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2Proportionality in NTU Games: Proportional Excess, Nucleolus and Prenucleolus
Sergei Pechersky
St. Petersburg Institute for Economics and Mathematics, RAS, and European University at St. Petersburg 1, Tchaikovskogo str., St. Petersburg, 191187, Russia E-mail address: specher@eu.spb.ru
Abstract. An axiomatic approach is developed to define the proportional excess function on the space of positively generated NTU games.
This excess generalizes to NTU games the proportional TU excess v(S)/x(S). Five axioms are proposed, and it is shown that the proportional excess is the unique excess function satisfying the axioms. The properties of proportional excess and corresponding nucleolus, prenucleolus and, in particular, status quo-proportional solution for bargaining games are studied.
Keywords: NTU games, excess function, proportional excess, Minkowski gauge function, nucleolus, prenucleolus, bargaining solution.
Introduction
In this paper we discuss the problem of definition of proportional solutions for NTU games. Of course, proportional allocation is not a new idea. It goes back to Aristotle (see, for example, [Moulin, 2002]). Young [8] writes that “proportionality is deeply rooted in law and custom as a norm of distributive justice”. Thompson [Thomson, 1998] puts proportionality “at the heart of equity theory”. It is also the commonly recognized standard of business practice. The relevant results on proportionality are found in the accounting and social choice literature. For two person positive TU games the proportional solution is defined in an obvious way:
xi=v({l,2}) *=1,2.
nUD + n!2})
Unfortunately, for TU games with more than two players the straightforward definition of proportional solution is not possible, since it is not clear what the proportionality should mean when different coalitions are possible. Hence, the definition of
proportional solution for positive TU games can be realized as some extensions of proportional solution for two person games.
The first such solution, the proportional nucleolus, was defined by Lemaire [Lema-ire, 1991]. He defined it in a usual manner, but he used the relative excess er (S, x, v) = = vlyS'l^s)S'1 instead of standard excess e(S,x,v) = v(S) — x(S). Note that the relative excess just mentioned is ordinally equivalent to the proportional excess ^§y-Another proportional solution defined independently by Feldman [Feldman, 1999] and Ortmann [Ortmann, 2000] can be considered as a consistent (in Hart-Mas-Colell sense) extension of the proportional solution for two-person positive games on n-person positive games.
Yanovskaya (see [Pechersky, 2004]) used the proportional excess to define the proportional solution, proportionality of solution being defined as follows.
A solution ^ on the set of positive TU games GN + with the set N of players is called proportional, if for any two games (N,v), (N,w) G GN + and any two payoff vectors x G X(N, v), y G X(N, w)
= I for a ll 5 c /V
x(S) y(S)
implies
x = ^(N, v) y = ty(N,w). (1)
For example, ^ defined by
^(N,v) = arg max TT x(S)w(s)v(S) (2)
xEX(N.v)
v ’ J SCN
for some non-negative numbers w(s),s < n — 1,n = \N\, where X(N,v) is the set of preimputations for v, is proportional solution. Clearly, (1) is an analogues of shift covariant solution for standard excess: let $ be an arbitrary shift covariant solution on Gn, and (N,v), (N,w) G GN. Then
v(S) — x(S) = w(S) — y(S) for all S C N (3)
implies
x G $(N,v) y G $(N,w). (4)
It is well-known that the generalization of solutions based on cardinal notion
of excess to NTU games creates difficulties. In particular, the problem is that there is no natural analogues of excesses in NTU case. In his survey Maschler [Maschler, 1992] noted that “research concerning the extension of the kernel and the nucleolus to games without side payments is still scarce, ... the main issue is to decide what the analogue of the excess function should be”. Although, there have been several suggestions (see, for example, [Kalai, 1973], [Kalai, 1975], [Billera, 1972], [Bondareva, 1989], [McLean, 1989], [Nakayama, 1983],...) that these proposals have
not yet achieved the status of a general theory similar to the one that exists for the side payment case.
Kalai [Kalai, 1975] defined a family of excess functions for cooperative NTU games. Using these excess functions he defined the e-core, the kernel and the nucleolus of a NTU game in a way that preserves a significant portion of the structure that these concepts exhibit in the TU case. These excess functions satisfy some natural conditions, which seem to be required for such functions.
In the frameworks of Kalai’s approach in [Pechersky, 2000, 2001] the gauge excess which generalizes the relative excess er(S,x,v) = to NTU games was
defined and axiomatically characterized.
Our aim in this paper is to define, axiomatically characterize the proportional excess for NTU games, and consider its property. In particular, we study the corresponding nucleolus and prenucleolus. For bargaining games the nucleolus defines the status quo-proportional solution. The axiomatic characterization of this solution is given.
The paper is organized as follows. Section 1 provides definitions and notations. In Section 2 we describe axioms, state the existence and uniqueness theorem, and study the properties of the proportional excess function and the properties of corresponding nucleolus and prenucleolus. In Section 3 we give an axiomatic characterization of the status quo-proportional solution for bargaining games. The proofs are given in Appendix 1. The geometric characterization of the nucleolus and prenucleolus is given in Appendix 2.
1. Definitions and Notations
Let N = {1, 2,...,n} be a non-empty finite set of players. A coalition is a nonempty subset S of N. For a subset S C N let RS denote \S\-dimensional Euclidean space with axes indexed by elements of S. A payoff vector for S is a vector x G RS. For z G RN and S C N, zS will denote the projection of z on the subspace
R[s] = {x G RN : x, = 0 for i G S},
and zS - the restriction of z on RS. To simplify notations, if \S\ = 1 or \S\ = 2, i.e. S = {i} or S = {i,j} for some i,j G N, we write RM and R[’] instead of R[{i}] and R[{’}] , respectively.
Let x,y G RN. We will write x > y, if xM > yM for all i G N; x > y, if xM > yM for all i G N. Denote
RN = {x G Rn : x > 0},
R++ = {x G RN : x > 0},
where 0 = (0,0,..., 0). We denote the coordinate-wise product by x * y, i. e. x * y = = (xiyi,..., xnyn), and coordinate-wise division by x : y = (x\jy\,..., xn/yn) for
y> 0.
Let A C RN. If x G RN, then x+A = {x + a : a G A} and XA = {Aa : a G A}. A is comprehensive, if x G A and x > y imply y G A. A is bounded above, if A fi (x+RN)
is bounded for every x G RN. The boundary of A is denoted by dA. The interior of a set A will be denoted by int A, and the relative interior by rel int A. The closed convex hull of a set A we denote with co A.
A nontransferable utility game (or shortly NTU game) is a pair (N,V), where N is the set of players, and V is the set-valued map that assigns to each coalition S C N a set V(S), that satisfies:
(1) V(S) C R[s] = {x C RN : xM = 0 for i j S};
(2) V(S) is closed, non-empty, comprehensive and bounded above.
(Usually V(0) = 0). The following particular cases should be mentioned.
TU game. A TU game v can be considered as a NTU game of the following form:
V(S) = {x G R[s] : x(S) < v(S)},
where x(S) = ieS xM. The boundary of V(S) is a hyperplane in R[s] with normal
eS, where e = (1,1,..., 1).
Hyperplane game. A hyperplane game V is defined as follows: for every S C N
V(S) = jx G R[s] : ^p(S)xH < r(S)
I ies
where pi(S') > 0 for every i G S, S C N (p(S) =0, i G S). The boundary of V(S) is a hyperplane in R[s] with a normal p(S). Clearly TU game is a hyperplane game with p(S) = 1 for all i G S, S C N.
Bargaining game. A n-person bargaining game is a pair (q,Q), where q G RN is the status quo point, Q C RN and N = {1, 2,...,n}. When interpreting this pair one can think as follows: if the players act separately the only possible outcome for the players is q giving utility qM to player i = 1, 2,...,n. If all players cooperate they can potentially agree on an arbitrary outcome x G Q. The corresponding NTU game can be defined as follows:
V(N) = {x G Rn : there is y G Q such that x < y},
V(S) = {x G R[s] : xM < qM for every i G S} for S = N.
2. Proportional Excess for NTU games
2.1. The Problem and the Space GN+
As we have mentioned above, at the last time there is a growing interest to proportional excess, defined for every positive TU game u (i. e. u(S) > 0 for every coalition S) by formula
i. < i “<s>
M“’I) = i(S)'
Our goal is to generalize this excess to NTU games. We restrict our attention to the space GN + of all normally generated NTU games. Roughly speaking (the formal definition will be given further) a game V belongs to GN +, if every set V(S) is compactly generated, contains 0 = (0,..., 0) as its relatively interior point and coincides with the comprehensive hull of its positive part V+(S) = V(S) f R+S].
To define the corresponding NTU excess we impose five axioms (continuity axiom, scale invariance axiom, MIN and MAX axioms and TU game axiom) which describe desirable properties of an excess function. Continuity axiom asserts that the excess (of a coalition) should be continuous jointly in x and V. Scale invariance asserts that excess does not depend on linear transformations of the game and payoff vector. MIN and MAX axioms state that the excess in the “intersection game” (V = Vi f V2) and in the “union game” (V = V1 U V2) should be equal to the minimum and to the maximum of two component games excesses, respectively. TU game axiom asserts that in the TU case the excess should coincide with the proportional excess.
These five axioms uniquely define the proportional excess hS(V, x), defined by formula
hs(V, x) = 1/y(V(S),xS),
where y(W,.) is the gauge (or the Minkowski gauge function) of a set W (see, for example, [Rockafellar, 1997]):
Y(W, x) = inf {A > 0 : x G AW}.
We study the properties of the proportional excess and corresponding nucleolus and prenucleolus.
Let us define the space GN +. A game V G GN + iff for every S:
(a) V(S) is positively generated (i. e. V(S) = (V(S) f R+) — R+S], and V+(S) =
[ S]
V(S) f R+ is compact), and every ray Lx = {Ax : A > 0}, x = 0 does not intersect the boundary of V(S) more than once;
(b) 0 is an interior point of the set VA(S) = V(S) +R[n\s].
For S C N a set V(S) C R[s] will be called a game subset, if it satisfies (a) and (b). The space consisting of all game subsets satisfying (a) and (b) will be denoted by GN +. Clearly every V G GN + is a game in Kalai’s sense (cf. [Kalai, 1975]).
It is often convenient to consider some modification of a game (N,V) defined by the set-valued map V with V(S) = Vs(S), where Vs(S) is the restriction of V(S) on Rs. Clearly V(S) = V(S) x 0^.
Obviously, every set V+ (S) is normal (or 0-comprehensive), i.e. if x G V+ (S), y G R[s] and 0 < y < x, then y G V+ (S). In the sense a game V G GN + is normally generated. It is clear also that every U G GN + is star-shaped (cf. [Rubinov, 1986]), i.e. U is closed, it contains 0 as a relatively interior point (in R[s]), and every ray Lx does not intersect the boundary dU of U more than once. (This definition is stronger than the usual one: a star-shaped subset of a real vector space contains a
distinguished member, the center, which can be connected with every other element by a line segment which is completely contained in the set.) Note also that for every
V(S) G GN + and every x G R+, x = 0 there is a unique A > 0 such that Ax G dV(S). It is clear that if V1 ,V2 G GN +, then the games V1 f V2 and V1 U V2, defined by
(Vi f V0(S) = V1(S) f V2(S), (V1U V‘2)(S) = V1(S) U v>(s),
also belong to GN +.
Next, if V(S) G GN +, A G R++ and A * V(S) = {A * y : y G V(S)}, then A * V(S) gGS +.
Remark 1. It is not difficult to prove that if a game V is such that for every S the set V(S) is normally generated, possesses (b) and satisfies traditional non-levelness condition:
x,y G dV(S) f R+S], x > y ^ x = y,
then V G Gn +.
Indeed, let V(S) satisfies these properties. It is sufficient to check that non-
levelness condition implies the second part of (a). Suppose the contrary, i. e. for some
[ s]
2 G R+ , 2 = 0 the ray Lz = {Az : A > 0} intersects the boundary of V(S) more than once. Hence, there are A1 > A2 > 0, such that u = A1z G dV(S), v = A2z G dV(S) and uS > vS, uS = vS, which contradicts the non-levelness condition.
Two particular cases should be mentioned: NTU games corresponding the TU games and hyperplane games. Of course, they are not exactly TU games and hyperplane games (in the standard sense), but we leave the usual name.
TU game. Let v be a positive TU game, i.e. v(S) > 0 for every S. Then the corresponding NTU game V G GN + can be defined by
V(S) = {x G R+S] : x(S) < v(S)}— R+1.
Hyperplane game. Let V be a hyperplane game, i.e. for every S C N,
V(S) = i x G R[s] : p(lS)xi < rS
I ieS
(S)
where pi ’ > 0 for every i G S, and rS > 0. Then corresponding NTU game V1 G GN+ can be defined by:
Let us recall some definitions from Kalai’s paper ([Kalai, 1975]) adopting them to the case under consideration and taking into account our goal to generalize the proportional excess. In particular, in property (C) we require that excess should be equal to 1 on the boundary, and not 0.
Define excess function for a coalition S = 0 as such function ES : Gn + x RN ^ R, that
(A) If x,y G Rn and xi = yi for every i G S, then for every V, ES(V, x) = ES(V, y).
(B) If x,y G Rn are such that xi < yi for every i G S, then for every V, ES(V, x) > >Es(V, y).
(C) For every game V, x G dV(S) ^ ES (V,x) = 1.
(D) ES(V, x) is continuous jointly in x and V.
The metric on GN + is the Hausdorff metric: for V,W G GN+
(Recall that the Hausdorff metric is defined as follows. Let A and B be the subsets of RS, then HS(A, B) = max(l(A, B), l(B, A)), where
and d is Euclidean metric.)
ES will be said to be independent of other coalitions if for every two games V and W such that V(S) = W(S), and for every x G RN, ES (V,x) = ES (W, x).
We restrict our attention to nonnegative vectors x only since any reasonable solution of a game V G GN +, should be, clearly, non-negative. We consider only those excesses which are independent of other coalitions. Taking into account property (A) and, of course, (B), we will consider in what follows excess functions as those functions
The following notations will be used throughout the paper ([Kalai, 1975]): IR(V) = {x G V(N) : yi G N xi > yi for every y G V(i)} - the set of individually rational points of a game V;
GR(V) = {x G V(N): there is no y G V(N), such that y > x} - the set of (weakly)
Pareto optimal points of a game V;
C(V) = {x G V(N): there is no S, y G V(S), such that yi > xi yi G S} - the core
of a game V.
Finally, we recall the definitions of the nucleolus and the prenucleolus of a game (cf., for example, [Kalai, 1975]).
Let {ES}S be a fixed family of excess functions, and let X be a closed subset of
Rn . For an arbitrary x G X and game V define a vector 0(x) to be:
where various excesses of all coalitions are arranged in decreasing (nonincreasing) order. The components of 0(x) are well defined and vary continuously for “good” excess functions. We say that 0(x) is lexicographically smaller than d(y), 0(x) ~<lex Q(y), if there is such a positive integer q that 0i(x) = di(y) for all i < q and Qq(x) <
p(V,W) = max HS (V (S ),W(S)).
l(A, B) = sup{d(x, B) : x G A}
0(x) = Q(V, x) = (Esj (V, x),..., Es2n (V, x)),
< Qq (y).
The nucleolus of V (with respect to X and given family of excess functions {ES}S) - we denote it by N(X, V) - is the set of vectors in X which Q’s are lexicographically least, i. e.
N(X, V) = {x G X : Q(x) ^lex Q(y) for all y G X}.
If X = IR(V) f GR(V), then N(X,V) := N(V) is called the nucleolus of V. If X = GR(V), then N(X, V) := PN(V) is called the prenucleolus of V.
2.2. Proportional Excess: Axioms, Existence and Solutions
Let HS be an excess function, i. e. HS : G‘N + x R+] ^ R. Let us impose the
following axioms (we write V instead of V(S)).
Continuity. HS(V, x) is continuous jointly in V and x = 0.
Scale invariance. If V G Gjv+, A G R++ and A * V = {A * y : y G V}, then
HS(A * V, A * x) = HS(V, x).
MIN. Let V1,V2 G GN +, then
Hs(V1 f V2,x) = min{Hs(V1 ,x), Hs(V*,x)}.
MAX. Let V1,V2 G GN +, then
Hs(V1 U V2,x) = max{Hs(V1,x), Hs(V*,x)}.
Proportionality for TU games. If V G GN + corresponds to TU game v, i.e.
V(S) = {x G R+1 : x(S) < v(S)} — R+S],
then HS(V, x) = v(S)/x(S).
Continuity axiom is self-explanatory, and there is no need to comment on it. Scale independence seems also to be clear, but the absence of translation invariance should be stressed. Note that in the TU case the translation invariance can be justified by the lack of “income effect”. However, in the NTU case, where the “income effect” can be of great importance, translation invariance seems to be not justified.
Note also that the proportional, or its ordinal equivalent relative excesses make good sense in economic environment, when the players are e.g. companies. If an excess at x is measure of dissatisfaction of a coalition of companies from x (satisfaction, if negative) it is reasonable to assume that a rich coalition will “tolerate” a large loss and a poor coalition will not “tolerate” a much smaller loss. Thus, a nucleolus based on the above and similar excess functions, is a reasonable solution concept. If a poor person looses, say, $ 1000 he will protest strongly, whereas a large conglomerate may not bother to even rise the issue. In such cases the fact that the excess is not invariant under translation is even a merit.2 MIN and MAX axioms seem to be natural conditions. The interpretation of games V1 f V2 and V1 U V2 is
2 I am grateful to Michael Maschler for this observation.
evident: corresponding set V1(S) f V2(S) represents the payoffs vectors feasible for coalition S in both games V1 and V2, and V1(S) U V2(S) is the set of payoffs vectors feasible for coalition S for at least in one of two given games V1 and V2. (Note that these two properties are trivially fulfilled in TU case for all excesses: the standard v(S) — x(S), the relative (v(S) — x(S))/v(S) and the proportional v(S)/x(S)).
Finally, proportionality axiom is the reformulation of our aim to generalize the TU proportional excess to NTU games.
Let V G GN + be an arbitrary game. Define a function hS : GN + x RN ^ R as follows:
hs(V, x) = 1/y(V(S),xS), (5)
where Y(W,y) = inf {A > 0 : y G AW} is the gauge (or Minkowski gauge) function ([Rockafellar, 1997]).
Theorem 1. There is a unique function Hs : GN + x RN ^ R, satisfying continuity, scale invariance, MIN, MAX, and proportionality for TU games axioms. Moreover, Hs = hs, where hs is defined by (5).
For the proof of the theorem we use four lemmas which we formulate for S = N (the proof for an arbitrary S is the same). We also omit index N to simplify notations. The proofs of lemmas are given in Appendix.
Let z G RN+ and Pz = {y G RN : y < z}, i.e. Pz = z — RN. Clearly Pz is star-shaped. Also star-shaped is every finite union of such sets, i. e. for every natural number M and arbitrary vectors zm G m = 1, 2,...,M the set P(z1,..., zM),
defined by
M
P (z 1,...,zM ) = U Pzm ,
m=1
is star-shaped. Obviously, P(z1,..., zM) G GN+, since zm > 0 for every m.
Lemma 1. The proportional excess defined by (5) satisfies all axioms mentioned.
Let us denote the metric on Gn+ by p.
Lemma 2. Let V G Gn+ be an arbitrary game subset. Then for every e > 0 there are such natural number M and points zm G RN+ m= 1, 2,...,M that
1) P(z1,...,zM) D V;
2) p(P(z1,..., zM), V) < en1!2.
Lemma 3. Let V G Gn+ be a hyperplane game subset, i. e.
V = z G RN : £)pizi < r\ — RN
i£N
for some pi > 0, i = 1,...,n, and r > 0. Then H(V, x) = r/ ^2^npixi = h(V,x).
Lemma 4. Let z1,..., zM G RN+, then H (P (z1,...,zM ),x) =
max min
m=1,...,M i=1,...,n
Moreover, H(P(z1, ..., zM), x) = 1/y(P(z1, ..., zM), x).
The proof of the Theorem follows from these lemmas and continuity of H and h.
We call this function the proportional excess though it is not an excess function in Kalai sense (recall that we replaced the equality ES(V, x) = 0 on the boundary by
2.3. Proportional Nucleolus and Prenucleolus
As we have just mentioned the proportional excess is not an excess in Kalai sense, nevertheless the assertions of Kalai’s Theorem [Kalai, 1975] hold. (It should be noted that the e-core in our case (proportional excess equals 1 on the boundary) can be defined in different ways. For example, hS(V,x) < 1/(1 — e) instead of
hS(V, x) < 1 + e can be required).
In the following proposition we state only the part of theorem mentioned concerning proportional nucleolus and prenucleolus.
Proposition 1. Let {hs }s be a family of proportional excesses, and V G Gn +. Then if IR(V) f GR(V) = 0, then N (V) = 0 and consists of a finite number of points. For every V PN(V) is non-empty and consists of a finite number of points.
Since hS — 1 is an excess in Kalai sense his proof can be applied directly. Though the non-emptiness of the prenucleolus is not discussed in Kalai’s theorem it follows in a similar manner to the non-emptiness of the nucleolus.
Corollary 1. If V G Gn + is a hyperplane game then the nucleolus (if not empty) consist of precisely one point. The prenucleolus consist of precisely one point.
Proposition 2. Let V,V G GN + be such games that V(N) = V (N) and V (S) = = aV(S), VS = N for some a > 0. Then x G PN(V) ^ x G PN(V ), where PN(V) is the prenucleolus of V.
Proof.
Let us consider an arbitrary coalition S C N and the corresponding set V(S). It is clear that for every x G RS
Es(V, x) = 1).
Proof.
y(V (S), x) = Y(aV (S), x) = inf {A > 0 : x G AaV (S)} = inf{i/a > 0 : x G fiV(S)} = (1/a)Y(V(S),x). Then hS(V (S),x) = 1/y(V (S),x) = a/Y(V(S),x). Therefore,
hs(V , x) = ahs(V, x), 0(V , x) = aO(V, x).
Note that this property is analogous to one of the characteristic property of the prenucleolus in sidepayments case for classical excess function (see, for example, [Pechersky, 1995]): let v and v' be two such side payment games that v(N) = v'(N) and for an arbitrary real number a, v'(S) = v(S) + a for every S = N. Then PN(v') = PN(v).
Remark 2. In the proof of the Proposition we have used the property
Y(aV (S),x) = (1/a)Y(V (S),x).
It is important, and it will be used to modify Maschler-Peleg-Shapley’s geometric characterization of the nucleolus in side payment case ([Maschler, 1979]) for NTU games. Of course, it is more cumbersome. We consider it in Appendix 2.
It is clear that Proposition 2 can be reformulated in the following manner.
Proposition 3. Let V, V G GN + be such games that V (N) = aV(N) and V (S) = = V(S), VS = N for some a > 0. Then x G PN(V) ^ ax G PN(V ).
Example [Bargaining game]. Let V G GN + be a bargaining game, i.e. for some qj R++
V(S) = {x G RS : xi < qi for every i G S},
for every S = N and q G intV(N). Let V(N) p|(q + RN) be nonlevelled. Then N(V) = PN(V) = Aq, where A is such that Aq G dV(N).
The following Proposition is a simple corollary of the definition of the proportional excess and equality y(V, x) = y(A * V,A * x) for every A G R++.
Proposition 4. The nucleolus and the prenucleolus are scale covariant, i. e. if A G Rn+, then for every V G GN+, N (AV) = A * N (V), and PN(AV) = A * PN(V), where AV is defined by
AV(S) = A * V(S) for every S.
The next Proposition is an analogue of the corresponding property of prenucleolus in TU case (cf., for example, [Pechersky, 1995]).
Proposition 5. Let U, V G GN + be such games that U(N) = V(N), and x G
PN(U) f PN(V). Let W = U U V, then x G PN(W).
Proof.
Suppose the contrary, i.e. x j PN(W). Then there is an y G N(W) such that 9(W,y) ~<iex d(W, x). However, since hS(W,z) = max{hS(U,z), hS(V, z)} for any
z G RN and every S, we have hS(U, z) < hS(W, z) and hS(V, z) < hS(W, z).
Therefore, 0(U,y) ~<lex 0(U,x) and 0(V,y) ~<lex 0(V,x). Hence, x j PN(U) f PN(V), and this contradiction proves the proposition.
3. Status quo Proportional Solution for Bargaining Games
In example we noted that the nucleolus (and prenucleolus) of a nonlevelled bargaining game is a Pareto optimal point proportional to the status quo point. Let us consider this solution in more details.
We consider bargaining games (q, Q) with positive status quo points q. Moreover, we suppose that the sets Q possess not only properties (a)-(b) characterizing the sets V+ (S) and GN+, but also that they are nonlevelled and normal, and for every bargaining game (q, Q) there is such x G Q that x > q.
For a bargaining game (q, Q) define solution R as follows: let
l(q, Q) = max{t G R+ : tq G Q},
and R(q,Q) = i(q,Q)q. This solution will be called sq-proportional (status quo proportional).
It can be shown easily that the solution defined in such a manner is proportional in the sense of (1). (Recall that we consider bargaining games as NTU games of special structure.) Of course, the proportional excess for NTU games defined in previous section should be used instead of v(S)/x(S).
Proposition 6. SQ-proportional solution is proportional in the sense of (1).
Proof.
Let (q,Q) and (q1,Q1) be two bargaining games, V and V1 be corresponding NTU games (note that VS = Pqs). Let x G dQ, y G dQ1, x = l(q,Q)q and hS(V,x) = hS(V 1,y) for every S C N. Clearly hS(V,x) = 1/i(q,Q). Then hs (V 1,y) = 1/i(q,Q) for every S C N .In particular, y1 = !(q,Q)q\,...,yn = = l(q,Q)qn. Since y G dQ1, then l(q,Q) = ^(q1 ,Q1), and y is sq-proportional solution of bargaining game (q1,Q1).
Let F be a bargaining solution. Consider following axioms.
Pareto optimality. F(q,Q) G nQ, where nQ denotes the set of Pareto optimal points of Q.
Scale covariance. Let A G RN+. Then for every bargaining game (q,Q)
F(A * q, A * Q) = A * F(q, Q),
where * denotes the coordinate-wise product.
Anonymity. If t is an arbitrary permutation of N, then
F(t*q, t*Q) = t*F(q,Q),
where t* is the transformation of RN, induced by t, i.e. t*(x) = (xT(1), ...,xT(n)).
Strong monotonicity. If Q D Q, then F(q,Q ) > F(q, Q).
The following proposition follows immediately from the definition.
Proposition 7. SQ-proportional solution satisfies Pareto optimality, scale covariance, anonymity and strong monotonicity axioms.
Theorem 2. SQ-proportional solution is the unique solution satisfying Pareto optimality, scale covariance, anonymity and strong monotonicity axioms.
Proof.
Let F be a bargaining solution satisfying axioms mentioned. Consider an arbitrary bargaining game (e, Q), where e = (1,1,..., 1). Let x = /ie, where i = max{t G R+ : te G Q}, and let F(e, Q) = x.
Consider the bargaining game (e, Qn), where
Qn = U t*Q,
t Gn
and n denotes the set of all permutations of N. Clearly the set Qn is invariant relative to any permutation of N, hence, F(e, Qn) = Ae for some A > 0. Therefore, Ae = x. (On the one hand, x G Qn, and on the other hand, it cannot be F(e, Qn) > x, since F(e, Qn) G t*Q for every permutation t, but fie G t*Q for every t, and, hence, 1 = max{t G R+ : te G Q}.)
Since Qn D Q, then x = F(e, Qn) > F(e,Q). However (by non-levelness condition) this is possible if F(e,Q) = x only. Since Q was an arbitrary set the scale covariance proves the statement.
SQ-proportional solution can be easily characterized in another manner. Namely, let us formulate Proposition 2 as an axiom.
Invariance under common change of status quo point. If the bargaining games (q, Q) and (q', Q) are such that q' = aq for some a > 0, then F(q', Q) = F(q, Q).
Proposition 8. SQ -proportional solution is the unique bargaining solution satisfying individual rationality, Pareto optimality and invariance under common change of status quo point axioms.
Proof.
Consider an arbitrary bargaining game (q, Q). If q G nQ, then by non-levelness condition F(q, Q) = q. Let now q j nQ, then (since Q is normal and 3 y G Q : y > q) there is such a > 0, that aq G nQ. Then F(aq,Q) = aq, and by invariance under common change of status quo point axiom F(q, Q) = aq.
The last proposition can be easily reformulated in the spirit of Proposition 3. Let us consider the following axiom.
Positive homogeneity with respect to changes of feasible set. If (q,Q) and (q,Q') be such that Q' = aQ for some a > 0, then F(q, Q') = aF(q, Q).
Proposition 9. SQ-proportional solution is the unique bargaining solution satisfying individual rationality, Pareto optimality and positive homogeneity with respect to changes of feasible set axioms.
The proof is obvious.
Acknowledgments
The author expresses his gratitude to Elena Yanovskaya for useful discussions on the subject, and to Michael Maschler for useful comments on the early versions of the paper.
Appendix
First Appendix
Here we give the proofs of the Lemmas.
Proof of Lemma 1.
Continuity, MIN and MAX follow immediately from the continuity of gauge function Y and its well-known properties:
Y(V1 f V2,x) = max{y(V,x), y(V2,x)},
and
Y(V1 U V2,x) = min{Y(V1,x),y(V2,x)}.
Now check the scale invariance. Let a = (a1, a2,..., an) G RN+. Since
Y (a * V,a * x) = inf {A > 0 : a * x G a * V} =
= inf {A > 0 : x G V} = y(V, x),
it is clear that h is scale invariant.
Let now V G GN + corresponds to a TU game v, i. e.
V(S) = {xj R+ : x(S) < v(S)} — R+l
Then
Y(V, x) = inf {A > 0 : xj AV} = inf {A > 0 : x(S) < Av(S)} =
= inf {A > 0 : x(S)/v(S) < A} = x(S)/v(S).
Hence, h(V,x) = 1/y(V,x) = v(S)/x(S).
Proof of Lemma 2.
Since V is normally generated it is sufficient to consider only the positive parts of V and P(z1,..., zM). Consider the covering of V+ by the cubes with the edge equal to e and vertices in the nodes of the e-lattice in RN. Since V+ is compact, it is contained in a finite union of such cubes, each having non-empty intersection with
V+. Denote the number of such cubes with M, and let z1,..., zM be the maximum vertices of these cubes (i.e. vertices with coordinate-wise maximum).
Consider P(z1,...,zM) = UM=1 Pzm. By construction, P(z1,...,zM) D V. Moreover, the following inclusion holds also by construction:
V + (en1/2)B D P(z1,...,zM), where B is the unit ball in RN. Hence, p(P(z1,..., zM), V) < en1/2.
Proof of Lemma 3.
Let p = (p1,... ,pn) grN+. Then p transforms the game subset V into the TU game subset p * V = {w G RN : $^ieN wi < r} — RN, where wi = pizi. By TU game axiom
r
H(p *V,p * x) = =—, wi
and by scale invariance
H(V, x) = H(p * V,p * x) = r/ ''y^pixi.
It is easy to check that y(V, x) = ^pixi/r. Indeed,
inf {A > 0 : x G AV} = inf {A > 0 : 53 pi xi < Ar} =
= inf {A > 0 : 53 pixi/r < A} = 53 pixi/r.
Therefore, H (V,x) = 1/y (V, x) = h(V,x).
Proof of Lemma 4.
Prove firstly that if 2 G R++, then H(Pz,x) = min* jfr j = h(Pz,x).
Clearly, the faces of Pz are the sets
Qi = {y G Rn : yei = zei, y < z}, i =1, 2,...,n,
where
ei = / 1 j = i
ej' I 0, j = i.
Consider for some natural k the hyperplane game subset
Vi = {yeR^ : yie1 + < z(e* + ie^W)} - R*
kk
Then by lemma 3,
H (Vi,x) =
+ ^z(N\ {»}) Xi + tx(N \ {*})
Now consider a game Vk with Vk = nn=1 Vi. Clearly VkG GN+, and we have by MIN axiom
Zi + lz(N\ {*})
H(Vk,x) = min<j ' *
xH + tx(N \ {*})
Besides, H(Vk,x) = 1/Y(Vk,x). Letting k ^ +ro, we get by continuity
H(Pz,x) = min ( — 1 = I/7(Pz,x) = h(Pz,x).
i xi
By MAX axiom
( z m
HfPfz1,..., zM), x) = max H(Pzm,,x)= max min I -J—
i y Xi
Y(V f V ,x) = max(Y(V, x), y(V , x)); y(V U V ,x) = min(Y(V, x), y(V , x)) for every V, V' G Gn+ , we have
H(P(z1,..., zM), x) = 1/y(P(z1,..., zM), x).
Second Appendix
In the Appendix we consider the geometric characterization of the proportional nucleolus in Maschler-Peleg-Shapley spirit (see [Maschler, 1979]).
To define e-core it is convenient to require hS(V,x) < 1/(1 — e). In this case it coincides with the e-core defined for NTU games by the gauge excess gS(V,x) = 1 — —y(V(S),xS) (cf., [Pechersky, 2000]), which is ordinally equivalent to the proportional excess and is an excess in Kalai sense. We start with the definition of lexicographic centers of a NTU game. Of course, it is more cumbersome, but the intuition behind it is transparent and very close to that by Maschler-Peleg-Shapley in TU case.
Let V be a game in GN +. It is convenient to introduce the function g(S,x) = gS(V,x) = 1 — 1/hS(V, x), since hS(V,x) < 1/(1 — e) is equivalent to g(S,x) < e.
Define X0 = IR(V) f GR(V) (we suppose it is not empty) and £0 = {S C N : S = N, 0}. Let
e1 = min max q(S, x), X1 = {x G X0 : max g(S, x) = e1}. (6)
xGX0SeE0 S£S°
Both e1 and X1 are well-defined, and X1 is a compact set. Let now x G X1. Define
S1(x) = {S G £° : g(S, x) = e1}.
Since S° is finite the set X1 is partitioned into finite family of compact sets X^Xj,..., X^ such that S1(x') = S1(x) for any x, x' G Xj, l = 1,...,r1. Denote such S1(x) by Ek, i.e.
E1l = {S G E° : g(S, x) = e1 for all x G Xj}.
Let 0-1 = mini=i...,n |Ei;| and E1 = {Ei( : |Ei;| = ai} = {En,..., Eim'}, M' > 0. Consider E^ = E° \ E1m = 0, m = 1,..., M. If all E^ = 0 we stop the process. If M > 1, we continue the process for each m.
Let e'L = minXi maxS^ 1 g(S, x), and take only such e'L, that
e2m =min{e2,...,eMM}. (7)
Denote such e'm by e2. Further we can define the set
XL = < x G XL : max q(S, x) = e2 m 1 m sgei g , ;
and corresponding sets X^, X^2,..., X;^ r2 and Em2i,..., Em2r2 for m, satisfying (7).
Let 02 = minmmax;=ii...ir2 |Em2(| and £2 = {Em2( : |Em2(| = 02}, and so on. Clearly, this process will stop after finite number of steps. It defines a finite set of points in IR(V) f GR(V), called lexicographic centers, and this set coincides with the nucleolus N(V) of a game V. The geometry of the process is akin to that in sidepayment case and can be described in the following way. We start with an e large enough so that the e-core is non-empty:
Ce(V) = {xj GR(V) f IR(V) : VS = N, gS(V, x) < e} = 0.
If gS(V, x) < e, then 1 — y(V(S), x) < e and y(V(S), x) > 1 — e. Consider aV(S) for a > 0. Since y(aV(S),x) = (1/a)Y(V(S),x), we have
Y(aV(S), x) > 1 ^ y(V(S), x) > a,
and we can take a = 1 — e. Therefore, if e is decreasing then a is increasing.
The process can be characterized by expanding (“blowing up”) the sets V(S): we start with sufficiently small a > 0 such that
X (a) = {xj GR(V) f IR(V) : y (V (S), x) > a, VS = N} =
= {xj GR(V) f IR(V) : xS j rel int aV (S), VS = N} = 0.
Then we begin to expand (“blow up”) all sets aV(S) (by increasing a) “pushing up” some x from GR(V) f IR(V). This expansion (blowing up) is performed at equal speed and is stopped either when the set X(a) becomes empty or would become disjoint from GR(V) f IR(V) (or C(V) if it is non-empty). This bring us to the sets X;1 . Any further increase in a will render the sets X;1 empty. We, therefore, continue to expand only those sets V(S), where S G E^ (there are M such E^). These we expand so long as the corresponding modification of the set X(a) is neither empty nor disjoint from GR(V) f IR(V) (or C(V) if it is non-empty). This bring us to the sets XLr, and so on. The process continues in the same manner until all the sets aV(S) have been expanded to their respective limits (i.e. corresponding to appropriate 1 — a = ek).
Remark 3. The same procedure can be used for the prenucleolus. We have to replace GR(V) f IR(V) by GR(V), and to consider only whether set X(a) is empty.
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