CC BY
5Theo S.H. Driessen1, Vito Fragnelli2, Ilya V. Katsev3 and Anna B. Khmelnitskaya4
1 University of Twente, Department of Applied Mathematics,
P.O. Box 217, 7500 AE Enschede, The Netherlands E-mail: t.s.h.driessen@ewi.utwente.nl 2 University of Eastern Piedmont, Department of Science and Advanced Technologies, Viale T. Michel 11, 15121, Alessandria, Italy E-mail: vito.fragnelli@mfn.unipmn.it
3 SPb Institute for Economics and Mathematics Russian Academy of Sciences,
1 Tchaikovsky St., 191187 St.Petersburg, Russia E-mail: katsev@yandex.ru
4 SPb Institute for Economics and Mathematics Russian Academy of Sciences,
1 Tchaikovsky St., 191187 St.Petersburg, Russia E-mail: a.khmelnitskaya@math.utwente.nl
Abstract The situation, in which an enormous risk is insured by a number of insurance companies, is modeled through a cooperative TU game, the so-called co-insurance game, first introduced in Fragnelli and Marina (2004).
In this paper we show that a co-insurance game possesses several interesting properties that allow to study the nonemptiness and the structure of the core and to construct an efficient algorithm for computing the nucleolus.
Keywords: cooperative game, insurance, core, nucleolus
Mathematics Subject Classification (2000): 91A12, 91A40, 91B30
JEL Classification Number: C71
1. Introduction
In many practical situations the risks are too large to be insured by only one company, for example environmental pollution risk. As a result, several insurance companies share the liability and premium. In such a risk sharing situation two important practical questions arise: which premium the insurance companies have to charge and how should the companies split the risk and the premium keeping themselves as much competitive as possible and at the same time obtaining a fair division? In Fragnelli and Marina (2004) the problem is approached from a game theoretic point of view through the construction of a cooperative game, the so-called co-insurance game. In this paper we study the nonemptiness and the structure of the core and the nucleolus of the co-insurance game subject to the premium value. If the premium is large enough, the core is empty. If the premium meets a critical upper bound, the nonemptiness of the core, being a single allocation composed of player’s marginal
* The research of Theo Driessen, Ilya Katsev, and Anna Khmelnitskaya was supported by NWO (The Netherlands Organization for Scientific Research) grant NL-RF 047.017.017. The research of Ilya Katsev was also supported by RFBR (Russian Foundation for Basic Research) grant 09-06-00155. The research was partially done during Anna Khmelnitskaya 2008 research stay at the Tilburg Center for Logic and Philosophy of Science (TiLPS, Tilburg University) whose hospitality and support are appreciated as well.
contributions, turns out to be equivalent to the so-called 1-convexity property of the co-insurance game. Moreover, if nonemptiness applies, the co-insurance game inherits the 1-convexity property while lowering the premium till a critical lower bound induced by the individual evaluations of the enormous risk. In addition, 1-convexity of the co-insurance game yields the linearity of the nucleolus which, in particular, appears to be a linear function of the variable premium. If 1-convexity does not apply, then for the premium below another critical number we present an efficient algorithm for computing the nucleolus.
The interest to the class of co-insurance games is not only because they reflect the well defined actual economic situations but also it is determined by the fact that any arbitrary nonnegative monotonic cooperative game may be represented in the form of a co-insurance game. This allows to glance into the nature of a nonnegative monotonic game from another angle and by that to discover its new properties and peculiarities. Further, a co-insurance game appears to be a very natural extension of the well-known bankruptcy game introduced by Aumann and Maschler (1985). Besides, the study of 1-convex/1-concave TU games possessing a nonempty core and for which the nucleolus is linear was initiated by Driessen and Tijs (1983) and Driessen (1985), but until recently appealing abstract and practical examples of these classes of games were missing. The first practical example of a 1-concave game, the so-called library cost game, and the 1-concave complementary unanimity basis for the entire space of TU games were introduced in Drioessen, Khmelnitskaya, and Sales (2005). A co-insurance game under some conditions provides a new practical example of a 1-convex game. Moreover, in this paper we also show that a bankruptcy game is not only convex but 1-convex as well when the estate is sufficiently large comparatively to the given claims.
The structure of the paper is as follows. Basic definitions and notation are given in Sect. 2.. Sect. 3. studies the nonemptiness and the structure of the core and the nucleolus of a co-insurance game with respect to the premium value. In Sect. 4. an algorithm for computing the nucleolus is introduced.
2. Preliminaries
Recall some definitions and notation. A cooperative game with transferable utility (TU game) is a pair (N, v), where N = {1,...,n} is a finite set of n > 2 players and v: 2n ^ IR is a characteristic function, defined on the power set of N, satisfying v(0) = 0. A subset S C N (or S € 2N) of s players is called a coalition, and the associated real number v(S) represents the worth of the coalition S; in particular,
N is call a grand coalition. The set of all games with a fixed player set N is denoted
2^_1
by GN and it can be naturally identified with the Euclidean space IR . For simplicity of notation and if no ambiguity appears, we write v instead of (N, v) when referring to a game. A value is an operator £: GN ^ IRn that assigns to any game v € GN a vector £(v) € IRn; the real number £i(v) represents the payoff to the player i in the game v. A payoff vector x € IRn is said to be efficient in the game v, if x(N) = v(N). Given a game v, the subgame v\T with the player set T C N, T = 0, is a game defined by v\T (S) = v(S) for all S C T .A game v is nonnegative if v(S) > 0 for all S C N .A game v is monotonic if v(S) < v(T) for all S C T C N. For the cardinality of a given set A we use a standard notation |A| along with lower case letters like n = \N\, m = \M\, nk = \Nk\, and so on. We also use standard notation x(S) =J2its Xi and xS = {'Xi}i^s, for all x € IRn, S C N.
The imputation set of a game v €GN is defined as a set of efficient and individually rational payoff vectors
while the preimputation set of a game v €GN is defined as a set of efficient payoff vectors
The core (Gillies, 1953) of a game v € GN is defined as a set of efficient payoff vectors that are not dominated by any coalition, i.e.,
C(v) = {x € IRn \ x(N) = v(N), x(S) > v(S), for all S C N}.
A game v € GN is balanced if C(v) = 0.
For any game v €GN, the excess of a coalition S C N with respect to a vector x € IRn is given by
The nucleolus (Schmeidler, 1969) is a value defined as a minimizer of the lexicographic ordering of components of the excess vector of a given game v € GN arranged in weakly decreasing order of their magnitude over the imputation set
The prenucleolus is a value defined as a minimizer of the lexicographic ordering of components of the excess vector of a given game v € GN arranged in weakly decreasing order of their magnitude over the preimputation set I * (v).
For a game v € GN with a nonempty core the nucleolus v(v) belongs to C(v).
For a game v €GN we consider the vector mv € IRn of marginal contributions to the grand coalition, the so-called marginal worth vector, defined as
i.e., the gap vector measures for every S C N the total coalitional surplus of marginal contributions to the grand coalition over its worth. In fact, gv (S) = -ev (S,mv), with ev (S,mv) being th excess vector of S in game v at payoff vector x = mv.
It is easy to check that in any game v €GN, the vector mv relates to the core being an upper bound in that xi < mv, for any x € C (v) and all i € N .In particular, the condition v(N) <5^i£N mv is a necessary (but not sufficient) condition for nonemptiness of the core of the arbitrary game v, i.e., a strictly negative gap of the grand coalition gv (N) < 0 implies C(v) = 0.
A game v € GN is convex if for all i € N and all S C T C N\{i},
I(v) = {x € IRn \ x(N) = v(N), xi > v(i), for all i € N},
I*(v) = {x € IRn \ x(N) = v(N)}.
ev(S, x) = v(S) — x(S).
I(v).
mv = v(N) — v(N\{i}),
for all i € N,
S C N,S = 0,
S = 0,
v(S U {i}) — v(S) < v(T U {i}) — v(T), or equivalently, if for all S, T C N,
v(S) + v(T) < v(S U T)+ v(S n T). Any convex game has a nonempty core (Shapley, 1971).
Proposition 1. For every convex game v €Gn it holds that
gv(N) > 0, and gv(N) > gv (S), for all S C N.
Proof. The inequality gv (N) > 0 follows directly from the nonemptiness of the core of any convex game.
Next notice that for any S C N,
gv(N) — gv(S)= £ [v(N) — v(N\{i})] — [v(N) — v(S)].
ieN\S
Denote elements of N\S by ii,i2,... in-s, i.e., N\S = {ii,i2,... in-s}. Then, v(N)— v(S) =
[v(N)—v(N\{ii})] + [v(N\{ii})—v(N\{ii, i2})]+.. .+ [v(SU{in-s})—v(S)].
Therefore, applying successively n — s times the inequality (1), we obtain that for all S C N, gv (N) — gv (S) > 0. □
A game v €GN is 1-convex if
0 < gv(N) < gv(S), for all S C N, S = 0. (2)
As it is shown in Driessen and Tijs (1983) and Driessen (1985), every 1-convex game has a nonempty core. In a 1-convex game v, for every efficient vector x € IRn, the inequalities xi < m^, for all i € N, guarantee that x € C(v). In particular, the characterizing property of a 1-convex game is that the replacement of any single coordinate mv in the vector mv by the amount of v(N) — mv (N\i) places the resultant vector mv(i) = {mv(i)}j£N, given by
v,., / v(N) — mv(N\i) = mv — gv(N), j = i,
mv (i) = < for all j e N,
j \ mv, j = i, j ,
into the core C(v). Moreover, in a 1-convex game the set of vectors {mv (i)}i£N creates a set of extreme points of the core which in turn coincides with their convex hull, i.e., C(v) = co({mv(i)}ieN). Besides, the nucleolus v(v) occupies the central position in the core coinciding with the barycenter of the core vertices, and is given by the formula
gv (N)
Vi(v) = m1--------------------------------------------------------, for all i G N. (3)
So, the nucleolus coincides with the equal allocation of nonseparable contribution the amount of gv (N) over the players, or in other terms, every player according to nucleolus gets its marginal contribution to the grand coalition minus an equal share in the gap gv (N) of the grand coalition. That presents a special advantage of the class of 1-convex games because the nucleolus, defined as a solution to a lexicographical optimization problem that in general is difficult to compute, for 1-convex games appears to be linear and thus simple to determine.
By definition of 1-convexity (2) and from Proposition 1 we easily obtain
Proposition 2. A convex game v € Gn is 1-convex, if and only if
gv(N )= gv (S),
for all S C N, S = 0.
In the next section we study the so-called co-insurance game that appears to be closely related to the well-known bankruptcy game. For a bankruptcy problem (E; d) given by an estate E € IR+ and a vector of claims d € IR+ assuming that the total claim of the creditors is greater than the remaining estate, i.e., d(N) = 2 ieN di > E, the corresponding bankruptcy game vE;d € GN is defined in Aumann and Maschler (1985) by
vE;d(S) = max{0, E — d(N\S)}, for all S C N. (4)
To conclude this section recall a few extra definitions that will be used below.
A set of coalitions B C 2N\{N} is called a set of balanced coalitions, if positive numbers XS, S €B exist such that
XS = 1, for all i € N.
SeB: S3i
A player i is a veto-player in the game v € GN, if v(S) = 0, for every S C N \ i. A game v € GN is a veto-rich game if it has at least one veto-player.
For a game v € GN, a coalition S C N, S = 0, and an efficient payoff vector x € IRn, the Davis-Maschler reduced game with respect to S and x is the game vS,x €GS defined in Davis and Maschler (1965) by
( 0, T = 0,
vS,x(T) = < v(N) — x(N\S), T = S, for all T C S.
[ maxQcN\S(v(T U Q) — x(Q)), otherwise,
3. Co-insurance Game and Its Core
Consider the problem in which a risk is evaluated too much heavy for a single insurance company, but it can be insured by the finite set N of companies that share a given risk R and premium n. First, it is assumed that every company i € N expresses the valuation of a random variable R through a real-valued nonnegative functional Hi(R) such that Hi(0) =0, for all i € N. For any nonempty subset S C N of companies, let ^(S) = {X € IRS \Y1 i£S X = R} represents the (non-empty) set of feasible decompositions of the given risk R. Second, by hypothesis, it is supposed, for every S C N, S = 0,, that an optimal decomposition of the risk exists, so that minx eA(S)J2 i£S Hi(Xi) := P (S) is well-defined. Here the real-valued set function P can be seen as the evaluation of the optimal decomposition of the risk R by the companies in coalition S as a whole.
To determine the evaluation function P may result in general not an easy task. However, under some reasonable assumptions borrowed from real-life applications it turns out that P can be easily computed for all coalitions. For instance, in case of constant quotas, when it is supposed that for each insurable risk R, for every S C N, S ^ 0, there exists the only one feasible decomposition ^s) *^). ^
specified by a priori given quotas qi > 0, i € N, ieN qi = 1, and moreover, for each
insurable risk R,
Hinz) =qiH(—), for all i e N,
qi
where H is some a priori fixed convex function, the evaluation function P for every S C N, S = 0, is given by
If insurance companies evaluate a risk R according to the variance principle, i.e.,
Hi(R) = E(R) + aiVar(R), ai > 0, for all i € N,
where E (R) and Var(R) denote the expectation and variance of a random variable R, then we are in case of constant quotas when the corresponding quotas may be
obtained as rji = where a(N) = (j2ieN (cf. Deprez and Gerber (1985),
Fragnelli and Marina (2004)). Later on we do not discuss the construction of the evaluation function P. The only important in what follows is that P is nonnegative and non-increasing, i.e., for all 0 = S C T C N, 0 < P(T) < P(S).
For a given premium n and an evaluation function P: 2N ^ IR, Fragnelli and Marina (2004) define the associated co-insurance game vn,p € GN as following
max{0, n — P(S)}, S C N,S = 0, , ,
0, S = 0. (5)
By definition, the co-insurance game vn,p is nonnegative and since P is nonincreasing it easily follows that v is monotonic, i.e., for all S C T C N, 0 <
vn,p(S) < vn,p(T).
Notice that the well-known bankruptcy game (4) presents an example of the co-insurance game (5). Indeed, if for each insurance company i € N there exists a fixed ”claim” di > 0 such that P(S) = ieN\S di, for all S C N, S = 0, then the co-insurance game reduces to the bankruptcy game with the estate equal to the premium n. This particular evaluation function P is nonnegative and nonincreasing, P (N )=0.
In the framework of the co-insurance game, we consider the evaluation function P being fixed, while the premium n as a variable quantity varying from small up to sufficiently large amounts. In order to avoid trivial situations, let the premium n be large enough so that n > P(N). The following results are already proved in Fragnelli and Marina (2004):
• If the premium n is small enough in that n < maxieN P(N\{i}), then the coinsurance game vn,p is balanced since the core C(vn,p) contains the efficient allocation £ = {£i}ieN, where £i* = vn,p(N) for some i* € argmaxieN P(N\{i}), and £i =0 for all i = i*.
• If n>av = £ ieN [P (N\{i}) — P(N)] + P (N), then C (vnp) = 0.
• For all n < a-p, under the hypothesis of reduced concavity of function P:
P(S) —P(SU{i}) >P(N\{i})—P(N), for all S g N and every i € N\S, (6)
C (vn,p) = 0.
To ensure strictly positive worth vn,p (S) > 0 for every coalition S C N, S = 0, we suppose that the premium n is strictly bounded from below by the critical number a-p = maXj£Ar P({*}). For all II > a-p, we have
mvin’r = vn,p(N) — vn,p(N\{i}) = P(N\{i}) — P(N), for all i € N, (7)
vn,p(S) = |
for any S C N, S = 0,
gvn,r (S) = £ - Vn v(S) = (N\{i}) -P(N)] + P(S) - n. (8)
ieS ieS
In what follows we distinguish the two cases ap > a-p and ap < a-p.
Notice that in the bankruptcy setting, ap = J2ieN ck and ap = J2ieN di — minjgAr di, i.e., it always holds that ap < ap.
First consider the case ap > ap. It turns out that in this case the nonemptiness of the core C(vn,v) for n = ap is equivalent to 1-convexity of the co-insurance game v &r ,p.
Theorem 1. Let ap > ap, then the following equivalences hold:
(i) the co-insurance game v &r ,p is balanced;
(ii) the core C(var ,p) is a singleton and coincides with the marginal worth vector mVar ;
(iii) the evaluation function P meets the so-called 1-concavity condition
P(S) - P(N) > ^ [P(N\{i}) - P(N)], for all S C N, S = 0; (9)
ieN\S
(iv) the co-insurance game v&r ,p is 1-convex.
Proof. From (8) it follows that for all II > ap,
ap = E [P(N\{i}) - P(N)] + P(N) = gvnp (N) + n.
ieN
By hypothesis ap > ap, therefore, applying the last equality to II = ap, we obtain that
gVaP (N) = 0. (10)
Since for any game v &Qn , the marginal worth vector mv provides upper bound for the core, a game v with zero gap gv (N) = 0 can possess at most one core allocation coinciding with mv, which is mVar 'P in case of the co-insurance game v&r ,p. Next notice that the 1-concavity condition (9) is equivalent to
J2[P(N\{i})-P(N)] >•£ [P(N\{i})-P(N)]+P(N)-P(S), VSCN, S = 0, (11)
ies ieN
which is the same as the marginal worth vector mvar 'P satisfies the core constraints
J2mvar,P > ap - P(S) = vap,p(S), for all S C N,S = 0.
ie s
Whence it follows that the marginal worth vector mvaP 'P G C(v&r ,p), if and only if the evaluation function P satisfies the 1-concavity condition (9). Moreover, because of (8), the inequality (11) is equivalent to
gvaPP(N) < gvaPp(S), for all S C N, S = 0,
which together with equality (10) is equivalent to 1-convexity of the co-insurance game v&v,p. □
Remark 1. Notice that our 1-concavity condition (9) is weaker then the condition of reduced concavity (6) used in Fragnelli and Marina (2004).
Theorem 2. If for some fixed premium II* > ap, the co-insurance game vjj* ,p is
1-convex, then for every premium II, a-p < II < II*, the corresponding co-insurance game vn,p is 1-convex as well.
Proof. For all II > ap, due to (8) it holds that for every S C N, S ^ 0, the gap gvn'P (S) is a decreasing linear function of the variable n, while the difference gvn,v (s) - gvnp (N) is constant for all n. Whence, it follows that if for some fixed premium II* > ap the co-insurance game vn*,p is 1-convex, i.e., for all S C N, S = 0, the inequality (2) holds, then this inequality remains valid for all premium ap < II < 27*, i.e., all games vn,p appear to be 1-convex as well. □
The next theorem follows easily from Theorem 1 and Theorem 2.
Theorem 3. Let a-p > ap. If the evaluation function V satisfies the 1-concavity condition (9), then for any premium ap < II < a-p,
(i) the corresponding co-insurance game vn,p is 1-convex;
(ii) the core C(vn,p) = 0;
(iii) the nucleolus v(vn,p) is the barycenter of the core C(vn,p) and is given by
Vi(vn r) = P(N\{i}) - V(N) + 11 ~ ar , for all % £ N. (12)
n
Proof. The first statement follows directly from Theorem 1 and Theorem 2. Next, recall already mentioned above results obtained in Driessen and Tijs (1983) and Driessen (1985), stating that every 1-convex game has a nonempty core and its nucleolus being the barycenter of the core is given by the formula (3). These facts, together with (7) and (8), complete the proof. □
In words, the third statement of Theorem 3 means that the nucleolus of these co-insurance games is a linear function of the variable premium such that each incremental premium is shared equally among the insurance companies. Geometrically, the nucleoli payoffs follow a straight line to end up at the marginal worth vector yielding payoff P(N\{i}) - P(N) to player i G N.
Remark 2. The statement of Theorem 3 remains in force if the 1-concavity condition (9) for the evaluation function P is replaced by any one of the equivalent conditions given by Theorem 1, in particular if C(v&r ,p) = 0 or if the co-insurance game vav p is 1-convex.
Remark 3. Formula (12) for nucleolus of a co-insurance game can be derived alternatively using the method for computing the nucleolus of the so-called compromise stable game introduced in Quant et al. (2005). Indeed, it is not difficult to check that every 1-convex game appears to be compromise stable.
Remark 4. In the bankruptcy setting Theorem 3 expresses the fact that the nucleolus provides equal losses to all creditors (insurance companies) with respect to their individual claims, if estate (premium) varies between £^N di - minieN di and S^n di, which agrees well with the Talmud rule for bankruptcy situations studied exhaustively in Aumann and Maschler (1985).
Consider now the case a-p < a-p. In this case, even if the co-insurance game vav p is 1-convex, for the co-insurance game vn,p corresponding to the premium n < ap the 1-convexity may be lost immediately while lowering the premium. This happens due to the fact that the co-insurance worth of at least one coalition turns out to be at zero level. For instance, consider the following example.
Example 1. Let the evaluation function P for 3 insurance companies be given by P({1}) = 5, P({2}) = 4, P({3}) = 3, P({1, 2}) = P({1, 3}) = P({2, 3}) = 2, and P({1, 2, 3}) = 1. In this case, A = ap < ap = 5.
• If the premium n = 4, then the co-insurance game v4,p:
v4,p({1}) = v4,p({2})=0, v4,p({3}) = 1, v4,p({12})= v4,p({13})= v4,p({23}) =
2, v4,p({123}) = 3,
is a 1-convex game with the minimal for a 1-convex game gap gv4'P ({123})=0 and, therefore, with the unique core allocation mvi-r = (1,1,1).
• If the premium n = 3, then the co-insurance game v3,p:
v3,p({1})= v3,p({2})= v3,p({3})=0, v3,p({12})=v3,p({13})= v3,p({23}) = 1, v3,p ({123}) = 2,
is a symmetric 1-convex and convex, since the gap gv3-r (S) = 1 is constant for all S C N, S = 0, while its core C(v3,p) is the triangle with three extreme points (1, 1, 0), (1, 0, 1), (0, 1, 1).
• For any premium 2 < n < 3, the corresponding co-insurance game vn,p is zero-normalized and symmetric: vn,p(i)=0, vn,p(ij) = n - 2, vn,p(123) = n - 1. However, the 1-convexity fails because the gap of singletons is strictly less than the gap of N: gvnp (i) = 1 < 4 - n = gvnp (123).
4. Algorithms for Computing Nucleolus
It is easy to compute the nucleolus of a co-insurance game when it is a linear function of a given premium as it is stated by Theorem 3. In this section we introduce a comparatively simple algorithm that allows to compute the nucleolus of a coinsurance game also in cases when it is nonlinear in the premium. To do that, we uncover first the relation between the class of co-insurance games, in particular bankruptcy games, and the class of Davis-Maschler reduced games of monotonic veto-rich games obtained by deleting a veto-player with respect to the nucleolus. Second, we provide an algorithm for computing the nucleolus for games of the latter class.
In what follows by Gm we denote the class of all monotonic games with a player set N. Let No := N U {0} and n0 = n +1. Consider the class Gm0 of monotonic veto-rich games with a player set N0 and the player 0 being a veto-player. Besides, we consider the class G+ of nonnegative veto-rich games with a player set N0 and the player 0 being a veto-player, that satisfies also the property v0(N0) > v0(S), for all S C N0. It is easy to see that Gm0 C G+. Define RN as a class of veto-removed games v G GN that are the Davis-Maschler reduced games of games v0 g Gm0 obtained by deleting the veto-player 0 in accordance to the nucleolus payoff. As it was already shown in Arin and Feltkamp (1997), for every veto-rich game from the class G+ the core is nonempty and the nucleolus payoff to a veto-player is larger than or equal to that of the other players. From where it easily follows that every veto-removed game is balanced because the Davis-Maschler reduced game inherits the core property and, moreover, in every nontrivial veto-rich game v0 G G+0 the
nucleolus payoff to a veto-player v0(v0) > 0 since in any nontrivial game v0 G G+ the worth of the grand coalition v0(N0) > 0.
Some extra notation. With every game v GG+ we associate the following veto-rich game v0 GG+0 defined as
v°<S> = { v(S\{0}), S I! 0, S C N0. <13>
For every veto-rich game v0 GG+0 let v0 denote the nucleolus payoff v0(v0) to the veto-player 0 in v0. Besides, for a game v gGN and a G IR + we define the game v-a gGN as follows
v-a(S) = max{0, v(S) - a}, for all S C N. (14)
Below for the facilitation of reading for any set of players M containing the veto-player 0, any coalition S0 C M with subindex 0 is assumed to contain the veto-player
0, and it is supposed that for any S0 C M holds the equality s0 = |S0| = s+1, where s = |S0 n M\{0}|. Furthermore, for any game w G Gm, for every S C M we define a number
' w{M)—w{S)
^
MM) S' = 0.
m ’
For M10, we define also a number K*(w)= min kw (S), and for M10, we define a
SgM
number K^(w)= min kw (S0).
S0 gM
Theorem 4. It holds that
(i) every game v G Rn can be presented in the form of a co-insurance game vn,p G
Gn ;
(ii) if vn*,p G Rn, then for every premium n < n*, vn,p G Rn as well;
(iii) for every evaluation functionP: 2N ^ IR, for every premium n,
n <n* = V{N) + n min V<yS') ~ V<yN'), (16)
SgN n - s + 1
the co-insurance game vn,p gRn.
Proof, (i). Consider v G Rn. By definition of Rn there exists v0 g Gmmo such that
v is the Davis-Maschler reduced game derived from v0 by deleting the veto-player 0 with respect to the nucleolus. By definition of the Davis-Maschler reduced game it holds
0, 5 = 0,
j(S)={ v0(N U{0}) - v0, S = N,
max{v0(S), v0 (S U {0}) - v0} = max{0, v0(S U {0}) - v0}, S C N, S = 0.
Take some positive k > v (^jvu^0^ — 1 and set
n = kv0,
P(S) = (k + 1)v0 - v0(S U{0}), for all S C N, S = 9.
Whence
v(S) = max{0, n- P(S)}, for all S C N, S = 9.
(ii). Recall first that every co-insurance game is monotonic and, moreover, for any co-insurance game vn,p, for any a G IR +, the game v—ap is also a co-insurance game with the premium equal to n — a, i.e, v^p = vn-ap. Therefore in view of (i) proved above, for proving (ii) it is sufficient to show that if for certain game v G GN it holds that v-a G RN for some a G IR+, then v-b G RN for all b G IR+, a < b. Moreover, notice that it is enough to prove that v-b G RN only for a < b < v(N) since due to (14), it holds v-b = 0 G RN for all b > v(N).
Consider now a game v G GNm together with its associated veto-rich game v0 G GN}0. From (13) and already mentioned above statement of Arin and Feltkamp (1997) concerning the nucleolus payoff to a veto-player, it follows easily that <vq< v(N). Set a := v0. It is not difficult to see that v-a G GN is the Davis-Maschler reduced game of the game v0 obtained by deleting the veto-player 0 with respect to the nucleolus payoff a. So, by definition of RN, v-a G RN. Recall that if a = v(N), v-a = 0 G Rn.
Next, we show that if a < v(N), then for all b, a < b < v(N), also v-b G RN. The above procedure of constructing a veto-removed game may be applied to any monotonic game, in particular to the just obtained monotonic game v-a G RN. Doing that, we get another monotonic game, say v-a G RN, with a1 = a + v0(v-a) > a when a < v(N). We show first that v-b G RN for all a < b < a1. Consider 0 < c < a and apply the above procedure for all monotonic games v-c G Gn. For c = 0 we start with v and obtain the monotonic game v-a G RN. For c = a we start with v-a and obtain the monotonic game v-a G RN. Due to the continuity of the nucleolus we obtain all v-b while c varies between 0 and a. Hence v-b G Rn for all a < b < a1.
When a1 < v(N) then applying the above procedure to the game v-a we obtain a game v-a G RN with a2 > a1 and so on. Since on each step k, ak — ak-1 =
vo(v~a ) > v<"N^l+1----, any number a<b<v(N) can be reached by not more than
v(N)-b(n + 1) stePs- Therefore, for every b, a <b < v(N), v~b G Rn-
(Hi). Take a co-insurance game vn,v with II > a-p, for simplicity of notation denote vn,p by v, and consider the corresponding veto-rich game v0 g Gm0 defined by (13). As it is shown in the proof of (ii), the Davis-Maschler reduced game of the game v0 obtained by deleting the veto-player 0 with respect to the nucleolus coincides with the game v-a, a = v0, which in turn coincides with the co-insurance game vni p with n' = n — v0. Hence, vn/ p gRn . From Proposition 3 below and the definition of a co-insurance game (5), since II > a-p, it follows that
n<n- P{N) - n min
sCn n — s +1
le" i7->m) + ,.mmP(S)-P(JV).
sCn n — s + 1
Then the validity of (iii) follows immediately from the just proved (ii). □
Notice that (16) provides rather rough estimation of n*. In fact, in the particular case of bankruptcy games, (16) guarantees that vE;d G RN only when E < 0. Next theorem imposes weaker conditions on the parameters of a bankruptcy game vE ; d to guarantee that vE;d G RN.
Theorem 5. For any estate E G IR + and any vector of claims d G IR + such that
E
S"=i di 2 ’
the corresponding bankruptcy game vE;d G Rn.
Proof. First take a bankruptcy game vE;d with E =£ieN di and let v be its coinsurance game representation, i.e., v = vn,p with n = d(N) and P(S) = d(N\S). For a co-insurance game v consider the corresponding veto-rich game v0 defined by (13),
f 0, S 3 0, v0(S) = I di, S 3 0, for all S C N0.
[i£S\{0}
We compute now the nucleolus payoff v0 to the veto player 0 in v0 applying Algorithm 2 yielding nucleolus for monotonic veto-rich games with a veto-player 0 introduced below. Without loss of generality we assume that d1 < d2 < ... < dn. Moreover, for every k =1, ...,n we define a veto-rich game vk on N0\{1, ...,k} as follows
( 0, s 3 0,
vk(S) = { k for all S C N0.
I E dt + Ef, ^o, - 0
I i£S\{0} i=1
For any coalition S0 C N0 it holds that
v0(No)-v°(So) d(No\So) d\(n — s) d\ ,ArWl1,
0 =-------------—-----=---------—- >-—- > — = ^o(AT0\{l}).
n — s +1 n — s + 1 n — s + 1 2
Whence it follows that k0(v°) = Kv0 (N0\{1}), and therefore, Step 1 of Algorithm 2 assigns the nucleolus payoff v\(v°) = 4r to the player 1. Moreover, the Davis-Maschler reduced game constructed in Step 1 is defined on the player set N0\{1} and coincides with the game v1. Using the similar reasoning it is not difficult to see that for any k = 2, ...,n, Algorithm 2 applied to the veto-rich game vk-1 defined on the player set Aro\{l,... ,k — 1} assigns the nucleolus payoff 4^ to the player k and goes to the next step with the Davis-Maschler reduced game coinciding with the game vk defined on the player set N0\{1,. ..,k}. Then applying the induction argument we obtain that vi(v°) = y for all i G N and vq = vo(v°) =
Next observe that if a co-insurance game vn,p represents a bankruptcy game vE;d, then for any a G IR +, the co-insurance game v—ap represents the bankruptcy game vE-a;d. Hence, we may complete the proof following the same arguments as in the proof of the statement (ii) of Theorem 4. □
Consider now the following algorithm for constructing a payoff vector, say x G IRn , in a game v G RN.
Algorithm 1
0. Set M = N and w = v.
1. Find a coalition S C M with minimal size such that Kw(S) = K*(w).
2. For i G M\S, set xi = Kw(S). If S = ft, then stop, otherwise go to Step 3.
3. Construct the Davis-Maschler reduced game ws,x G Gs . Set M = S and w = ws,x and return to Step 1.
Theorem 6. For any veto-removed game v G Rn, Algorithm 1 yields the nucleolus payoff, i.e., x = v(v).
The proof of Theorem 6 is obtained by comparing the outputs of two algorithms yielding nucleoli - Algorithm 1 applied to a veto-removed game v G RN and another Algorithm 2, applied to the associated monotonic veto-rich game v° g Gm0. Algorithm 2 is closed conceptually to the algorithm for computing the nucleolus for veto-rich games suggested in Arin and Feltkamp (1997). It is worth noting that for the application of Algorithm 1 to a veto-removed game v G RN there is no need in construction of the associated monotonic veto-rich game v0 G Gn0 which is only necessary for proving Theorem 6. The proof of Theorem 6 is given after the proof of Theorem 7.
The following Algorithm 2 constructs a payoff vector, say y G IRNo, in a game v0 G G+ . Since Gn0 C G+ , Algorithm 2 is applicable to any game v0 G Gn0 as well.
Algorithm 2
0. Set M = N° and w = v0.
1. Find a coalition S° C M with minimal size such that kw(S°) = K*(w).
2. For i G M\S°, set yi = Kw(S°). If S° = {0}, set y° = v°(N°) — £ yi and stop,
i£N
otherwise go to Step 3.
3. Construct the Davis-Maschler reduced game ws0,y G Gs0 . Set M = S° and w = ws0,y and return to Step 1.
Theorem 7. For any veto-rich game v0 G G+ , Algorithm 2 yields the nucleolus payoff, i.e., y = v(v0).
Proof. Let v° G G+0 . For the simplification of notation denote the nucleolus v(v0) by x, x G IRn+1, and let e*(v0) denote the maximal excess with respect to the nucleolus in the game v0, i.e., e*(v0) = max ev (S,x). As a corollary to the Kohlberg’s
SgNo
characterization of the prenucleolus (Kohlberg, 1971) it holds that the collection of coalitions with maximal excess values with respect to the nucleolus is balanced. Due to the balancedness, among the coalitions having the maximal excess there exists S0 C n°. We show that every singleton {i}, i G S°, also has the maximal excess. Let i G S0. Again due to the balancedness, there exists S C N°, S 3 i, S 3 0, with maximal excess. Observe that since S 3 0, then by definition of a veto-rich game v0(S) = v0(S\{i}) = v0({i}) = 0. If \S| > 1 then
e(S, x) = v0(S) — x(S) = —x(S) = —x({i}) — x(S\{i}) = e({i}, x) + e(S\{i}, x).
Since the core of every veto-rich game in GN0 is nonempty, the nucleolus belongs to the core and all excesses with respect to the nucleolus are nonpositive, in particular,
e(S\{i},x) < 0. From where it follows that e({i},x) > e(S,x), i.e., every singleton {i}, i G S, possesses the maximal excess as well.
For every S0 C N0 with maximal excess with respect to the nucleolus from the efficiency of the nucleolus and the equality v0({i}) = 0 for all iGN0\S0, it follows that
v0(So)—v0(N°) = v0(So)—x(N°) = v°(S° )— x(S°)— ^ x(i) =
i£No\So
= ev (So,x)+ ^ ev ({i},x) = e*(v°)*(n° s°+1) = e*(v°)• (n s+1).
i£No\So
Moreover, for every TO C N° it holds that
v°(T°) —v0(N°) = evo (T°,x)+ ^ evo ({i},x) < e*(v0) • (n — t +1).
i*ENo\To
Whence, for every T0 C N0
)<g _v°m- »°w,) v n° — t° + 1
while for S0 C N0 with maximal excess with respect to the nucleolus holds the equality ko (S0) = —e*(v0). Then it follows that S0 C N° has the maximal excess with respect to the nucleolus if and only if k o (S°) = k* (v0).
Therefore, on the first iteration of Algorithm 2 when M = N° and w = v0, Step 1 provides a coalition S0 C N0 with maximal excess with respect to the nucleolus. Then Step 2 assigns to every i G N0\S0 its nucleolus payoff because the assigned payoff yi = k o (So) coincides with xi = vi(v0) since
yi = Kvo (S°) = —e*(v0) = —(v0({i}) — xi) = xi, for all i G No\So.
In every veto-rich game from the class G+ with M containing the veto-player 0 the nucleolus coincides with the prenucleolus due to the nonemptiness of the core which was already mentioned above with reference to Arin and Feltkamp (1997). Then, because of the Davis-Maschler consistency of the prenucleolus (Sobolev, 1975), the nucleolus payoffs to the players in the Davis-Maschler reduced game wSo,y G GSo constructed in Step 3 of Algorithm 2 are the same as the nucleolus payoffs to the players in S0 in the game w G G+. Thus in order to complete the proof, it only remains to show that the Davis-Maschler reduced game wSo,y G GSo of a game w G Gm with M containing the veto-player 0 is itself a veto-rich game belonging to
G+o.
Take T C S0\{0}. Then
wso,y(T)= max (w(T U Q) — y(Q)) = max {0 — y(Q)} = 0,
QCM\So Q^M\So
because w(TUQ) = 0 for every w G G++, since TUQ C M\{0} for every Q C M\S°. Thus, 0 is a veto-player in wSo,y G GSo as well. Further, it is evident that wSo,y is nonnegative. Hence, it remains to show that wSo,y(S°) > wSo,y(T) for every
T С So. When T С S0\{G}, wSo,y(T) = G К wSo,y(S). Consider now To С So and let Q С M\S0. Since To П Q = 0,
w(M) - w(Tq U Q) _ w(M) — w(Tq U Q)
Kw[ ° 4) m-\T0UQ\ + l m-to-q+l '
Moreover, To U Q С M and To U Q Э G. By Step 1 of Algorithm 2, Kw(S0) is the
minimal among all coalitions in M containing the veto-player G. From where and also because of the obvious inequality s0 >t0 — 1, it holds that
w(M) — w(To U Q) w(M) — w(To U Q)
------------------- > ------------t---— > Kw{b0).
m — so — q m — to — q +1
Hence,
w(M) — (m — so) • Kw (So) > w(To U Q) — q • Kw(So),
and therefore, since at Step 2 every player’s i Є M\S0 payoff yi = Kw (So), it holds that
w(M) — y(M\So) > w(To U Q) — y(Q).
Then by definition of the Davis-Maschler reduced game we obtain wSo y(So) >
wso,y (To) for every To С So. □
From the proof of Theorem 7 also the upper bound for the nucleolus payoff v0
'+o
to the veto-player G in a veto-rich game v0 є G+ easily follows.
Proposition 3. For any veto-rich game v0 G G+
v° < v°(N°) — n • k* (v°), (18)
with the equality, if and only if vi(v0) = Vj(v0) holds for all i,j G N.
Proof. Since v0(i) = 0 for all i G N, every excess of a singleton coalition {i}, i = 0, with respect to the nucleolus v(v0) is equal to —vi(v0). From (17) in the proof of Theorem 7 it follows that the maximal excess in v0 with respect to the nucleolus is equal to —k*(v°). Therefore from the efficiency of the nucleolus we obtain that
v° = v°(N°) — ^ vi(v0) = v°(N°) + ^(—vi(v0)) < v0(N°) + n • (—k°(v°)),
ieN ieN
where the equality hold, if and only if vi(v0) = k*(v°) for all i,j G N. □
Remark 5. The inequality (18) can be equivalently presented in the form
K*(v°) < ^0(Ar°)~i/0, (19)
0n
with the equality, if and only if vi(v0) = vj(v0) holds for all i,j G N. Inequality (19) will be used later in the proof of Theorem 6.
We are ready now to prove Theorem 6.
Proof. of Theorem 6 Consider a veto-removed game v G . By definition of RN there exists a monotonic veto-rich game v° G GNo such that v is the Davis-Maschler reduced game of v0 obtained by deleting the veto-player 0 in accordance to the nucleolus payoff. Then because of the already mentioned above the Davis-Maschler consistency of the nucleolus in a veto-rich game in GN0, vi(v) = vi(v0) for all i G N. Since Algorithm 2 yields v(v0), for proving Theorem 6 it is sufficient to show that the payoff vector x produced by Algorithm 1 for the game v coincides on N with the payoff vector y produced by Algorithm 2 for the game v0. For giving evidence for that we show first that either in Step 1 of Algorithm 1 S = 0 is chosen and the algorithm yields the nucleolus, or it holds that
K*(v) = K* (v0). (20)
From Theorem 7, y = v(v0). v is the Davis-Maschler reduced game of v0 obtained by deleting the veto-player 0 in accordance to the nucleolus v(v0). Hence by definition of the Davis-Maschler reduced game, v(S) = max{0,v°(S U {0}) — y0},
i.e., either v(S) = v0(S U {0}) — y0 or v(S) = 0. Let S C N be the coalition chosen in Step 1 at the first iteration of Algorithm 1. It turns out that either S = 0, or v(S) = v0(S U {0}) — y0. Indeed if we assume that S = 0 and v(S) = 0, then
„,(*) S V(N)-V(S) = > HOT s
n — s + 1 n — s + 1 n
i.e., kv(0) < kv(S) while |0| =0 < s, which contradicts the choice of S.
Let now S0 C N0 be the coalition chosen in Step 1 at the first iteration of Algorithm 2 and let S = S0\{0}. Similarly to the paragraph above, it turns out that either S = 0, or v(S) = v0(S U {0}) — y0. Indeed if S = 0 and v(S) = max{0,v°(S U {0}) — yo} = 0, i.e., v0(S U {0}) — y° < 0, then
fc, , def v°(No)-v°(So) v°(No)-yo ^ v°(No)-yo
Kvo{S0) = -------—j > ------------—j— > ---------------,
v n° — s° + 1 n — s +1 n
which contradicts Proposition 3 restated in the form (19). Thus, for the coalition S0 chosen in Step 1 at the first iteration of Algorithm 2 it holds that either S = S°\{0} = 0, or v(S) = v0(S U {0}) — y°.
Hence, due to the Davis-Maschler reduced game relationship between v and v0, in both algorithms for all S C N with the assumption that S = S0\{0} for S0 chosen in Step 1 of Algorithm 2, it holds that either S = 0 or v(S) = v0(S U {0}) — y0. Thus, for proving (20) it is sufficient to prove that for all S C N it holds that Kv(S) = Kvo (S U {0}).
First consider the case S C N, S = 0. Then v0(S0) = v(S)+y0, and in particular, v0(N0) = v(N) +y0. Therefore,
.. , def v0(No)-v°(So) v(N)-v(S) def ..
^ o( 0/ , 1 , 1
v n0 — s0 + 1 n — s + 1
i.e.,
Ko (S0) = Kv(S), for all S C N, S = 0. (21)
Consider now the case S = 0. Then S0 = S U {0} = {0} and
= .))g',|№-|-,(WI
v v n°
Again there are two options possible, namely k 0 ({0}) = k*(v°) or k 0 ({0}) >
K°(v°)-
If k 0 ({0}) = k°(v°), then Algorithm 2 terminates at the first iteration and in Step 2 every player i G N gets the same payoff yi = k 0 ({0}). Due to the coincidence of nucleoli in games v and v° on N, it holds that every player i G N in v has the
v(N)
same nucleolus payoff that by efficiency is equal to --. Hence, with i G N,
n
*„({»» = m = — "*.(8).
u n
From where together with (21) it follows that (20) holds true when k 0 ({0}) =
K*(v°).
If K0 ({0}) >K°(v°), then there exists SCN, S = 0, such that k0 (S°) = k°(v°). Because of (21), k 0 (S°) = Kv(S), and hence,
nn
where the second equality is due to v being the Davis-Maschler reduced game of v°. Whence, either kv (0)=Kv (S), or kv (0) >Kv (S).
If kv(0) = kv (S), then
°
n
and from Proposition 3 and Remark 5 it follows that vi(v°) = Vj (v°) for all i,j G N, and therefore, for all i G N,
/ ^ D —M concist / x eff v(N) def
Vi{V ) = vi{v) = ----- = Kv (</)),
n
i.e., in Step 1 of Algorithm 1 the empty set is chosen and Algorithm 1 in fact yields the nucleolus.
If kv(0) > kv(S), then there exists S' C N, S1 = 0, such that kv(S') = K*(v) (possibly, S' = S). Hence, due to (21), for S° = S' U {0} C N°, kv(S') = k0 (S°),
i.e., in this case K*(v) = k*(v°) as well. Thus, it is proved that either in Step 1 of Algorithm 1 either S = 0 is chosen and the algorithm yields the nucleolus, or
K*(v) = K*(v°).
For completing the proof it remains to consider the situation when in Step 1 of Algorithm 1 a coalition S C N, S = 0, is chosen. As it is shown above in such a case in Step 1 of Algorithm 2 we always can chose S° C N°, S° = S U {0}, for which k 0 (S°) = kv (S). Thus, at the first iteration both algorithms at Step 2 assign Xi = y for every i G N\S = N°\S°. It is easy to see that the Davis-Maschler reduced game ws,x constructed in Step 3 of Algorithm 1 is the Davis-Maschler reduced game of the Davis-Maschler reduced game ws0,y constructed in Step 3 of Algorithm 2. Then observe that the situation at all next iterations of both algorithms remains the same. Therefore, repeating the same reasoning as above we obtain that both algorithms assign the same payoffs to all players in N. □
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