Convexity and positivity in partially defined cooperative games
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Convexity and positivity in partially definedcooperative games
Jan Bok · Martin ˇCern´y · DavidHartman · Milan Hlad´ık
Received: date / Accepted: date
Abstract
Partially defined cooperative games are a generalisation of classicalcooperative games in which payoffs for some of the coalitions are not known. Inthis paper we perform a systematic study of partially defined games, focusingon two important classes of cooperative games: convex games and positivegames.In the first part, we focus on convexity and give a polynomially decidablecondition for extendability and a full description of the set of symmetric convexextensions. The extreme games of this set, together with the lower game and
Jan BokComputer Science Institute, Faculty of Mathematics and Physics, Charles University,Prague, Malostransk´e n´am. 25, 118 00 Praha 1, Czech Republic.ORCID: 0000-0002-7973-1361E-mail: [email protected]ff.cuni.czThis author is the corresponding author.Martin ˇCern´yDepartment of Applied Mathematics, Faculty of Mathematics and Physics, Charles Univer-sity, Prague, Malostransk´e n´am. 25, 118 00 Praha 1, Czech Republic.ORCID: 0000-0002-0619-4737E-mail: [email protected]ff.cuni.czMilan Hlad´ıkDepartment of Applied Mathematics, Faculty of Mathematics and Physics, Charles Univer-sity, Prague, Malostransk´e n´am. 25, 118 00 Praha 1, Czech Republic.ORCID: 0000-0002-7340-8491E-mail: [email protected]ff.cuni.czDavid HartmanComputer Science Institute, Faculty of Mathematics and Physics, Charles University,Prague, Malostransk´e n´am. 25, 118 00 Praha 1, Czech Republic and Institute of Com-puter Science of the Czech Academy of Sciences, Pod Vod´arenskou vˇeˇz´ı 271/2, Prague 8,Czech RepublicORCID: 0000-0003-3566-8214E-mail: [email protected]ff.cuni.cz a r X i v : . [ c s . G T ] O c t Jan Bok et al. the upper game, are also described. In the second part, we study positivity.We characterise the non-extendability to a positive game by existence of acertificate and provide a characterisation for the extreme games of the setof positive extensions. We use both characterisations to describe the positiveextensions of several classes of incomplete games with special structures.Our results complement and extend the existing theory of partially definedcooperative games. We provide context to the problem of completing partialfunctions and, finally, we outline an entirely new perspective on a connectionbetween partially defined cooperative games and cooperative interval games.
Keywords
Cooperative games · Partially defined games · Upper game · Lower game · Convex games · Totally monotonic games
Mathematics Subject Classification (2010)
Developing various approaches to deal with uncertainty is heavily intertwinedwith decision making and game theory. It is only natural since inaccuracy indata is an everyday problem in real-world situations, be it a lack of knowledgeon the behaviour of others, corrupted data, signal noise or a prediction ofoutcomes such as voting or auctions. Since the degree of applications is sowide, an abundance of models of various complexity and use-case scenariosexist. Among those models most relevant to the theory of partially definedcooperative games, there are fuzzy cooperative games [7,19,20], multi-choicegames [7], cooperative interval games [1,2,5], fuzzy interval games [18], gamesunder bubbly uncertainty [25], ellipsoidal games [32], and games based on greynumbers [24].In the classical cooperative game theory, every group of players, called coali-tion , knows the precise reward (or payoff) for the cooperation of its members.In partially defined cooperative games, this is generally no longer true, sinceonly some of the coalitions know their values, while the others do not. Thismodels the uncertainty over data and its consistency. The theme was first in-troduced in literature by Wilson [33] in 1993. Wilson gave the basic notionof incomplete game and a generalised definition of the Shapley value for suchgames. After two decades, Inuiguchi and Masuya revived the research. In [23],they focused mainly on the class of superadditive games (and also briefly men-tioned particular cases of convex and positive games in which precisely thevalues of singleton coalitions and grand coalition are known. Subsequently,Masuya pursued this line of research in [21] where he discussed a more generalsubclass of superadditive games than in the previous paper and concentratedefforts on the Shapley value. Finally, in [22], Masuya focused on superadditivegames and its Shapley value in full generality, with special attention given toconsistency axioms. Apart from that, Yu [34] introduced a generalisation of onvexity and positivity in partially defined cooperative games 3 incomplete games to the games with coalition structures and studied the pro-portional Owen value (which is a generalisation of the Shapley value for thesegames). Unfortunately, for the general public the paper of Yu is published onlyin Chinese.We note that partially defined cooperative games can also be viewed as partialfunctions defined on a power set and, indeed, such structures were also stud-ied, yet without highlighting the connection to the theory of partially definedcooperative games. We will base some of our results on several publicationson partial functions, namely on [28,30,31]. We refer to the excellent book ofGrabisch [14] which discusses in a great detail connections of various typesof set functions to entirely different parts of mathematics, with game theorybeing one of them.Our results concern two important classes of games: convex games and theirsubclass, games with even stronger properties – positive games . Let us nowhighlight the structure and main contributions of this paper. – In Section 2, we outline the necessary background of the cooperative gametheory and also introduce fundamental definitions of partially defined co-operative games, both needed further in the text. – Section 3 is dedicated to convexity. We first study extensions of convexgames in general by using the existing results in [30]. Then we proceed toanalyse symmetric convex extensions. We characterise under which condi-tions an incomplete game is extendable into a symmetric convex extension,provide the range of each coalition’s worth over all such possible extensionsand fully describe the set of symmetric convex extensions as a set of convexcombinations of its extreme games . We also provide a geometrical point ofview on the set of symmetric convex extensions. – In Section 4 we focus on positivity. We give a characterisation of non-extendability by existence of a certificate and show it can be used to decidethe extendability in linear time in the number of players N for incompletegames with a special structure. As one of our main results, we prove acharacterisation of extreme games (Theorem 8) of the set of positive ex-tensions and apply it for two special cases of incomplete games, obtaining adescription of the extreme games, the lower game and the upper game. Weconclude this section with our results on symmetric positive extensions. – Finally, in Section 5 we draw our attention to connections between the the-ory of partially defined cooperative games and cooperative interval games.
Jan Bok et al.
Definition 1 A cooperative game is an ordered pair ( N, v ), where N is a finiteset of players { , , . . . , n } and v : 2 N → R is a characteristic function of thecooperative game. We further assume that v ( ∅ ) = 0.Subsets of N are called coalitions and N itself is called the grand coalition .We often write v instead of ( N, v ) whenever there is no confusion over whatthe player set is.We note that the presented definition assumes transferable utility (shortlyTU). Therefore, by a cooperative game or a game we mean in fact a cooperativeTU game.To avoid cumbersome notation, we use the following abbreviations. Instead of S ∪ { i } , we use simply S + i and analogously, instead of S \ { i } , we use S − i .Also, we often replace singleton set { i } with just i . We use ⊆ for the relationof ”being a subset of” and (cid:40) for the relation ”being a proper subset of”.Now we focus on definitions of several important classes of cooperative games. Definition 2
A cooperative game (
N, v ) is – monotonic if for every T ⊆ S ⊆ N we have v ( T ) ≤ v ( S ), – superadditive if for every S, T ⊆ N such that S ∩ T = ∅ holds that v ( S ) + v ( T ) ≤ v ( S ∪ T ), or – convex if its characteristic function is supermodular. The characteristicfunction is supermodular if for every S, T ⊆ N , v ( T ) + v ( S ) ≤ v ( S ∪ T ) + v ( S ∩ T ) . Each of these classes incorporates a different approach to formalising the con-cept of bigger coalitions being stronger. Clearly, supermodularity implies su-peradditivity. Since the characteristic function of convex games is supermodu-lar, convex games are also often called supermodular games in literature. Theclass of convex games is maybe the most prominent class in cooperative gametheory since it has many applications and it enjoys both elegant and powerfulcharacterisations. Among them, the following characterisation by Shapley willbe necessary for proving our results. onvexity and positivity in partially defined cooperative games 5
Theorem 1 [29] A cooperative game ( N, v ) is convex if and only if for every i ∈ N and every S ⊆ T ⊆ N \{ i } , it holds that v ( S + i ) − v ( S ) ≤ v ( T + i ) − v ( T ) . Another important class is that of totally monotonic games , also called as pos-itive games (from now on, we will refer to them by the latter name). As thefirst name of this class suggests, the concept generalises the notion of mono-tonicity in cooperative games. Positive games are also a subclass of convexgames. The results concerning positive games are relatively sparse and scat-tered. We would like to refer the reader to [8,12,15,27]. Most importantly, inthe definition of this class, we shall employ the notion of unanimity games and
Harsanyi dividend . Definition 3
For any T ⊆ N, T (cid:54) = ∅ , the unanimity game ( N, u T ) is definedas u T ( S ) := (cid:40) T ⊆ S, Definition 4
For every finite set N and T ⊆ N , the Harsanyi dividend d v ( T )of a game ( N, v ) is defined as d v ( T ) := (cid:88) S ⊆ T ( − | T |−| S | v ( S ) . Observe that according to the definition, d v ( ∅ ) is always equal to zero.The set of all games on a player set N forms a vector space. The importanceof unanimity games is that they form a basis of this space and for a givencooperative game, the coefficients of the linear combination with respect tothis basis which gives the game correspond to its Harsanyi dividends, i.e. v = (cid:80) T ⊆ N,T (cid:54) = ∅ d v ( T ) u T .The dividend d v ( T ) has another interesting property; it is also the value cor-responding to T in M¨obius transform of value v ( T ) (see [14] for further dis-cussion). Another way to represent d v is the following. Let d v ( T ) = , if T = ∅ ,v ( i ) , if T = i for i ∈ N,v ( T ) − (cid:80) S (cid:40) T d v ( S ) , if T ⊆ N, | T | > . We can finally define positivity for cooperative games.
Definition 5
A cooperative game (
N, v ) is said to be positive if the Harsanyidividend d v ( T ) is non-negative for all T ⊆ N .As we will see later, the general case of both convex and positive games yieldsvery complex situations and so we shall occasionally restrict ourselves to sim-pler subclasses of these games. One of our approaches are games with the Jan Bok et al. additional property of symmetry. A game (
N, v ) is said to be symmetric if forevery
S, T ⊆ N such that | S | = | T | holds that v ( S ) = v ( T ). It is immediateto see that such games can be described in a succinct way. This will help ouranalysis and will provide an interesting view-point on our results; while theset of convex symmetric extensions is easy to describe, it is not the case forsymmetric positive extensions. We shall provide examples showing that tryingto describe such extensions is a difficult problem to tackle.2.2 Partially defined cooperative gamesThe following definitions are inspired by [23]. Definition 6 (Incomplete game)
An incomplete game can be characterised bya set of players N = { , , . . . , n } , a set of coalitions whose values are known,that is K ⊆ N , and a function v : K → R , where we assume that ∅ ∈ K and v ( ∅ ) = 0. We associate with such an incomplete game the triple ( N, K , v ).Compared to the definitions in [23], we have slightly more general assumptionson the incomplete games. Most importantly, we do not a priori assume thatthe profits of singleton coalitions and the grand coalition are among the knownvalues, i.e. that they are in K .The fundamental definition and object of our study is extension of an incom-plete game. Definition 7 (Extension and extendability)
We say that a cooperative game(
N, w ) is an extension of an incomplete game ( N, K , v ) if w and v coincideon values from K , i.e., v ( S ) = w ( S ) for every S ∈ K . We further say that( N, K , v ) is extendable if there exists at least one extension to it.Here we note the crucial fact that we are mostly not interested just in anarbitrary extension, but often in an extension with a further property, e.g. anextension which is either superadditive, convex, positive, symmetric or anycombination of these. Thus we often just say that an incomplete game isextendable if it is clear in what kind of extensions we are interested to.Questions and problems connected to extensions are central for the theory ofpartially defined cooperative games as is also the search for nice descriptionsof lower game and upper game . Definition 8 (The lower game and the upper game of a set of extensions)
Let( N, K , v ) be an incomplete game and let E ( v ) be particular a set of extensionsof ( N, K , v ). Then the lower game ( N, v ) of E ( v ) and the upper game ( N, v )of E ( v ) are games such that for any ( N, w ) ∈ E ( v ) and any S ⊆ N holds v ( S ) ≤ w ( S ) ≤ v ( S ) , onvexity and positivity in partially defined cooperative games 7 and for each S ⊆ N there exist ( N, w ) , ( N, w ) ∈ E ( v ) such that v ( S ) = w ( S ) and v ( S ) = w ( S ) . We remark that it is important to distinguish between lower and upper gamesof different sets of extensions. For example, lower games of superadditive andconvex extensions do not coincide in general. There are also examples of incom-plete games where the set of convex extensions might be empty and therefore,the lower game of convex extensions might not exist. However, the same in-complete game can have the lower game of superadditive extensions.When a set of extensions E ( v ) forms a convex polytope, then we are interested,as is often the case in combinatorial optimization, in its extreme games . Definition 9 (Extreme game of a set of extensions)
Let E ( v ) be a particularset of extensions of an incomplete game ( N, K , v ). If E ( v ) forms a convexpolytope, then a game which is a vertex of E ( v ) is called an extreme game ofthe extension set E ( v ).For superadditive extensions of incomplete games, the extreme games werestudied in [22,23]. In this paper, we study extreme games of convex symmet-ric extensions and of positive extensions. As we already mentioned, we areinterested in the property of symmetry in games. The following is the gener-alisation of this property to incomplete games. Definition 10
An incomplete game ( N, K , v ) is partially symmetric if for ev-ery K , K ∈ K such that | K | = | K | , the equality v ( K ) = v ( K ) holds. N , the function v : 2 N → R is submodular if forevery S, T ⊆ N , v ( T ) + v ( S ) ≥ v ( S ∪ T ) + v ( S ∩ T ). Observe that the function − v is supermodular (and compare this with Definition 2).Extendability of submodular functions was first studied in 2014 by Seshadhriand Vondr´ak [28]. They introduced path certificate , a combinatorial structurewhose existence certifies that a submodular function is not extendable. Theyalso showed an example of a partial function defined on almost all coalitionswhich is not extendable, but by removing a value for any coalition, the gamebecomes extendable. Later in 2018, Bhaskar and Kumar studied extendabilityof several classes of set functions, including submodular functions (see [30]).Inspired by the results of Seshadhri and Vondr´ak, they introduced a more nat-ural combinatorial certificate of non-extendability — square certificate . Usingthis concept, they were able to show that a submodular function is extendable Jan Bok et al. on the entire domain if and only if it is extendable on its lattice closure. In2019, the same authors showed that the problem of extendability for a sub-class of submodular functions, so-called coverage functions , is NP-complete.For more information, see [31]. Formally, the following theorem will be neededfor our results.We remind the reader that the lattice closure LC ( K ) of a set of points K ⊆ N in a partially ordered set (poset) (2 N , ⊆ ) is the inclusion-minimal subset of 2 N that contains K and is closed under the operation of union and intersection ofsets. Theorem 2 [30] Let f : K → R be a partial function on a power set N .Let F := LC ( K ) ∩ { S : ∃ T i , T j ∈ K s.t. T j ⊆ S ⊆ T i } be the sets obtainedby the union and intersection of sets in K that are also both contained inand contained by sets in K . If the partial function f can be extended to asubmodular function on F , then it can be extended to a submodular functionon N . We may apply the theorem to the case of incomplete games in which the setof known coalitions forms a chain. We recall that a set Y forms a chain in aposet ( X, ≤ ) if the subposet of ( X, ≤ ) induced by Y is a total ordering, i.e. iffor every x, y ∈ Y , it holds that either x ≤ y or y ≤ x . Proposition 1
Let ( N, K , v ) be an incomplete game with K forming a chainin the poset (2 N , ⊆ ) . Then ( N, K , v ) has a convex extension.Proof We shall first prove the theorem for the case when K is of size | N | +1. Inthat case, the lattice closure of K is the set K itself. Furthermore, any inequalityof the form v ( A ) + v ( B ) ≤ v ( A ∪ B ) + v ( A ∩ B ) where A, B ∈ K is satisfied.That is because, without loss of generality, A ⊆ B and thus, A ∪ B = B and A ∩ B = A . In this manner, the aforementioned inequality is always satisfied.Furthermore, it is clear that an analogous result to Theorem [30] holds if wereplace submodularity by supermodularity and hence, since the assumptionsare met, the proof of this case follows.Now if the chain is of size less than | N | + 1, we can arbitrarily choose a chainof size | N | + 1 which contains K and set the values for the coalitions in thechain not included in K arbitrarily. This can be done without loss of generalityand we can proceed as in the previous part. That finishes the proof. (cid:117)(cid:116) onvexity and positivity in partially defined cooperative games 9 extensions. The additional property of symmetry yields compact (and by ouropinion elegant) descriptions.The main ingredient for our results will be the following characterisation ofsymmetric convex games. Proposition 2
Let ( N, v ) be a symmetric cooperative game. For every S (cid:40) N, i ∈ S, j ∈ N \ S , it holds that v ( S − i ) + v ( S + j )2 ≥ v ( S ) , if and only if the game is convex.Proof “ ⇒ ” Suppose that the assumptions hold and the game is not convex. ByTheorem 1, there is a player i (cid:48) ∈ N and coalitions T (cid:40) T ⊆ N \ i (cid:48) such that v ( T + i (cid:48) ) − v ( T ) > v ( T + i (cid:48) ) − v ( T ) . (1)For minimality, let us take player i (cid:48) and coalitions T , T such that the differ-ence | T | − | T | is minimal. We distinguish three possible cases.1. If | T | − | T | = 0, then the game is not symmetric.2. If | T | − | T | = 1, then by symmetry of v , we have that v ( T ) = v ( T + i (cid:48) ).In that case, we get v ( T ) > v ( T ) + v ( T + i (cid:48) )2 . Furthermore, there exists a unique i (cid:48)(cid:48) ∈ T \ T such that T + i (cid:48)(cid:48) = T .Thus we can write v ( T ) > v ( T − i (cid:48)(cid:48) ) + v ( T + i (cid:48) )2 , which is in contradiction with our assumption on v .3. If | T | − | T | >
1, then there is a coalition T such that T (cid:40) T (cid:40) T . Byminimality, we know that v ( T + i (cid:48) ) − v ( T ) ≤ v ( T + i (cid:48) ) − v ( T )and v ( T + i (cid:48) ) − v ( T ) ≤ v ( T + i (cid:48) ) − v ( T ) . By adding these two inequalities together, we get v ( T + i (cid:48) ) − v ( T ) ≤ v ( T + i (cid:48) ) − v ( T ) , which is a contradiction with (1). S { } { } { } { , } { , } { , } { , , } v ( S ) 1 1 1 4 6 4 9 Table 1
The game (
N, v ) from Example 1 with its characteristic function given in the table. “ ⇐ ” If the game is symmetric convex, we take the inequality v ( S + j ) − v ( S + j − i ) ≥ v ( S ) − v ( S − i ) , and because | S + j − i | = | S | , we have v ( S + j − i ) = v ( S ) by symmetry. Bysubstituting and rearranging the inequality, we get v ( S − i ) + v ( S + j )2 ≥ v ( S ) , which concludes the proof. (cid:117)(cid:116) We note that regarding the general convex games, the characterisation fromProposition 2 does not hold in general. This can be seen in the followingexample.
Example 1 (A convex game not satisfying conditions from Proposition 2.)
The game (
N, v ) given in Table 1 is convex, as can be easily checked. However,the inequality v ( { } ) + v ( { , , } )2 ≥ v ( { , } )is not satisfied, as (cid:3) s ( k ) the value of v ( S ) of any S ⊆ N such that the size of S is equal to k . This allows us to formulate the followingcharacterisation of symmetric convex games. Theorem 3
A game ( N, v ) is symmetric convex if and only if for all k =1 , . . . , n − , s ( k −
1) + s ( k + 1)2 ≥ s ( k ) . Hence we can associate every symmetric convex game (
N, v ) with a function s : { , . . . , n } → R having the above property. Similarly, we can apply this to( N, K , v ) with a function σ : X → R where X ⊆ { , . . . , n } is constructed from K . For an easier use, we define reduced forms of games ( N, v ) and ( N, K , v ). Definition 11
Let (
N, v ) be a symmetric game and ( N, K , v ) a partially sym-metric incomplete game. – The reduced form of a game (
N, v ) is an ordered pair (
N, s ), where thefunction s : { , . . . , n } → R is a reduced characteristic function such that s ( i ) := v ( S ) for any S ⊆ N with | S | = i . onvexity and positivity in partially defined cooperative games 11 v a l u e s o ff un c t i o n s coalition size v a l u e s o ff un c t i o n s coalition size Fig. 1
Examples of line charts of symmetric convex games in reduced form. The figure onthe left depicts a game (
N, s ) where s (1) >
0, the figure on the right a situation where s (1) <
0. The slopes of the line segments are bounded by convexity of the function. – The reduced form of an incomplete game ( N, K , v ) is an ordered triple( N, X , σ ) where X = { i | i ∈ { , . . . , n } , ∃ S ∈ K : | S | = i } and the function σ : X → R is defined as σ ( i ) := v ( S ) for any S ∈ K such that | S | = i .We also call ( N, s ) and ( N, X , σ ) the reduced game and the reduced incompletegame , respectively.Since always ∅ belongs to K , for every reduced incomplete game ( N, X , σ ) italso holds that 0 ∈ X and σ (0) = 0.Notice that a game ( N, v ) is symmetric convex if and only if the function s ofits reduced form ( N, s ) satisfies the property from Theorem 3.We can visualize the reduced form (
N, s ) of a symmetric convex game (
N, v )by a graph in R . On the x -axis we put the coalition sizes and on the y -axis the values of s . The point (0 ,
0) is fixed for all reduced games. Now byTheorem 3, the conditions for k = 1 , . . . , n − i, s ( i ))(where i ∈ { , . . . , n } ) lie in a convex position. More precisely, if we connectthe neighbouring pairs ( i, s ( i )) , ( i +1 , s ( i +1)) (where i ∈ { , . . . , n − } ) by linesegments, we obtain a graph of a convex function. The graph is illustrated onexample in Figure 1. Further in this text, we refer to this function as the linechart of ( N, s ). Similarly, for ( N, X , σ ), the line chart is obtained by connectingconsecutive elements from X by line segments. If n / ∈ X , the rightmost linesegment is extended to end at x -coordinate n . The values of s are then set tolie on the union of these line segments.Now if we have an incomplete game in reduced form, i.e. ( N, X , σ ), the firstquestion that arises is that of extendability. For X = { , i } with i ∈ { , . . . , n } ,the game is always extendable (a possible extension is when we set the valuesof each coalition size to lie on the line coming through (0 , σ (0)) and ( i, σ ( i ))),therefore in the following theorem, we consider |X | > x x x x σ ( x ) v a l u e s o ff un c t i o n s coalition size ...... σ ( x ) σ ( x ) σ ( x ) ks ( k ) Fig. 2
Construction of a symmetric convex extension of ( N, X , σ ) where X = { x , x , x , x } , using the line chart of ( N, X , σ ). The value s ( k ) lies on the line segmentconnecting ( x , σ ( x )) and ( x , σ ( x )). Theorem 4
Let ( N, X , σ ) be a reduced form of a partially symmetric incom-plete game ( N, K , v ) where |X | > . The game is extendable if and only if σ ( k ) ≤ σ ( k ) + ( k − k ) σ ( k ) − σ ( k ) k − k , for all consecutive elements k < k < k from X .Proof If the game is extendable, there is a symmetric convex extension inreduced form (
N, s ). By Theorem 3, the line chart of (
N, s ) is a convex functionwhich coincides with σ on the values of X . Therefore, for any consecutiveelements k , k , k from X , the inequality must hold.For the other implication, we construct a symmetric convex extension by set-ting the values of s to lie on the line chart of ( N, X , σ ) The construction isillustrated in Figure 2.Notice that s ( k ) = σ ( k ) for k ∈ X , therefore the constructed game ( N, s ) is anextension of ( N, X , σ ). Also, because the inequalities for consecutive elements k , k , k from X hold, the line chart represents a convex function. Therefore,the convex property s ( k −
1) + s ( k + 1)2 ≥ s ( k )holds for all k ∈ { , . . . , n − } . By Theorem 3, the game ( N, s ) is a symmetricconvex extension of ( N, X , σ ). (cid:117)(cid:116) As a direct consequence of the previous theorem, we get that the problem ofextendability of partially symmetric incomplete games into symmetric convexgames can be decided in linear time with respect to the size of the original onvexity and positivity in partially defined cooperative games 13 game (i.e. the size of the characteristic function). It is important to understandthe situation when the set of symmetric convex extensions is bounded. Thefollowing proposition addresses this problem.
Proposition 3
Let ( N, X , σ ) be a reduced form of a partially symmetric in-complete game ( N, K , v ) with | N | ≥ that has a symmetric convex extension.The set of symmetric convex extensions is bounded if and only if |X | > and n ∈ X .Proof Let ( N, X , σ ) be a reduced form of an extendable incomplete game. If n / ∈ X , clearly, from Theorem 3 there is no upper bound on the profit of n . Let n ∈ X and suppose for a contradiction that there is k ∈ N such that there is noupper bound on its profit. Choose an extension ( N, s ) such that s ( k ) > k σ ( n ) n .The line chart of ( N, s ) is not a convex function (the property is violated for(0 , s (0)) , ( k, s ( k )) , ( n, s ( n ))), therefore ( N, s ) cannot be a symmetric convexgame.If |X | ≤
2, then X = { , n } (otherwise the set of extensions is not boundedfrom above). Let (cid:96) be a negative value less or equal to σ ( n ). Any game( N, s (cid:96) ) with s (cid:96) ( k ) = (cid:96) for k ∈ { , . . . , n − } and s (cid:96) ( n ) = σ ( n ) and s (cid:96) (0) = σ (0) , s (cid:96) ( n ) = σ ( n ) is a symmetric convex extension of ( N, X , σ ). Thus, thereis no lower bound on values of 1 , . . . , n − |X | >
2, then let i ∈ X \ { , n } . For k = 1 , . . . , i −
1, the point ( k, s ( k ))must lie on or above the line coming through points ( i, σ ( i )) , ( n, σ ( n )), oth-erwise the convexity of line chart of ( N, s ) is violated. Similarly, for k = i + 1 , . . . , n − s ( k ) must lie on or above the line coming throughpoints (0 , σ (0)) , ( i, σ ( i )), otherwise the convexity is violated, again. The profitof every k is therefore bounded from below. (cid:117)(cid:116) Theorem 5
Let ( N, X , σ ) be a reduced form of a partially symmetric incom-plete game that is extendable. Suppose that the set of symmetric convex exten-sions is bounded. Further, for every k ∈ { , , . . . , n }\X , denote by i , i , j , j the closest distinct elements from X such that i < i < k < j < j , if theyexist. Then the lower game has the following form: s ( k ) := σ ( i ) + ( k − i ) σ ( i ) − σ ( i ) i − i if j does not exist, σ ( j ) + ( k − j ) σ ( j ) − σ ( j ) j − j if i does not exist, max (cid:40) σ ( i ) + ( k − i ) σ ( i ) − σ ( i ) i − i ,σ ( j ) + ( k − j ) σ ( j ) − σ ( j ) j − j (cid:41) if i , i , j , j exist,for k / ∈ X and s ( k ) := σ ( k ) for k ∈ X . The upper game has the followingform: s ( k ) := (cid:40) σ ( k ) if k ∈ X ,σ ( i ) + ( k − i ) σ ( j ) − σ ( i ) j − i otherwise. Proof
To prove that (
N, s ) is the lower game, we start by showing that forevery symmetric convex extension (
N, w ) and every coalition size k ∈ N , itholds that s ( k ) ≤ w ( k ). If k ∈ X , trivially s ( k ) = σ ( k ) = w ( k ). If k / ∈X , since any extension must have a convex line chart, the value w ( k ) mustlie on or above the lines coming through pairs of points ( i , σ ( i )) , ( i , σ ( i ))and ( j , σ ( j )) , ( j , σ ( j )). The three cases in the definition of the lower gamecapture this fact by setting the value of s ( k ) so that it lies on either one of thelines (if the other one does not exist) or on the maximum of both of them.Now it remains to show that for every k ∈ N , the value s ( k ) is attained forat least one symmetric convex extension. We employ the extension ( N, s { a,b } )defined for consecutive a, b ∈ X such that a < b as s { a,b } ( (cid:96) ) := σ ( (cid:96) ) if (cid:96) ∈ X ,s ( (cid:96) ) if (cid:96) / ∈ X and a < (cid:96) < b,s ( (cid:96) ) if (cid:96) / ∈ X and either (cid:96) < a , or b < (cid:96). For (cid:96) ∈ X , s { a,b } ( (cid:96) ) = σ ( (cid:96) ), therefore the game is an extension of ( N, X , σ ). For i ∈ { , . . . , n − } such that s { a,b } ( i − , s { a,b } ( i ) , s { a,b } ( i + 1) attain the valuesof upper game s , s { a,b } ( i ) ≤ s { a,b } ( i − s { a,b } ( i +1)2 because ( N, s ) is a symmetricconvex game (as we show further in this proof) so by Theorem 3, the same in-equality holds for values of s . In the rest of the cases, either all the three points( i − , s { a,b } ( i − , ( i, s { a,b } ( i )) , ( i + 1 , s { a,b } ( i + 1)) lie on the same line, there-fore the inequality holds with equal sign, or the three points lie on the maxi-mum of two lines coming through ( a , σ ( a )) , ( a, σ ( a )) and ( b, σ ( b )) , ( b , σ ( b ));pairs of consecutive coalition sizes with defined values (coalition sizes from X )such that a < a < b < b . If s { a,b } ( i ) > s { a,b } ( i − s { a,b } ( i +1)2 , then either σ ( a ) > σ ( a ) + ( a − a ) σ ( b ) − σ ( a ) b − a or σ ( b ) > σ ( a ) + ( b − a ) σ ( b ) − σ ( a ) b − a , both re-sulting in a contradiction with extendability of ( N, X , σ ) by Theorem 4. Nowfor k ∈ X we choose ( N, s { a,b } ) such that a = k and for k / ∈ X game ( N, s { a,b } )such that a < k < b are the closest coalition sizes with defined value.For the upper game ( N, s ), suppose for a contradiction that there is a sym-metric convex extension (
N, w ) of ( N, X , σ ) such that for k ∈ N , s ( k ) An example of a reduced game ( N, X , σ ) with X = { , , , , } where the condition s (3)+ s (5)2 (cid:3) s (4) from Theorem 3 is not satisfied. This implies that ( N, s ) is not a convexsymmetric extension. The game ( N, s ) is always a symmetric convex extension, however, this is nottrue for ( N, s ) in general, as can be seen in the example in Figure 3. Theprofit s ( i ) is achieved for each coalition size i ∈ { , . . . , n } \ X by a specialgame ( N, s { a,b } ) where a < i < b . The games ( N, s { a,b } ) are actually moreimportant because they are extreme games of the set of symmetric convexextensions. Proposition 4 Let ( N, X , σ ) be an extendable partially symmetric incompletegame in a reduced form. Games ( N, s { a,b } ) for consecutive a, b ∈ X (where a < b ) and ( N, s ) are extreme games of the set of symmetric convex extensions CE ( σ ) .Proof For a contradiction, suppose that ( N, s { a,b } ) is not an extreme game.It means that there are two symmetric positive extensions of ( N, X , σ ), say( N, s ) and ( N, s ), such that ( N, s { a,b } ) is their nontrivial convex combina-tion and without loss on generality, there is i ∈ { , . . . , n } such that s ( i ) In general, ( N, s ) and ( N, s { a,b } ) for all consecutive a < b from X are not theonly extreme games. In the following theorem, we describe all the extremegames of symmetric convex extensions. Theorem 6 Let ( N, X , σ ) be an extendable partially symmetric incompletegame such that its set of symmetric convex extensions CE ( σ ) is a bounded polytope. For k ∈ { , . . . , n } \ X and i, j ∈ X closest to k such that i < k < j ,the games ( N, s k ) defined by s k ( m ) := σ ( m ) if m ∈ X ,s ( m ) if m / ∈ X and either m < i or j < m,s ( m ) if m = k,σ ( j ) + ( m − j ) σ ( j ) − s ( k ) j − k if m / ∈ X and k < m < j,σ ( i ) + ( m − i ) s ( k ) − σ ( i ) k − i if m / ∈ X and i < m < k together with ( N, s ) form all the extreme games of CE ( σ ) .Proof We divide the proof into two parts. In the first part, we show that anyextension ( N, s ) ∈ CE ( σ ) is a convex combination of games ( N, s ) and ( N, s k )for k ∈ { , . . . , n } \ X . In the second part, we show that every game ( N, s k )is an extreme game, thus (together with the upper game ( N, s )) they form allthe extreme games.We prove the first part by induction on the size of gap of unknown values (thatis those from { , . . . , n } \ X ) between two consecutive coalition sizes i, j ∈ X .First, suppose that there is only one gap (i.e. all the other coalition sizes havedefined profit except for the gap). If the gap size is 1, there is only one game( N, s k ) which is equal to ( N, s { k − ,k +1 } ), together with the upper game ( N, s ).Any extension ( N, s ) ∈ CE ( σ ) can be expressed as a convex combination ofthese two games as s = αs k + (1 − α ) s with α = s ( k ) − s ( k ) s k ( k ) − s ( k ) ∈ [0 , i , j ∈ X is (cid:96) .Hence we have (cid:96) games( N, s i +1 ) , ( N, s i +2 ) , . . . , ( N, s j − ) together with ( N, s ) . We construct a new system of (cid:96) − N, ( s i +2 ) (cid:48) ) , ( N, ( s i +3 ) (cid:48) ) , . . . , ( N, ( s j − ) (cid:48) ) together with ( N, ( s i +1 ) (cid:48) ) , where ( s m ) (cid:48) := αs m + (1 − α ) s with α = s ( i + 1) − s ( i + 1) s i +1 ( i + 1) − s ( i + 1) . These games correspond to the extreme games of an incomplete game ( N, X (cid:48) , σ (cid:48) )where X (cid:48) := X ∪ { i + 1 } , and the function σ (cid:48) is defined as σ (cid:48) ( m ) := σ ( m ) for m ∈ X and σ (cid:48) ( i + 1) := s ( i + 1). The game ( N, ( s i +1 ) (cid:48) ) represents the uppergame of the set of symmetric convex extensions of this incomplete game. Sincethe new system of (cid:96) games forms the extreme games of CE ( σ (cid:48) ), the game( N, s ) (which is also an extension of ( N, X (cid:48) , σ (cid:48) )) is, by induction hypothesis,their convex combination. And as each game ( N, ( s m ) (cid:48) ) is a convex combina-tion of ( N, s ) and ( N, s m ), the game ( N, s ) is also a convex combination of theformer system ( N, s i +1 ) , ( N, s i +2 ) , . . . , ( N, s j − )together with ( N, s ). onvexity and positivity in partially defined cooperative games 17 i k m v a l u e s o ff un c t i o n s coalition size j s ( m ) i m k v a l u e s o ff un c t i o n s coalition size j s ( m ) Fig. 4 Examples of violation of convexity of the line chart of both ( N, s ) and ( N, s ). Thefull lines depict the line chart of ( N, s ) and the dotted lines the line charts of ( N, s ) and( N, s ). On the left, the situation where k < m is shown. We have values s k ( i ) = s ( k ) and s k ( k ) = s ( k ), yet s ( m ) is too small. Similarly, on the right, the situation where m < k isshown, with s k ( i ) = s ( k ), s k ( k ) = s ( k ). However, in this case, the value s ( m ) is too big. Notice that if there is more than one gap between the coalition sizes in X , thenwe can follow a similar construction as in the situation with a single gap. Thisis because any two extreme games parametrized by two coalition sizes fromone gap assign the same profit to any coalition size from a different gap. Thus,we can start our construction by filling in the first gap, after that, taking theextreme games of the extended incomplete game and so on, until there is nogap left.As for the second part of the proof, suppose for a contradiction that ( N, s k )for k ∈ { , . . . , n } \ X is not an extreme game of CE ( σ ). Then there are twogames ( N, s ) , ( N, s ) ∈ CE ( σ ) and m ∈ N such that s ( m ) < s k ( m ) < s ( m ).Clearly, m / ∈ X (since s ( m ) = s k ( m ) = s ( m ) = σ ( m )) and if m is such that s k ( m ) = s ( k ) or s k ( m ) = s ( m ), we arrive into a contradiction, so the only casethat remains is m / ∈ X together with i < m < j and m (cid:54) = k . For any such m ,the convexity of the line chart is violated for ( i, s ( i )) , ( k, s ( k )) , ( m, s ( m )) (if k < m ), or ( i, s ( i )) , ( m, s ( m )) , ( k, s ( k )) (if m < k ). Both cases are depictedin Figure 4. (cid:117)(cid:116) For an extendable partially symmetric incomplete game in a reduced form( N, X , σ ) with the set of symmetric convex extensions bounded and of sizelarger than one, the number of extreme games is always |{ , . . . , n }|−|X | +1 = n − |X | + 2, no matter what the values of σ are.Algebraically, we can describe the set of symmetric convex extensions CE ( σ )of a partially symmetric incomplete game in a reduced form ( N, X , σ ) as CE ( σ ) = (cid:40)(cid:16) N, α s + (cid:88) k ∈ N \X α k s k (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) α + (cid:88) k ∈ N \X α k = 1 , α, α k ≥ , k ∈ N \ X (cid:41) , namely as the set of convex combinations of extreme games s k , s .Geometrically, we can describe the set of extensions as follows. Denote by CE ( σ ) the set of all symmetric convex extensions of ( N, X , σ ) where X = { , n } and σ (0) = σ ( n ) = 0. According to Theorem 3, we can describe CE ( σ )by a system of n − n − Ay ≤ 0, where A = − − − − . Matrix A is an M-matrix [16], therefore it is nonsingular and A − ≥ 0. Non-singularity of A implies that CE ( σ ) is a pointed polyhedral cone, which istranslated such that its vertex is not necessarily in the origin of the coordinatesystem. Further, because A − ≥ 0, the normal cone CE ( σ ) ∗ of CE ( σ ) (see[6]) contains the whole nonnegative orthant. Thus, the vertex of polyhedralcone CE ( σ ) is the biggest element of CE ( σ ) when restricted to each coordi-nate (this corresponds with the statement that the upper game is a symmetricconvex extension). Therefore, geometrically, the set CE ( σ ) looks like squeezed negative orthant. For an incomplete game ( N, X (cid:48) , σ (cid:48) ) where { , n } ⊆ X (cid:48) , theset of extensions is CE ( σ ) ∩ k ∈X (cid:48) { s ( k ) = σ ( k ) } , in other words, we take CE ( σ ) and fix some coordinates. The following basic result is in our opinion of independent interest. Proposition 5 Let ( N, K , v ) be a positively extendable incomplete game. Theset of positive extensions P ( v ) is bounded if and only if N ∈ K .Proof If N ∈ K , then for any positive extension ( N, w ), (cid:80) T ⊆ N d w ( T ) = v ( N ),and since for all T ⊆ N , d w ( T ) ≥ 0, we can deduce that d w ( T ) ∈ [0 , v ( N )].This yields a bound (possibly overestimation) for all possible values of d w ( T ).Since the dividends are bounded, the set of positive extensions is also bounded.If N / ∈ K , then the value of coalition N can be possibly infinitely large, sincethere is no upper bound on d w ( N ). Thus, the set P ( v ) is unbounded. (cid:117)(cid:116) Lemma 1 (Farkas’ lemma, [11]) Let A ∈ R m × n and b ∈ R n . Then exactlyone of the following two statements is true. onvexity and positivity in partially defined cooperative games 19 1. There exists x ∈ R n such that Ax = b and x ≥ .2. There exists y ∈ R m such that A T y ≥ and b T y ≤ − . The next theorem employs duality of linear systems to provide a certificatefor non-extendability of an incomplete game. This approach was motivatedby the aforementioned work by Seshadhri and Vondr´ak [28] and the so-called path certificate for non-extendability of submodular functions. Although itssize is exponential in the number of players in general, for special cases, thesolvability of the dual system is polynomial and therefore, the extendability ispolynomially decidable for such cases. Theorem 7 Let ( N, K , v ) be an incomplete game. The game is positively ex-tendable if and only if the following system of linear equations is not solvable:1. ∀ T ⊆ N, T (cid:54) = ∅ : (cid:80) S ∈K ,T ⊆ S y ( S ) ≥ ,2. (cid:80) S ∈K v ( S ) y ( S ) ≤ − .Proof Let M = 2 | N | − 1. Let further U (cid:48) ∈ R M × M be a matrix with characteris-tic vectors of unanimity games u T as its columns. Then it holds that U (cid:48) d = w for every game ( N, w ) and its vector of Harsanyi dividends d .For a partially defined cooperative game ( N, K , v ), we reduce matrix U (cid:48) bydeleting the rows corresponding to coalitions with undefined utility, reachingthe system U d = v . This adjustment eliminates unknowns on the right handside of the equation, yet no information about the complete game is lost, sincethe vector of Harsanyi dividends carries full information.The game ( N, K , v ) is extendable if and only if the system of linear equationsabove is solvable for d ≥ 0. By Farkas’ lemma (Lemma 1) this happens if andonly if the following system has no solution, U T y ≥ v T y ≤ − . (2)The conditions given by the system (2) correspond to those from the statementof the theorem. (cid:117)(cid:116) Notice that even though the number of inequalities (cid:80) S ∈ K,T ⊆ S y ( S ) ≥ | N | − T ⊆ N ), the actualnumber of distinct inequalities is not larger than 2 |K| − 1, because each in-equality sums over a subset of K . Depending on the structure of K , the actualnumber might be even smaller as is shown in the following result. Proposition 6 Let ( N, K , v ) be an incomplete game with the sizes of coali-tions in K bounded by a fixed constant c . Then the problem of extendabilityinto a positive extension is polynomially-time solvable in the size of N . Proof Let ( N, K , v ) be the incomplete game from the claim. The number ofcoalitions with a defined value is at most (cid:80) ci =1 (cid:0) ni (cid:1) , which is polynomial in n . Also, if we consider the linear system from Theorem 7, every T ⊆ N suchthat | T | > c yields an empty sum in its corresponding inequality. Therefore,the number of unique conditions in the problem is bounded by the numberof coalitions with defined value, that is by the sum (cid:80) ci =1 (cid:0) ni (cid:1) . We concludethat the linear system can be solved in polynomial time by means of linearprogramming. (cid:117)(cid:116) Lemma 2 Let P be a convex subset of R n , A ∈ R n × n a nonsingular matrix,and x ∈ P an extreme point of P . Then Ax is an extreme point of A ( P ) = { Au | u ∈ P } .Proof Suppose that x ∈ P is an extreme point of P and the image Ax is not anextreme point of A ( P ). Therefore, there are Au, Av ∈ A ( P ) and α ∈ (0 , 1) suchthat αAu +(1 − α ) Av = Ax . But then αAu +(1 − α ) Av = A ( αu +(1 − α ) v ) = Ax ,and therefore, x is not an extreme point of P , as it is a nontrivial convexcombination of u, v ∈ P . This is a contradiction. (cid:117)(cid:116) Let us denote the set of positive extensions of ( N, K , v ) by P ( v ) := (cid:26) ( N, w ) (cid:12)(cid:12)(cid:12) ∀ S ∈ K : w ( S ) = v ( S ) and ∀ T ⊆ N : d w ( T ) ≥ (cid:27) , or equivalently in terms of dividends, P (cid:48) ( v ) := (cid:26) ( d w ( T )) T ⊆ N (cid:12)(cid:12)(cid:12) ∀ S ∈ K : (cid:88) T ⊆ S d w ( T ) = v ( S ) , ∀ T ⊆ N : d w ( T ) ≥ (cid:27) . Notice that P ( v ) (cid:54) = P (cid:48) ( v ) since the former is a set of ordered pairs and thelatter is a set of vectors of dividends.We see that both sets are closed convex polytopes since they are formed byintersections of closed half-spaces. If we suppose that ( N, K , v ) is positivelyextendable, then the sets are nonempty. Furthermore, the sets are bounded ifand only if N ∈ K . Bounded, closed and convex polytopes are convex hulls oftheir extreme points. onvexity and positivity in partially defined cooperative games 21 Let M := 2 | N | − U ∈ R M × R M be a matrix with vectors of unanimitygames u T ∈ R M as columns. It holds that U d ( w ) = w where w ∈ R M is acharacteristic vector of game ( N, w ) and d ( w ) ∈ R M represents a vector ofHarsanyi dividends of the game. Since unanimity games form a basis of R M ,the matrix U is nonsingular and thus, by Lemma 2, the extreme points of P ( v )correspond to those of P (cid:48) ( v ).With this notation, we can rewrite the description of the set P (cid:48) ( v ) as follows: P (cid:48) ( v ) = (cid:26) d ( w ) ∈ R M (cid:12)(cid:12)(cid:12) (cid:88) T ⊆ S d w ( T ) u T = w where ( N, w ) ∈ P ( v ) , d ( w ) ≥ (cid:27) . We will follow the terminology of Peleg and S¨udholter [26] and define balancedcollections for incomplete games. Definition 12 A collection B ⊆ N is balanced for ( N, K , v ) if there is apositive extension ( N, ν ) of ( N, K , v ) such that (cid:88) T ∈B d ν ( T ) u T = ν and d ν ( T ) > T ∈ B .Furthermore, a balanced collection B is minimal if B does not contain a properbalanced subcollection. Formally, if for all B ∗ (cid:40) B and for all positive exten-sions ( N, w ) of ( N, K , v ) holds that (cid:88) T ∈B ∗ d w ( T ) u T (cid:54) = w. We sometimes call the vector of coefficients d ( w ) balanced vector (of B ) toemphasize its connection to a specific balanced collection B .For a minimal collection B , there exists a positive extension such that d w ( T ) > T ∈ B but by fixing any dividend of any coalition from B tozero, there is no positive extension of ( N, K , v ).The following two lemmata will help us make a connection between minimalcollections and extreme points. Lemma 3 A collection B ⊆ N is minimal for ( N, K , v ) if and only if thereis a unique positive extension ( N, w ) satisfying (cid:80) T ∈B d w ( T ) u T = w and d w ( T ) > if T ∈ B .Proof “ ⇐ ” If B ∗ is a proper balanced subcollection of a balanced collection B , thenthere are positive extensions, ( N, w ∗ ) and ( N, w ), whose balanced vectors cor-respond to B ∗ and B , respectively. It holds that (cid:88) T ∈B ∗ d w ∗ ( T ) u T = w ∗ . We define d w α ( T ) := (cid:40) αd w ( T ) + (1 − α ) d w ∗ ( T ) if T ∈ B ∗ ,αd w ( T ) if T ∈ B \ B ∗ . Now it is easy to verify that for every α ∈ [0 , N, w α ) are different positiveextensions of ( N, K , v ) with balanced vectors corresponding to B .“ ⇒ ” Assume that ( N, w ) and ( N, w ) are two distinct positive extensions of( N, K , v ) corresponding to the collection B . Then there is S ∈ B such that d w ( S ) > d w ( S ), so β := min S ∈B : d w ( S ) >d w ( S ) (cid:26) d w ( S ) d w ( S ) − d w ( S ) (cid:27) is well defined. Let d w ∗ be defined as follows: d w ∗ := (cid:40) (1 + β ) d w ( S ) − βd w ( S ) for all S ∈ B , . This clearly implies that there is S ∈ B such that d w ∗ ( S ) = 0. Thus, B \ { S } is a proper subcollection of B and therefore B is not minimal. (cid:117)(cid:116) The following lemma can be proved analogously to Peleg and Sudh¨olter [26]and thus we omit the proof. Lemma 4 Let P be a convex polyhedral set in R k given by P := x ∈ R k (cid:12)(cid:12) k (cid:88) j =1 a tj x j ≥ b t , t = 1 , . . . , m . For x ∈ P , let S ( x ) := (cid:8) t ∈ { , . . . , m }| (cid:80) kj =1 a tj x j = b t (cid:9) . The point x ∈ P isan extreme point of P if and only if the system of linear equations k (cid:88) j =1 a ij y j = b t for all t ∈ S ( x ) has x as its unique solution. By a direct application of Lemmata 3 and 4, we can describe the extremepoints by minimal balanced vectors d ( w ), obtaining the following theorem. Theorem 8 Let ( N, K , v ) be a positively extendable incomplete game and let P ( v ) be the set of its positive extensions. If N ∈ K , then P ( v ) is a convex hullof its extreme points. A positive extension ( N, w ) is an extreme game of P ( v ) if and only if its balanced vector d ( w ) is minimal. onvexity and positivity in partially defined cooperative games 23 K First, we consider the case in which the coalitions with known values excludingthe grand coalition are pairwise-disjoint and the value of the grand coalitionis known. Theorem 9 Let ( N, K , v ) be an incomplete game with K = { S , . . . , S k − , N } such that for all i, j ∈ { , . . . , k − } , it holds that S i ∩ S j = ∅ . Then the extremegames v T , lower game v , and upper game v of its positive extensions can bedescribed as follows: v T ( S ) := if (cid:64) T ∈ K : T ⊆ S, (cid:80) i : T i ⊆ S v ( S i ) if ∃ T ∈ K : T ⊆ S and T N (cid:42) S,v ( N ) − (cid:80) i : T i (cid:42) S v ( S i ) if ∃ T ∈ K : T ⊆ S and T N ⊆ S,v ( S ) := v K ( S ) = , if (cid:64) T ∈ K : T ⊆ S, (cid:80) i : S i ⊆ S v ( S i ) , if ∃ T ∈ K : T ⊆ S and N (cid:54) = S,v ( N ) , if ∃ T ∈ K : T ⊆ S and N = S,v ( S ) := (cid:40) v ( S i ) , if S ⊆ S i ,v ( N ) − (cid:80) i : S i (cid:42) S v ( S i ) , otherwise,where T = { T , . . . , T k − , T N } such that T i ⊆ S i , T N ⊆ N and T N (cid:42) S (cid:96) for any (cid:96) ∈ { , . . . , k − } . Furthermore, ( N, K , v ) is extendable to a positiveextension if and only if v ( N ) ≥ (cid:80) ≤ i ≤ k − v ( S i ) .Proof Let ( N, K , v ) be an incomplete game with the properties above. Forany positive extension ( N, w ), from the fact that the coalitions in K \ { N } are disjoint, at least one subcoalition T i of each coalition S i ∈ K \ { N } musthave a nonzero dividend d w ( T i ), otherwise v ( S i ) = 0. By Theorem 8 on theminimal number of nonzero dividends, there is at most one such subcoalitionif we consider an extreme game. If there were two nonzero dividends d w ( T i ), d w ( T i ) for one S i , then the corresponding balanced collection of the exten-sion would not be minimal. Setting the dividend of T i to d w ( T i ) + d w ( T i )yields a smaller collection. By this, d w ( T i ) = v ( S i ). We further see, since v ( N ) = (cid:80) T ⊆ N d w ( T ), that it must hold that v ( N ) ≥ (cid:80) S i ∈K\{ N } v ( S i ). If theinequality does not hold, it means that there is no extreme game of positiveextensions of ( N, K , v ) and hence, since the game is bounded, it is not extend-able. Now, if the inequality is strict, there has to be another nonzero dividendof a coalition T N ⊆ N such that T N (cid:42) S i for S i ∈ K \ { N } , otherwise T i , T N are two distinct subsets of S i and the corresponding collection is not minimal.Again, since we are interested in extreme games, by Theorem 8 there is onlyone such coalition T N , resulting in d w ( T N ) = v ( N ) − (cid:80) S i ∈K\{ N } v ( S i ). Anygame parametrized by a collection T = { T , . . . , T k − , T N } and expressed as v T from the statement of the theorem is clearly an extreme game. Notice the correspondence between minimal collections and collections T . From the factthat d w ( T i ) = v ( S i ), this game is equivalent to the one above.We will now proceed to show that the game v K is the lower game. For acoalition S with no subcoalition contained in K , v K ( S ) = 0 which is clearlythe value of the lower game. For a coalition S and those coalitions in K whichare subsets of S , we see that the value of w ( S ) in any positive extension of( N, K , v ) cannot be smaller than the sum (cid:80) T : T ∈K ,T ⊆ S v ( T ). Observe that thisis precisely the value v K ( S ). For N , the value v T ( N ) = v ( N ) is attained forall possible T . Therefore, v K ( N ) = v ( N ) is the value of the lower game.Finally, we show that each value of the upper game is achieved by a differentextreme game. If S is a proper subcoalition of S i , the value v ( S i ) is, thanks tothe non-negativity of dividends, an upper bound for the value of S . For anyextreme game v T such that S ∈ T , this bound is tight. If S is not a subcoalitionof any S i , its value cannot exceed v ( N ) − (cid:80) T i ∈T \ T N d w ( T i ), otherwise thecharacterisation of extandability is not satisfied for the grand coalition N . Bytaking an extreme game with T N = S , we see that this bound is tight. (cid:117)(cid:116) In the next special case that we are interested in, the set K contains the grandcoalition and for every other coalition in K , all its subcoalitions are also in K .The understanding of this case will help us in the study of symmetric positivegames. Theorem 10 Let ( N, K , v ) be an incomplete game with a positive extensionsuch that N ∈ K and S ∈ K \ { N } implies that every T ⊆ S also lies in K .Furthermore, let D = { ∆ S ≥ | S ∈ K} such that ∆ { i } = v ( { i } ) and ∆ S = v ( S ) − (cid:80) T (cid:40) S ∆ T . Then the extreme games v C , lower game v , and upper game v of positive extensions can be described as follows: v C ( S ) := (cid:40) ∆ N + (cid:80) T ∈K ,T ⊆ S ∆ T , if C ⊆ S, (cid:80) T ∈K ,T ⊆ S ∆ T , otherwise,where C / ∈ K \ { N } , and v ( S ) := v N ( S ) = (cid:40) ∆ N + (cid:80) T ∈K ,T ⊆ S ∆ T , if S = N, (cid:80) T ∈K ,T ⊆ S ∆ T , otherwise, v ( S ) := (cid:40) v ( S ) , if S ∈ K ,v S ( S ) , otherwise.Proof Let us suppose that ( N, K , v ) is extendable to a positive extension.Therefore, thanks to the structure of K , the dividends d w ( S ) for S ∈ K \{ N } are the same for any positive extension ( N, w ) of ( N, K , v ), therefore wedenote them by ∆ S . As a consequence, for any positive extension, its balancedcollection has to contain K \ N . Now if the uniquely defined value v ( N ) − onvexity and positivity in partially defined cooperative games 25 (cid:80) S ∈K\{ N } ∆ S is positive, there has to be at least one C / ∈ K \ { N } such thatits dividend d w ( C ) is not equal to zero. By Theorem 8, following a similarargument as in the proof of the previous theorem, the collection is minimal ifand only if there is only one such C , otherwise we could set one dividend tozero and add its original value to the other to get a smaller balanced collection.Thus setting the dividend of coalition C to v ( N ) − (cid:80) S ∈K\{ N } ∆ S yields anextreme game v C of positive extensions for any C / ∈ K \ { N } .For any coalition S , its value in any positive extension has to be larger orequal to (cid:80) T ∈K ,T ⊆ S ∆ S . Notice that v N ( S ) is equal to this number for any S ,thus being the lower game.For any coalition S , its maximal value is either v ( S ) if S ∈ K or at most v ( N ) − (cid:80) T ∈K\{ N } : T (cid:42) S ∆ T = ∆ N + (cid:80) T ∈K ,T ⊆ S ∆ T , which is equal to v S ( S )and thus such bound is tight. (cid:117)(cid:116) Note that in both previous special cases, v ( S ) belongs to the set of positiveextensions.Also notice that the number of extreme games v C equals the number of coali-tions C such that C / ∈ K \{ N } , that is 2 N −| K | +1 if v ( N ) − (cid:80) S ∈K\{ N } ∆ S > v ( N ) − (cid:80) S ∈K\{ N } ∆ S = 0) or no game at all (if v ( N ) − (cid:80) S ∈K\{ N } ∆ S < reduced forms ( N, s ) and ( N, X , σ ) of games ( N, v )and ( N, K , v ), respectively, which are defined in Definition 11. We can easilyobtain the following result as a corollary of Theorem 10. Proposition 7 Let ( N, K , v ) be a partially symmetric incomplete game with K consisting of N and all coalitions of size at most k . Furthermore, let Γ bethe value of the characteristic function v for the coalitions of size k . Then thelower and the upper game of symmetric positive extensions can be describedas follows v ( S ) := (cid:40) v ( S ) for S ∈ K ,Γ otherwise, and v ( S ) := (cid:40) v ( S ) for S ∈ K ,v ( N ) otherwise. The following game illustrates that even in the symmetric scenario, the lowergame of symmetric positive extensions does not have to be symmetric positive. Example 2 (Example of a game with the lower game of symmetric positiveextensions not being symmetric positive.) Let ( N, X , σ ) be an incomplete game of a symmetric positive game in a reduced form such that | N | = 4 and X = { , } . From the properties of symmetric positive games we know that anyextension ( N, s ) is given by 4 nonnegative dividends with corresponding values d , d , d , d such that – s (1) = d , – s (2) = 2 d + d , – s (3) = d + 3 d + 3 d , – s (4) = d + 4 d + 6 d + 4 d .By setting d := 0 , d := σ (2) , d := 0, and d := σ (4) − d we get anextension where s (1) = 0 (clearly the minimum) and it is achieved if and onlyif d = 0. Setting d = 0 yields s (3) = 3 σ (2). However, to minimize s (3),we can choose d := σ (2)2 , d := 0, d := 0, and d := σ (4) − d , obtaining s (3) = 3 d = σ (2). We cannot minimize both values simultaneously and thusthe lower game is not symmetric positive.For similar reasons, even the lower game of positive extensions is not neces-sarily positive. Therefore, in general, the lower game is not included in theset of positive extensions, contrary to what we showed for the special cases inTheorem 9 and 10. We would like to conclude our paper with observations on a connection betweenpartially defined cooperative games and cooperative interval games. This con-nection has not been mentioned in literature so far and we think it providesa nice bridge between the two approaches to uncertainty in cooperative gametheory.We remind the reader that a cooperative interval game is a pair ( N, w ), where N is a finite set of players and w : 2 N → IR is the characteristic function ofthis game with IR being the set of all real closed intervals. We further set v ( ∅ ) := [0 , Proposition 8 For a given incomplete game ( N, K , v ) and a set of its ex-tensions E , the associated lower game and upper game induce a cooperativeinterval game ( N, w ) containing the set E of extensions of ( N, K , v ) . Further-more, this game is inclusion-wise minimal, i.e., for every S ⊆ N , there is anextension from E attaining the lower bound of w ( S ) and an extension from E attaining the upper bound of w ( S ) . We can also take another view-point. We can generalise the definition of in-complete game to the interval setting by allowing the partial game to be aninterval game. We can then ask what are the extensions having some desired onvexity and positivity in partially defined cooperative games 27 property, for example being selection superadditive interval games. Indeed,this aligns with the main motivation behind some of the results in [5] and [4].We think that this is just the first step towards unifying both theories together.The fact that so far, no connection has been made between these two areas inliterature seems surprising to us. Some of the issues raised here are work inprogress. Acknowledgements All authors were supported by the Czech Science Foundation GrantP403-18-04735S. The second author acknowledges the support of Student Faculty Grant byFaculty of Mathematics and Physics of Charles University. The first author was supportedby the grant SVV–2020–260578. 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