Type-Based Least-Core Payoff Allocation

• The paper introduces a type-based least-core payoff allocation scheme, a cooperative game construct designed for stable benefit distribution under coalition size constraints in heterogeneous agent systems.
• It leverages a least-core relaxation and linear programming formulation to compute uniform payoffs across agent types while ensuring near-core stability despite coalition structure restrictions.
• Empirical results demonstrate high stability (often above 92%) and improved fleet benefits compared to baseline allocation methods in mixed-energy transportation settings.

A type-based least-core payoff allocation scheme is a cooperative game-theoretic construct specifically adapted to heterogeneously-typed agent systems with coalitional size constraints, as exemplified by mixed-energy truck platooning scenarios. This scheme assigns payoffs based exclusively on agent types, ensuring all members of a given type receive an identical benefit. It leverages least-core concepts to achieve stable, approximately core-like allocations even when strict core allocations are infeasible due to coalition structure restrictions. The method is computationally efficient and ensures strong stability against coalitional deviations under bounded coalition sizes (Bai et al., 6 Dec 2025).

1. Mathematical Framework and Game Definition

The underlying coalitional game is defined on a set $N = \{1,2,...,N\}$, partitioned into electric truck set $N_e$ and fuel truck set $N_f$, so that $N = N_e \cup N_f$ and $|N| = N_e + N_f$. Each agent $i \in N$ is assigned a type $T_i \in \{e, f\}$, denoting either electric or fuel-propulsion. Feasible coalitions are required to satisfy a platoon-size constraint: $2 \leq |S| \leq M$, where $M$ is the maximum coalition size ($2 \leq M < N$).

The characteristic value function is prescribed as: $N_e$0 where $N_e$1, $N_e$2, and $N_e$3 are type-specific per-unit-time savings for electric and fuel followers, respectively, with $N_e$4. Leaders receive zero, while the formulae reflect the total platooning benefit available to each coalition under the imposed size cap [(Bai et al., 6 Dec 2025), Eq. (2)].

2. Type-Based Allocation Principle

In contrast to individualistic payoff schemes, type-based allocation restricts the form of the allocated payoff vector $N_e$5: $N_e$6 for scalars $N_e$7 [(Bai et al., 6 Dec 2025), Eq. (13)]. This uniformity reflects the symmetry of cost savings postulated for each powertrain type and greatly reduces the dimensionality of the allocation space, enabling tractable computation of stability-preserving payoffs in large-scale fleet settings.

3. Coalition-Structure Core and Stability Constraints

The coalition-structure core (CS-core) is adapted to games with imposed coalition partitions $N_e$8, typically reflecting the optimal set of size-constrained platoons. A payoff vector $N_e$9 lies in $N_f$0 if and only if:

• Efficiency on each coalition: For all $N_f$1, $N_f$2,
• No profitable deviation: For any $N_f$3 with $N_f$4, $N_f$5 [(Bai et al., 6 Dec 2025), Definition 5].

Non-emptiness of the CS-core depends on the feasibility of these constraints. When infeasible (e.g., due to coalition overlaps or partition size mismatches), a least-core formulation becomes necessary.

4. Least-Core Relaxation and Linear Programming Formulation

The least-core relaxes blocking constraints via a uniform slack $N_f$6 and seeks the minimum $N_f$7 rendering an allocation feasible. An allocation is $N_f$8-feasible if:

• It satisfies partition efficiency: $N_f$9 for all $N = N_e \cup N_f$0,
• For all $N = N_e \cup N_f$1 with $N = N_e \cup N_f$2, $N = N_e \cup N_f$3 [(Bai et al., 6 Dec 2025), Definition 6].

Within the type-based restriction, the optimal $N = N_e \cup N_f$4 solve: $N = N_e \cup N_f$5 where $N = N_e \cup N_f$6, $N = N_e \cup N_f$7 are the number of electric and fuel platoon leaders in $N = N_e \cup N_f$8 [(Bai et al., 6 Dec 2025), Eq. (14)]. Due to type symmetry, blocks of constraints collapse to $N = N_e \cup N_f$9, ensuring manageable computational overhead.

5. Algorithmic Realization and Computational Complexity

The procedure first enumerates all possible type compositions $|N| = N_e + N_f$0 of feasible coalitions up to size $|N| = N_e + N_f$1. For each composition, it encodes the blocking constraint in the LP. The full system thus involves:

• $|N| = N_e + N_f$2 constraint inequalities
• 3 real variables $|N| = N_e + N_f$3.

The LP can be solved using standard optimization packages. For practical values ($|N| = N_e + N_f$4), run-time is negligible [(Bai et al., 6 Dec 2025), Algorithm, Section IV]. This efficiency is a direct consequence of the type-based reduction in variable count.

6. Stability Characterization and Least-Core Radius

If the optimal $|N| = N_e + N_f$5, the scheme achieves a CS-core allocation—i.e., exact coalition-wise stability. Otherwise, $|N| = N_e + N_f$6 quantifies the minimal total deficit tolerated in blocking constraints, often interpreted as the smallest uniform subsidy necessary to stabilize the allocation within the restricted family [(Bai et al., 6 Dec 2025), Propositions 1(a,b), Remark 2]. The type-based least-core allocation always minimizes the worst-case violation possible through type-symmetric payoffs.

7. Empirical Evaluation and Comparative Performance

A numerical case with $|N| = N_e + N_f$7 ($|N| = N_e + N_f$8, $|N| = N_e + N_f$9), $i \in N$0, $i \in N$1, $i \in N$2 produces optimal $i \in N$3, leading to a stability index of $i \in N$4 [(Bai et al., 6 Dec 2025), Section V A].

Comparison against equal-split, follower-only, type-proportional, and leader-subsidy baselines demonstrates a superior stability index (generally $i \in N$592\%) and higher overall fleet benefit under the type-based least-core scheme across all tested platoon sizes ($i \in N$6) [(Bai et al., 6 Dec 2025), Section V B, Figures 3–4].

Allocation Scheme	Stability Index $i \in N$7 (typical)	Notes
Type-based least-core	$i \in N$892%	Highest stability, efficient
Equal-split	Lower	Less fair for heterogeneity
Follower-only	Lower	Ignores leader compensation
Type-proportional	Lower	Proportional, lacks stability
Leader-subsidy	Lower	Incomplete stability

A plausible implication is that this approach is well suited to large-scale, mixed-fleet logistical optimization and benefit-sharing tasks where computational scalability and stability are critical. The method provides a closed-form, highly efficient framework for fair and robust benefit distribution in practical transportation networks (Bai et al., 6 Dec 2025).