Bel Coalitional Games

• Bel coalitional games are cooperative games characterized by finite players, state-contingent TU-games, and non-additive Dempster–Shafer belief functions to model ambiguity.
• They generalize classical expected-value games by recovering the traditional core under additivity while using Choquet integrals to evaluate contracts.
• Ex-ante solution concepts like the core, nucleolus, and kernel are extended to incorporate statewise allocations and ambiguity, providing robust stability criteria.

Bel coalitional games are cooperative games with uncertainty in which a finite player set $N$ faces a finite set of states of the world $\Omega$, each state $\omega$ carrying a classical transferable-utility game $v_\omega$, while each player evaluates state-contingent contracts through a belief function and the associated Choquet integral. In the formulation developed in the survey literature, they generalize classical coalitional games by introducing uncertainty through Dempster–Shafer belief functions, so that the expected-value model is recovered when the common belief is additive, while non-additivity permits the representation of ambiguity attitudes in blocking and core notions (Grabisch et al., 27 Feb 2026).

1. Primitive objects and formal definition

A Bel coalitional game is specified by the tuple

$$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$

where $N=\{1,\dots,n\}$ is the finite player set and $\Omega=\{\omega_1,\dots,\omega_d\}$ is the finite state space. For each player $j\in N$, $m_{\nu_j}:2^\Omega\to[0,1]$ is a mass function, or Möbius inverse, whose support covers $\Omega$. It induces the belief function

$\Omega$0

with $\Omega$1, $\Omega$2, monotonicity, and total-monotonicity. For each state $\Omega$3, $\Omega$4 is a classical TU-game with $\Omega$5 (Grabisch et al., 27 Feb 2026).

The Dempster–Shafer vocabulary is intrinsic to the model. A capacity $\Omega$6 is a non-decreasing set function with $\Omega$7 and $\Omega$8; a belief function satisfies the inclusion–exclusion inequalities, and its dual plausibility is

$\Omega$9

The focal elements are those $\omega$0 such that $\omega$1. In this framework, player $\omega$2's a priori knowledge is $\omega$3, not necessarily a probability measure and not necessarily identical across players (Grabisch et al., 4 Feb 2026).

A coalition does not choose a deterministic transfer vector but a state-contingent contract. For $\omega$4, an $\omega$5-contract is a mapping

$\omega$6

Feasibility at state $\omega$7 requires

$\omega$8

A grand contract $\omega$9 is efficient if

$v_\omega$0

This setup replaces the single characteristic function of a deterministic TU-game by a family of state-indexed TU-games together with player-specific non-additive beliefs. A plausible implication is that the source of uncertainty is external to coalition formation itself: uncertainty concerns the realized state $v_\omega$1, while coalition worths remain TU within each realized state.

2. Contract evaluation and the ex-ante core

Each player evaluates a payoff distribution $v_\omega$2 by the Choquet integral with respect to $v_\omega$3. In ordered form,

$v_\omega$4

where $v_\omega$5 orders the states so that

$v_\omega$6

and $v_\omega$7. Player $v_\omega$8 prefers $v_\omega$9 to $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$0 if $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$1 (Grabisch et al., 27 Feb 2026).

An equivalent integral representation used in the ex-ante analysis is

$$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$2

Thus the utility of player $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$3 from an $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$4-contract is $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$5 (Grabisch et al., 4 Feb 2026).

The central stability notion is the ex-ante core. A coalition $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$6 ex-ante blocks a feasible grand contract $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$7 if there exists a feasible $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$8-contract $$G_{\rm Bel} =\bigl(N,\Omega,\,(m_{\nu_j})_{j\in N},\,(v_\omega)_{\omega\in\Omega}\bigr),$$9 such that

$N=\{1,\dots,n\}$0

The ex-ante core is the set of all efficient grand contracts that no coalition blocks (Grabisch et al., 27 Feb 2026).

When all players share the same belief function $N=\{1,\dots,n\}$1, the ex-ante core admits linear core conditions analogous to the classical case. A grand contract $N=\{1,\dots,n\}$2 belongs to the ex-ante core if and only if:

1. efficiency:

$N=\{1,\dots,n\}$3

2. coalition-rationality:

$N=\{1,\dots,n\}$4

Under a common prior probability $N=\{1,\dots,n\}$5 with full support, the Choquet integral reduces to expectation: $N=\{1,\dots,n\}$6 Defining the expected game by

$N=\{1,\dots,n\}$7

the ex-ante core of the Bel game is characterized by efficiency in expectation and coalitional rationality in expectation. The ex-ante core then coincides with the set of state-contingent allocations $N=\{1,\dots,n\}$8 satisfying

$N=\{1,\dots,n\}$9

and, since $\Omega=\{\omega_1,\dots,\omega_d\}$0 has full support, efficiency in expectation forces statewise efficiency $\Omega=\{\omega_1,\dots,\omega_d\}$1 for each $\Omega=\{\omega_1,\dots,\omega_d\}$2 (Grabisch et al., 4 Feb 2026).

A recurrent misconception is to identify Bel coalitional games with ordinary expected-value cooperative games. This identification is valid only in the additive case $\Omega=\{\omega_1,\dots,\omega_d\}$3. Outside that case, evaluation is Choquet rather than linear expectation, and coalition blocking depends on non-additive beliefs.

3. Core conditions, necessity, sufficiency, and geometry

The ex-ante core admits a necessity statement in full generality, even when players hold different belief functions. If $\Omega=\{\omega_1,\dots,\omega_d\}$4 belongs to the ex-ante core, then for every coalition $\Omega=\{\omega_1,\dots,\omega_d\}$5 there exists $\Omega=\{\omega_1,\dots,\omega_d\}$6 such that

$\Omega=\{\omega_1,\dots,\omega_d\}$7

where $\Omega=\{\omega_1,\dots,\omega_d\}$8 is the dual plausibility. This condition uses plausibility rather than belief on the right-hand side, which marks a structural departure from the classical deterministic core (Grabisch et al., 27 Feb 2026).

A clean sufficiency statement is available under identical priors. If all players share the same belief $\Omega=\{\omega_1,\dots,\omega_d\}$9, then

$j\in N$0

implies $j\in N$1. In that regime, coalition inequalities written in terms of Choquet values are sufficient for ex-ante stability (Grabisch et al., 27 Feb 2026).

When the common prior is an additive probability $j\in N$2, the ex-ante core has a precise geometric description. The corresponding system defines a convex polyhedron in $j\in N$3. Its recession cone equals its lineality space of dimension $j\in N$4; in particular, it has no pointed cone and no vertices, only pseudo-vertices. If $j\in N$5 is the classical core of the expected game $j\in N$6, then

$j\in N$7

Moreover, each vertex $j\in N$8 lifts to a pseudo-vertex $j\in N$9 by solving

$m_{\nu_j}:2^\Omega\to[0,1]$0

and conversely any pseudo-vertex projects to a vertex of $m_{\nu_j}:2^\Omega\to[0,1]$1 via $m_{\nu_j}:2^\Omega\to[0,1]$2. Hence, by Weyl–Minkowski,

$m_{\nu_j}:2^\Omega\to[0,1]$3

(Grabisch et al., 4 Feb 2026).

This geometry clarifies the relation between statewise contingent contracts and classical expected allocations. The projected object is the usual core of $m_{\nu_j}:2^\Omega\to[0,1]$4, while the lineality directions encode statewise transfers that sum to zero in expectation. This suggests that state contingency enlarges the allocation space without changing the expected-game obstruction to nonemptiness.

4. Convexity and other ex-ante solution concepts

Ex-ante convexity is defined by the Choquet-supermodularity condition

$m_{\nu_j}:2^\Omega\to[0,1]$5

Under this condition, for every permutation $m_{\nu_j}:2^\Omega\to[0,1]$6 of $m_{\nu_j}:2^\Omega\to[0,1]$7, the contract whose $m_{\nu_j}:2^\Omega\to[0,1]$8-column is the classical marginal vector $m_{\nu_j}:2^\Omega\to[0,1]$9 lies in the ex-ante core. The proof sketch given in the survey relies on total monotonicity of $\Omega$0, which implies superadditivity of $\Omega$1, and then adapts the standard marginal-vector construction to the state-contingent setting (Grabisch et al., 27 Feb 2026).

A parallel ex-ante convexity result in the probabilistic identical-prior case places the Shapley contract in the core: $\Omega$2 In that setting, convexity also yields coincidence between the core and a strengthened bargaining set: $\Omega$3 The strengthening requires that a counter-blocking coalition $\Omega$4 satisfy $\Omega$5 (Grabisch et al., 4 Feb 2026).

The ex-ante excess of coalition $\Omega$6 at a grand contract $\Omega$7 is

$\Omega$8

Ordering the $\Omega$9 excesses into a nonincreasing vector

$\Omega$00

one defines the ex-ante nucleolus as the set of feasible contracts minimizing $\Omega$01 in lexicographic order, and the ex-ante prenucleolus as the same minimization over efficient grand contracts. The prenucleolus is always nonempty (Grabisch et al., 4 Feb 2026).

The ex-ante kernel is defined by pairwise surpluses

$\Omega$02

with the kernel consisting of feasible $\Omega$03 such that

$\Omega$04

The ex-ante bargaining set, in Mas-Colell style, is based on ex-ante objections and counter-objections formulated through feasible contingent contracts. The inclusion relations follow the classical pattern: $\Omega$05

$\Omega$06

(Grabisch et al., 4 Feb 2026).

These results indicate that Bel coalitional games preserve much of the solution-concept architecture of deterministic cooperative game theory, but relocate it to a contract space evaluated through Choquet integrals.

5. Recovery of classical games and illustrative examples

The additive-probability case shows explicitly how the classical expected-value model is recovered. Consider $\Omega$07, $\Omega$08, and a common prior $\Omega$09 with $\Omega$10, $\Omega$11. Let

$\Omega$12

$\Omega$13

Then the expected game is

$\Omega$14

Its classical core is

$\Omega$15

namely the singleton $\Omega$16. Interpreting this as a Bel contract by setting

$\Omega$17

one obtains a contract in the ex-ante core, demonstrating that additive Bel games recover the expected-value model (Grabisch et al., 27 Feb 2026).

A second two-player example with $\Omega$18 takes

$\Omega$19

$\Omega$20

The expected game then satisfies

$\Omega$21

Its classical core is the segment

$\Omega$22

or equivalently $\Omega$23. The ex-ante core of the Bel game consists of the lifts of this segment, together with a one-dimensional lineality generated by statewise transfers summing to zero in expectation. In the same example, the Shapley contract is

$\Omega$24

hence $\Omega$25, and the ex-ante nucleolus and kernel also select $\Omega$26 (Grabisch et al., 4 Feb 2026).

These examples exhibit two structural features. First, additive common priors collapse Bel coalitional games to expected TU-games. Second, contingent contracts create additional directions in the feasible allocation space even when the projected expected-game solution is unique.

6. Relation to other uncertainty models and interpretive boundaries

The survey places Bel coalitional games within a broader literature on cooperative games with uncertainty beginning with the seminal paper of Charnes and Granot (1976). Stochastic games such as the Charnes–Granot and Suijs–Borm models use quantiles or expected values of random variables $\Omega$27 under additive probabilities and allocate risk by sharing rules. Bel coalitional games replace the probability prior by a non-additive belief and use the Choquet integral in place of expectation or quantile, thereby capturing ambiguity attitudes. Bayesian coalitional games, such as those of Ieong–Shoham, use a common probability prior and information partitions; Bel coalitional games drop public knowledge of the prior, may allow different $\Omega$28, eliminate partitions, and still distinguish ex-ante, ex-interim, and ex-post through a sequence of shrinking information sets. Interval and fuzzy-valued games model imprecision in $\Omega$29 through intervals or fuzzy numbers; Bel coalitional games instead model uncertainty in the state of the world, with arbitrary correlations across coalitions, while allowing both imprecision and ambiguity attitudes through the shape of $\Omega$30 (Grabisch et al., 27 Feb 2026).

A useful comparison is with partition function form games with probabilistic beliefs. There, each coalition $\Omega$31 holds a probability distribution $\Omega$32 over partitions of outsiders and induces a standard characteristic function

$\Omega$33

after which one studies the usual core of $\Omega$34. That approach is belief-based, but the beliefs are probabilistic conjectures over outsider partitions rather than Dempster–Shafer beliefs over states, and the resulting object is reduced to an ordinary characteristic-function game (Lekeas et al., 2 May 2026).

Another neighboring line is coalitional games with opinion exchange. In that framework players begin with heterogeneous opinions about the characteristic function $\Omega$35, update those opinions through a weighted consensus dynamic, and, once a common estimate $\Omega$36 is reached, classical solution concepts such as the Shapley value, core, or nucleolus are applied to the agreed-upon game. Under truth-telling, consensus and payoff division are fully decoupled; under strategic lying, they become coupled but may still admit an efficient pure-strategy Nash equilibrium under the stated risk-aversion condition (Jiang et al., 2017).

These comparisons delimit the scope of Bel coalitional games. They are not merely stochastic coalitional games with expectation, not merely Bayesian games with information partitions, and not merely consensus models over a single unknown characteristic function. Their distinguishing feature is the joint use of state-contingent coalition values, player-specific belief functions, and Choquet evaluation of contracts. A plausible implication is that Bel coalitional games occupy a unifying position among uncertainty models: they recover the classical core as a special case, preserve many standard cooperative solution concepts in the ex-ante scenario, and incorporate ambiguity directly into coalition rationality.