Advances in Decision Sciences

Advances in Decision Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 342089 | 14 pages | https://doi.org/10.1155/2009/342089

Convex Interval Games

Academic Editor: Graham Wood
Received23 Oct 2008
Accepted24 Mar 2009
Published23 Jun 2009

Abstract

Convex interval games are introduced and characterizations are given. Some economic situations leading to convex interval games are discussed. The Weber set and the Shapley value are defined for a suitable class of interval games and their relations with the interval core for convex interval games are established. The notion of population monotonic interval allocation scheme (pmias) in the interval setting is introduced and it is proved that each element of the Weber set of a convex interval game is extendable to such a pmias. A square operator is introduced which allows us to obtain interval solutions starting from the corresponding classical cooperative game theory solutions. It turns out that on the class of convex interval games the square Weber set coincides with the interval core.

1. Introduction

In classical cooperative game theory payoffs to coalitions of players are known with certainty. A classical cooperative game is a pair 𝑁,𝑣 where 𝑁={1,2,,𝑛} is a set of players and 𝑣2𝑁 is a map, assigning to each coalition 𝑆2𝑁 a real number, such that 𝑣()=0. Often, we also refer to such a game as a (transferable utility) TU game. We denote by 𝐺𝑁 the family of all classical cooperative games with player set 𝑁. The class of convex games [1] is one of the most interesting classes of cooperative games from a theoretical point of view as well as regarding its applications in real-life situations. A game 𝑣𝐺𝑁 is convex (or supermodular) if and only if the supermodularity condition 𝑣(𝑆𝑇)+𝑣(𝑆𝑇)𝑣(𝑆)+𝑣(𝑇) for each 𝑆,𝑇2𝑁 holds true. Many characterizations of classical convex games are available in literature ([2], Biswas et al. [3], Brânzei et al. [4], Martinez-Legaz [5, 6]). On the class 𝐶𝐺𝑁 of classical convex games solution concepts have nice properties; for details we refer the reader to Brânzei et al. [4]. Classical convex games have many applications in economic and real-life situations. It is well-known that classical public good situations [7], sequencing situations (Curiel et al. [8]), and bankruptcy situations ([9], Aumann and Maschler [10], Curiel et al. [11]) lead to convex games.

However, there are many real-life situations in which people or businesses are uncertain about their coalition payoffs. Situations with uncertain payoffs in which the agents cannot await the realizations of their coalition payoffs cannot be modelled according to classical game theory. Several models that are useful to handle uncertain payoffs exist in the game theory literature. We refer here to chance-constrained games (Charnes and Granot [12]), cooperative games with stochastic payoffs (Suijs et al. [13]), cooperative games with random payoffs (Timmer et al. [14]. In all these models probability and stochastic theory plays an important role.

This paper deals with a model of cooperative games where only bounds for payoffs of coalitions are known with certainty. Such games are called cooperative interval games. Formally, a cooperative interval game in coalitional form (Alparslan Gök et al. [15]) is an ordered pair 𝑁,𝑤 where 𝑁={1,2,,𝑛} is the set of players, and 𝑤2𝑁𝐼() is the characteristic function such that 𝑤()=[0,0], where 𝐼() is the set of all nonempty, compact intervals in . For each 𝑆2𝑁, the worth set (or worth interval) 𝑤(𝑆) of the coalition 𝑆 in the interval game 𝑁,𝑤 is of the form [𝑤(𝑆),𝑤(𝑆)]. We denote by 𝐼𝐺𝑁 the family of all interval games with player set 𝑁. Note that if all the worth intervals are degenerate intervals, that is, 𝑤(𝑆)=𝑤(𝑆) for each 𝑆2𝑁, then the interval game 𝑁,𝑤 corresponds in a natural way to the classical cooperative game 𝑁,𝑣 where 𝑣(𝑆)=𝑤(𝑆) for all 𝑆2𝑁.

Cooperative interval games are very suitable to describe real-life situations in which people or firms that consider cooperation have to sign a contract when they cannot pin down the attainable coalition payoffs, knowing with certainty only their lower and upper bounds. Such contracts should specify how the interval uncertainty with regard to the coalition values will be incorporated in the allocation of the worth 𝑤(𝑁) before its uncertainty is resolved, and how the realization of the payoff for the grand coalition 𝑅𝑤(𝑁) will be finally distributed among the players. Interval solution concepts for cooperative interval games are a useful tool to settle cooperation within the grand coalition via such (binding) contracts.

An interval solution concept on 𝐼𝐺𝑁 is a map assigning to each interval game 𝑤𝐼𝐺𝑁 a set of 𝑛-dimensional vectors whose components belong to 𝐼(). We denote by 𝐼()𝑁 the set of all such interval payoff vectors. An interval allocation obtained by interval solution concept commonly chosen by the players before the interval uncertainty with regard to the coalition values is removed offers at this ex-ante stage an estimation of what individual players may receive, between two bounds, when the uncertainty on the reward of the grand coalition is removed in the ex post stage. We notice that the agreement on a particular interval allocation (𝐼1,,𝐼𝑛) based on an interval solution concept merely says that the payoff 𝑥𝑖 that player 𝑖 will receive in the interim or ex post stage is in the interval 𝐼𝑖. This is a very weak contract to settle cooperation. Therefore, writing down in the contact how to transform the interval allocation of 𝑤(𝑁) into an allocation (𝑥1,,𝑥𝑛) in 𝑛 of the realization 𝑅 of 𝑤(𝑁), 𝑖𝑁𝑥𝑖=𝑅, in a consistent way with (𝐼1,,𝐼𝑛), that is, 𝐼𝑖𝑥𝑖𝐼𝑖 for each 𝑖𝑁, is mandatory (see Brânzei et al. [16]).

In this paper, we introduce the class of convex interval games and extend classical results regarding characterizations of convex games and properties of solution concepts to the interval setting. Some classical 𝑇𝑈-games associated with an interval game 𝑤𝐼𝐺𝑁 will play a key role, namely, the border games 𝑁,𝑤, 𝑁,𝑤 and the length game 𝑁,|𝑤|, where |𝑤|(𝑆)=𝑤(𝑆)𝑤(𝑆) for each 𝑆2𝑁. Note that 𝑤=𝑤+|𝑤|.

The paper is organized as follows. In Section 2 we recall basic notions and facts from the theory of cooperative interval games. In Section 3 we introduce supermodular and convex interval games and give basic characterizations of convex interval games. In Section 4 we introduce for size monotonic interval games the notions of interval marginal operators, the interval Shapley value and the interval Weber set and study their properties for convex interval games. Moreover, we introduce the notion of population monotonic interval allocation scheme (pmias) and prove that each element of the interval Weber set of a convex interval game is extendable to such a pmias. In Section 5 we introduce the square operator and describe some interval solutions for interval games that have close relations with existing solutions from the classical cooperative game theory. It turns out that on the class of convex interval games the interval core and the square interval Weber set coincide. Finally, in Section 6 we conclude with some remarks on further research.

2. Preliminaries on Interval Calculus and Interval Games

In this section some preliminaries from interval calculus and some useful results from the theory of cooperative interval games are given (Alparslan Gök et al. [17]).

Let 𝐼,𝐽𝐼() with 𝐼=[𝐼,𝐼], 𝐽=[𝐽,𝐽], |𝐼|=𝐼𝐼 and 𝛼+. Then,

(i)𝐼+𝐽=[𝐼,𝐼]+[𝐽,𝐽]=[𝐼+𝐽,𝐼+𝐽];(ii)𝛼𝐼=𝛼[𝐼,𝐼]=[𝛼𝐼,𝛼𝐼].

By (i) and (ii) we see that 𝐼() has a cone structure.

In this paper we also need a partial substraction operator. We define 𝐼𝐽, only if |𝐼||𝐽|, by 𝐼𝐽=[𝐼,𝐼][𝐽,𝐽]=[𝐼𝐽,𝐼𝐽]. Note that 𝐼𝐽𝐼𝐽. We recall that 𝐼 is weakly better than 𝐽, which we denote by 𝐼𝐽, if and only if 𝐼𝐽 and 𝐼𝐽. We also use the reverse notation 𝐼𝐽, if and only if 𝐼𝐽 and 𝐼𝐽. We say that 𝐼 is better than 𝐽, which we denote by 𝐼𝐽, if and only if 𝐼𝐽 and 𝐼𝐽.

For 𝑤1,𝑤2𝐼𝐺𝑁 we say that 𝑤1𝑤2 if 𝑤1(𝑆)𝑤2(𝑆), for each 𝑆2𝑁. For 𝑤1,𝑤2𝐼𝐺𝑁 and 𝜆+ we define 𝑁,𝑤1+𝑤2 and 𝑁,𝜆𝑤 by (𝑤1+𝑤2)(𝑆)=𝑤1(𝑆)+𝑤2(𝑆) and (𝜆𝑤)(𝑆)=𝜆𝑤(𝑆) for each 𝑆2𝑁. So, we conclude that 𝐼𝐺𝑁 endowed with is a partially ordered set and has a cone structure with respect to addition and multiplication with nonnegative scalars described above. For 𝑤1,𝑤2𝐼𝐺𝑁 with |𝑤1(𝑆)||𝑤2(𝑆)| for each 𝑆2𝑁, 𝑁,𝑤1𝑤2 is defined by (𝑤1𝑤2)(𝑆)=𝑤1(𝑆)𝑤2(𝑆).

Now, we recall that the interval imputation set (𝑤) of the interval game 𝑤, is defined by 𝐼(𝑤)=1,,𝐼𝑛𝐼()𝑁𝑖𝑁𝐼𝑖=𝑤(𝑁),𝐼𝑖𝑤(𝑖),𝑖𝑁,(2.1) and the interval core 𝒞(𝑤) of the interval game 𝑤, is defined by 𝐼𝒞(𝑤)=1,,𝐼𝑛(𝑤)𝑖𝑆𝐼𝑖𝑤(𝑆),𝑆2𝑁{}.(2.2)

Here, 𝑖𝑁𝐼𝑖=𝑤(𝑁) is the efficiency condition and 𝑖𝑆𝐼𝑖𝑤(𝑆), 𝑆2𝑁{}, are the stability conditions of the interval payoff vectors.

A game 𝑤𝐼𝐺𝑁 is called -balanced if for each balanced map 𝜆2𝑁{}+ we have 𝑆2𝑁{}𝜆(𝑆)𝑤(𝑆)𝑤(𝑁). We recall that a map 𝜆2𝑁{}+ is called a balanced map [18] if 𝑆2𝑁{}𝜆(𝑆)𝑒𝑆=𝑒𝑁. Here, 𝑒𝑁=(1,,1), and for each 𝑆2𝑁, (𝑒𝑆)𝑖=1 if 𝑖𝑆 and (𝑒𝑆)𝑖=0 otherwise. It is easy to prove that if 𝑁,𝑤 is -balanced then the border games 𝑁,𝑤 and 𝑁,𝑤 are balanced. A game 𝑤𝐼𝐺𝑁 is -balanced if and only if 𝒞(𝑤) (in Alparslan Gök et al. [17, Theorem 3.1]). We denote by 𝐵𝐼𝐺𝑁 the class of -balanced interval games with player set 𝑁.

Let 𝑤𝐼𝐺𝑁, 𝐼=(𝐼1,,𝐼𝑛),𝐽=(𝐽1,,𝐽𝑛)(𝑤) and 𝑆2𝑁{}. We say that 𝐼 dominates 𝐽 via coalition 𝑆, denoted by 𝐼dom𝑠𝐽, if

(i)𝐼𝑖𝐽𝑖 for all 𝑖𝑆;(ii)𝑖𝑆𝐼𝑖𝑤(𝑆).

For 𝑆2𝑁{} we denote by 𝐷(𝑆) the set of those elements of (𝑤) which are dominated via 𝑆. 𝐼 is called undominated if there does not exist 𝐽 and a coalition 𝑆 such that 𝐽dom𝑠𝐼. The interval dominance core 𝒟𝒞(𝑤) of 𝑤𝐼𝐺𝑁 consists of all undominated elements in (𝑤). For 𝑤𝐼𝐺𝑁 a subset 𝐴 of (𝑤) is an interval stable set if the following conditions hold.

(i)(Internal stability ) there does not exist 𝐼,𝐽𝐴 such that 𝐼 dom 𝐽 or 𝐽 dom 𝐼.(ii)(External stability ) for each 𝐼𝐴 there exist 𝐽𝐴 such that 𝐽 dom 𝐼.

It holds 𝒞(𝑤)𝒟𝒞(𝑤)𝐴 for all 𝑤𝐼𝐺𝑁 and 𝐴 a stable set of 𝑤.

3. Supermodular and Convex Interval Games

We say that a game 𝑁,𝑤 is supermodular if 𝑤(𝑆)+𝑤(𝑇)𝑤(𝑆𝑇)+𝑤(𝑆𝑇)𝑆,𝑇2𝑁.(3.1)

From formula (3.1) it follows that a game 𝑁,𝑤 is supermodular if and only if its border games 𝑁,𝑤 and 𝑁,𝑤 are supermodular (convex). We introduce the notion of convex interval game and denote by CIG𝑁 the class of convex interval games with player set 𝑁. We call a game 𝑤𝐼𝐺𝑁convex if 𝑁,𝑤 is supermodular and its length game 𝑁,|𝑤| is also supermodular. We straightforwardly obtain characterizations of games 𝑤CIG𝑁 in terms of 𝑤, 𝑤 and |𝑤|𝐺𝑁.

Proposition 3.1. Let 𝑤𝐼𝐺𝑁 and its related games |𝑤|,𝑤,𝑤𝐺𝑁. Then the following assertions hold. (i)A game 𝑁,𝑤 is convex if and only if its length game 𝑁,|𝑤| and its border games 𝑁,𝑤, 𝑁,𝑤 are convex.(ii)A game 𝑁,𝑤 is convex if and only if its border game 𝑁,𝑤 and the game 𝑁,𝑤𝑤 are convex.

We notice that the nonempty set CIG𝑁 is a subcone of 𝐼𝐺𝑁 and traditional convex games can be embedded in a natural way in the class of convex interval games because if 𝑣𝐺𝑁 is convex then the corresponding game 𝑤𝐼𝐺𝑁 which is defined by 𝑤(𝑆)=[𝑣(𝑆),𝑣(𝑆)] for each 𝑆2𝑁 is also convex. The next example shows that a supermodular interval game is not necessarily convex.

Example 3.2. Let 𝑁,𝑤 be the two-person interval game with 𝑤()=[0,0], 𝑤(1)=𝑤(2)=[0,1] and 𝑤(1,2)=[3,4]. Here, 𝑁,𝑤 is supermodular, but |𝑤|(1)+|𝑤|(2)=2>1=|𝑤|(1,2)+|𝑤|(). Hence, 𝑁,𝑤 is not convex.

The next example shows that an interval game whose length game is supermodular is not necessarily convex.

Example 3.3. Let 𝑁,𝑤 be the three-person interval game with 𝑤(𝑖)=[1,1] for each 𝑖𝑁, 𝑤(𝑁)=𝑤(1,3)=𝑤(1,2)=𝑤(2,3)=[2,2], and 𝑤()=[0,0]. Here, 𝑁,𝑤 is not convex, but 𝑁,|𝑤| is supermodular, since |𝑤|(𝑆)=0, for each 𝑆2𝑁.

Interesting examples of convex interval games are unanimity interval games. First, we recall the definition of such games. Let 𝐽𝐼() with 𝐽[0,0] and let 𝑇2𝑁{}. The unanimity interval game based on 𝐽 and 𝑇 is defined by 𝑢𝑇,𝐽[](𝑆)=𝐽,𝑇𝑆,0,0,otherwise,(3.2) for each 𝑆2𝑁.

Clearly, 𝑁,|𝑢𝑇,𝐽| is supermodular. The supermodularity of 𝑁,𝑢𝑇,𝐽 can be checked by considering the following case study: 𝑢𝑇,𝐽(𝐴𝐵)𝑢𝑇,𝐽(𝐴𝐵)𝑢𝑇,𝐽(𝐴)𝑢𝑇,𝐽[]𝐽[][]𝐽[].(𝐵)𝑇𝐴,𝑇𝐵𝐽𝐽𝐽𝐽𝑇𝐴,𝑇𝐵𝐽0,00,0𝑇𝐴,𝑇𝐵𝐽0,0][0,0𝑇𝐴,𝑇𝐵𝐽or0,0][0,0][0,0][0,0(3.3)

For convex TU-games various characterizations are known. In the next theorem we give some characterizations of convex interval games inspired by Shapley [1].

Theorem 3.4. Let 𝑤𝐼𝐺𝑁 be such that |𝑤|𝐺𝑁 is supermodular. Then, the following three assertions are equivalent: (i)𝑤𝐼𝐺𝑁 is convex;(ii)for all 𝑆1,𝑆2,𝑈2𝑁 with 𝑆1𝑆2𝑁𝑈 one has 𝑤𝑆1𝑆𝑈𝑤1𝑆𝑤2𝑆𝑈𝑤2;(3.4)(iii)for all 𝑆1,𝑆22𝑁 and 𝑖𝑁 such that 𝑆1𝑆2𝑁{𝑖} one has 𝑤𝑆1𝑆{𝑖}𝑤1𝑆𝑤2𝑆{𝑖}𝑤2.(3.5)

Proof. We show (i)(ii), (ii)(iii), (iii)(i).
Suppose that (i) holds. To prove (ii) take 𝑆1,𝑆2,𝑈2𝑁 with 𝑆1𝑆2𝑁𝑈. From (3.1) with 𝑆1𝑈 in the role of 𝑆 and 𝑆2 in the role of 𝑇 we obtain (3.4) by noting that 𝑆𝑇=𝑆2𝑈, 𝑆𝑇=𝑆1. Hence, (i) implies (ii).
That (ii) implies (iii) is straightforward (take 𝑈={𝑖}).
Now, suppose that (iii) holds. To prove (i) take 𝑆,𝑇2𝑁. Clearly, (3.1) holds if 𝑆𝑇. In case 𝑇𝑆, suppose that 𝑆𝑇 consists of the elements 𝑖1,,𝑖𝑘 and let 𝐷=𝑆𝑇. Then, 𝑤𝑖(𝑆)𝑤(𝑆𝑇)=𝑤𝐷1+𝑤(𝐷)𝑘𝑠=2𝑤𝑖𝐷1,,𝑖𝑠𝑖𝑤𝐷1,,𝑖𝑠1𝑖𝑤𝑇1+𝑤(𝑇)𝑘𝑠=2𝑤𝑖𝑇1,,𝑖𝑠𝑖𝑤𝑇1,,𝑖𝑠1=𝑤(𝑆𝑇)𝑤(𝑇),foreach𝑆2𝑁,(3.6)where the inequality follows from (iii).

Next we give as a motivating example a situation with an economic flavour leading to a convex interval game.

Example 3.5. Let 𝑁={1,2,,𝑛} and let 𝑓[0,𝑛]𝐼() be such that 𝑓(𝑥)=[𝑓1(𝑥),𝑓2(𝑥)] for each 𝑥[0,𝑛] and 𝑓(0)=[0,0]. Suppose that 𝑓1[0,𝑛], 𝑓2[0,𝑛] and (𝑓2𝑓1)[0,𝑛] are convex monotonic increasing functions. Then, we can construct a corresponding interval game 𝑤2𝑁𝐼() such that 𝑤(𝑆)=𝑓(|𝑆|)=[𝑓1(|𝑆|),𝑓2(|𝑆|)] for each 𝑆2𝑁. It is easy to show that 𝑤 is a convex interval game with the symmetry property 𝑤(𝑆)=𝑤(𝑇) for each 𝑆,𝑇2𝑁 with |𝑆|=|𝑇|.
We can see 𝑁,𝑤 as a production game if we interpret 𝑓(𝑠) for 𝑠𝑁 as the interval reward which 𝑠 players in 𝑁 can produce by working together.

Before closing this section we indicate some other economic and OR situations related to supermodular and convex interval games. In case the parameters determining sequencing situations are not numbers but intervals, under certain conditions also convex interval games appear (Alparslan Gök et al. [17, 19]). Bankruptcy situations when the estate of the bankrupt firm and the claims are intervals, under restricting conditions, give rise in a natural way to supermodular interval games which are not necessarily convex [20].

4. The Shapley Value, the Weber Set and Population Monotonic Allocation Schemes

We call a game 𝑁,𝑤  size monotonic if 𝑁,|𝑤| is monotonic, that is, |𝑤|(𝑆)|𝑤|(𝑇) for all 𝑆,𝑇2𝑁 with 𝑆𝑇. For further use we denote by SMIG𝑁 the class of size monotonic interval games with player set 𝑁. We notice that size monotonic games may have an empty interval core. In this section we introduce interval marginal operators on the class of size monotonic interval games, define the Shapley value and the Weber set on this class of games, and study their properties on the class of convex interval games.

Denote by Π(𝑁) the set of permutations 𝜎𝑁𝑁. Let 𝑤SMIG𝑁. We introduce the notions of interval marginal operator corresponding to 𝜎, denoted by 𝑚𝜎, and of interval marginal vector of 𝑤 with respect to 𝜎, denoted by 𝑚𝜎(𝑤). The marginal vector 𝑚𝜎(𝑤) corresponds to a situation, where the players enter a room one by one in the order 𝜎(1), 𝜎(2),,𝜎(𝑛) and each player is given the marginal contribution he/she creates by entering. If we denote the set of predecessors of 𝑖 in 𝜎 by 𝑃𝜎(𝑖)={𝑟𝑁𝜎1(𝑟)<𝜎1(𝑖)}, where 𝜎1(𝑖) denotes the entrance number of player 𝑖, then 𝑚𝜎𝜎(𝑘)(𝑤)=𝑤(𝑃𝜎(𝜎(𝑘)){𝜎(𝑘)})𝑤(𝑃𝜎(𝜎(𝑘))), or 𝑚𝜎𝑖(𝑤)=𝑤(𝑃𝜎(𝑖){𝑖})𝑤(𝑃𝜎(𝑖)). We notice that 𝑚𝜎(𝑤) is an efficient interval payoff vector for each 𝜎Π(𝑁). For size monotonic games 𝑁,𝑤, 𝑤(𝑇)𝑤(𝑆) is well defined for all 𝑆,𝑇2𝑁 with 𝑆𝑇 since |𝑤(𝑇)|=|𝑤|(𝑇)|𝑤|(𝑆)=|𝑤(𝑆)|. Now, we notice that for each 𝑤SMIG𝑁 the interval marginal vectors 𝑚𝜎(𝑤) are defined for each 𝜎Π(𝑁), because the monotonicity of |𝑤| implies 𝑤(𝑆{𝑖})𝑤(𝑆{𝑖})𝑤(𝑆)𝑤(𝑆), which can be rewritten as 𝑤(𝑆{𝑖})𝑤(𝑆)𝑤(𝑆{𝑖})𝑤(𝑆). So, 𝑤(𝑆{𝑖})𝑤(𝑆) is defined for each 𝑆𝑁 and 𝑖𝑆.

The following example illustrates that for interval games which are not size monotonic it might happen that some interval marginal vectors do not exist.

Example 4.1. Let 𝑁,𝑤 be the interval game with 𝑁={1,2}, 𝑤(1)=[1,3],𝑤(2)=[0,0] and 𝑤(1,2)=[2,3(1/2)]. This game is not size monotonic. Note that 𝑚(12)(𝑤) is not defined because 𝑤(1,2)𝑤(1) is undefined since |𝑤(1,2)|<|𝑤(1)|.

A characterization of convex interval games with the aid of interval marginal vectors is given in the following theorem.

Theorem 4.2. Let 𝑤𝐼𝐺𝑁. Then, the following assertions are equivalent: (i)𝑤 is convex;(ii)|𝑤| is supermodular and 𝑚𝜎(𝑤)𝒞(𝑤) for all 𝜎Π(𝑁).

Proof. (i)(ii) Let 𝑤CIG𝑁, let 𝜎Π(𝑁) and take 𝑚𝜎(𝑤). Clearly, we have 𝑘𝑁𝑚𝜎𝑘(𝑤)=𝑤(𝑁). To prove that 𝑚𝜎(𝑤)𝒞(𝑤) we have to show that for 𝑆2𝑁, 𝑘𝑆𝑚𝜎𝑘(𝑤)𝑤(𝑆). Let 𝑆={𝜎(𝑖1),𝜎(𝑖2),,𝜎(𝑖𝑘)} with 𝑖1<𝑖2<<𝑖𝑘. Then, 𝑤𝜎𝑖(𝑆)=𝑤1+𝑤()𝑘𝑟=2𝑤𝜎𝑖1𝑖,𝜎2𝑖,,𝜎𝑟𝜎𝑖𝑤1𝑖,𝜎2𝑖,,𝜎𝑟1𝑖𝑤𝜎(1),,𝜎1𝑖𝑤𝜎(1),,𝜎1+1𝑘𝑟=2𝑤𝑖𝜎(1),𝜎(2),,𝜎𝑟𝑖𝑤𝜎(1),𝜎(2),,𝜎𝑟=1𝑘𝑟=1𝑚𝜎𝜎(𝑖𝑟)(𝑤)=𝑘𝑆𝑚𝜎𝑘(𝑤),(4.1) where the inequality follows from Theorem 3.4. Further, by convexity of 𝑤, |𝑤| is supermodular.
(ii)(i) From 𝑚𝜎(𝑤)𝒞(𝑤) for all 𝜎Π(𝑁) follows that 𝑚𝜎(𝑤)𝐶(𝑤) and 𝑚𝜎(𝑤)𝐶(𝑤) for all 𝜎Π(𝑁). Now, by the well-known characterization of classical convex games with the aid of marginal vectors we obtain that 𝑁,𝑤 and 𝑁,𝑤 are convex games. Since 𝑁,|𝑤| is convex by hypothesis, we obtain by Proposition 3.1(i) that 𝑁,𝑤 is convex.

Now, we straightforwardly extend for size monotonic interval games two important solution concepts in cooperative game theory which are based on marginal worth vectors: the Weber set [21] and the Shapley value [22].

The interval Weber set 𝒲 on the class of size monotonic interval games is defined by 𝒲(𝑤)=conv{𝑚𝜎(𝑤)𝜎Π(𝑁)} for each 𝑤SMIG𝑁. We notice that for traditional TU-games we have 𝑊(𝑣) for all 𝑣𝐺𝑁, while for interval games it might happen that 𝒲(𝑤)= (in case none of the interval marginal vectors 𝑚𝜎(𝑤) is defined). Clearly, 𝒲(𝑤) for all 𝑤SMIG𝑁. Further, it is well-known that 𝐶(𝑣)=𝑊(𝑣) if and only if 𝑣𝐺𝑁 is convex. However, this result cannot be extended to convex interval games as we prove in the following proposition.

Proposition 4.3. Let 𝑤CIG𝑁. Then, 𝒲(𝑤)𝒞(𝑤).

Proof. By Theorem 4.2 we have 𝑚𝜎(𝑤)𝒞(𝑤) for each 𝜎Π(𝑁). Now, we use the convexity of 𝒞(𝑤).

The following example shows that the inclusion in Proposition 4.3 might be strict.

Example 4.4. Let 𝑁={1,2} and let 𝑤2𝑁𝐼() be defined by 𝑤(1)=𝑤(2)=[0,1] and 𝑤(1,2)=[2,4]. This game is convex. Further, 𝑚(1,2)(𝑤)=([0,1],[2,3]) and 𝑚(2,1)(𝑤)=([2,3],[0,1]), belong to the interval core 𝒞(𝑤) and 𝒲(𝑤)=conv{𝑚(1,2)(𝑤),𝑚(2,1)(𝑤)}. Notice that ([1/2,1(3/4)],[1(1/2),2(1/4)])𝒞(𝑤) and there is no 𝛼[0,1] such that 𝛼𝑚(1,2)(𝑤)+(1𝛼)𝑚(2,1)(𝑤)=([1/2,1(3/4)],[1(1/2),2(1/4)]). Hence, 𝒲(𝑤)𝒞(𝑤) and 𝒲(𝑤)𝒞(𝑤).

In Section 5 we introduce a new notion of Weber set and show that the equality between the interval core and that Weber set still holds on the class of convex interval games.

The interval Shapley value   ΦSMIG𝑁𝐼()𝑁 is defined by 1Φ(𝑤)=𝑛!𝜎Π(𝑁)𝑚𝜎(𝑤),foreach𝑤SMIG𝑁.(4.2)

Since Φ(𝑤)𝒲(𝑤) for each 𝑤SMIG𝑁, by Proposition 4.3 we have Φ(𝑤)𝒞(𝑤) for each 𝑤CIG𝑁. Without going into details we note here that the Shapley value Φ on the class of size monotonic interval games, and consequently on CIG𝑁, satisfies the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties for classical TU-games.

Proposition 4.5. Let 𝑤𝐼𝐺𝑁. If 𝑁,𝑤 is convex, then it is size monotonic.

Proof. Let 𝑤CIG𝑁. This assures that 𝑁,|𝑤| is supermodular which implies that 𝑁,|𝑤| is monotonic because for each 𝑆,𝑇2𝑁 with 𝑆𝑇 we have |𝑤|(𝑇)+|𝑤|()|𝑤|(𝑆)+|𝑤|(𝑇𝑆),(4.3) and from this inequality follows |𝑤|(𝑆)|𝑤|(𝑇) since |𝑤|(𝑇𝑆)0. So, CIG𝑁SMIG𝑁.

In the next two propositions we provide explicit expressions of the interval marginal vectors and of the interval Shapley value on SMIG𝑁.

Proposition 4.6. Let 𝑤SMIG𝑁 and let 𝜎Π(𝑁). Then, 𝑚𝜎𝑖(𝑤)=[𝑚𝜎𝑖(𝑤),𝑚𝜎𝑖(𝑤)] for all 𝑖𝑁.

Proof. By definition, 𝑚𝜎𝑤=𝑤(𝜎(1)),𝑤(𝜎(1),𝜎(2))𝑤(𝜎(1)),,𝑤(𝜎(1),,𝜎(𝑛))𝑤(,𝑚𝜎(1),,𝜎(𝑛1))𝜎𝑤=𝑤(𝜎(1)),𝑤(𝜎(1),𝜎(2))𝑤(𝜎(1)),,𝑤(𝜎(1),,𝜎(𝑛)).𝑤(𝜎(1),,𝜎(𝑛1))(4.4) Now, we prove that 𝑚𝜎(𝑤)𝑚𝜎(𝑤)0. Since |𝑤|=𝑤𝑤 is a classical convex game we have for each 𝑘𝑁𝑚𝜎𝜎(𝑘)𝑤𝑚𝜎𝜎(𝑘)𝑤=𝑤𝑤(𝜎(1),,𝜎(𝑘))𝑤𝑤(𝜎(1),,𝜎(𝑘1))=|𝑤|(𝜎(1),,𝜎(𝑘))|𝑤|(𝜎(1),,𝜎(𝑘1))0,(4.5) where the inequality follows from the monotonicity of |𝑤|. So, 𝑚𝜎𝑖(𝑤)𝑚𝜎𝑖(𝑤) for all 𝑖𝑁, and 𝑚𝜎𝑖(𝑤),𝑚𝜎𝑖(𝑤)𝑖𝑁=(𝑤(𝜎(1)),,𝑤(𝜎(1),,𝜎(𝑛))𝑤(𝜎(1),,𝜎(𝑛1)))=𝑚𝜎(𝑤).(4.6)

Since CIG𝑁SMIG𝑁 we obtain from Proposition 4.6 that 𝑚𝜎𝑖(𝑤)=[𝑚𝜎𝑖(𝑤),𝑚𝜎𝑖(𝑤)] for each 𝑤CIG𝑁, 𝜎Π(𝑁) and for all 𝑖𝑁.

Proposition 4.7. Let 𝑤SMIG𝑁 and let 𝜎Π(𝑁). Then, Φ𝑖(𝑤)=[𝜙𝑖(𝑤),𝜙𝑖(𝑤)] for all 𝑖𝑁.

Proof. From (4.2) and Proposition 4.6 we have for all 𝑖𝑁Φ𝑖1(𝑤)=𝑛!𝜎Π(𝑁)𝑚𝜎𝑖1(𝑤)=𝑛!𝜎Π(𝑁)𝑚𝜎𝑖𝑤,𝑚𝜎𝑖𝑤=1𝑛!𝜎Π(𝑁)𝑚𝜎𝑖𝑤,1𝑛!𝜎Π(𝑁)𝑚𝜎𝑖𝑤=𝜙𝑖𝑤,𝜙𝑖𝑤(4.7)

From Proposition 4.7 we obtain that for each 𝑤CIG𝑁 we have Φ𝑖(𝑤)=[𝜙𝑖(𝑤),𝜙𝑖(𝑤)] for all 𝑖𝑁.

In the sequel we introduce the notion of population monotonic interval allocation scheme (pmias) for totally -balanced interval games, which is a direct extension of pmas for classical cooperative games [23]. A game 𝑤𝐼𝐺𝑁 is called totally -balanced if the game itself and all its subgames are -balanced.

We say that for a game 𝑤𝑇𝐵𝐼𝐺𝑁 a scheme 𝐴=(𝐴𝑖𝑆)𝑖𝑆,𝑆2𝑁{} with 𝐴𝑖𝑆𝐼()𝑁 is a pmias of 𝑤 if

(i)𝑖𝑆𝐴𝑖𝑆=𝑤(𝑆) for all 𝑆2𝑁{};(ii)𝐴𝑖𝑆𝐴𝑖𝑇 for all 𝑆,𝑇2𝑁{} with 𝑆𝑇 and for each 𝑖𝑆.

Notice that the total -balancedness of an interval game is a necessary condition for the existence of a pmias for that game. A sufficient condition is the convexity of the interval game. We notice that all subgames of a convex interval game are also convex. In what follows we focus on pmias on the class of convex interval games.

We say that for a game 𝑤CIG𝑁 an imputation 𝐼=(𝐼1,,𝐼𝑛)(𝑤) is pmias extendable if there exist a pmias 𝐴=(𝐴𝑖𝑆)𝑖𝑆,𝑆2𝑁{} such that 𝐴𝑖𝑁=𝐼𝑖 for each 𝑖𝑁.

Theorem 4.8. Let 𝑤CIG𝑁. Then, each element 𝐼 of 𝒲(𝑤) is extendable to a pmias of 𝑤.

Proof. Let 𝑤CIG𝑁. First, we show that for each 𝜎Π(𝑁), 𝑚𝜎(𝑤) is extendable to a pmias. We know that the interval marginal operator 𝑚𝜎SMIG𝑁𝐼()𝑁 is efficient for each 𝜎Π(𝑁). Then, for each 𝑆2𝑁, 𝑖𝑆𝑚𝜎𝑖(𝑤𝑆)=𝑘𝑆𝑚𝜎𝜎(𝑘)(𝑤𝑆)=𝑤(𝑆) holds, where (𝑆,𝑤𝑆) is the corresponding (convex) subgame.
Further, by convexity, 𝑚𝜎𝑖(𝑤𝑆)𝑚𝜎𝑖(𝑤𝑇) for each 𝑖𝑆𝑇𝑁, where (𝑆,𝑤𝑆) and (𝑇,𝑤𝑇) are the corresponding subgames.
Second, each 𝐼𝒲(𝑤) is a convex combination of 𝑚𝜎(𝑤), 𝜎Π(𝑁), that is, 𝛼𝐼=𝜎𝑚𝜎(𝑤) with 𝛼𝜎[0,1] and 𝜎Π(𝑁)𝛼𝜎=1. Now, since all 𝑚𝜎(𝑤) are pmias extandable, we obtain that 𝐼 is pmias extendable as well.

From Theorem 4.8 we obtain that the total interval Shapley value, that is, the interval Shapley value applied to the game itself and all its subgames, generates a pmias for each convex interval game. We illustrate this in Example 4.9, where the calculations are based on Proposition 4.7.

Example 4.9. Let 𝑤CIG𝑁 with 𝑤()=[0,0], 𝑤(1)=𝑤(2)=𝑤(3)=[0,0], 𝑤(1,2)=𝑤(1,3)=𝑤(2,3)=[2,4] and 𝑤(1,2,3)=[9,15]. It is easy to check that the interval Shapley value generates for this game the pmias depicted as 𝑁{[][][][][][][][]1,2}{1,3}{2,3}{1}{2}{3}1233,5][3,5][3,51,2][1,21,21,21,2][1,20,00,00,0.(4.8)

5. Interval Solutions Obtained with the Square Operator

Let 𝑎=(𝑎1,,𝑎𝑛) and 𝑏=(𝑏1,,𝑏𝑛) with 𝑎𝑏. Then, we denote by 𝑎𝑏 the vector ([𝑎1,𝑏1],,[𝑎𝑛,𝑏𝑛])𝐼()𝑁 generated by the pair (𝑎,𝑏)𝑁. Let 𝐴,𝐵𝑁. Then, we denote by 𝐴𝐵 the subset of 𝐼()𝑁 defined by 𝐴𝐵={𝑎𝑏𝑎𝐴,𝑏𝐵,𝑎𝑏}.

With the use of the operator, we give a procedure to extend classical multisolutions on 𝐺𝑁 to interval multisolutions on 𝐼𝐺𝑁.

For a multisolution 𝐺𝑁𝑁 we define 𝐼𝐺𝑁𝐼()𝑁 by =(𝑤)(𝑤) for each 𝑤𝐼𝐺𝑁.

Now, we focus on this procedure for multisolutions such as the core and the Weber set on interval games. We define the square interval core 𝒞𝐼𝐺𝑁𝐼()𝑁 by 𝒞(𝑤)=𝐶(𝑤)𝐶(𝑤) for each 𝑤𝐼𝐺𝑁. We notice that a necessary condition for the non-emptiness of the square interval core is the balancedness of the border games.

Proposition 5.1. Let 𝑤𝐵𝐼𝐺𝑁. Then, 𝒞(𝑤)=𝒞(𝑤).

Proof. (𝐼1,,𝐼𝑛)𝒞(𝑤) if and only if (𝐼1,,𝐼𝑛)𝐶(𝑤) and (𝐼1,,𝐼𝑛)𝐶(𝑤) if and only if (𝐼1,,𝐼𝑛)=(𝐼1,,𝐼𝑛)(𝐼1,,𝐼𝑛)𝒞(𝑤).

Since CIG𝑁𝐵𝐼𝐺𝑁 we obtain that 𝒞(𝑤)=𝐶(𝑤)𝐶(𝑤) for each 𝑤CIG𝑁.

We define the square Weber set 𝒲𝐼𝐺𝑁𝐼()𝑁 by 𝒲(𝑤)=𝑊(𝑤)W(𝑤) for each 𝑤𝐼𝐺𝑁. Note that 𝒞(𝑤)=𝒲(𝑤) if 𝑤CIG𝑁.

The next two theorems are very interesting because they extend for interval games, with the square interval Weber set in the role of the Weber set, the well-known results in classical cooperative game theory that 𝐶(𝑣)𝑊(𝑣) for each 𝑣𝐺𝑁 [21] and 𝐶(𝑣)=𝑊(𝑣) if and only if 𝑣 is convex [24].

Theorem 5.2. Let 𝑤𝐼𝐺𝑁. Then, 𝒞(𝑤)𝒲(𝑤).

Proof. If 𝒞(𝑤)= the inclusion holds true. Suppose 𝒞(𝑤) and let (𝐼1,,𝐼𝑛)𝒞(𝑤). Then, by Proposition 5.4, (𝐼1,,𝐼𝑛)𝐶(𝑤) and (𝐼1,,𝐼𝑛)𝐶(𝑤), and, because 𝐶(𝑣)𝑊(𝑣) for each 𝑣𝐺𝑁, we obtain (𝐼1,,𝐼𝑛)𝑊(𝑤) and (𝐼1,,𝐼𝑛)𝑊(𝑤). Hence, we obtain (𝐼1,,𝐼𝑛)𝒲(𝑤).

From Theorem 5.2 and Proposition 4.3 we obtain that 𝒲(𝑤)𝒲(𝑤) for each 𝑤CIG𝑁. This inclusion might be strict as Example 4.4 illustrates.

Theorem 5.3. Let 𝑤𝐵𝐼𝐺𝑁. Then, the following assertions are equivalent: (i)𝑤 is convex;(ii)|𝑤| is supermodular and 𝒞(𝑤)=𝒲(𝑤).

Proof. By Proposition 3.1(i), 𝑤 is convex if and only if |𝑤|,𝑤 and 𝑤 are convex. Clearly, the convexity of |𝑤| is equivalent with its supermodularity. Further, 𝑤 and 𝑤 are convex if and only if 𝑊(𝑤)=𝐶(𝑤) and 𝑊(𝑤)=𝐶(𝑤). These equalities are equivalent with 𝒲(𝑤)=𝒞(𝑤). By Proposition 5.1 this is equivalent to 𝒞(𝑤)=𝒲(𝑤).

With the aid of Theorem 5.3 we will show that the interval core is additive on the class of convex interval games, which is inspired by Dragan et al. [25].

Proposition 5.4. The interval core 𝒞CIG𝑁𝐼()𝑁 is an additive map.

Proof. The interval core is a superadditive solution concept for all interval games (Alparslan Gök et al. [17]). We need to show the subadditivity of the interval core. We have to prove that 𝒞(𝑤1+𝑤2)𝒞(𝑤1)+𝒞(𝑤2). Note that 𝑚𝜎(𝑤1+𝑤2)=𝑚𝜎(𝑤1)+𝑚𝜎(𝑤2) for each 𝑤1,𝑤2CIG𝑁. By definition of the square interval Weber set we have 𝒲(𝑤1+𝑤2)=𝑊(𝑤1+𝑤2)𝑊(𝑤1+𝑤2). By Theorem 5.3 we obtain 𝒞𝑤1+𝑤2=𝒲𝑤1+𝑤2𝒲𝑤1+𝒲𝑤2𝑤=𝒞1𝑤+𝒞2.(5.1)

Finally, we define 𝒟𝒞(𝑤)=𝐷𝐶(𝑤)𝐷𝐶(𝑤) for each 𝑤𝐼𝐺𝑁 and notice that for convex interval games we have 𝒟𝒞(𝑤)=𝐷𝐶(𝑤)𝐷𝐶(𝑤)=𝐶(𝑤)𝐶(𝑤)=𝒞(𝑤)=𝒞(𝑤), where the second equality follows from the well-known result in the theory of TU-games that for convex games the core and the dominance core coincide, and the last equality follows from Proposition 5.1. From 𝒟𝒞(𝑤)=𝒞(𝑤) for each 𝑤CIG𝑁 and 𝒞(𝑤)𝒟𝒞(𝑤) for each 𝑤𝐼𝐺𝑁 we obtain 𝒟𝒞(𝑤)𝒟𝒞(𝑤) for each 𝑤CIG𝑁. We notice that this inclusion might be strict (Alparslan Gök et al. [17], Example 4.1).

6. Concluding Remarks

In this paper we define and study convex interval games. We note that the combination of Theorems 3.4, 4.2 and 5.3 can be seen as an interval version in (Brânzei et al. [4], Theorem 96). In fact these theorems imply (Brânzei et al. [4], Theorem 96) for the embedded class of classical TU-games. Extensions to convex interval games of the characterizations of classical convex games where exactness of subgames and superadditivity of marginal (or remainder) games play a role (Biswas et al. [3], Brânzei et al. [26] and Martinez-Legaz [5, 6]) can be found in (Brânzei et al. [27]).

There are still many interesting open questions. For further research it is interesting to study whether one can extend to interval games the well-known result in the traditional cooperative game theory that the core of a convex game is the unique stable set [1]. It is also interesting to find an axiomatization of the interval Shapley value on the class of convex interval games. An axiomatic characterization of the interval Shapley value on a special subclass of convex interval games can be found in Alparslan Gök et al. [15]. Other topics for further research could be related to introducing new models in cooperative game theory by generalizing cooperative interval games. For example, the concepts and results on (convex) cooperative interval games could be extended to cooperative games in which the coalition values 𝑤(𝑆) are ordered intervals of the form [𝑢,𝑣] of an (infinite dimensional) ordered vector space. Such generalization could give more applications to the interval game theory. Also to establish relations between convex interval games and convex games in other existing models of cooperative games could be interesting. One candidate for such study could be convex games in cooperative set game theory [28].

Acknowledgments

Alparslan Gök acknowledges the support of TUBITAK (Turkish Scientific and Technical Research Council) and hospitality of Department of Mathematics, University of Genoa, Italy. Financial support from the Government of Spain and FEDER under project MTM2008-06778-C02-01 is gratefully acknowledged by R. Branzei and S. Tijs. The authors thank Elena Yanovskaya for her valuable comments. The authors greatfully acknowledge two anonymous referees.

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