Uncertainty and inaccurate data are an everyday issue in real-world situations. Therefore it is important to be able to make decisions even when the exact data are not available and only bounds on them are known.
In classical cooperative game theory, every group of players (coalition) knows the precise reward for their cooperation; in cooperative interval games, only the worst and the best possible outcome is known. Such situations can be naturally modeled with intervals encapsulating these outcomes.
Cooperative games under interval uncertainty were first considered by Branzei, Dimitrov and Tijs in 2003 to study bankruptcy situations  and later further extensively studied by Alparslan Gök in her PhD thesis  and in follow-up papers (see the references section of  ).
We note that there are several other models incorporating a different level of uncertainty, namely fuzzy cooperative games, multichoice games, crisp games (see  for more), or games under bubbly uncertainty .
There are several reasons why it is interesting to study cooperative interval game. From the aforementioned models of cooperative games, it is a quite simple model but it is easier to analyze and it is suitable for situations where we do not have any other assumptions on data we get. There are already a few applications of this model, namely on forest situations , airport problems , bankruptcy  or network design .
Here is a summary of our results and also how our paper is organized.
Section 4 investigates a problem of core coincidence, i.e. when the two different versions of generalized core for interval games coincide. We partially solve this problem.
2.1 Classical cooperative games
Comprehensive sources on classical cooperative game theory are for example [12, 14, 16, 22]. For more on applications, see e.g. [7, 13, 18]. Here we present only the necessary background theory for studying interval games. We examine only the games with transferable utility (TU) and therefore by a cooperative game or a game we mean a cooperative TU game.
The set of all cooperative games with a player set is denoted by . Subsets of are called coalitions and itself is called the grand coalition. We often write instead of because we can identify a game with its characteristic function.
To further analyze players’ gains, we need a payoff vector
payoff vectorwhich can be interpreted as a proposed distribution of rewards between players.
(Payoff vector) A payoff vector for a cooperative game is a vector with being a reward given to the th player.
An imputation ofis a vector such that and for every .
The set of all imputations of a given game is denoted by .
(Core) The core of is the set
The last solution concept we will write about is the Shapley value. It was introduced by Lloyd Shapely in 1952 . It has many interesting properties; namely, it is a one-point solution concept, it always exists and it can be axiomatized by very natural axioms. We refer to  for a survey of results on the Shapley value.
(Shapley, 1952, ) There exists a unique function , satisfying the following properties for every .
(Efficiency) It holds that .
(Dummy player) It holds for every , such that for every , equality holds.
(Symmetry) If for every ,
holds, then .
(Additivity) For every two games , holds.
This unique function is called the Shapley value () and it is defined as
There are many important classes of cooperative games. Here we show the most important ones.
(Monotonic game) A game is monotonic if for every we have
Informally, in monotonic games, bigger coalitions are stronger.
Another important type of game is a convex game.
(Convex game) A game is convex if its characteristic function is supermodular. The characteristic function is supermodular if for every ,
Clearly, supermodularity implies superadditivity. The class of convex games is maybe the most prominent class, it has many applications and theoretical properties. We present the most important one for this paper.
(Shapley, 1971, ) Every convex game has a nonempty core.
2.2 Interval analysis
(Interval) An interval is a set
with being the lower bound and being the upper bound of the interval. The length of an interval is defined as .
From now on, by an interval we mean a closed interval. The set of all real closed intervals is denoted by .
The following definition (from ) shows how to do basic arithmetics with intervals.
(Interval arithmetics) For every , and , define
For our purpose, we need to have a slightly different definition of subtraction. The aforementioned subtraction operator is known as Moore’s subtraction operator.
(Partial subtraction operator) For every , such that , define
In other words, the length of the subtracted interval has to be lesser or equal to the length of the interval we subtract from.
Take two intervals and . Then . Notice, however, that is undefined.
We note that this notation is not common in interval analysis. The minus sign is used for Moore’s subtraction operator there. Also, in our previous paper [8, 9] we used minus sign for Moore’s subtraction.
(Weakly better operator ) An interval is weakly better than interval () if and . Interval is better than () if and only if and .
Naturally, we also use and for and , respectively.
2.3 Cooperative interval games
Now we review basics of cooperative games with interval uncertainty. The following is the main definition of this paper.
(Cooperative interval game) A cooperative interval game is an ordered pair , where is a set of players and is the characteristic function of the cooperative game. We further assume that . The set of all interval cooperative games on a player set is denoted by .
We often write instead of and instead of .
Every cooperative interval game in which its characteristic function maps to degenerate intervals only can be associated with a classical cooperative game. The converse holds as well.
(Border games) For every , border games (lower border game) and (upper border game) are given by and for every .
(Length game) The length game of is the game with
(Degenerated game) We call a game degenerated if its length game is everywhere zero, that is, for every . A non-degenerated game is a game which is not degenerated.
The basic notion of our approach will be a selection and consequently a selection imputation and a selection core.
(Selection) A game is a selection of if for every we have . The set of all selections of is denoted by .
Note that border games are examples of selections and also of degenerated games.
There are many possibilities how to define imputations and core for interval games. We present the following two. The first one is based on selections, the second one on the weakly better operator.
The set of interval selection imputations (or just selection imputations) of is defined as
The interval selection core (or just selection core) of is defined as
In an analogous way as in classical games, we have a term for games with nonempty selection core for all selections.
 An interval game is called strongly balanced if every selection of this game has a nonempty core. The set of all strongly balanced games on a player set is denoted by BIG.
The set of interval imputations of is defined as
The interval core of is defined as
An important difference between the definitions of interval and selection core and imputation is that selection concepts yield payoff vectors from , while and yield vectors from . Thus they both possess a different degree of uncertainty.
Classes of interval games.
(Size monotonic interval game) A game is size monotonic if for every we have
That is, its length game is monotonic. The class of size monotonic games on a player set is denoted by .
As we can see, size monotonic games capture situations in which an interval uncertainty grows with the size of a coalition.
We should be careful with the following analogy of a convex game since unlike for classical games, supermodularity is not the same as convexity.
(Supermodular interval game) An interval game is supermodular interval if for every holds
We get immediately that an interval game is supermodular interval if and only if its border games are convex.
(Convex interval game) An interval game is convex interval if its border games and length game are convex. We write for a set of convex interval games on a player set .
A convex interval game is supermodular as well but the converse does not hold in general. See  for characterizations of convex interval games and discussion of their properties.
Finally, we define selection based classes of interval games. The paper  discusses their properties and relations with the previous classes.
(Selection monotonic interval game) An interval game is selection monotonic if all its selections are monotonic games. The class of such games on a player set is denoted by .
(Selection convex interval game) An interval game is selection convex if all its selections are convex games. The class of such games on a player set is denoted by .
We will use relation on real vectors. For every we write if holds for every .
We do not use symbol in this paper. Instead, and are used for the subset and the proper subset relation, respectively, to avoid ambiguity.
We also use instead of occasionally.
Throughout the papers on cooperative interval games, notation, especially of core and imputations, is not unified. It is, therefore, possible to encounter different notation from ours. Also, in some papers the selection core is called the core of interval game. We consider that confusing and that is why we use the term selection core instead. The term selection imputation is then used because of its connection with the selection core.
For every interval game , the following assertions are equivalent.
The game is a selection convex interval game.
For every nonempty , such that , and ,
For every coalition , such that , and is nonempty,
For every coalition , and for every , such that ,
This proof is very similar to the proof of Theorem 2 in .
Suppose for a contradiction that there exist , nonempty, such that , and
Define , and . Both and are nonempty sets and they are incomparable. Furthermore:
And we obtained a contradiction.
Straightforward; take .
Suppose that (4) holds and (3) does not. Take that violates (3) of minimal cardinality. If , we get a contradiction. If , we can construct , with , such that it violates (3) as well. This contradicts the minimality of .
For a contradiction, take and which violate (2). Define ; this must be nonempty since S and T are nonempty and incomparable. Define and . As for the conditions on and , we see that , since is nonempty and . Now:
A contradiction. ∎
4 Core coincidence
“A difficult topic might be to analyze under which conditions the set of payoff vectors generated by the interval core of a cooperative interval game coincides with the core of the game in terms of selections of the interval game.”
The main thing to notice is that while the interval core gives us a set of interval vectors, selection core gives us a set of real numbered vectors. To be able to compare them, we need to assign to a set of interval vectors a set of real vectors generated by these interval vectors. That is exactly what the following function does.
The function maps to every set of interval vectors a set of its selections. It is defined as
The core coincidence problem can be formulated in the following way.
(Core coincidence problem) What are the necessary and sufficient conditions so that an interval game satisfies ?
To avoid a cumbersome notation we define the following property.
Let be a cooperative interval game. We call the game core-coincident if . Also, we say that a set of interval games is core-coincident if all games in this set are core-coincident.
Our results in this section are an important step towards a complete classification of core-coincident games.
4.1 Positive results
Every cooperative interval game with empty selection core is core-coincident.
This easily follows from [8, Theorem 7]. ∎
Every degenerated cooperative interval game is core-coincident.
We present the following example, showing there exist infinitely many core- coincident non-degenerated games with a nonempty player set. But first, we need one more result.
(Core coincidence technical lemma, ) For every interval game we have , if and only if for every , there exist nonnegative vectors and , such that
There are infinitely many non-degenerated core-coincident interval games.
Define a game , , if , and further .
Clearly, consists exactly of vectors , such that , and .
Take any such vector . Define , and , and , for every . It is now straightforward to check that all inequalities of Theorem 4.1 hold and, therefore, this game is core-coincident. ∎
4.2 Negative results
Let be an interval game such that:
a game , defined by
has a nonempty core, and
Then is not core-coincident.
We define an excess function as .
If for every , and every player , there is a coalition , such that , then we claim that the core of the upper border game of is empty.
To see this, observe that . But, if every has the aforementioned property then, by Theorem 4.1 none of those can be in ; a contradiction with .
So the other option is that there exists a vector , and a player , such that for every , . We define
and the set on which this minimum is attained. We pick an arbitrary player . Such player must exist. Then we construct a new vector :
It can be checked that and by a similar argument as in the previous case, does not satisfy the mixed system of inequalities in Theorem 4.1 and we are done. ∎
This theorem has several important corollaries.
Every interval game with for every , is not core-coincident.
Classes and are not core-coincident for . Furthermore, every game in with every interval non-degenerated and is not core-coincident.
Observe that , so we immediately obtain that all these sets are not core-coincident as well for nontrivial player sets. Also, , so superadditive interval games are not core-coincident either.
From this we conclude that selection core and interval core behave differently on many important and widely used classes with nontrivial uncertainty. Therefore, to further develop theory and solve problems regarding both versions of cores of interval games is an important task.
5 The Shapley value
Preliminaries and definitions.
Before we list the axioms we need in this section, a few definitions are needed.
Every function is called interval value function. We omit interval when context is clear.
Two intervals are said to be indifferent if . We denote it by .
Let and . Then, is called a null player in if for every .
Let and . Then, is called a total null player in if for every . In other words, is total null player if it is a dummy player in every selection.
Let and . Then, and are symmetric players in , if for every .
We can state a few axioms.
Indifference efficiency (IEFF): for all .
Efficiency (EFF): for all .
Indifference null player property (INP): There exists a unique such that for any and all null players in .
Total null player property (TNP): For every total null player in a game , .
Symmetry (SYM): for all and all symmetric players and in .
Additivity (ADD): for all with .
The interval Shapley value extension is a value function ,
 The function satisfies axioms IEFF, INP, SYM, and ADD. Furthermore, it is the only function satisfying these axioms.
We now prove an important, yet never noted and proved property of the interval Shapley value extension.
For every interval game , we have
In other words, the interval Shapley value extension contains exactly all possible Shapley values that can be attained when uncertainty is settled. We find this property very important.
However, as is noted in , efficiency is not always satisfied. Let us explain this issue. From properties of interval arithmetics, we see that is not equal to in general for . In fact, for every interval , . An analogous fact holds for Moore’s subtraction as well. Since, by definition of the interval Shapley value extension, in are some intervals added and subtracted multiple times, the resulting value does not satisfy efficiency if we first compute for every , and only then add them together. This is the reason why EFF is not satisfied in general. We can first simplify and only then add it together. Then we would get the efficiency by the same reasoning as we get efficiency for the Shapley value in classical games.
The following theorem shows a different axiomatization of the interval Shapley value extension than . We show that the axiom TNP can be interchanged with the axiom INP, which is, from our point of view, more natural.
There is a unique value function satisfying axioms IEFF, TNP, SYM and ADD. Furthermore, it equals .
If a value function satisfies IEFF, INP, SYM, and ADD, then it is equal to . From its formula, we conclude that TNP is satisfied.
Now in the other direction, if a value function satisfies IEFF, TNP, SYM and ADD we want to show that it satisfies INP as well.
Our goal is to prove that in every game with a null player , is an interval symmetric around zero.
It suffices to prove that:
If , then , and
if , and , then .
Both of these claims can be proved by using ADD axiom and the fact, that on degenerated game, coincides with . We omit technical details here.
We know that every null player gets a symmetrical interval under a value function satisfying IEFF, TNP, SYM and ADD. So the only remaining option is that there must exist a game in which two null players get a different symmetrical interval. Let us denote such game as and the two null players as and .
Observe from the definition of null player that
holds for every , and thus, specially, for every . Following the same reasoning, we arrive on conclusion that
holds for every . Combining this, we get that
That means that and are symmetrical and from the axiom SYM, should be equal to , a contradiction.
For every , we have .
From Theorem 5.2, the Shapley value of every selection is in . Since every selection of is a convex game, its Shapley value lies in its core and thus also in ∎
On the improved interval Shapley-like value.
In Han et al. , an improved Shapley-like value satisfying EFF is presented.
(The improved interval Shapley-like value) For any with , the improved interval Shapley like value is defined by
For every interval game , we have
We believe that this is a big downside of the improved interval Shapley-like value. We borrow a game from  to illustrate the theorem.
Let be a three-person interval game where , and . Then . However, .
By Theorem 5.5, there must be a selection of , such that . But this value is not contained in .
6 Conclusion and future research
We investigated convexity in interval games, core coincidence problem and interval Shapley value. To this end, we would like to summarize our results.
We showed a Shapley-like characterization of selection convex interval games in Theorem 3.1.
We tried to characterize all core-coincident games. Our main contribution is Theorem 4.3, saying that a large class of interval games is not core-coincident. This result implies that many classes, including CIG, SeCIG, and strongly balanced games are not core-coincident.
We analyzed interval Shapley value extension for interval games. We emphasized several facts which speak in favor of using this solution concept. Also, we showed a different, from our point of view more natural axiomatization of this value function in Theorem 5.3.
Apart from the open problems presented in the papers from the references we think it could be interesting to define prekernel for interval games and axiomatically characterize it, analogously to Peleg . Also, interval games with communication structures were not studied yet. See Bilbao’s book  for a theoretical background.
The author would like to acknowledge the support of GAČR P403-18-04735S of the Czech Science Foundation and the support of the grant SVV–2017–260452.
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