In the theory of (cooperative) games, uncertainty is a long studied and s very important problem. The reasons are both practical and theoretical. Regarding applications, inaccuracy in data is relatively common in real-world situations. The sources of such inaccuracies can be for example the lack of knowledge on the behaviour of others, corrupted data, signal noise or events such as voting or auctions for which we do not know full information.
Regarding the theoretical aspects, there is clearly no single ideal way how to tackle such problems under every possible setting. This stemmed various approaches to uncertainty, resulting in models differing in complexity, applicability, and other qualitative criteria. To name a few such models, let us highlight fuzzy cooperative games branzei2008models ; mares2013fuzzy ; fuzzy , multi-choice games branzei2008models , cooperative interval games gokphd ; alparslan2011set ; BH15 , fuzzy interval games fuzzyinterval , games under bubbly uncertainty palanci2014cooperative , ellipsoidal games weber2010cooperative , and games based on grey numbers palanci2015cooperative .
In the classical cooperative game theory, groups of players, called coalitions, know the precise reward for the cooperation of its members. In partially defined cooperative games, this is no longer true, since only some of the coalitions know their values, while the others do not. This model was first introduced by Willson willson1993value in 1993. He gave the basic notion of incomplete games and a generalised definition of the Shapley value for such games. After more than two decades, Inuiguchi and Masuya continued in this line of research Masuya2016 . Their main focus was on superadditivity of possible extensions of the underlying incomplete game. Their research focused on a class of incomplete games with minimal information. Subsequently, Masuya further extended the results in masuya2016shapley ; masuya2017axioms , where he discussed more general classes of incomplete games and concentrated his efforts on generalisations of the Shapley value. Apart from that, Yu Yu2019Owen introduced a generalisation of incomplete games to games with coalition structures and studied the proportional Owen value (which is in some sense a generalisation of the Shapley value for these games). Unfortunately for the general public, the paper of Yu was published only in Chinese. Very recently, Bok, Černý, Hartman, and Hladík BCHH20 initiated study of convexity and positivity of extensions of incomplete games.
We shall now introduce fundamentals of the theory of convex sets, classical cooperative games, and partially defined cooperative games. We present only the necessary background needed for our study of 1-convexity in the framework of partially defined cooperative games. We invite the interested reader to consult the following comprehensive sources on the theory of cooperative games: branzei2008models ; driessen1988cooperative ; gilles2010cooperative ; peleg2007introduction . For more on applications of cooperative games, see e.g. bilbao2012cooperative ; combopt ; insurance .
We denote a real closed interval from to , , by . For an inequality , where is the left-hand side in variable and is the right-hand side in variable , we distinguish two cases. For , the inequality is strict (at ) if and it is tight or binding at if . For the sake of brevity, we write (or ) in one inequality instead of two inequalities with and , e.g. instead of and . Notice the difference between and .
2.1 Convex sets
In partially defined cooperative games, we study subsets of complete games, so called -extensions. All of the classes of -extensions studied to date form convex sets. In this section, we revise the theory of convex sets and introduce tools for elegant and compact descriptions of -extensions. We state all the results as facts and refer the reader to the book by Soltan Soltan2015 with exhaustive analysis of convex sets.
A set is called convex provided whenever and .
The convex sets we study are of a special form as they are intersections of closed halfspaces. A closed halfspace is the set where and . The hyperplane is the set .
A set is called polyhedron if it is an intersection of finitely many closed halfspaces, say : The sets and are polyhedrons. A bounded convex polyhedron is called polytope.
We say that a hyperplane corresponding to is supporting the set . An important example of polyhedrons are sets for some and . Faces are the intersections of the convex set and its supporting hyperplanes. Extreme faces satisfy that whenever for , then . Extreme points and extreme rays are extreme faces of a special importance as they fully characterise the polyhedrons.
Let be a convex set. A point is an extreme point (or vertex) of if there is no way to express as a convex combination such that and , except for taking .
Let be a convex polyhedron. A point is an extreme point (or vertex) if and only if for every
To define extreme rays, we use halflines. A closed halfline (from to ) is the set and an open halfline (from to ) is the set . Closed and open halflines form together halflines. We call to be the endpoint of the halfline. Notice that halflines form a special case of halfspaces.
An extreme ray of a convex set is a halfline which is an extreme face of .
We shall make use of an alternative definition of extreme rays arising from the connection of extreme rays of a polyhedron and extreme rays of a specific polyhedral cone (defined further in the text as recession cone).
A nonempty set is a cone with apex provided whenever and . A convex cone is a cone which is a convex set. Further, a polyhedral cone is a convex cone which can be expressed as for some .
Setting in the definition of the cone, we can observe that the apex is always a part of the cone. One can reformulate this definition, stating that a nonempty set is a cone with apex if and only if every halfline from to lies in whenever . Hence a cone with apex is either the singleton or a union of closed halflines with the common endpoint .
We will now introduce a connection between polyhedral cone and its extreme rays.
Consider a nonempty polyhedron and . The recession cone of (at ) is the set
From its definition, the recession cone consists of all directions along which we can move indefinitely from and still remain in . Notice that for all if and only if for all and this holds if and only if . Thus does not depend on a specific vector .
We call the extreme rays of associated with the extreme rays of . Note that the recession cone allows us to associate extreme rays of polyhedrons with extreme rays of convex cones. Any characterisation of extreme rays of polyhedral cones (including the following one) can be therefore also applied to extreme rays of polyhedrons.
A nonzero element of a polyhedral cone is an extreme ray if and only if there are linearly independent constraints binding at .
The following theorem gives a full description of pointed polyhedron (i.e. unbounded convex set with at least one extreme point) based only on extreme points and extreme rays.
Let be a nonempty polyhedron with at least one extreme point. Let be its extreme points and be its extreme rays. Then
2.2 Classical cooperative games
This subsection introduces the definition of cooperative games and the definition of the class of 1-convex games. We proceed with an introduction of solution concepts, namely the -value, the nucleolus and the Shapley value and we review their properties for 1-convex games.
2.2.1 Main definitions and notation
We denote the set of -person cooperative games by . Subsets of are called coalitions and itself is called the grand coalition. We often write instead of whenever there is no confusion over what the player set is. We shall often associate the characteristic functions with vectors . This will be more convenient for viewing sets of cooperative games as (possibly convex) sets of points.
We note that the presented definition assumes transferable utility (shortly TU). Therefore, by a cooperative game or a game we mean in fact a cooperative TU game.
To avoid cumbersome notation, we use the following abbreviations. We often replace singleton set with just . Analogously, we use instead of and instead of . We use for the relation of “being a subset of” and for the relation “being a proper subset of”. To denote the sizes of coalitions e.g. , we often use , respectively.
We note that incomplete games can also be viewed as partial functions defined on a power set and, indeed, such structures were also extensively studied seshadhri2014submodularity ; BK2018partextension ; BK2019coverextension (although in these cases without highlighting the connection to the theory of partially defined cooperative games). We refer to the excellent book of Grabisch grabisch2016setfunctions which discusses in a great detail connections of various types of set functions to entirely different parts of mathematics, with cooperative game theory being one of them.
The definition of 1-convex games relies on the notion of utopia (or upper) vector . It captures each player’s marginal contribution to the grand coalition, i.e. . When there is no ambiguity, we use instead of . The value is considered to be the maximal rightful value that player can claim when is distributed among players. If he claims more, it is more advantageous for the rest of the players to form a coalition without player .
A cooperative game is called 1-convex game, if for all coalitions , the inequality
holds and also
The set of 1-convex -person games is denoted by .
From (1), is 1-convex if even after every player outside the coalition gets paid his utopia value, there is still more left of the value of the grand coalition for players from than if they decided to cooperate on their own. This condition challenges the players to remain in the grand coalition and try to find a compromise in the payoff distribution. Also, in (2), the utopia vector sums to a value at least as large as the value of the grand coalition . This was motivated by the idea that the study of possible distributions is not interesting if every player can obtain his maximal rightful (utopia) value.
An equivalent formulation of 1-convexity is in terms of the gap function, defined as . It captures the gap between the utopia distribution for coalition and a possible distribution of the profit of .
Driessen1985 A game is 1-convex if and only if for all coalitions .
2.2.2 Solution concepts
The main task of cooperative game theory is to distribute the payoff of the grand coalition between the players. To be able to work with individual payoffs more easily, payoff vectors are introduced. Those are vectors where represents the individual payoff of player .
The definition of payoff vector is quite general, therefore, a suitable subset of payoff vectors, so called imputations are defined. Those are payoff vectors such that is efficient, i.e. and individually rational, i.e. for all
. This means an imputation distributes the worth of the grand coalitionbetween the players and only those payoffs where each player is at least as better off as he would be on his own are considered.
To choose between payoff vectors, different solution concepts are defined.
Let be a class of -person cooperative games. Then a function is a solution concept (on class ).
If the image of every cooperative game is exactly one vector, we write and we say is a one-point solution concept. Otherwise, we say is a multi-point solution concept.
We shall consider a generalization of two (actually three) solution concepts: the -value, the nucleolus, and the Shapley value. We now introduce these solution concepts, stating their properties and different characterisations, which will be used for our generalisations to incomplete games.
The -value is a known solution concept for 1-convex games (actually defined for a more general class of quasi-balanced games) defined originally by Tijs in 1981 Tijs1981 . We will follow his definition where he defines the -value as a compromise between the utopia vector and the minimal right vector that is defined as where is the so called concession vector.
The class of quasi-balanced games is defined as
It holds that .
Let . Then the -value of game is defined as the unique convex combination of and such that .
For class , the -value can be expressed by a simple formula depending on the utopia vector and the gap function. The formula can be interpreted as follows. Every player receives his utopia value minus an equal share of the loss represented by the gap .
Driessen1985-thesis Let be a 1-convex game. Then the -value can be expressed as
There are two known axiomatic characterisations of the -value.
Tijs1987 The -value is a unique function satisfying
(minimal right property) ,
(restricted proportionality property)
where and it the zero-normalisation of .
The second, axiomatic characterisation can be found in paper of Tijs Tijs1995 . It consists of five axioms, namely efficiency, translation equivalence, bounded aspirations, convexity, and restricted linearity.
On top of that, there are further results concerning axioms of the -value, thus providing an even better comparison with other solution concepts. In the next theorem, we state several of them.
Tijs1981 For a 1-convex game , the -value satisfy
(individual rationality) ,
(symmetry) For each permutation we have ,
(dummy player) ,
(S-equivalence property) .
We note the -value does not satisfy additivity which is crucial in our generalisation of this concept. Surprisingly, we show that our generalisation of the -value satisfy a certain form of additivity on the class of incomplete games with minimal information.
The second solution concept is the nucleolus. Essential component of its definition is the excess which is a function dependent on a coalition and an imputation (a payoff vector which is both individually rational and efficient). It computes the remaining potential of when the payoff is distributed according to , i.e. . Further, is a vector of excesses with respect to which is arranged in non-increasing order.
The nucleolus, , is the solution concept which assigns to a given game the minimal imputation with respect to the lexicographical ordering , defined as:
It is a basic result in cooperative game theory that the nucleolus is a one-point solution concept peleg2007introduction
. In general, the nucleolus can be computed by means of linear programmingKopelowitz1967 . For 1-convex games, however, the notion of the nucleolus and the -value coincide.
Driessen1985-thesis Let be 1-convex game. Then .
In this text, we shall consider a generalisation of the -value for -extendable incomplete games, however, thanks to the theorem it can be also considered as a generalisation of the nucleolus.
Finally, we define the Shapley value.
The Shapley value is the unique function such that
There are alternative formulas for the Shapley value, including the one from the next theorem.
Peters2008 The Shapley value for can be expressed as
The Shapley value can be also characterised by means of axioms. The following is the characterisation proposed and proved by Shapley.
Shapley1953 The Shapley value is a unique function with
(null player) ,
Since the original introduction of the Shapley value, many alternative axiomatic characterisations of the Shapley value were given. Let us pin point the following few: Brink1995 ; Brink2002 ; Roth1977 ; Young1989 . As it would be an exhaustive task to investigate all of them at once, we considered only several of them (namely the second and the fourth mentioned). The Shapley value also satisfies all of the axioms from Theorem 2.7 except for individual rationality.
2.3 Partially defined cooperative games
(Incomplete game) An incomplete game is a tuple where is a finite set of players (in this text ), is the set of coalitions with known values and is the characteristic function of the incomplete game. We further assume that and .
We denote the set of -person incomplete games with by . An incomplete game can be viewed from several perspectives. In one of the views, there is an underlying complete game from a class of -person games . The presence of in implies further properties of the characteristic function, e.g. superadditivity. Unfortunately, only partial information (captured by ) is known and there is no way to acquire more knowledge. The goal is then to reconstruct as accurately as possible. This leads to the definition of -extensions.
Let be a class of -person games. A cooperative game is a -extension of an incomplete game if for every .
The set of all -extensions of an incomplete game is denoted by . We write -extension whenever we want to emphasize the game . Also, if there is a -extension, we say is -extendable. Finally, the set of all -extendable incomplete games with fixed is denoted by . In this text, we are mainly interested in -extensions.
The sets of -extensions studied in this text are always convex. One of the main goals of the model of partially defined cooperative games is to describe these sets using their extreme points and extreme rays whenever the description is possible. We refer to the extreme points as to extreme games.
If the structure of is too difficult to describe and it is bounded from either above or from below, we introduce the lower and the upper game.
(The lower game and the upper game of a set of -extensions) Let be a -extendable incomplete game. Then the lower game of and the upper game of are complete games such that for every and every , it holds
and for each , there are such that
These games delimit the area of that contains the set of -extensions. Even if we know the description of , the lower and the upper game are still useful as they encapsulate a range of possible profits of coalition across all possible -extensions.
In many situations in the cooperative game theory, full information on a cooperative game is not necessary for a satisfiable answer. For example, the -value of a 1-convex cooperative game depends only on values and for . What if there are other satisfiable ways to distribute the payoff between players that can be computed only from partial information encoded by an incomplete game? Based on this question, we can generalise solution concepts to incomplete games.
Let be a class of -extendable -person incomplete games. Then function is a solution concept (on class ).
If the image of every cooperative game is exactly one vector, we write and we say is a one-point solution concept. Otherwise, we say is a multi-point solution concept.
The model of partially defined cooperative games is still in its beginnings. One of the most significant downsides of classical cooperative games is the complexity of information required. For an -person game, we have to consider different coalitions with corresponding values of the characteristic function and to be able to apply the model, we often need all this information (the -value of 1-convex games is rather an exception).
3 Incomplete games with minimal information
In this section, we focus on the subclass of incomplete games with minimal information and their -extensions. An incomplete game is a game with minimal information if it contains (apart from ) only the grand coalition and singletons, i.e. . We also define the total excess as which will be widely used in this text.
In the first subsection, we derive a description of the set of -extensions. In the second subsection, we define different solution concepts and show they coincide on incomplete games with minimal information. Introduced as the average value, we investigate different axiomatisations of this solution concept in the third subsection.
3.1 Description of the set of -extensions
The first step towards understanding the set of -extensions is to characterise when it is empty.
An incomplete game with minimal information is -extendable if and only if .
Let . Since it is 1-convex, it must hold for each ,
We sum the inequalities over all players to get
We now expand expressions and rearrange the inequality into
Since is equivalent to , we bound the right-hand side of (5) by and by rearranging, we conclude that .
For the opposite direction, let us consider -extensions for defined as
Notice that such games coincide on values of . We claim that for any , the game . The condition holds since
Furthermore, and hence the condition is clearly satisfied.
Now to verify the condition for each , we distinguish two cases based on presence of in .
For , , which is equal to . Therefore, the condition is satisfied and in fact, its upper bound is attained.
For , and . Again, the condition holds and the upper bound is attained. ∎
We note that if , the set of -extensions is rather simple and consists only of (the upper game defined in Theorem 3.2). Therefore, we are naturally more interested in situations when .
The set of -extensions is not bounded from below. For a -extendable incomplete game and its -extension , we can construct yet another -extension dependent on a coalition such that . We set the characteristic functions of the two games to differ only in values of , so that . The 1-convexity of is easy to check from 1-convexity of and it can be immediately seen that any arbitrarily small number satisfying could be chosen for the worth of coalition in . While not bounded from below, the set of -extensions is bounded from above.
Let be a -extendable game with minimal information. Then the upper game has the following form:
To show that this is an upper bound for the value of each coalition , we formulate the following optimization problem:
Clearly, the optimal value of the optimization problem (if it exists) is the value . Also notice that from the condition for , i.e. , that the upper bound of is dependent only on value (which is a constant since ) and values for (which are variables). The sum of these variables is bounded from above by (since ). From below, we have to consider only conditions , because for , we can always choose a -extension such that the value is small enough to satisfy .
Therefore, we can simplify the optimization problem by:
removing conditions for ,
removing variables for , and
substituting objective function for .
By these simplifications, we get the following optimization problem:
The set of feasible solutions is now where and together with (for ) are constants. A feasible solution of problem (8) is equivalent to a feasible solution of problem (7) by setting . Notice that the optimal values for both problems with corresponding feasible solutions equal.
We restate the problem in terms of the characteristic function and we substitute for , arriving at
Problem (9) is an instance of linear programming. Therefore, we can construct its dual program:
Let us define the vector as
We deduce that
for all ,
Hence is a feasible solution of (10). Furthermore, the value of the objective function for equals . This means (from the duality of linear programming) that the primal program is feasible and the value of its objective function is bounded from above by this value.
To see that this upper bound is attained, take the game (from the proof of Theorem 3.1) such that . ∎
It is important (and by our opinion interesting) that the upper game of the set of superadditive extensions of non-negative incomplete games with minimal information coincides with the upper game of -extensions from Theorem 3.2.
The upper game is not 1-convex in general. For example, a 3-person incomplete game with minimal information with and for all is -extendable since . However, . From the condition , we can derive that , and hence . For and conditions , we can easily derive from the definition of the upper game that
Let be an incomplete game with minimal information. Then it holds that the upper game if and only if
So far, we showed that the set is a convex polyhedron since it can be described by a set of inequalities. It is bounded from above by and unbounded from below. Such polyhedrons (provided that they have at least one vertex) can be characterised by a set of extreme points and a cone of extreme rays (see Theorem 2.3).
We begin the derivation of the full description of the set of -extensions by proving that games (defined as (6)) are actually extreme points of the set. To prove this, we use a characterisation of extreme points from Theorem 2.1.
For -extendable game with minimal information, the games are extreme games of .
Let be a vector such that both . We will show that in such case, inevitably for all , thus by Theorem 2.1 is an extreme game.
Define and . For , clearly . It remains to show that for , .
For and , for the sake of contradiction, suppose w.l.o.g. that . Then , therefore , a contradiction.
For and , again, suppose . Because and are both 1-convex, conditions
must hold. We can rewrite both of the inequalities (and aggregate them by ) as
which is equivalent to
Since both inequalities hold, we conclude . We already derived that if if and only if . But since we conclude , which is a contradiction.
We proved that is necessarily a vector of zeroes and thus we conclude the proof by taking Theorem 2.1 into account. ∎
Not only are games for the extreme games of , they are also the only extreme games.
For a -extendable game with minimal information, the games are the only extreme games of .
We will prove this theorem by showing that any extreme game has the form of one of the games. Since there are different games, we have to enforce that the game coincides with for a specific .
To do so, realise there is such that . If there was no such , then . The sum of these conditions leads to
But this is a contradiction, because the opposite inequality holds. Now we proceed to prove that .
First, we show is the unique coalition of size with its coalition value different from , i.e. there is no such that . For a contradiction, if there is such , denote and . We define a non-trivial game such that both and , contradicting (by Theorem 2.1) that is an extreme game. The game can be described as
Since , it is equivalent to the respective condition of . The rest of the cases for non-empty can be dealt with in a similar manner, therefore both games . But this leads to a contradiction, because