When is the fuzzy core non-empty for exchange economies with discontinuous utility?

The fuzzy core relaxes full coalition participation

The fuzzy core is a cooperative solution concept for pure exchange economies that generalizes the classical core. In the standard Edgeworth and Debreu-Scarf formulation, a coalition can block an allocation only if every member participates fully. Aubin relaxed this all-or-nothing requirement by letting each agent join a blocking coalition with an arbitrary fraction of resources, indexed by a coefficient in [0,1]. A fuzzy coalition is a vector of participation rates, and the fuzzy core collects all feasible allocations that no such fractional coalition can block.

The paper works with a pure exchange economy defined by consumption sets, initial endowments, and utility functions for each agent. Two standing assumptions are used: each consumption set is closed and convex and contains the agent's endowment, and the set of feasible commodity allocations is compact. Under these assumptions the feasible set is compact and convex, which is what allows the later existence arguments to go through.

Discontinuity alone does not empty the core

The paper's Example 2.1 is a two-agent, two-commodity economy with endowments (1,0) and (0,1) and consumption sets [0,2] x [0,2]. Agent 1's utility equals 1 exactly at (1,0) and 0 otherwise; agent 2's utility equals 1 at (0,1), -1 at (0,0), and 0 otherwise. Agent 2's utility is neither upper semi-continuous nor lower semi-continuous on its consumption set, yet both utility functions are quasi-concave.

Computing the fuzzy core directly shows it equals the single allocation ((1,0),(0,1)), the initial endowment point. At that allocation both agents attain utility 1, and no fractional coalition can push either agent strictly above 1 because each utility attains its maximum only at the endowment. Conversely, any other feasible allocation gives both agents utility at most 0, and the equal-split coalition with weights (1/2,1/2) blocks it. So an economy with genuinely discontinuous utility can still have a non-empty fuzzy core, which is the phenomenon the rest of the paper tries to characterize.

Independent-pi-balanced families generalize classical balancedness

To connect the fuzzy core to game-theoretic cores, the paper introduces an independent-pi-balanced family of subsets of the agent set. For a finite index set K and, for each coalition S and index k, a vector pi in the simplex over S, a family beta of coalitions is independent-pi-balanced if there exist nonnegative weights such that the weighted sum of the pi vectors equals the barycenter of the unit simplex, the weights sum to one, and the pi vectors carrying positive weight are affinely independent.

This notion generalizes the classical balancedness conditions. When only one fixed pi is chosen for each coalition, an independent-pi-balanced family reduces to a pi-balanced family in the sense of Shapley. The affine independence requirement is what makes the construction usable for payoff-dependent cores of NTU games, because it controls how many coalitions can be active at a boundary payoff.

Two payoff-dependent NTU games bridge the cores

The paper constructs two payoff-dependent NTU games from the exchange economy, using the independent-pi-balanced family. These games serve as bridges linking the fuzzy core, the newly defined TU-fuzzy core, and the core of NTU games. The core of a characteristic function game is defined as the feasible payoff set of the grand coalition minus the interior of the union of all coalition payoff sets, following Bonnisseau and Iehle.

Under assumptions that each coalition's payoff set is non-empty, closed, and comprehensive, and that payoffs outside the interiors of singleton sets are bounded above, the paper shows the interior of the union of coalition payoff sets equals the union of their interiors. This lets the core be written as the grand coalition's payoff set minus the union of interiors, which is the form used in the payoff-dependent balancedness condition. A payoff-dependent balanced game is then guaranteed to have a core payoff satisfying an additional selection requirement.

Local stability conditions deliver non-emptiness

The paper proposes two stability conditions for exchange economies. Local fuzzy-coalition stability says that for every feasible allocation outside the fuzzy core, there is a collection of blocking threats, one for each coalition, together with an open neighborhood of that allocation, such that every allocation in the neighborhood outside the fuzzy core is strictly improved for all members of at least one coalition by the corresponding threat. Local TU-fuzzy-coalition stability is the analogous condition for the TU-fuzzy core, where blocking is evaluated in terms of weighted utility sums rather than individual utilities.

Under local fuzzy-coalition stability, the paper establishes non-emptiness of the fuzzy core for economies with quasi-concave utility functions. Under local TU-fuzzy-coalition stability, it establishes non-emptiness of the TU-fuzzy core for economies with concave utility functions. The two theorems therefore pair a stability condition with a concavity assumption: quasi-concavity for the fuzzy core, and the stronger concavity for the TU-fuzzy core. When the economy has only two agents, the paper notes that the quasi-concavity assumption for the fuzzy core and the concavity assumption for the TU-fuzzy core can be dispensed with entirely.

The TU-fuzzy core is a new transferable-utility object

The paper introduces the TU-fuzzy core for pure exchange economies, which the authors describe as the first attempt to define a transferable-utility version of the fuzzy core in this setting. This links the fuzzy-core literature to the classical TU-game approach. In the TU-fuzzy core, blocking is assessed through weighted sums of utilities rather than through each member's individual utility, so the object is distinct from the fuzzy core and answers a different question about coalitional stability.

The distinction matters because the two cores can behave differently on the same economy. The paper's Example 3.2 uses three agents with quasi-concave but not concave utility functions. That economy satisfies local TU-fuzzy-coalition stability and has a non-empty TU-fuzzy core, yet it fails local fuzzy-coalition stability while still having a non-empty fuzzy core. So the sufficient conditions proposed in the paper are not necessary, and the gap between the sufficient conditions and actual existence of fuzzy cores for discontinuous economies remains substantial.

What the stability conditions do and do not settle

The paper's contribution is a set of sufficient conditions, not a characterization. Local fuzzy-coalition stability guarantees a non-empty fuzzy core for quasi-concave utilities, and local TU-fuzzy-coalition stability guarantees a non-empty TU-fuzzy core for concave utilities, but Example 3.2 shows an economy can have a non-empty fuzzy core and a non-empty TU-fuzzy core without satisfying the corresponding local stability condition. Readers should therefore treat these conditions as tools for proving existence in specific economies, not as tests that a discontinuous economy must pass to have a non-empty core.

The most decision-relevant split in the paper is between the two cores and their assumptions. If the question is whether a discontinuous exchange economy has a non-empty fuzzy core, the relevant condition is local fuzzy-coalition stability paired with quasi-concave utilities, and with two agents the quasi-concavity requirement drops out. If the question concerns the transferable-utility version, the relevant condition is local TU-fuzzy-coalition stability paired with concave utilities, again with the concavity requirement dropping out for two agents. The three-agent example is the clearest warning that satisfying one condition does not imply satisfying the other, and that a non-empty core can occur even when neither local stability condition is met.

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