Two-bound core games and the nucleolus

This paper introduces the new class of two-bound core games , where the core can be described by a lower bound and an upper bound on the payoffs of the players. Many classes of games turn out to be two-bound core games. We show that the core of each two-bound core game can be described equivalently by the pair of exact core bounds, and study to what extent the exact core bounds can be stretched while retaining the core description. We provide explicit expressions of the nucleolus for two-bound core games in terms of all pairs of bounds describing the core, using the Talmud rule for bankruptcy problems, and study to what extent these expressions are robust against game changes.


Introduction
In the theory of cooperative games (with transferable utility), players collaborate in coalitions to generate profits.Cooperative game theory analyzes how to allocate profits generated by the grand coalition among the players in a fair way, and provides several significant solution concepts.
A central solution concept is the core, which consists of all coalitionally stable preimputations, that is, no coalition will obtain more by deviating from cooperation in the grand coalition.Bondareva (1963) and Shapley (1967) showed that the core is nonempty if and only if the corresponding cooperative game is balanced.Another important solution concept is the nucleolus (cf.Schmeidler 1969), which lexicographically minimizes the excesses of coalitions.The nucleolus selects from the core in each balanced game.Quant et al. (2005) studied the class of compromise stable games where the core coincides with the core cover (cf.Tijs and Lipperts 1982), and provided an explicit expression of the nucleolus for this class using the Talmud rule for bankruptcy problems.The core cover is the set of pre-imputations between a specific pair of bounds.In this paper, we generalize the approach of Quant et al. (2005) to all games where the core equals the set of pre-imputations between an arbitrary pair of bounds, which we call two-bound core games.
We show that the core of each two-bound core game can be described equivalently by the pair of exact core bounds (cf.Bondareva and Driessen 1994), which are defined by the minimum and maximum individual payoffs within the core.Inspired by Quant et al. (2005), we provide conditions to check whether a game is a two-bound core game, and describe the extreme points of the core for each such game.All balanced games with at most three players are two-bound core games, but this does not hold for more players.
We study to what extent the exact core bounds of a two-bound core game can be stretched while retaining the core description.It turns out that only three possible cases exist.In the first case, only the lower bounds are decreased for players who obtain their lower exact core bounds when all other players obtain their upper exact core bounds, while keeping all other bounds fixed.In the second case, only the upper bounds are increased for players who obtain their upper exact core bounds when all other players obtain their lower exact core bounds, while keeping all other bounds fixed.In the third case, both the lower bound is decreased and the upper bound is increased for only a single player who obtains the lower exact core bound when all other players obtain their upper exact core bounds and obtains the upper exact core bound when all other players obtain their lower exact core bounds.
In line with Quant et al. (2005), we provide an explicit expression of the nucleolus for two-bound core games in terms of the exact core bounds using the Talmud rule.In fact, the nucleolus of these games can be equivalently expressed by each pair of bounds describing the core.We study to what extent these expressions are robust against game changes.
The remainder of this paper is organized as follows.Section 2 introduces preliminary definitions and notation about cooperative games and bankruptcy problems.In Section 3, we formally introduce two-bound core games.The nucleolus for two-bound core games is studied in Section 4. Finally, we conclude this paper with some remarks in Section 5.

Preliminaries
Let N be a nonempty and finite set of players and let 2 N be the collection of all subsets of N .An order of N is a bijection σ : {1, . . ., |N |} → N , where |N | denotes the cardinality of N , and σ(i) represents the player at position i.The set of all orders of N is denoted by Π(N ).Denote by R + the set of all non-negative real numbers.
Let x, y ∈ R N .We denote x + y = (x i + y i ) i∈N , x − y = (x i − y i ) i∈N , and λx = (λx i ) i∈N for all λ ∈ R.Moreover, x ≥ y denotes x i ≥ y i for all i ∈ N , and x > y denotes x i > y i for all i ∈ N .The notations ≤ and < are defined analogously.We denote A cooperative game with transferable utility (a game, for short) is a pair (N, v), where v : 2 N → R is the characteristic function with v(∅) = 0, representing the worth v(S) for each coalition S ⊆ N when the players in S cooperate.The set of all games with player set the imputation set of v is and the core of v is Note that C(v) ⊆ I(v) ⊆ X(v), and C(λv + a) = λC(v) + a for all λ ∈ R + and a ∈ R N , where λv + a ∈ Γ N is defined by (λv + a)(S) = λv(S) + i∈S a i for all S ⊆ N .Bondareva (1963) and Shapley (1967) showed that a game v ∈ Γ N is balanced if and The set of all balanced games with player set N is denoted by Γ A value φ on a domain of games assigns to each game v in this domain a pre-imputation φ(v) ∈ X(v).The nucleolus (cf.Schmeidler 1969) is the value η that assigns to each game A bankruptcy problem is a triple (N, E, c), where E ∈ R + is the estate to be divided and c ∈ R N + is the vector of claims satisfying i∈N c i ≥ E. The set of all bankruptcy problems with player set N is denoted by B N .For simplicity, we write (E, c) ∈ B N rather than (N, E, c) ∈ B N .
A bankruptcy rule f : Aumann and Maschler 1985) if for all (E, c) ∈ B N , The Talmud (TAL) rule assigns to each bankruptcy problem (E, c) ∈ B N and each player Aumann and Maschler (1985) showed that the Talmud rule is self-dual and invariant under claims truncation.Curiel et al. (1987) showed that bankruptcy games are convex games.Aumann and Maschler (1985) showed that for each bankruptcy problem, the payoff vector assigned by the Talmud rule coincides with the nucleolus of the corresponding bankruptcy game.

Two-bound core games
In this section, we introduce two-bound core games, where the core equals the set of preimputations between a lower bound and an upper bound.
consists of all pre-imputations between lower bound l and upper bound u, i.e., it is the intersection of the pre-imputation set and the |N |-dimensional hypercube restricted by l and u, so it is a convex set.If this set is nonempty, then its extreme points can be described as follows.Similar to Quant et al. (2005), we define m l,u,σ (v) ∈ R N for all σ ∈ Π(N ) and all k ∈ {1, . . ., |N |} by Thus, m l,u,σ (v) assigns to the first players in σ their upper bound payoffs in such a way that the last players in σ are assigned their lower bound payoffs.The pivot player of m l,u,σ (v) is the first player in σ who is not assigned the upper bound payoff.If all the players receive their upper bound payoffs, then the last player is the pivot player of m l,u,σ (v).These definitions are straightforward generalizations of concepts in Quant et al. (2005) to arbitrary lower and upper bounds, which can be used to describe the l,u-efficient set.
Lemma 1 Let x ∈ R N be an arbitrary extreme point of [l, u] ∩ X(v), i.e., for each 0 < λ < 1 and We claim that there exists at most one player i ∈ N such that l i < x i < u i and [x j = l j or x j = u j for all j ∈ N \ {i}].
Assume, to the contrary, that there exist i, j ∈ N with i ̸ = j such that l i < x i < u i and x k for all k ∈ N \{i, j}, and let x ′′ be defined by The l,u-efficient set and the core are both convex subsets of the pre-imputation set.We are interested in l,u-efficient sets that contain the core.Many well-known sets are of this type, such as the imputation set and the core cover (cf.Tijs and Lipperts 1982).
Example 1 for all i ∈ N .Then [l, u] ∩ X(v) defines the core cover (cf.Tijs and Lipperts 1982), which contains the core.Quant et al. (2005) defined compromise stable games as games where the core cover coincides with the core.△ To check whether a core-containing l,u-efficient set coincides with the core, we only need to verify a specific inequality for each nonempty coalition.
Theorem 1 (1) Theorem 1, the proof of which is in the Appendix, generalizes the work of Quant et al. (2005), where this result was proven for the specific pair of bounds in Example 3. If the l,u-efficient set does not contain the core, then expression (1) may hold even when the core does not coincide with the l,u-efficient set.This is shown by the following example.

Example 4
Let N = {1, 2} and let v ∈ Γ N be given by v It is easy to verify that expression (1) holds for each nonempty coalition.However, We focus on games where the core coincides with some l,u-efficient set.These games are called two-bound core games.
Definition 1 The set of all two-bound core games with player set N is denoted by Γ N t .It is worthwhile mentioning that many classical games are two-bound core games.For example, additive games, unanimity games, bankruptcy games (cf.O'Neill 1982), 1-convex games (cf.Driessen 1986), big boss games (cf.Muto et al. 1988), clan games (cf.Potters et al. 1989), compromise stable games (cf.Quant et al. 2005) and reasonable stable games (cf.Dietzenbacher 2018).
It turns out that the core of each two-bound core game can be described by the following specific pair of bounds.Let v ∈ Γ N b .The lower exact core bound is defined by x i for all i ∈ N .
The upper exact core bound is defined by x i for all i ∈ N .
The lower and upper exact core bounds were also studied by Bondareva and Driessen (1994).
Lemma 2 Proof.The if-part follows directly from the definition of two-bound core games.For the only- All balanced games with at most three players are two-bound core games, but this does not hold for more players.
In what follows next, we study to what extent the exact core bounds of a two-bound core game can be stretched while retaining the core description.It turns out that the exact core bounds can be stretched in only three different ways.
Proposition 2 , then exactly one of the following cases holds: , and l j = l * j (v) and u j = u * j (v) for all j ∈ N \ {i}.
Proof.In view of l ≤ l * (v) and u ≥ u * (v), it suffices to prove that if l ̸ = l * (v) and u ̸ = u * (v), then case (iii) arises.Assume to the contrary that there exist i, j ∈ N with i ̸ = j such that Moreover, we show that the first case in Proposition 2 arises only if the players whose lower bounds are decreased obtain their lower exact core bounds when all other players obtain their upper exact core bounds.The second case in Proposition 2 arises only if the players whose upper bounds are increased obtain their upper exact core bounds when all other players obtain their lower exact core bounds.The third case in Proposition 2 arises only if the player whose exact core bounds are stretched obtains the lower exact core bound when all other players obtain their upper exact core bounds and obtains the upper exact core bound when all other players obtain their lower exact core bounds.

Theorem 2
Let v ∈ Γ N t and let l, u ∈ R N .Then the following statements hold: It follows that there exists j ∈ N \ {i} such that Together with (ii) The proof is analogous to the proof of (i).
(iii) For the only-if part, assume that Then, analogous to the proofs of (i) and (ii), it follows that For the if-part, assume that there exists i ∈ N such that l i < l * i (v), u i > u * i (v), l j = l * j (v) and u j = u * j (v) for all j ∈ N \ {i}, and expression (2) holds.We show that and x j ≥ l j = l * j (v) for all j ∈ N \ {i}, so x ≥ l * (v).Similarly, Proposition 2 and Theorem 2 directly imply the following result, which shows exactly under which condition two-bound core games can be described by different pairs of bounds.

The nucleolus
In this section, we consider the nucleolus of two-bound core games.Quant et al. (2005) provided an explicit expression of the nucleolus for compromise stable games in terms of the pair of bounds in Example 3, using the Talmud rule for bankruptcy problems.On the one hand, we extend their approach by providing an explicit expression of the nucleolus for all two-bound core games in terms of the exact core bounds.On the other hand, we show that the nucleolus can be equivalently expressed by each pair of bounds describing the core.
Lemma 3 The proof of Lemma 3, which is in the Appendix, is similar to the proof of Theorem 4.2 of Quant et al. (2005).However, as the following example shows, the expression obtained by Quant et al. (2005) in terms of the pair of bounds in Example 3 is not valid for all two-bound core games.

△
More generally, the nucleolus of two-bound core games can be equivalently expressed in terms of each pair of bounds describing the core.
Theorem 3 Proposition 2, exactly one of the following cases holds.
Applying Lemma 3, invariance under claims truncation, and self-duality, (ii) l = l * (v) and u ≥ u * (v).By Theorem 2, v(N ) = u * i (v) + j∈N \{i} l * j (v) for all i ∈ N with u i > u * i (v).This implies that v(N ) Applying Lemma 3, invariance under claims truncation, and self-duality, and c j = c * j for all j ∈ N \ {i}.Moreover, for all j ∈ N \ {i},

Concluding remarks
In this paper, we introduced the large class of two-bound core games and provided explicit expressions of the nucleolus in terms of all pairs of bounds describing the core, using the Talmud rule for bankruptcy problems.Other values for two-bound core games are directly obtained by replacing the role of the Talmud rule in these expressions by any other bankruptcy rule.Quant et al. (2006) studied these extensions from a general point of view and paid particular attention to the specific random arrival rule (cf.O'Neill 1982).González-Díaz et al. (2005) followed a similar approach with a focus on the adjusted proportional rule (cf.Curiel et al. 1987).Future research could study extensions of these and other bankruptcy rules to the class of two-bound core games.
prove that [l, u] ∩ X(v) ⊆ C(v).In view of the convexity of the core, together with Lemma 1, it suffices to show that m l,u,σ (v) ∈ C(v) for all σ ∈ Π(N ).For all S ∈ 2 N \ {∅} and all σ ∈ Π(N ), This implies that