBP, i.e., the minimum payoff needed to satisfy all proper subcoalitions): the core has a non-empty R-interior if and only if the grand coalition's payoff is greater than the MNBP. The second condition is based on the intuition that the core of a game or a surplus-sharing problem is non-empty (empty) whenever the grand coalition has enough (too little) to split. This argument provides a new and intuitive method in proving core existence, which has at least two advantages over the balancedness method: the concept of balancedness is quite abstract and unfamiliar to most applied scholars, while the MNBP method is intuitive and can be readily used by applied scholars; the balanced approach tells nothing about the R-interior of the core, while the MNBP method automatically reveals the status of the core's R-interior. The conditions have applications in the sensitivity of the core, in general, and in cartel stability, in particular. The core with a non-empty R-interior remains non-empty against small perturbations. Consequently, a cartel agreement will be stable against small shocks if it specifies an allocation that is in the R-interior of the core. On the other hand, a non-empty core with an empty R-interior could be made empty by certain perturbations. Therefore, a cartel agreement will be unstable if the cartel has an empty core or has a non-empty core with an empty interior. The rest of the paper is organized as follows. Section 2 describes the problem and definitions; Section 3 establishes two conditions on the R-interior of the core and the base polyhedron, and discusses their implications. Section 4 studies the geometric structure of the core. Section 5 concludes, and the appendix provides all proofs. 2 2. Description of the Problem Let N = {1, 2, ..., n} be a finite set, and N = 2N denote the collection of subsets of N. A set function v: N ®R+ with v(Æ) = 0 is submodular if v(S) + v(T) ³ v(SÈT) + v(SÇT) for every S, T Î N , and it is supermodular (or a convex game) if the inequality is reversed. A submodular function defines a cost-sharing (CS) problem, which characterizes those situations like carpooling in which the costs exhibit scale/scope economies: for any partition D = {S1, S2, ..., Sk }of S Î N, it costs $v(Si) for the travelers in a coalition Si to take a taxicab alone, it costs only k v(S) £ S i=1v(Si) if all travelers in S share one taxicab. On the other hand, a supermodular function (or a convex game in coalition value form) defines a surplus-sharing (SS) problem, which characterizes those situations like networks in which returns exhibit scale/scope economies: for any partition D = {T1, ...,Tk} of T Î N, the network providers in a coalition Ti alone can generate a revenue of $v(Ti), the full cooperation by k all providers in T can generate v(T) ³ Si=1v(Ti). In both CS and SS problems, it is assumed that the gains (savings) of full cooperation are k positive. Precisely, we assume v(N) > Si=1v(Ti) for any non-trivial partition {T1, ...,Tk} of N in a k SS problem, and v(N) < Si=1v(Si) for any non-trivial partition {S1, ..., Sk } of N in a CS problem. How to split the joint returns (costs) v(N) is the main subject in cooperative game theory, and the core (base polyhedron) provides a natural solution to the SS (CS) problems. Now, interpret the set function v as a general SS problem without requiring the supermodularity, let it be given as a TU (transferable utility) game in coalition value form (1) G = {N, v(.)}, or alternatively, as a (2n-1)–dimensional non-negative vector i 2n–1 (2) t = {t = v(S) | S ¹ Æ and S Í N}Î R+ , 3 where the components are arranged decreasingly by coalition size and then lexicographically decreasing by player indices1. For example, for n = 3, t = {t1, t2, …, t7} = {v(123), v(23), v(13), v(12), v(3), v(2), v(1)}. Thus, one always has ti = v(N) for i = 1, and ti = v(1) for i = 2n-1. 1 n n i A payoff vector is any x = (x ,…, x )Î R+ specifying a payoff x ³ 0 to each player i n n S i Î N. An allocation of v(N) is any x Î D (v(N)) = {x Î R+ | iÎNx = v(N)}. The base polyhedron and the core are defined below: Definition 1: The base polyhedron of a set function v (or the vector (2)) is given by (3) n i i B(v) = B(t) = {yÎR+ | S iÎTy £ v(T) for all T ¹ Æ, T Í N, SiÎNy = v(N)}; the core of the game (1) (or the vector (2)) is given by n S i S i (4) C(G) = C(t) = {xÎR+ | iÎS x ³ v(S) for all S ¹ Æ, S Í N, iÎN x = v(N)}. An allocation y in the base polyhedron splits the total costs v(N) in such a way that no coalition T pays more than its own cost v(T), while a core allocation x splits the total payoff v(N) in such a way that no coalition S receives less than its own payoff v(S). Because no coalition alone can do better than the proposed allocations, coalitional rationality is satisfied for all coalitions, so the allocations in the core and base polyhedron are socially stable. Since base polyhedron and core (and CS and SS problems) are dual to each other, we shall focus the discussions, like the literature in economics and game theory, on the core of a SS problem and will not repeat the dual results for base polyhedron. It has been shown that the core (base polyhedron) of a convex game (submodular function) is always non-empty (see Edmonds, 1970; Shapley, 1971; Fujishige, 1991). However, the core of a general game (1) (the base polyhedron of a general set function) might be empty or non-empty. As already discussed, the characterizations of the core's R-interior provides a new 1 i j n Let |S| denote the number of players in S, and t = v(Si), t = v(Sj) for any 1 £ i < j £ 2 -1, then |Si| ³ |Sj|, and the sum of the entries of Si is lexigraphically greater than that of Sj if |Si| = |Sj|. 4 method of proving core existence, which is more intuitive than the balanced2 method (Bondareva, 1962; Shapley, 1967). n 2 n 2 n For any xÎ R , let ||x|| = Si=1 xi , Bd(x) = {yÎ R | ||x–y|| < d} be its open d- n n S i neighborhood (d > 0). Recall that D (v(N)) = {x Î R+ | iÎNx = v(N)} is the set of all allocations for v(N). The R-interior of C(t) can be given as n R.IntC(t) = {xÎC(t) | Bd(x)ÇD (v(N)) Ì C(t) for some small d>0}. The first condition for R.IntC(t) ¹ Æ uses the concept of prenucleolus defined below: given a payoff vector x in the game (1), the excess of a coalition S against x is the difference between its payoff v(S) and those allocated by x: i (5) e(S, x) = v(S) – SiÎS x . Let q(x) denote the (2n-1)–dimensional excess vector whose components are arranged in a decreasing order, i.e., (6) qi(x) = e(Si, x) for some Si ¹ Æ, Si ÎN , and n qi(x) ³ qj(x), 1 £ i < j £ 2 -1. The tie-breaking rule is the same as the earlier "decreasing by coalition size and then lexicographically decreasing by player indices" of (2). In other words, if qi(x) = e(Si, x) = e(Sj, x) = qj(x) for some i < j, then |Si| ³ |Sj|, and the sum of the entries of Si is lexicographically greater than that of Sj if |Si| = |Sj|.
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