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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.

In classical cooperative game theory payoffs to coalitions of players are known with certainty. A classical cooperative game is a pair

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 [

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 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

An interval solution concept

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

The paper is organized as follows. In Section

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. [

Let

By (i) and (ii) we see that

In this paper we also need a partial substraction operator. We define

For

Now, we recall that the interval imputation set

Here,

A game

Let

For

(

(

It holds

We say that a game

From formula (

Let

A game

A game

We notice that the nonempty set

Let

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

Let

Interesting examples of convex interval games are unanimity interval games. First, we recall the definition of such games. Let

Clearly,

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

Let

for all

for all

We show (i)

Suppose that (

That (ii) implies (iii) is straightforward (take

Now, suppose that (iii) holds. To prove (i) take

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

Let

We can see

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. [

We call a game

Denote by

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

Let

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

Let

(i)

(ii)

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 [

The

Let

By Theorem

The following example shows that the inclusion in Proposition

Let

In Section

The

Since

Let

Let

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

Let

By definition,

Since

Let

From (

From Proposition

In the sequel we introduce the notion of

We say that for a game

Notice that the total

We say that for a game

Let

Let

Further, by convexity,

Second, each

From Theorem

Let

Let

With the use of the

For a multisolution

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

Let

Since

We define the square Weber set

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

Let

If

From Theorem

Let

By Proposition

With the aid of Theorem

The interval core

The interval core is a superadditive solution concept for all interval games (Alparslan Gök et al. [

Finally, we define

In this paper we define and study convex interval games. We note that the combination of Theorems

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 [

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.