1 Introduction

Developing various approaches to deal with uncertainty is heavily intertwined with decision-making and is thus also related with game theory. While sources of this uncertainty can be very different in nature, be it a lack of knowledge on the behaviour of others, corrupted data, signal noise, or a prediction of outcomes such as voting or auctions, there is a strong need to have procedures in place so that the best possible choice is made in spite of these challenges. Given the wide range of applications, numerous models of various complexities and use-case scenarios exist to remedy these inaccuracies. Among those models most relevant to cooperative game theory are fuzzy cooperative games (Branzei et al., 2008), Mareš 2001, Mareš and Vlach (2004), multi-choice games (Branzei et al., 2008), cooperative interval games (Alparslan Gök, 2014; Alparslan Gök, 2009; Lardon, 2017; Alparslan Gök et al., 2009; Bok & Hladík, 2015; Bok, 2021), fuzzy interval games (Mallozzi et al., 2011), games under bubbly uncertainty (Palancı et al., 2014), ellipsoidal games (Weber et al., 2010; Alparslan Gök & Weber, 2013), and games based on grey numbers (Palancı et al., 2015).

In the theory of classical (transferable utility) cooperative games, we know the precise value (or worth) of the cooperation of every group of players, referred to as a coalition. In incomplete cooperative games, this is generally not the case, as only some of the coalition values are known. Incomplete cooperative games can also be viewed as partial set functions. Indeed, such structures were already studied, yet without highlighting the relation to the theory of incomplete games. In particular, extension of partial set functions into so-called submodular functions was studied e.g. in Bhaskar and Kumar (2020, 2019); Seshadhri and Vondrák (2014). We refer to the exhaustive book by Grabisch (2016), which discusses in detail connections of various types of set functions to entirely different parts of mathematics, with classical cooperative games being one of them.

The model of incomplete cooperative games was first introduced in the literature by Willson (1993) in 1993. Willson provided the basic notion of an incomplete game (referred to as partially defined games) and generalized the definition of the Shapley value for such games. Two decades later, Inuiguchi and Masuya revived the research. In Masuya and Inuiguchi (2016), they focused mainly on the class of superadditive games (and also briefly mentioned particular cases of convex and positive games in which precisely the values of singleton coalitions and grand coalition are known). Masuya continued the research (Masuya, 2021a, b), focusing on approximations of the Shapley value and its interval estimations. Further, Yu (2021) introduced a generalization of incomplete games to games with coalition structures and studied the proportional Owen value (a generalization of the Shapley value). In 2022, Albizuri et al. (2022) defined a value for a general incomplete cooperative game defined recursively by a generalization of Harsanyi dividends. Bok and Černý considered the property of 1-convexity and related solution concepts (values) for incomplete games Bok and Černý (2024), and Černý and Grabisch (2024) focused on the so-called player-centered incomplete cooperative games. In 2024, Masuya (2024) investigated a simplified notion of convexity and Úradník et al. (2024) investigated the question of the learnability of an optimal strategy for revealing coalition values in order to minimize the so-called gap of the Shapley value.

There are two natural questions about incomplete games. The first question is: What could be the unknown values of coalitions? This question makes sense only if some restrictions are imposed on the game; otherwise, any value for a given coalition could be taken. A particular assignment of these values defines an extension of the incomplete game. The issue of uncertainty can actually be reversed. We can reverse the uncertainty in incomplete games to understand which information can be deduced by omitting some values. This approach is valuable because, as the size of cooperative games increases exponentially with the number of players, it allows us to simplify complex tasks, albeit with some loss of precision. The second question is: Since the usual solution concepts of TU-games (the core, the Shapley value, etc.) cannot be calculated for incomplete games, considering all possible extensions in a given family of games, what would these solutions be for each extension? To answer the second question, we must first analyze the former one. This is exactly the focus of our paper. While we do not discuss specific scenarios, we lay the theoretical groundwork for such an analysis. Our results concern two significant classes of cooperative games: convex games, which have remarkable properties in both game theory and decision theory, and their subclass, positive games, forming the basis of the Dempster-Shafer theory (Shafer, 1976) for modelling uncertainty.

Section 2 consists of preliminaries to classical and incomplete cooperative games In Sect. 3, we generally study the positivity of incomplete games. We tackle questions concerning the extendability to a positive extension, the boundedness of the set of positive extensions, and provide a description of this set using its extreme games in case it is bounded. Section 4 is focused on three different classes with a special structure of the known values. We analyze these classes as an application of the characterization of extreme games from the previous section. Section 5 is dedicated to convexity and to symmetric convex extensions. We characterize the conditions under which an incomplete game is extendable into a symmetric convex extension. We provide the range of each coalition’s worth over all such possible extensions and fully describe the set of symmetric convex extensions as a set of convex combinations of its extreme games. We also provide a geometrical point of view on the set of symmetric convex extensions. The model of incomplete games also has strong connections to other uncertainty models in cooperative game theory. In particular, we discuss some natural connections to cooperative interval games in Sect. 6, which concludes the paper.

2 Preliminaries

Comprehensive sources on classical cooperative game theory include (Branzei et al., 2008; Driessen, 1988; Gilles, 2010; Peleg & Sudhölter, 2007), with further applications discussed in works by Bilbao (2012), Curiel (2013), Molina et al. (2022), Omrani et al. (2019), and Lemaire (1991). To simplify our discussion, we introduce the following notational conventions: single-element sets i are denoted by i; \(\subseteq \) and \(\subsetneq \) indicate subset and proper subset relations, respectively. The notation \(\emptyset \ne S \subseteq N\) signifies that S is a non-empty subset of N. Lastly, n, s, t represent the sizes of N, S, T, accordingly.

2.1 Classical cooperative games

A cooperative game is an ordered pair (N, v) where \(N=\{1,\ldots ,n\}\) represents the set of players and the characteristic function \(v:2^N \rightarrow \mathbb {R}\) satisfying \(v(\emptyset )=0\) represents the worth of each coalition \(S \subseteq N\). We write v instead of (N, v) when the player set is clear from the context and associate \(v:2^N\rightarrow \mathbb {R}\) with vector \(v\in \mathbb {R}^{2^{|N |}-1}\). We denote the set of n-person cooperative games by \(\Gamma ^n\). A cooperative game (N, v) is

  1. 1.

    monotone if it satisfies \(v(S) \le v(T)\),              \(S\subseteq T \subseteq N\);

  2. 2.

    superadditive if \(v(S) + v(T) \le v(T\cup S)\),         \(S, T \subseteq N, S\cap T=\emptyset \);

  3. 3.

    convex if \(v(S) + v(T) \le v(T\cup S) + v(T\cap S)\),    \(S, T \subseteq N\).

There is also another characterization of convex games: a game (N, v) is convex if and only if \(v(S\cup i) - v(S) \le v(T\cup i) - v(T)\) for every \(S \subseteq T \subseteq N {\setminus } i\). A positive game (also known as totally monotonic) is defined as a non-negative combination of unanimity games \((N,u_T)\), parametrized by \(\emptyset \ne T \subseteq N\), where \(u_T(S) = 1\) if \(T \subseteq S\) and \(u_T(S)=0\) otherwise. Unanimity games form a basis of the vector space of cooperative games. Therefore, every cooperative game can be expressed as a linear combination of these games, with coefficients \(d_v(T)\), known as Harsanyi dividends. The Harsanyi dividend can be expressed either as \(d_v(T) = \sum _{S \subseteq T}(-1)^{|T {\setminus } S |}v(S)\) or recursively, as \(d_v(i) = v(i)\) for \(i \in N\) and \(d_v(T) = v(T) - \sum _{S \subsetneq T}d_v(S)\) for every other T. The sets of n-person superadditive, convex, and positive games are denoted by \(S^n\), \(C^n\), and \(P^n\), respectively. These classes of games form a hierarchy, specifically \(P^n \subseteq C^n\subseteq S^n\). A cooperative game (N, v) is symmetric if, for every \(S,T \subseteq N\) with \(|S |= |T |\), \(v(S) = v(T)\) holds. The sets of symmetric convex and symmetric positive n-person games are denoted by \(C^n_\sigma \) and \(P^n_\sigma \), respectively.

2.2 Incomplete cooperative games

An incomplete cooperative game is a tuple \((N,\mathcal {K},v)\) where \(N = \{1,\ldots ,n\}\) represents the set of players, \(\mathcal {K}\subseteq 2^N\) represents the set of coalitions with known values and the partial characteristic function \(v :\mathcal {K}\rightarrow \mathbb {R}\) represents the worth of each coalition \(S \in \mathcal {K}\). Further, it always holds \(\emptyset \in \mathcal {K}\) and \(v(\emptyset )=0\). An incomplete cooperative game \((N,\mathcal {K},v)\) is symmetric if for every \(S, T \in \mathcal {K}\) such that \(|S |= |T |\), it holds that \(v(S) = v(T)\). In the literature [see (Algaba et al., 2000, 2004; Myerson, 1977)], incomplete cooperative games are also called cooperative games with restricted cooperation. The main distinction in our approach is in the meaning of set \(\mathcal {K}\). Under the scope of restricted cooperation, coalitions outside \(\mathcal {K}\) cannot form, whereas in incomplete games, they can form but their worth is unknown. Therefore, the questions connected to each of the lines of research are fundamentally different. Compared to definitions in Masuya and Inuiguchi (2016), our assumptions on incomplete cooperative games are slightly more general. Most importantly, we do not assume a priori that the values of singleton coalitions and the grand coalition are among the known values, that is, that they are in \(\mathcal {K}\).

For a class of n-person games \(C \subseteq \Gamma ^n\), a cooperative game \((N,w) \in C\) is a C-extension of \((N,\mathcal {K},v)\) if \(w(S)=v(S)\) for every \(S \in \mathcal {K}\). The set of all C-extensions of an incomplete game \((N,\mathcal {K},v)\) is denoted by C(v). We use the term C(v)-extension to emphasize the game \((N,\mathcal {K},v)\). Also, if there is a C(v)-extension, we say \((N,\mathcal {K}, v)\) is C-extendable. Finally, the set of all C-extendable incomplete games with fixed \(\mathcal {K}\) is denoted by \(C(\mathcal {K})\). C-extensions are a fundamental tool for analyzing incomplete cooperative games. The smaller the set of C-extensions, the more is known about the underlying game, and thus about its solution concepts.

If the set of C-extensions is bounded, the area in \(\mathbb {R}^{2^n-1}\) which contains the set of C-extensions is delimited by the lower game \((N,\underline{v})\) and the upper game \((N,\overline{v})\). These games satisfy the condition that for every \(S \subseteq N\) and every C(v)-extension (N, w)

$$\begin{aligned} \underline{v}(S) \le w(S) \le \overline{v}(S) \end{aligned}$$

and for every \(S \subseteq N\), there exist \((N,w_1), (N,w_2) \in C(v)\) satisfying

$$\begin{aligned} \underline{v}(S) = w_1(S) \text {, and } \overline{v}(S) = w_2(S). \end{aligned}$$

If the set of C-extensions is unbounded, then either only one or none of the games is defined. The interval \(\left[ \underline{v}(S),\overline{v}(S)\right] \) represents the range of possible profits of coalition S across all possible C-extensions. It is important to note that distinguishing between the lower and upper games of different sets of extensions is crucial. For example, lower games of superadditive and convex extensions do not coincide in general. There are also examples of incomplete games where the set of convex extensions might be empty, and therefore, the lower game of convex extensions might not exist. However, the same incomplete game can have the lower game of superadditive extensions.

If we further assume that the incomplete cooperative game represents partial knowledge of a complete underlying game, then the set of C-extensions contains all the possible candidates for what the underlying game might be. In this scenario, it is important to be able to express the set exactly. The sets of C-extensions studied in this text are always convex. This allows us to describe them using their extreme points and extreme rays. We refer to the extreme points of sets of C-extensions as extreme games.

3 Positive extensions

In Masuya and Inuiguchi (2016), Masuya and Inuiguchi studied \(P^n\)-extensions of incomplete games with a special structure, namely \((N,\mathcal {K},v)\) with \(\mathcal {K}= \{\emptyset , N\} \cup \{\{i\} \mid i \in N\}\) and \(v(S) \ge 0\) for \(S \in \mathcal {K}\). In this text, we refer to these games as non-negative minimal incomplete games. In their work, due to an approach slightly different from ours, they do not address the question of \(P^n\)-extendability. To characterize \(P^n\)-extendability of non-negative minimal incomplete games, we denote \(\Delta = v(N) - \sum _{i \in N}v(i)\) and \(N_1 = \{T \subseteq N \mid |T |> 1\}\).

Theorem 1

Let \((N,\mathcal {K},v)\) be a non-negative minimal incomplete game. It is \(P^n\)-extendable if and only if \(\Delta \ge 0\).

Proof

If \(\Delta \ge 0\), it immediately follows that game \((N,w^*)\) defined using its dividends as

$$\begin{aligned} d_{w^*}(S) :={\left\{ \begin{array}{ll} v(i) &{} \textit{if } S = \{i\},\\ \Delta &{} \textit{if } S = N,\\ 0 &{} \textit{otherwise},\\ \end{array}\right. } \end{aligned}$$

is \(P^n(v)\)-extension. If \(\Delta < 0\), it follows for any \(P^n(v)\)-extension (N, w) that

$$\begin{aligned} \Delta = \sum _{\emptyset \ne S \subseteq N}d_w(S) - \sum _{i \in N}\delta _w(i) = \sum _{S \in N_1}d_w(S) < 0. \end{aligned}$$

As \(d_w(S) \ge 0\) for every \(S \in N_1\), this leads to a contradiction. \(\square \)

In Masuya and Inuiguchi (2016), it was shown that the lower game \((N,\underline{v})\) and the upper game \((N,\overline{v})\) of \(P^n\)-extensions coincide with those of \(S^n\)-extensions. These games are defined by formulas

$$\begin{aligned} \underline{v}(S) :={\left\{ \begin{array}{ll} v(S), &{} \text {if }S \in \mathcal {K},\\ \sum _{i \in S}v(i), &{} \text {if }S \notin \mathcal {K},\\ \end{array}\right. }\text { and } \overline{v}(S) :={\left\{ \begin{array}{ll} v(S), &{} \text {if }S \in \mathcal {K},\\ v(N), &{} \text {if }S \notin \mathcal {K}.\\ \end{array}\right. } \end{aligned}$$
(1)

The set of \(P^n\)-extensions of non-negative minimal incomplete games is described in Masuya and Inuiguchi (2016) as a set of all convex combinations of its extreme points. Those correspond to games \((N,v^T)\), parametrized by coalitions \(\emptyset \ne T \subseteq N\),

$$\begin{aligned} v^T(S) = {\left\{ \begin{array}{ll} 0, &{} S = \emptyset ,\\ \Delta + \sum _{i \in S}v(i), &{} S \notin \mathcal {K}\text { and }T \subseteq S,\\ \sum _{i \in S}v(i), &{} S \notin \mathcal {K}\text { and }T \subsetneq S.\\ \end{array}\right. } \end{aligned}$$
(2)

Observe the similarity between\((N,v^T)\) and the unanimity game \((N,u_T)\).

Theorem 2

Masuya and Inuiguchi (2016) Let \((N,\mathcal {K},v)\) be a non-negative minimal incomplete game and let \((N,v^T)\) for \(T \in N_1\) be games from (2). The set of \(P^n\)-extensions can be expressed as

$$\begin{aligned} P^n(v)=\left\{ \sum _{T \in N_1}\alpha _Tv^T \mid \sum _{T \in N_1}\alpha _T=1, \alpha _{T} \ge 0\right\} . \end{aligned}$$
(3)

In the rest of this section, we generalize the discussed results to a general setting. In Sect. 3.1, we provide a characterization of \(P^n\)-extendability based on the duality of linear programming and give an example of its application in the time-complexity analysis of \(P^n\)-extendability of incomplete games with special structures. We further establish the necessary and sufficient conditions for the boundedness of \(P^n(v)\). In Sect. 3.2, we investigate a description of the set of \(P^n(v)\)-extensions if the set is bounded. We do so by characterizing extreme games of the set by following and slightly modifying the proof of the sharp form of Bondareva-Shapley theorem (the theorem was introduced independently by Bondareva (1963) and Shapley (1967).

3.1 \(P^n\)-extendability and boundedness of \(P^n\)

To provide a certificate for non-\(P^n\)-extendability of an incomplete game, we employ the duality of linear systems. This approach was motivated by a work by Seshadhri and Vondrák (2014) and the so-called path certificate for the non-extendability of submodular functions (corresponding to convex games). Although its size is exponential in the number of players in general, for special cases, the solvability of the dual system is polynomial in n, and therefore, the \(P^n\)-extendability is polynomially decidable in n for such cases. In the proof of the characterization, we use the seminal result by Farkas (1902).

Lemma 3

(Farkas’ lemma, Farkas 1902) Let \(A \in \mathbb R^{m \times n}\) and \(b \in \mathbb R^n\). Then exactly one of the following two statements is true.

  1. 1.

    There exists \(x \in \mathbb R^n\) such that \(Ax = b\) and \(x \ge 0\).

  2. 2.

    There exists \(y \in \mathbb R^m\) such that \(A^Ty \ge 0\) and \(b^Ty \le -1\).

Theorem 4

Let \((N,\mathcal {K},v)\) be an incomplete game. The game is \(P^n\)-extendable if and only if the following system of linear equations is not solvable:

  1. 1.

    \(\forall \,T \subseteq N, T \ne \emptyset : \sum _{S \in \mathcal {K}, T \subseteq S} y(S) \ge 0\),

  2. 2.

    \(\sum _{S \in \mathcal {K}}v(S)y(S) \le -1\).

Proof

Let \(M :=2^{n}-1\) and let \(U' \in \mathbb {R}^{M \times M}\) be a matrix whose columns are the characteristic vectors of unanimity games \(u_T\). It then holds that \(U'd = w\) for every game (N, w) and its corresponding vector of Harsanyi dividends d.

For an incomplete cooperative game \((N,\mathcal {K},v)\), we reduce matrix \(U'\) by deleting the rows corresponding to coalitions with unknown values, thereby obtaining a system \(Ud=v\). This adjustment eliminates unknowns on the right hand side of the equation, yet no information about the complete game is lost since the vector of Harsanyi dividends carries full information.

Game \((N,\mathcal {K},v)\) is \(P^n\)-extendable if and only if \(Ud=v\) is solvable for \(d \ge 0\). By Farkas’ lemma (Lemma 3) this happens if and only if the following system has no solution,

$$\begin{aligned} U^Ty \ge 0 \text { and } v^Ty \le -1. \end{aligned}$$
(4)

The conditions given by (4) correspond to those from the statement of the theorem. \(\square \)

Example 1

(Non-extendability of an incomplete game) Let \((N,\mathcal {K},v)\) be an incomplete cooperative game on 10 players, where \(\mathcal {K}= \left\{ \emptyset , \{1,2\},\{2,4\},\{1,2,3,4\}\right\} \) and

$$\begin{aligned} v(\{1,2\})=1\text {, }v(\{2,4\})=2\text { and }v(\{1,2,3,4\})=2. \end{aligned}$$

According to Theorem 4, the game \((N,\mathcal {K},v)\) is \(P^n\)-extendable if and only if the following system is not solvable:

$$\begin{aligned} y_{\{1,2,3,4\}}&\ge 0, \\ y_{\{1,2\}} + y_{\{1,2,3,4\}}&\ge 0, \\ y_{\{2,4\}} + y_{\{1,2,3,4\}}&\ge 0, \\ y_{\{1,2\}} + y_{\{2,4\}} + y_{\{1,2,3,4\}}&\ge 0, \\ y_{\{1,2\}} + 2y_{\{2,4\}} + 2y_{\{1,2,3,4\}}&\le -1. \end{aligned}$$

By adding the third and fourth inequalities, one gets \(y_{\{1,2\}} + 2y_{\{2,4\}} + 2y_{\{1,2,3,4\}} \ge 0\), which cannot be simultaneously satisfied with the fifth inequality. The system is not solvable; thus, confirming that \((N,\mathcal {K},v)\) is \(P^n\)-extendable.

Observe that although the total number of inequalities \(\sum _{S \in \mathcal {K}, T \subseteq S} y(S) \ge 0\) equals \(2^{n}-1\) (with one inequality for each non-empty subset \(T \subseteq N\)), the number of distinct inequalities does not exceed \(2^{|\mathcal {K}|-1}\), as each inequality aggregates over a subset of \(\mathcal {K}\). The actual count may be further reduced depending on the structure of \(\mathcal {K}\), as demonstrated in Example 1. The subsequent results present a necessary condition for the polynomial-time solvability of the \(P^n\)-extendability question.

Theorem 5

Let \((N,\mathcal {K},v)\) be an incomplete game such that sizes of all \(S \in \mathcal {K}\) are bounded by a fixed constant c. Then the problem of \(P^n\)-extendability is polynomially-time solvable in n.

Proof

In \((N,\mathcal {K},v)\), the number of coalitions with a defined value is at most \(\sum _{i = 1}^{c}{n\atopwithdelims ()i}\), which is a polynomial in n. Furthermore, when considering the linear system from Theorem 4, every \(T \subseteq N\) such that \(|T |> c\) yields an empty sum in its corresponding inequality. Therefore, the number of unique conditions in the problem is bounded by the number of coalitions with a defined value, namely, the sum \(\sum _{i = 1}^{c}{n\atopwithdelims ()i}\). Consequently, we deduce that the linear system is solvable in polynomial time using linear programming techniques. \(\square \)

Now we address the question of boundedness of \(P^n(v)\). We now address the boundedness of \(P^n(v)\). Observe that the set of \(P^n\)-extensions is always bounded from below, because for every \(P^n\)-extension (N, w), we have \(w(S) = \sum _{\emptyset \ne T \subseteq S}d_w(T)\), and \(d_w(T) \ge 0\) for all non-empty subsets \(T \subseteq N\). Therefore, 0 serves as a lower bound for the value of any coalition \(S \subseteq N\). Determining a lower bound that restricts the profit of every coalition, as well as identifying the corresponding upper bound (thus defining the lower and upper bounds of the game), remains an open problem.

Theorem 6

Let \((N,\mathcal {K},v)\) be a \(P^n\)-extendable incomplete game. The set of positive extensions \(P^n(v)\) is bounded if and only if \(N \in \mathcal {K}\).

Proof

If \(N \in \mathcal {K}\), then for any \(P^n(v)\)-extension (N, w), we have \(\sum _{T \subseteq N}d_w(T) = v(N)\), and since \(d_w(T) \ge 0\) for all non-empty subsets \(T \subseteq N\), it follows that \(d_w(T) \in \left[ 0,v(N)\right] \). This establishes a bound (potentially an overestimation) for all possible values of \(d_w(T)\). Given that the dividends are bounded, \(P^n(v)\) is also bounded.

If \(N \notin \mathcal {K}\), the value of coalition N can become arbitrarily large, as there is no upper bound on \(d_w(N)\) for any \(P^n(v)\)-extension (N, w). Therefore, \(P^n(v)\) is not bounded. \(\square \)

3.2 Description of the set of \(P^n\)-extensions

For an incomplete game \((N,\mathcal {K},v)\), the set of \(P^n(v)\)-extensions is described by

$$\begin{aligned} P^n(v) = \bigg \{(N,w) \Big |\, \forall S \in \mathcal {K}: w(S) = v(S) \text { and } \forall T\subseteq N: d_w(T) \ge 0\bigg \}, \end{aligned}$$

or equivalently in terms of dividends and \(M :=2^{n}-1\) by

$$\begin{aligned} P^n_d(v) :=\bigg \{ d_w \in \mathbb {R}^{M}\Big |\, \forall S \in \mathcal {K}: \sum _{T \subseteq S} d_w(T) = v(S),\forall T \subseteq N: d_w(T) \ge 0\bigg \}. \end{aligned}$$

Observe that \(P^n(v) \ne P^n_d(v)\), as the former represents a set of cooperative games, while the latter denotes a set of vectors of dividends. Both sets are formed by the intersections of closed half-spaces, and thus, they constitute convex polyhedrons. Assuming \((N,\mathcal {K},v)\) is \(P^n\)-extendable, then both sets are nonempty. Moreover, these sets are bounded if and only if \(N \in \mathcal {K}\). Bounded convex polyhedrons are the convex hulls of their extreme points. To justifiably neglect the distinction between the extreme points of both sets, we revisit a fundamental result from linear algebra. For the sake of completeness, we include the proof.

Lemma 7

Let P be a convex subset of \(\mathbb {R}^{n}\), \(A \in \mathbb {R}^{n \times n}\) a nonsingular matrix, and \(x \in P\) an extreme point of P. Then Ax is an extreme point of the convex set \(A(P) :=\left\{ Au |u \in P\right\} \).

Proof

Suppose that \(x \in P\) is an extreme point of P, but the image Ax is not an extreme point of A(P). Therefore, there exist \(Au, Av \in A(P)\) and \(\alpha \in (0,1)\) such that \(\alpha Au + (1 - \alpha ) Av = Ax\). However, \(\alpha Au + (1 - \alpha ) Av = A(\alpha u + (1 - \alpha )v) = Ax\), thereby implying that x is not an extreme point of P because it is a nontrivial convex combination of \(u, v \in P\). \(\square \)

Let \(U \in \mathbb R^{M\times M}\) be a matrix with vectors of unanimity games, \(u_T \in \mathbb {R}^{M}\) as columns. It holds that \(Ud_w = w\), where \(w \in \mathbb {R}^{M}\) is a characteristic vector of game (N, w), and \(d_w \in \mathbb {R}^{M}\) represents a vector of Harsanyi dividends of the game. Since unanimity games form a basis of \(\mathbb R^M\), the matrix U is nonsingular. Thus, by Lemma 7, the extreme points of \(P^n(v)\) correspond to those of \(P^n_d(v)\), allowing us to consider the latter instead of the former.

Following the proof of the sharp form of Bondareva-Shapley theorem from Peleg and Sudhölter (2007), we give an insight into the description of the extreme games of \(P^n(v)\). We show that for these games, the set of coalitions zero dividends is inclusion-wise maximal.

Our result is based on the following characterization of extreme points of polyhedrons.

Lemma 8

Peleg and Sudhölter (2007) Let P be a polyhedron given by

$$\begin{aligned} P :=\left\{ x \in \mathbb {R}^k \big |\, \sum _{j=1}^k a_{tj}x_j \ge b_t, t = 1,\dots , m\right\} . \end{aligned}$$

For \(x \in P\), let \(S(x) :=\big \{t \in \{1,\dots ,m\}|\, \sum _{j=1}^ka_{tj}x_j=b_t\big \}\). The point \(x \in P\) is an extreme point of P if and only if the system of linear equations

$$\begin{aligned} \sum _{j=1}^ka_{ij}y_j = b_t \text { for all } t \in S(x) \end{aligned}$$

has x as its unique solution.

Applying Lemma 8, \(d_e \in \mathbb {R}^M\) is an extreme game of \(P^n_d(v)\) if and only if there is no \(d_w\ne d_e\) such that \(d_w(T) = 0 \iff d_e(T)=0\) for every nonempty \(T \subseteq N\). For any \(P^n(v)\)-extension (N, w), we denote by E(w) the set of negligible coalitions, defined as \(E(w):=\{T \subseteq N \mid d_w(T)=0\}\). This set proves to be useful in the following lemma. The lemma states that the inclusion-maximality of E(e) across E(x) for \(d_x \in P^n_d(v)\) is equivalent to the uniqueness of E(e) across E(x) for \(d_x \in P^n_d(v)\). Together with Lemma 8, this connection establishes the relationship between the extremality of games and the inclusion-maximality of sets E(e).

Lemma 9

Let \((N,\mathcal {K},v)\) be a \(P^n\)-extendable incomplete game and \(d_e \in P^n_d(v)\). Then the following are equivalent:

  1. 1.

    There is no \(d_x\in P^n_d(v)\) such that \(E(e) \subsetneq E(x)\),

  2. 2.

    There is no \(d_y\in P^n_d(v)\) different from \(d_e\), such that \(E(e) = E(y)\).

Proof

First, suppose that there is \(d_x \in P^n_d(v)\) such that \(E(e) \subsetneq E(x)\). We show that there are not only one, but infinitely many vectors \(d_{y^\alpha } \in P_d(v)\) different from \(d_e\) such that \(E(e)=E(y^\alpha )\). The idea is to take any non-trivial convex combination \(d_{y^\alpha }:=\alpha d_e + (1-\alpha )d_x\) for \(0< \alpha < 1\). Such a game is clearly positive (a convex combination of non-negative dividends remains non-negative) as it is also an extension of \((N,\mathcal {K},v)\), because for every \(S \in \mathcal {K}\),

$$\begin{aligned} \sum _{T \subseteq S} d_{y^\alpha }(T) = \alpha \sum _{T \subseteq S} d_{e}(T) + (1-\alpha ) \sum _{T \subseteq S} d_{x}(T) = \alpha v(S) + (1-\alpha )v(S) = v(S). \end{aligned}$$

And since \(d_x \ne d_e\), there is \(S \notin \mathcal {K}\) such that \(x(S)\ne e(S)\) for which

$$\begin{aligned} y^\alpha (S) = \sum _{T \subseteq S}d_{y^\alpha }(T) = \alpha \sum _{T \subseteq S}d_{x}(T) + (1-\alpha ) \sum _{T \subseteq S}d_{e}(T) = \alpha x(S) + (1-\alpha )e(S). \end{aligned}$$

Therefore, any two parameters \(\alpha _1,\alpha _2\) such that \(0< \alpha _1< \alpha _2 < 1\) yield different values \(y^{\alpha _1}(S) \ne y^{\alpha _2}(S)\), thus \(d_{y^{\alpha _1}}\ne d_{y^{\alpha _2}}\).

Now, suppose that there is \(d_y \in P^n_d(v)\) different from \(d_e\) such that \(E(e)=E(y)\). We consider a combination \(d_z=d_e - \beta (d_y - d_e)\) with \(\beta \) such that for at least one \(S \notin E(e)\), \(d_z(S)=0\). Thus, \(E(e) \subseteq E(z)\) and still, \(d_z \in P^n_d(v)\). For such S, it must hold

$$\begin{aligned} d_z(S) = d_e(S) - \beta (d_y(S) - d_e(S)) = 0, \end{aligned}$$

therefore \(\beta = \frac{d_e(S)}{d_y(S)-d_e(S)}\). We have to choose S such that \(d_y(S)\ne d_e(S)\). Furthermore, we have to secure that for every \(T \notin E(e)\), \(d_z(T) \ge 0\), or equivalently

$$\begin{aligned} d_z(T)&= d_e(T) - \beta (d_y(T) - d_e(T))\\&= d_e(T) - \frac{d_e(S)}{d_y(S)-d_e(S)} (d_y(T) - d_e(T)) \ge 0 \end{aligned}$$

This can be done by taking the minimum for S over all such coalitions T, i.e.,

$$\begin{aligned} \beta :=\min \limits _{T \notin E(e): d_e(T)\ne d_y(T)}\frac{d_e(T)}{d_y(T)-d_e(T)}. \end{aligned}$$

Then for \(T \notin E(e)\), \(d_z(T)\ge 0\), since it is equal to

$$\begin{aligned} d_e(T) - \frac{d_e(S)}{d_y(S)-d_e(S)} (d_y(T) - d_e(T)) \ge d_e(T) - \frac{d_e(T)}{d_y(T)-d_e(T)} (d_y(T) - d_e(T)). \end{aligned}$$

Clearly, the last expression is equal to zero. Finally, for \(K \in \mathcal {K}\),

$$\begin{aligned} z(K) = \sum _{C \subseteq K}d_z(K) = \sum _{C \subseteq K}d_e(K) - \beta \left( \sum _{C \subseteq K} d_y(K) - \sum _{C \subseteq K}d_e(K)\right) , \end{aligned}$$

and since all the three sums in the last expression are equal to v(K), we conclude that \(z(K)=v(K)\) and thus, \(d_z \in P^n_d(v)\). \(\square \)

The following characterization of extreme points follows as a direct application of Lemmas 8 and 9.

Theorem 10

For \(P^n\)-extendable incomplete game \((N,\mathcal {K},v)\), it holds (N, e) is an extreme game of \(P^n(v)\) if and only if its set of negligible coalitions E(e) is inclusion-maximal, i.e., there is no \((N,w)\in P^n(v)\) such that \(E(e) \subsetneq E(w)\).

Example 2

(Extreme point of \(P^n(v)\)) Let \((N,\mathcal {K},v)\) be an incomplete cooperative game, where

  • \(N = \{1,2,3,4\}\),

  • \(\mathcal {K}= \{\emptyset , \{1\},\{1,2\},\{1,2,3\},\{1,2,3,4\}\}\), and

  • \(v(\{1\})=1,v(\{1,2\}) = 2, v(\{1,2,3\}) = 3, v(\{1,2,3,4\}) = 5\).

Consider two \(P^n\)-extensions \((N,w_1)\) and \((N,w_2)\) with all the dividends equal to zero except to

  • \(d_{w_1}(\{1\}) = d_{w_2}(\{1\}) = 1\),

  • \(d_{w_1}(\{1,2\}) = d_{w_2}(\{1,2\}) = 1\),

  • \(d_{w_1}(\{1,2,3\}) = d_{w_2}(\{1,2,3\}) = 1\),

  • \(d_{w_1}(\{1,2,4\}) = 2\), and

  • \(d_{w_2}(\{1,2,4\}) = d_{w_2}(\{2,3,4\}) = 1\),

Clearly, \((N,w_2)\) is not an extreme point since \(E(w_2) \subsetneq E(w_1)\). Through exhaustive analysis, it can be shown that for \(E(w_1) \subsetneq E\), there are no \(P^n\)-extensions with E as its set of negligible coalitions; thus, \((N,w_1)\) is an extreme point.

4 Application to analysis of positive extensions for special cases

This section presents an analysis of \(P^n\)-extensions of several classes of incomplete games. We demonstrate the direct application of Theorem 10 in describing the set of \(P^n\)-extensions for three classes of incomplete games. We do not show only a derivation of extreme games but also a derivation of the lower and the upper game together with a characterization of \(P^n\)-extendability.

4.1 Pairwise disjoint coalitions of known values

For the first class of incomplete games it holds that the coalitions with known values (excluding N) are pairwise-disjoint.

Theorem 11

Let \((N,\mathcal {K},v)\) be a \(P^n\)-extendable incomplete game, where \(\mathcal {K}= \left\{ S_1,\dots ,S_{k-1},N\right\} \) and for all \(i,j \in \{1,\ldots ,k-1 \}\), it holds that \(S_i \cap S_j = \emptyset \). Then the extreme games \(v^{\mathcal {T}}\), the lower game \(\underline{v}\), and the upper game \(\overline{v}\) can be described as follows:

$$\begin{aligned} v^{\mathcal {T}}(S):={\left\{ \begin{array}{ll} 0, &{} \text {if } \not \exists T \in \mathcal {K}: T \subseteq S,\\ \sum _{i:T_i \subseteq S}v(S_i), &{} \text {if } \exists T \in \mathcal {K}: T \subseteq S \text { and } T_N \nsubseteq S\\ v(N) - \sum _{i: T_i \nsubseteq S}v(S_i), &{} \text {if } \exists T \in \mathcal {K}: T \subseteq S \text { and } T_N \subseteq S,\\ \end{array}\right. } \\ \underline{v}(S) :=v^{\mathcal {K}}(S)={\left\{ \begin{array}{ll} 0, &{} \text {if } \not \exists T \in \mathcal {K}: T \subseteq S,\\ \sum _{i:S_i \subseteq S}v(S_i), &{} \text {if } \exists T \in \mathcal {K}: T \subseteq S \text { and } N \ne S,\\ v(N), &{} \text {if } \exists T \in \mathcal {K}: T \subseteq S \text { and } N = S,\\ \end{array}\right. } \\ \overline{v}(S) :={\left\{ \begin{array}{ll} v(S_i), &{} \text {if } S \subseteq S_i,\\ v(N) - \sum _{i: S_i \nsubseteq S}v(S_i), &{} \text {otherwise},\\ \end{array}\right. } \end{aligned}$$

where \(\mathcal {T} :=\left\{ T_1,\dots ,T_{k-1},T_N\right\} \) such that \(T_i \subseteq S_i\), \(T_N \subseteq N\) and \(T_N \nsubseteq S_\ell \) for any \(\ell \in \{1,\ldots ,k-1\}\). Furthermore, the \(P^n\)-extendability of \((N,\mathcal {K},v)\) is characterized by a condition

$$\begin{aligned} v(N) \ge \sum _{i=1}^{k-1} v(S_i). \end{aligned}$$

Proof

Let \((N,\mathcal {K},v)\) be an incomplete game with the properties above. For any \(P^n(v)\)-extension (N, w), given that the coalitions in \(\mathcal {K}{\setminus } \left\{ N\right\} \) are disjoint, at least one subcoalition \(T_i\) of each coalition \(S_i \in \mathcal {K}\setminus \left\{ N\right\} \) must have a nonzero dividend \(d_w(T_i)\); otherwise, \(v(S_i) = 0\). By Theorem 10, there is at most one such subcoalition if we consider an extreme game. If there were two nonzero dividends \(d_w(T_i^1)\) and \(d_w(T_i^2)\) for one \(S_i\), then the corresponding set of negligible coalitions would not be maximal. Setting the dividend of \(T_i^1\) to \(d_{w}(T_i^1) + d_{w}(T_i^2)\) yields a set E, such that \(E(w) \subsetneq E\). By this, for the extreme game, it holds that \(d_{w}(T_i) = v(S_i)\). Further, since \(v(N) = \sum _{T \subseteq N} d_{w}(T)\), it holds that \(v(N) \ge \sum _{S_i \in \mathcal {K}{\setminus } \left\{ N\right\} } v(S_i)\). If the inequality does not hold, then there is no extreme game of \(P^n(v)\), and hence, since the set is bounded (\(N \in \mathcal {K}\)), it is not \(P^n\)-extendable. Now, if the inequality is strict, there must be another nonzero dividend of a coalition \(T_N \subseteq N\) such that \(T_N \nsubseteq S_i\) for \(S_i \in \mathcal {K}{\setminus } \left\{ N\right\} \); otherwise, \(T_i\) and \(T_N\) would be two distinct subsets of \(S_i\), and E(w) would not be maximal. Again, since we are interested in extreme games, by Theorem 10, there is only one such coalition \(T_N\), resulting in \(d_{w}(T_N) = v(N) - \sum _{S_i \in \mathcal {K}{\setminus } \left\{ N\right\} } v(S_i)\). Any game parametrized by a collection \(\mathcal {T} :=\left\{ T_1,\dots ,T_{k-1},T_N\right\} \) and expressed as \(v^\mathcal {T}\) from the statement of the theorem is thus an extreme game of \(P^n(v)\).

Now let us show that the game \(v^\mathcal {K}\) is the lower game. For a coalition S with no subcoalition contained in \(\mathcal {K}\), \(v^\mathcal {K}(S) = 0 = \underline{v}(S)\). For a coalition S such that there exists \(T \in \mathcal {K}\), \(T \subseteq S\), the value of w(S) in any \(P^n(v)\)-extension cannot be smaller than the sum \(\sum _{T: T \in \mathcal {K}, T \subseteq S} v(T) = v^\mathcal {K}(S)\). And since \(N \in \mathcal {K}\), \(v^{\mathcal {T}}(N) = v(N) = \underline{v}(N)\).

Finally, we show that each value of the upper game is achieved by a different extreme game. If S is a proper subcoalition of \(S_i\), the value \(v(S_i)\) is, thanks to the non-negativity of dividends, an upper bound for the value of S. For any extreme game \(v^{\mathcal {T}}\) such that \(S \in \mathcal {T}\), this bound is tight. If S is not a subcoalition of any \(S_i\), its value cannot exceed \(v(N) - \sum _{T_i \in \mathcal {T}\setminus {T_N}} d_{w}(T_i)\); otherwise, the characterization of \(P^n\)-extendability is not satisfied for the grand coalition N. By taking an extreme game with \(T_N = S\), we ascertain that this bound is tight. \(\square \)

4.2 Set of known values \(\mathcal {K}\) closed on subsets

The second class of incomplete games satisfies that the set \(\mathcal {K}{\setminus } \{N\}\) is closed on subsets, i.e., \(S \in K, T \subseteq S \implies T \in \mathcal {K}\). The analysis of this case helps us further in the study of symmetric positive extensions (\(P^n_\sigma \)-extensions).

Theorem 12

Let \((N,\mathcal {K},v)\) be a \(P^n\)-extendable incomplete game such that \(N \in \mathcal {K}\) and for every \(S\in \mathcal {K}{\setminus } \left\{ N\right\} , T \subseteq S \implies T \in \mathcal {K}\). Furthermore, for \(S \in \mathcal {K}\), let \(\delta _S\) be defined as \(\delta _{\{i\}} = v(\{i\})\) and \(\delta _S = v(S) - \sum _{T \subsetneq S}\delta _T\). Then the extreme games \(v^{C}\), the lower game \(\underline{v}\), and the upper game \(\overline{v}\) can be described as follows:

$$\begin{aligned} v^{C}(S):={\left\{ \begin{array}{ll} \delta _N + \sum _{T \in \mathcal {K}, T \subseteq S} \delta _T, &{} \text {if } C \subseteq S,\\ \sum _{T \in \mathcal {K}, T \subseteq S} \delta _T, &{} \text {otherwise,}\\ \end{array}\right. } \end{aligned}$$

for \(C \notin \mathcal {K}\setminus \left\{ N\right\} \), and

$$\begin{aligned} \underline{v}(S) :=v^{N}(S) ={\left\{ \begin{array}{ll} \delta _N + \sum _{T \in \mathcal {K}, T \subseteq S} \delta _T, &{} \text {if } S = N,\\ \sum _{T \in \mathcal {K}, T \subseteq S} \delta _T, &{} \text {otherwise,}\\ \end{array}\right. } \\ \overline{v}(S) :={\left\{ \begin{array}{ll} v(S), &{} \text {if } S \in \mathcal {K},\\ v^{S}(S), &{} \text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Furthermore, \((N,\mathcal {K},v)\) is \(P^n\)-extendable if and only if \(\delta _S \ge 0\) for all \(S \in \mathcal {K}\).

Proof

Let \((N,w) \in P^n(v)\). Thanks to the structure of \(\mathcal {K}\), the dividends \(d_w(S)\) for \(S \in \mathcal {K}{\setminus } \left\{ N\right\} \) are the same for any \((N,w) \in P^n(v)\), and they are equal to \(\delta _S\). Consequently, for any S such that \(\delta _S = 0\), it holds that \(S \in E(w)\), and this is true for any \(P^n(v)\)-extension. Now, if the uniquely defined value \(\delta _N = v(N) - \sum _{S \in \mathcal {K} {\setminus } \left\{ N\right\} } \delta _S > 0\), there must be at least one \(C \notin \mathcal {K}\setminus \left\{ N\right\} \) with a nonzero dividend \(d_w(C)\). By Theorem 10, following a similar argument as in the proof of the previous theorem, E(w) is maximal if and only if there is only one such C. Otherwise, if there are \(C_1 \ne C_2\) such that \(d_w(C_1) \ne 0\) and \(d_w(C_2) \ne 0\), by taking \((N,x) \in P^n(v)\) such that \(d_x(C_1) = 0\) and \(d_x(C_2) = d_w(C_1) + d_w(C_2)\), we arrive at a contradiction with maximality, since \(E(w) \subsetneq E(x)\). Thus, choosing \((N,w) \in P^n(v)\), such that \(d_w(C) = \delta _N\), yields an extreme game \(v^C\) of \(P^n(v)\) for any \(C \notin \mathcal {K}\setminus {N}\).

For any coalition S, its value in any \(P^n(v)\)-extension must be greater than or equal to \(\sum _{T \in \mathcal {K}, T \subseteq S} \delta _S\). Observe \(v^{N}(S)\) is equal to this sum for any S, thus being the lower game.

For any coalition S, its maximal value is either v(S) if \(S \in \mathcal {K}\), or at most \(v(N) - \sum _{T \in \mathcal {K}{\setminus } \left\{ N\right\} : T \nsubseteq S} \delta _T = \delta _N + \sum _{T \in \mathcal {K}, T \subseteq S} \delta _T\), which is equal to \(v^S(S)\), thus defining it as the upper game. \(\square \)

For both classes of studied incomplete games, it holds that \(\underline{v}(S)\in P^n(v)\). Additionally, note that the number of extreme games \(v^\mathcal {C}\) is equal to the number of coalitions C such that \(C \notin \mathcal {K}{\setminus } \{N\}\), that is \(2^n-|\mathcal {K}|+ 1\) if \(v(N) - \sum _{S \in \mathcal {K}{\setminus } \left\{ N\right\} }\delta _S > 0\); otherwise, \(P^n(v)\) contains precisely one game (in case \(v(N) - \sum _{S \in \mathcal {K}{\setminus } \left\{ N\right\} }\delta _S = 0\)) or no game at all (if \(v(N) - \sum _{S \in \mathcal {K}{\setminus } \left\{ N\right\} }\delta _S < 0\)).

4.3 Symmetric positive extensions

We denote the set of symmetric positive extensions of \((N,\mathcal {K},v)\) by \(P^n_\sigma (v)\). Analogously to study of \(C^n_\sigma (v)\), we make use of the reduced forms (N, s) and \((N,\mathcal {X},\sigma )\) of games (N, v) and \((N,\mathcal {K}, v)\), respectively, which are defined in Definition 1. We can easily obtain the following result as a corollary of Theorem 12.

Theorem 13

Let \((N,\mathcal {X}, s)\) be the reduced form of a symmetric incomplete game such that \(n \in \mathcal {X}\) and \( i \in N, i \le k \implies i \in \mathcal {X}\). Then the lower game and the upper game of \(P^n_\sigma (v)\) can be described as

$$\begin{aligned} \underline{s}(i) :={\left\{ \begin{array}{ll} s(i), &{} \text {for } i \in \mathcal {X},\\ s(k), &{} \text {otherwise,}\\ \end{array}\right. } \text {\quad and\quad } \overline{s}(i) :={\left\{ \begin{array}{ll} s(i), &{} \text {for } i \in \mathcal {K},\\ s(n), &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

The following game illustrates that even in the symmetric scenario, there is \((N,\mathcal {X},\sigma )\) such that \((N,\underline{s}) \not \in P^n_\sigma (v)\).

Example 3

(The lower game is not necessarily a \(P^4_\sigma \)-extension) Let \((N,\mathcal X,\sigma )\) be the reduced form of a symmetric 4-person incomplete game such that \(\mathcal X = \left\{ 2,4\right\} \). From the properties of symmetric positive games we know that any \((N,s) \in P^4_\sigma (v)\) is given by 4 non-negative dividends with corresponding values \(d_1,d_2,d_3,d_4\) such that

  • \(s(1) = d_1\),

  • \(s(2) = 2d_1 + d_2\),

  • \(s(3) = d_3 + 3d_2 + 3d_1\),

  • \(s(4) = d_4 + 4d_3 + 6d_2 + 4d_1\).

By setting \(d_1 :=0, d_2 :=\sigma (2),d_3 :=0\), and \(d_4 :=\sigma (4) - 6d_2\) we get a \(P^4_\sigma \)-extension where \(s(1) = 0\) (clearly the minimum) and it is achieved if and only if \(d_1=0\). Setting \(d_1=0\) yields \(s(3) = 3\sigma (2)\). However, to minimize s(3), we can choose \(d_1 :=\frac{\sigma (2)}{2}\), \(d_2 :=0\), \(d_3 :=0\), and \(d_4 :=\sigma (4) - 4d_1\), obtaining \(s(3) = 3d_1 = \frac{3}{2}\sigma (2)\). We cannot minimize both values simultaneously and thus \((N,\underline{s})\notin P^n_\sigma (v)\).

Generalizing this example to symmetric n-person games is straightforward. For similar reasons, the lower game of the (non-symmetric) \(P^n\)-extensions of non-symmetric incomplete games is not contained in \(P^n(v)\). This contrasts with the findings for the classes of incomplete games in Theorems 11 and  12.

5 Convex extensions

For non-negative incomplete games with minimal information, the sets of \(S^n\)-extensions and \(P^n\)-extensions are described in Masuya and Inuiguchi (2016). For the sake of completeness, in this section we derive similar results for the set of \(C^n\)-extensions.

Theorem 14

Let \((N,\mathcal {K},v)\) be a non-negative incomplete game with minimal information. It is \(C^n\)-extendable if and only if \(\Delta \ge 0\).

Proof

If \(\Delta \ge 0\), it immediately follows that game \((N,w^*)\) defined using its dividends as

$$\begin{aligned} d_{w^*}(S) :={\left\{ \begin{array}{ll} v(i) &{} \textit{if } S = \{i\},\\ \Delta &{} \textit{if } S = N,\\ 0 &{} \textit{otherwise},\\ \end{array}\right. } \end{aligned}$$

is \(C^n\)-extension. If \(\Delta < 0\), it follows \(v(N)<\sum _{i \in N}v(i)\), thus any extension of \((N,\mathcal {K},v)\) cannot be convex. \(\square \)

In Masuya and Inuiguchi (2016), it is shown that the lower and upper games of \(P^n\)-extensions coincide with those of \(S^n\)-extensions, thus they must also coincide with the lower and upper games of \(C^n\)-extensions [see 1]. Finally, we derive a description of the set of \(C^n\)-extensions. We denote \(N_1= \{T \subseteq N \mid |T |> 1\}\).

Theorem 15

Let \((N,\mathcal {K},v)\) be a non-negative incomplete game with minimal information, and let \((N,v^T)\) for \(T \in N_1\) be games from (2). The set of \(C^n\)-extension can be expressed as

$$\begin{aligned} C^n(v)=\left\{ \sum _{T \in N_1}\alpha _Tv^T \mid \sum _{T \in N_1}\alpha _T=1, \forall S_1,S_2 \subseteq N: \sum _{T \in E(S_1,S_2)}\alpha _{T} \ge 0\right\} , \end{aligned}$$
(5)

where \(E(S_1,S_2):=\{T \subseteq S_1 \cup S_2 \,|\, T \nsubseteq S_1 \text { and } T \nsubseteq S_2\}\).

Proof

The proof follows from the proof of Theorem 6 in Masuya and Inuiguchi (2016). The only difference lies in the condition for coefficients \(\alpha _T\). For the description of the set of \(S^n\)-extensions, a condition \(\sum _{T \in E(S_1,S_2)}\alpha _{T} \ge 0\) is enforced for every pair of conditions where \(S_1\cap S_2 =\emptyset \). This condition corresponds to the fact that for \(S_1,S_2 \subseteq N\) such that \(S_1 \cap S_2 = \emptyset \), it holds \(v(S_1) + v(S_2) \le v(S_1 \cup S_2)\). In terms of Harsanyi dividends, it is equivalent to \(\sum _{T \in E(S_1,S_2)}\delta _v(T) \ge 0\). For convex games and \(S_1,S_2 \subseteq N\) (not necessarily disjoint coalitions), the conditions \(v(S_1) + v(S_2) \le v(S_1 \cap S_2) + v(S_1 \cup S_2)\) can be equivalently expressed in terms of Harsanyi dividends as

$$\begin{aligned} \sum _{T \subseteq S_1 \cup S_2, T \nsubseteq S_1, T \nsubseteq S_2}\alpha _T \ge 0. \end{aligned}$$

Notice that coalitions T are exactly those from the set \(E(S_1,S_2)\). \(\square \)

In our attempt to derive similar results for a more general setting, we surveyed existing results regarding submodular set functions (a set function \(v :2^N \rightarrow \mathbb {R}\) is submodular if and only if \(-v\) is supermodular).

The study of extendability of submodular functions initiated Seshadhri and Vondrák in Seshadhri and Vondrák (2014). They introduced path certificate, a combinatorial structure whose existence certifies that a submodular function is not extendable. They also showed an example of a partial function defined on almost all coalitions that is not extendable, but by removing a value for any coalition, the game becomes extendable. Later in 2018, Bhaskar and Kumar (2020) studied extendability of several classes of set functions, including submodular functions. Inspired by the results of Seshadhri and Vondrák, they introduced a more natural combinatorial certificate of non-extendability—square certificate. Using this concept, they demonstrated that a submodular function is extendable on the entire domain if and only if it is extendable on the lattice closure of the sets with defined values. The lattice closure \(LC(\mathcal {K})\) of a set of points \(\mathcal {K}\subseteq 2^N\) in a partially ordered set \((2^N,\subseteq )\) is the inclusion-minimal subset of \(2^N\) that contains \(\mathcal {K}\) and that is closed under the operation of union and intersection of sets. Following is a modification of Theorem 7 from Bhaskar and Kumar (2020).

Theorem 16

Bhaskar and Kumar (2020) Let \((N,\mathcal {K},v)\) be an incomplete cooperative game and

$$\begin{aligned} \mathcal {F} :=LC(\mathcal {K}) \cap \{S \subseteq N \mid \underline{S},\overline{S} \in \mathcal {K}\text { s.t. } \underline{S} \subseteq S \subseteq \overline{S}\}. \end{aligned}$$

Incomplete game \((N,\mathcal {K},v)\) is \(C^n\)-extendable if and only if there is supermodular \(w:2^\mathcal {F} \rightarrow \mathbb {R}\) such that \(w(S)=v(S)\) for \(S \in \mathcal {K}\).

In 2019, the same authors showed that the problem of extendability for a subclass of submodular functions, termed coverage functions, is NP-complete [see Bhaskar and Kumar (2019)]. Thus, the question of \(C^n\)-extendability is in general NP-complete as well.

The rest of the questions concerning \(C^n\)-extensions of general incomplete games remain open problems. From now on, we focus on extensions that are both convex and symmetric. The reasons are twofold. First, symmetry facilitates a simpler analysis of the set of \(C^n\)-extensions. Second, symmetric \(C^n\)-extensions constitute an important subset of \(C^n\)-extensions and can be considered an approximation of the set.

5.1 Symmetric convex extensions

Given the complexity of describing the set of convex extensions in full generality, we focus on a subset of \(C^n\)-extensions that are symmetric, denoting this set by \(C^n_\sigma \). The additional property of symmetry yields compact-and in our opinion, elegant-descriptions of the set of \(C^n_\sigma \)-extensions. Since \(C^n_\sigma (v) \subseteq C^n(v)\) for symmetric incomplete games, the set of symmetric convex extensions can be regarded as an approximation of the set \(C^n(v)\).

The main ingredient for our results is the following characterization of symmetric convex games. For completeness, the proof of this folklore result is provided in the appendix.

Proposition 17

Let (N, v) be a symmetric cooperative game. Then for every \(S \subsetneq N {\setminus } j\) and \(i \in S\), it holds that

$$\begin{aligned} v(S) \le \frac{v(S\setminus i) + v(S\cup j)}{2} \end{aligned}$$
(6)

if and only if the game is convex.

We note that the characterization from Proposition 17 does not hold for general convex games. This can be seen in the following example.

Example 4

(A convex game not satisfying conditions from Proposition 17) The game (N, v) given in Table 1 is convex, as can be easily checked. However, the inequality

$$\begin{aligned} v(\{1,3\}) \le \frac{v(\{1\}) + v(\{1,2,3\})}{2} \end{aligned}$$

is not satisfied, as \(6 \nleq \frac{1 + 9}{2}\).

Table 1 The game (N, v) from Example 4 with its characteristic function given in the table
Full size table

For symmetric games, we can denote by s(k) the value of v(S) of any \(S \subseteq N\) such that \(|S |= k\). This allows us to formulate the following characterization of symmetric convex games.

Theorem 18

A game (N, v) is symmetric convex if and only if for all \(k \in \{1, \dots , n-1\},\)

$$\begin{aligned} s(k)\le \frac{s(k-1)+s(k+1)}{2}. \end{aligned}$$
(7)

Hence we can associate every symmetric convex game (N, v) with a function \(s:\left\{ 0,\dots , n\right\} \rightarrow \mathbb {R}\) having the above property. Similarly, we can apply this to \((N,\mathcal {K},v)\) with a function \(\sigma :\mathcal X \rightarrow \mathbb {R}\) where \(\mathcal X \subseteq \{0,\dots ,n\}\) is constructed from \(\mathcal {K}\). To formalise these constructions, we define reduced forms of games (N, v) and \((N,\mathcal {K},v)\).

Definition 1

Let (N, v) be a symmetric game and \((N,\mathcal {K},v)\) a symmetric incomplete game.

  • The reduced form of a game (N, v) is an ordered pair (N, s), where the function \(s:\left\{ 0,\dots ,n\right\} \rightarrow \mathbb {R}\) is a reduced characteristic function such that \(s(k) :=v(S)\) for any \(S \subseteq N\) with \(|S |= k\).

  • The reduced form of an incomplete game \((N,\mathcal {K},v)\) is a tuple \((N,\mathcal X,\sigma )\) where \(\mathcal X = \{i |\, i \in \left\{ 0,\dots ,n\right\} , \exists S \in \mathcal {K}: |S |= i\}\) and the function \(\sigma :\mathcal X \rightarrow \mathbb {R}\) is defined as \(\sigma (k) :=v(S)\) for any \(S \in \mathcal {K}\) such that \(|S |= k\).

We refer to (N, s) and \((N, \mathcal X, \sigma )\) as the reduced game and the reduced incomplete game, respectively. Since \(\emptyset \) is always included in \(\mathcal {K}\), for every reduced incomplete game \((N,\mathcal X, \sigma )\), it follows that \(0 \in \mathcal X\) and \(\sigma (0) = 0\). For brevity, when considering a reduced game (N, s) of a \(C^n_\sigma (v)\)-extension, we often denote this as \((N,s) \in C^n_\sigma (v)\). We denote the complement of \(\mathcal {X}\) in \({0,\dots ,n}\) by \(\overline{X}\), i.e., \(\overline{\mathcal {X}} :={0,\dots ,n}\setminus \mathcal {X}\).

Note that a game (N, v) is symmetric convex if and only if the function s in its reduced form (N, s) satisfies property (7) from Theorem 18.

The reduced form (N, s) of a symmetric convex game (N, v) can be visualized as a graph in \(\mathbb {R}^2\). On the x-axis, we place the coalition sizes, and on the y-axis, the values of s. The point (0, 0) is fixed for all reduced games. According to Theorem 18, the conditions for \(k\in {1,\dots , n-1}\) dictate that for \(i \in {0,\dots ,n}\), the points (i, s(i)) lie in a convex position. More precisely, connecting the neighboring pairs \((i,s(i)), (i+1,s(i+1))\) (where \(i \in {0,\ldots ,n-1}\)) with line segments yields a graph of a convex function. The graph is illustrated with an example in Fig. 1. Further in this text, this function is referred to as the line chart of (N, s). Similarly, for \((N,\mathcal {X},\sigma )\), the line chart results from connecting consecutive elements of \(\mathcal {X}\) with line segments. If \(n \in \overline{\mathcal {X}}\), the rightmost line segment is extended to end at x-coordinate n. The values of s are then set to lie on the union of these line segments.

Fig. 1
figure 1

Examples of line charts of symmetric convex games in their reduced forms. The figure on the left depicts a game (N, s) where \(s(1) > 0\), the graph on the right a situation where \(s(1) < 0\). The slopes of the line segments are bounded by convexity of the function

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5.1.1 \(C^n_\sigma \)-extendability

For an incomplete game in reduced form, i.e., \((N,\mathcal X, \sigma )\), the first question that arises is that of \(C^n_\sigma \)-extendability. For \( \mathcal X = \left\{ 0,i\right\} \) with \(i \in \left\{ 1,\dots ,n\right\} \), the game is invariably \(C^n_\sigma \)-extendable (a possible \(C^n_\sigma \)-extension is the one in which the values of each coalition size lie on the line coming through \((0,\sigma (0))\) and \((i,\sigma (i))\)). Consequently, we consider cases where \(|\mathcal X |> 2\).

Theorem 19

Let \((N,\mathcal X,\sigma )\) be a reduced form of a symmetric incomplete game \((N,\mathcal {K},v)\) where \(|\mathcal X |> 2\). The game is \(C^n_\sigma \)-extendable if and only if

$$\begin{aligned} \sigma (k_2) \le \sigma (k_1) + (k_2 - k_1) \frac{\sigma (k_3) - \sigma (k_1)}{k_3-k_1}, \end{aligned}$$

for all consecutive elements \(k_1< k_2 < k_3\) from \(\mathcal X\).

Proof

If the game is \(C^n_\sigma \)-extendable, let (N, s) represent the reduced form of any \(C^n_\sigma \)-extension. According to Theorem 18, the line chart of (N, s) constitutes a convex function coinciding with \(\sigma \) on the values of \(\mathcal {X}\). Consequently, for any consecutive elements \(k_1, k_2, k_3\) in \(\mathcal {X}\), the inequality must be satisfied.

For the converse implication, we construct a \(C^n_\sigma (v)\)-extension by aligning the values of s with the line chart of \((N, \mathcal X, \sigma )\). The construction is illustrated in Fig. 2.

Fig. 2
figure 2

The construction of a \(C^n_\sigma \)-extension of \((N,\mathcal {X},\sigma )\) where \(\mathcal X = \{x_1,x_2,x_3,x_4\}\), using the line chart of \((N,\mathcal X,\sigma )\). The value s(k) lies on the line segment connecting \((x_3,\sigma (x_3))\) and \((x_4,\sigma (x_4))\)

Full size image

Notice that \(s(k) = \sigma (k)\) for \(k \in \mathcal X\) and also, because the inequalities for consecutive elements \(k_1,k_2,k_3\) from \(\mathcal X\) hold, the line chart represents a convex function. Thus for all \(k \in \{1,\dots ,n-1\}\), the following inequality holds:

$$\begin{aligned} s(k)\le \frac{s(k-1)+s(k+1)}{2}. \end{aligned}$$

According to Theorem 18, the game (N, s) is in \(C^n_\sigma (v)\). \(\square \)

As a direct consequence of the previous theorem, the problem of \(C^n_\sigma \)-extendability of symmetric incomplete games can be decided in linear time with respect to the size of the original game (i.e., the size of the characteristic function).

5.1.2 The lower game and the upper game

The following proposition addresses the boundedness of the set of \(C^n_\sigma \)-extensions. The restriction to \(|N |\ge 3\) is without loss of generality, because for \(|N |\le 2\), when the game \((N,\mathcal {X},\sigma )\) is not complete and is \(C^n_\sigma \)-extendable, the set of \(C^n_\sigma \)-extensions is always unbounded.

Proposition 20

Let \((N,\mathcal {X},\sigma )\) be the reduced form of a \(C^n_\sigma \)-extendable symmetric incomplete game \((N,\mathcal {K}, v)\) with \(|N |\ge 3\). The \(C^n_\sigma (v)\) is bounded if and only if \(|\mathcal {X}|\ge 3\) and \(n \in \mathcal {X}\).

Proof

Let \((N,\mathcal {X},\sigma )\) be the reduced form of a \(C^n_\sigma \)-extendable incomplete game. If \(n \in \overline{\mathcal {X}}\), it follows from Theorem 18 that there is no upper bound on the profit of n. Assume \(n \in \mathcal {X}\) and, for a contradiction, that there exists \(k \in N\) with no upper bound on its profit. Choose a \(C^n_\sigma (v)\)-extension (N, s) such that \(s(k) > k\frac{\sigma (n)}{n}\). The line chart of (N, s) is not a convex function (violating the property for (0, s(0)), (k, s(k)), (n, s(n))); hence, \((N,s)\not \in C^n_\sigma (v)\).

If \(|\mathcal {X} |\le 2\), then \(\mathcal {X} = {0,n}\); otherwise, the set of \(C^n_\sigma \)-extensions is unbounded from above. Let \(\ell \) be a value smaller than or equal to \(\sigma (n)\), and negative. Any game \((N,s_{\ell })\) with \(s_{\ell }(k) = \ell \) for \(k \in {1,\dots ,n-1}\) and \(s_{\ell }(0) = \sigma (0), s_{\ell }(n) = \sigma (n)\) constitutes a \(C^n_\sigma (v)\)-extension of \((N,\mathcal {X},\sigma )\). Consequently, there is no lower bound on the values of \(1,\dots ,n-1\).

If \(|\mathcal {X} |\ge 3\), let \(i \in \mathcal {X} {\setminus } {0,n}\). For \(k\in {1,\dots ,i-1}\), the point (k, s(k)) must lie on or above the line through points \((i,\sigma (i))\) and \((n,\sigma (n))\) to maintain the convexity of the line chart of (N, s); a violation leads to a contradiction. Similarly, for any \(k\in {i+1,\dots ,n-1}\), the value s(k) must be on or above the line through points \((0,\sigma (0))\) and \((i,\sigma (i))\) to avoid violating convexity. Hence, the profit of every k is bounded from below. \(\square \)

Theorem 21

Let \((N,\mathcal X,\sigma )\) be the reduced form of a \(C^n_\sigma \)-extendable symmetric incomplete game. Suppose that \(C^n_\sigma (v)\) is bounded. Furthermore, for every \(k \in \overline{\mathcal {X}}\), denote by \(i_1,i_2,j_1,j_2\) the closest distinct elements from \(\mathcal X\) such that it holds \(i_1< i_2< k< j_1 < j_2\), if they exist. Then the lower game has the following form:

The upper game has the following form:

$$\begin{aligned} \overline{s}(k) :={\left\{ \begin{array}{ll} \sigma (k), &{} \text {if } k \in \mathcal {X},\\ \sigma (i_2) + (k-i_2)\frac{\sigma (j_1) - \sigma (i_2)}{j_1-i_2}, &{}\text {otherwise.}\\ \end{array}\right. } \end{aligned}$$

Proof

To demonstrate that \((N,\underline{s})\) is the lower game, we begin by establishing that for every \(C^n_\sigma \)-extension (N, w) and every coalition size \(k \in N\), the inequality \(\underline{s}(k) \le w(k)\) holds. If \(k \in \mathcal {X}\), then \(\underline{s}(k) = \sigma (k) = w(k)\) trivially. For \(k \notin \mathcal {X}\), given that any \(C^n_\sigma \)-extension must exhibit a convex line chart, the value w(k) must be on or above the lines drawn through pairs of points \((i_1,\sigma (i_1)),(i_2,\sigma (i_2))\) and \((j_1,\sigma (j_1)),(j_2,\sigma (j_2))\). The definition of the lower game encapsulates this by determining \(\underline{s}(k)\) to be on one of these lines (if the other does not exist) or on the higher of the two.

To prove that \(\underline{s}(k)\) is achieved in at least one \(C^n_\sigma \)-extension for every \(k \in N\), we introduce a \(C^n_\sigma \)-extension \((N,s^{{a,b}})\) for consecutive \(a,b \in \mathcal {X}\) with \(a < b\), defined as

$$\begin{aligned} s^{\{a,b\}}(\ell ) :={\left\{ \begin{array}{ll} \sigma (\ell ), &{} \text {if } \ell \in \mathcal {X},\\ \underline{s}(\ell ), &{} \text {if } \ell \notin \mathcal {X} \text { and } a< \ell< b,\\ \overline{s}(\ell ), &{} \text {if } \ell \notin \mathcal {X} \text { and either } \ell< a \text {, or } b < \ell .\\ \end{array}\right. } \end{aligned}$$

This configuration confirms the game as an extension of \((N,\mathcal {X},\sigma )\). For \(i \in {2,\dots ,n-1}\), where all three values \(s^{{a,b}}(i-1),s^{{a,b}}(i),s^{{a,b}}(i+1)\) match the respective values of the upper game \(\overline{s}\), the relation \(s^{{a,b}}(i) \le \frac{s^{{a,b}}(i-1) + s^{{a,b}}(i+1)}{2}\) is maintained because \((N,\overline{s})\) represents a symmetric convex game. Thus, by Theorem 18, the same inequality applies to \(\overline{s}\). In other cases, either all three points \((i-1,s^{{a,b}}(i-1)),(i,s^{{a,b}}(i)),(i+1,s^{{a,b}}(i+1))\) align on a single line, satisfying the inequality with equality, or the three points are on the higher of two lines formed through points \((a_2,\sigma (a_2)),(a,\sigma (a))\) and \((b,\sigma (b)),(b_2,\sigma (b_2))\), with \(a_2 < a\) and \(b < b_2\) being consecutive pairs in \(\mathcal {X}\). If \(s^{{a,b}}(i) > \frac{s^{{a,b}}(i-1) + s^{{a,b}}(i+1)}{2}\), then it contradicts the \(C^n_\sigma \)-extendability of \((N,\mathcal {X},\sigma )\) either because \(\sigma (a) > \sigma (a_2) + (a - a_2) \frac{\sigma (b) - \sigma (a_2)}{b-a_2}\) or because \(\sigma (b) > \sigma (a) + (b - a) \frac{\sigma (b_2) - \sigma (a)}{b_2- a}\). For \(k \in \mathcal {X}\), we select \((N,s^{{a,b}})\) with \(a = k\), and for \(k \notin \mathcal {X}\), we choose \((N,s^{{a,b}})\) such that \(a< k < b\) to be the nearest coalition sizes with defined values.

Assuming for contradiction that there exists a reduced form (N, s) of a \(C^n_\sigma (v)\)-extension where \(\overline{s}(k) < s(k)\) for some \(k \in N\), and given that for \(k \in \mathcal {X}\), \(\overline{s}(k) = \sigma (k) = s(k)\), it implies \(k \notin \mathcal {X}\). If \(k \notin \mathcal {X}\) and \(\overline{s}(k) = \sigma (i_2) + (k-i_2)\frac{\sigma (j_1) - \sigma (i_2)}{j_1-i_2} < s(k)\), this violates the convexity of the line chart, evidenced by (k, s(k)) positioning above the line segment between \((i_2,\sigma (i_2))\) and \((j_1,\sigma (j_1))\), constituting a contradiction.

Finally, we validate that \((N,\overline{s})\) qualifies as a \(C^n_\sigma \)-extension of \((N,\mathcal {X},\sigma )\). It is evidently an extension, and the values of \((N,\overline{s})\) align with the line chart of \((N,\mathcal {X},\sigma )\). Given \(C^n_\sigma \)-extendability of the game and the resulting convexity of the line chart, the inequalities from Theorem 18 are applicable, confirming that \((N,\overline{s})\in C^n_\sigma (v)\). \(\square \)

The game \((N,\overline{s})\) is always a \(C^n_\sigma \)-extension, however, this is not true for \((N,\underline{s})\) in general, as can be seen in the example in Fig. 3.

Fig. 3
figure 3

An example of a reduced game \((N,\mathcal X,\sigma )\) with \(\mathcal X = \left\{ 0,1,2,4,6\right\} \) where the condition \(\frac{\underline{s}(3) + \underline{s}(5)}{2} \ngeq \underline{s}(4)\) from Theorem 18 is not satisfied. This implies that \((N,\underline{s})\) is not a \(C^n_\sigma \)-extension of \((N,\mathcal {X},\sigma )\)

Full size image

5.1.3 Extreme games

Games \((N,s^{\{a,b\}})\) are actually even more important because they are extreme games of \(C^n_\sigma (v)\).

Proposition 22

Let \((N,\mathcal {X},\sigma )\) be the reduced form of a \(C^n_\sigma \)-extendable symmetric incomplete game \((N,\mathcal {K},v)\). Games \((N,s^{\left\{ a,b\right\} })\) for consecutive \(a,b \in \mathcal {X}\), where \(a < b\), and \((N,\overline{s})\), are extreme games of \(C^n_\sigma (v)\).

Proof

Assume, for the sake of contradiction, that \((N,s^{{a,b}})\) is not an extreme game of \(C^n_\sigma (v)\) for some a, b. According to the definition of extreme points, there must exist two \(C^n_\sigma (v)\)-extensions \((N,s_1)\) and \((N,s_2)\), with \((N,s^{{a,b}})\) forming their nontrivial convex combination. Assuming, without loss of generality, there exists an \(i \in {0,\dots ,n}\) for which \(s_1(i)< s^{{a,b}}(i) < s_2(i)\). However, for \(i \in \mathcal {X}\), this situation cannot occur as \(s_1(i) = s_2(i) = s^{{a,b}}(i)\). Moreover, for \(i \notin \mathcal {X}\) with \(a< i < b\), it contradicts the condition \(s_1(i) < s^{{a,b}}(i) = \underline{s}(i)\), and for \(i \notin \mathcal {X}\) with either \(i < a\) or \(b < i\), a contradiction arises again since \(\overline{s}(i) = s^{{a,b}}(i) < s_2(i)\). Similarly, we can deduce that the upper game \((N,\overline{s})\) must also be an extreme game. \(\square \)

In general, \((N,\overline{s})\) and \((N,s^{\{a,b\}})\) are not the only extreme games. In the following theorem, we describe all the extreme games of \(C^n_\sigma (v)\).

Theorem 23

Let \((N,\mathcal {X},\sigma )\) be the reduced form of a \(C^n_\sigma \)-extendable symmetric incomplete game such that \(C^n_\sigma (v)\) is bounded. For \(k \in \left\{ 0,\dots ,n\right\} {\setminus } \mathcal {X}\) and \(i,j \in \mathcal {X}\) closest to k such that \(i< k < j\), games \((N,s^k)\) defined as

$$\begin{aligned} s^k(m) :={\left\{ \begin{array}{ll} \sigma (m), &{} \text {if } m \in \mathcal {X},\\ \overline{s}(m), &{} \text {if } m \notin \mathcal {X} \text { and either } m< i \text { or } j< m,\\ \underline{s}(m), &{} \text {if } m = k,\\ \sigma (j) + (m - j)\frac{\sigma (j)-\underline{s}(k)}{j - k}, &{} \text {if } m \notin \mathcal {X} \text { and } k< m< j,\\ \sigma (i) + (m - i)\frac{\underline{s}(k) - \sigma (i)}{k - i}, &{} \text {if } m \notin \mathcal {X} \text { and } i< m < k\\ \end{array}\right. } \end{aligned}$$

together with \((N,\overline{s})\) form all the extreme games of \(C^n_\sigma (v)\).

Proof

We divide the proof into two parts. Initially, we demonstrate that any \(C^n_\sigma (v)\)-extension (N, s) is a convex combination of the games \((N,\overline{s})\) and \((N,s^k)\) for \(k \in \overline{\mathcal {X}}\). Subsequently, we establish that each game \((N,s^k)\) is an extreme game, and thus, they collectively, along with the upper game \((N,\overline{s})\), constitute all the extreme games.

Defining a gap as an inclusion-wise maximal nonempty sequence of consecutive coalition sizes with unknown value, a gap exists between i and j if \(i,j \in \mathcal {X}\), \(i < j\), \(j-i > 1\), and for every \(i'\) such that \(i< i' < j\), we have \(i' \in \overline{\mathcal {X}}\). The size of a gap between i and j is \(j-i-1\), representing the count of coalition sizes with undefined values in that gap, which is at least one.

In the first part of the theorem, assuming a single gap in \((N,\mathcal {X},\sigma )\), we proceed with an induction based on the gap size. For a gap of size 1, only one game \((N,s^k)\), identical to \((N,s^{{k-1,k+1}})\), exists. Any \(C^n_\sigma \)-extension (N, s) can be represented as a convex combination of this game and the upper game \((N,\overline{s})\), expressed as \(s = \alpha s^k + (1-\alpha )\overline{s}\), where

$$\begin{aligned} \alpha = \frac{s(k)-\overline{s}(k)}{s^k(k) - \overline{s}(k)} \in [0,1]. \end{aligned}$$

In the induction step, for a gap between i and j of size \(\ell > 1\), there exist \(\ell \) games

$$\begin{aligned} (N,s^{i+1}),(N,s^{i+2}),\dots ,(N,s^{j-1}) \text { and } (N,\overline{s}). \end{aligned}$$

We construct a new system of \(\ell -1\) games

$$\begin{aligned} (N,(s^{i+2})'),(N,(s^{i+3})'),\dots ,(N,(s^{j-1})')\text { and } (N,(s^{i+1})'), \\ \text { where } (s^{m})' :=\alpha s^{m} + (1-\alpha )\overline{s}\text { with }\alpha = \frac{s(i+1)-\overline{s}(i+1)}{s^{i+1}(i+1) - \overline{s}(i+1)}. \end{aligned}$$

These games match the extreme games of an incomplete game \((N,\mathcal {X}',\sigma ')\) with \(\mathcal {X}' :=\mathcal {X} \cup {i+1}\) and \(\sigma '\) defined by \(\sigma '(m) = \sigma (m)\) for \(m \in \mathcal {X}\) and \(\sigma '(i+1) = s(i+1)\). Here, \((N,(s^{i+1})')\) serves as the upper game of \(C^n_\sigma (v)\). As this new system of \(\ell \) games forms the extreme games of \(C^n_{\sigma '}\)-extensions of \((N,\mathcal {X}',\sigma ')\), the game (N, s), being a \(C^n_{\sigma '}\)-extension, is their convex combination by induction. Since each \((N,(s^{m})')\) is a convex combination of \((N,\overline{s})\) and \((N,s^{m})\), the game (N, s) is also a convex combination of the initial system

$$\begin{aligned} (N,s^{i+1}),(N,s^{i+2}),\dots ,(N,s^{j-1}) \text { and } (N,\overline{s}). \end{aligned}$$

For more than one gap in \(\mathcal {X}\), a similar method applies, with each pair of extreme games corresponding to two sizes from one gap assigning identical profits to coalition sizes from another gap. This process begins by addressing the first gap and proceeds iteratively, employing the extreme games of the augmented incomplete game until no gaps remain.

For the second part, assuming that \((N,s^k)\) for \(k \in \overline{\mathcal {X}}\) is not an extreme game of \(C^n_\sigma (v)\) contradicts the definition of extreme points, which would necessitate \(C^n_\sigma \)-extensions \((N,s_1),(N,s_2)\) and some \(m \in N\) with \(s_1(m)< s^k(m) < s_2(m)\). Given \(s_1(m) = s^k(m) = s_2(m) = \sigma (m)\) for \(m \in \mathcal {X}\), we deduce \(m \notin \mathcal {X}\). If \(s^k(m) = \overline{s}(k)\) or \(s^k(m) = \underline{s}(m)\), a contradiction arises. The remaining scenario involves \(m \notin \mathcal {X}\) with \(i< m < j\) and \(m \ne k\), where the convexity of the line chart is compromised either for \((i,s_1(i)),(k,s_1(k)),(m,s_1(m))\) if \(k < m\), or \((i,s_2(i)),(m,s_2(m)),(k,s_2(k))\) if \(m < k\), leading to a contradiction. For an example of convexity violation, see (Fig. 4). \(\square \)

Fig. 4
figure 4

Examples of a violation of convexity of the line chart of both \((N,s_1)\) and \((N,s_2)\). The full lines depict the line chart of \((N,\underline{s})\) and the dotted lines depict the line charts of \((N,s_1)\) and \((N,s_2)\). On the left, the situation where \(k < m\) is shown. We have values \(s^k(i) = s_1(k)\) and \(s^k(k)=s_1(k)\), yet \(s_1(m)\) is too small. Similarly, on the right, the situation where \(m < k\) is shown, with \(s^k(i) = s_2(k)\), \(s^k(k) = s_2(k)\). However, in this case, the value \(s_2(m)\) is too big

Full size image

For a \(C^n_\sigma \)-extendable symmetric incomplete game in a reduced form \((N,\mathcal X,\sigma )\) with \(C^n_\sigma (v)\) bounded and \(|C^n_\sigma (v) |> 1\), the number of extreme games is always \(|\overline{\mathcal {X}} |+ 1 = n - |\mathcal {X} |+ 2\), no matter what the values of \(\sigma \) are.

Algebraically, we can describe the set \(C^n_\sigma (v)\) as

$$\begin{aligned} C^n_\sigma (v) = \Bigg \{\Big (N,\overline{\alpha } \text { }\overline{s} + \sum _{k \in \overline{\mathcal {X}}} \alpha _k s^k\Big ) \bigg \vert \,\, \overline{\alpha } + \sum _{k \in \overline{\mathcal {X}}} \alpha _k = 1, \overline{\alpha }, \alpha _k \ge 0, k \in \overline{\mathcal {X}}\Bigg \}, \end{aligned}$$
(8)

namely as the set of convex combinations of extreme games \(\overline{s}\) and \(s^k\) for \(k \in \overline{\mathcal {X}}\).

Geometrically, we can describe the set \(C^n_\sigma (v)\) when we slightly restrict the game \((N,\mathcal {X},\sigma )\). First, suppose \(\mathcal {X} = \{0,n\}\) and \(\sigma (0) = \sigma (n) = 0\). According to Theorem 18, we can describe \(C^n_\sigma (v)\) by a system of \(n-1\) inequalities with \(n-1\) unknowns, \(Ay \le 0\), where

The matrix A is an M-matrix (Horn & Johnson, 1991); therefore, it is nonsingular and \(A^{-1} \ge 0\). The nonsingularity of A implies that \(C^n_\sigma (v)\) is a pointed polyhedral cone, which is translated such that its vertex is not necessarily at the origin of the coordinate system. Furthermore, because \(A^{-1} \ge 0\), the normal cone \(C^n_\sigma (v)^{*}\) of \(C^n_\sigma (v)\) [see Boyd and Vandenberghe (2004)] contains the whole nonnegative orthant. Thus, the vertex of polyhedral cone \(C^n_\sigma (v)\) is the biggest element of \(C^n_\sigma (v)\) when restricted to each coordinate (this corresponds with the statement that the upper game is a \(C^n_\sigma \)-extension). Therefore, geometrically, the set \(C^n_\sigma (v)\) looks like squeezed negative orthant. For an incomplete game \((N,\mathcal {X'},\sigma ')\) where \({0,n} \subseteq \mathcal {X'}\) and \(\sigma '(0)=\sigma (n)=0\), the set of \(C^n_\sigma \)-extensions is \(C^n_\sigma (v)\) with some of the coordinates fixed, i.e.,

$$\begin{aligned} C^n_\sigma (v) \cap _{k\in \mathcal {X'}}\{s(k)=\sigma (k)\}. \end{aligned}$$

6 Conclusion

In this paper, the foundations for the analysis of positive and convex extensions were laid. Results on positivity, derived in Sect. 3, may be applied in scenarios where a special class \(\mathcal {K}\) is considered, as was illustrated in Sect. 4.

Convexity proves to be much more challenging to analyze. We extended the results of Masuya and Inuiguchi (2016), revised the literature on \(C^n\)-extendability, and analyzed the special class of symmetric convex extensions. Even though these extensions are much simpler than their general counterparts, they provide important insights that future research on non-symmetric extensions might build upon.

A potential area for future research that the authors plan to focus on is the connection between incomplete cooperative games and cooperative interval games. For cooperative interval game, each coalition is assigned a real closed interval w(S) with \(w(\emptyset ) :=[0,0]\). The uncertainty is thus represented by a range of values within which the worth of the coalition is guaranteed to lie. An incomplete cooperative game \((N,\mathcal {K},v)\) naturally gives rise to a cooperative interval game via bound games, i.e., \(w(S) = [\underline{v}(S),\overline{v}(S)]\) for every coalition S. The unification of both theories is thus at hand. One might view the analysis via incomplete cooperative games as a refinement of cooperative interval games. Another viewpoint is to generalize the definition of an incomplete game to the interval setting by allowing the partial game to be an interval game. We can then ask what extensions have some desired property, for example, being selection superadditive interval games. Indeed, this aligns with the main motivation behind some of the results in Bok and Hladík (2015) and Bok (2021). The fact that, so far, no connection has been made between these two areas in the literature seems surprising to us.