Imputation (Game Theory)¶
A payoff allocation in a transferable-utility cooperative game that distributes the grand coalition's entire worth while giving every player at least its stand-alone coalition value.
Core Idea¶
For a transferable-utility coalitional game ((N,v)), an imputation is a payoff vector \(x\in\mathbb R^N\) satisfying two baseline admissibility conditions:
The first condition is efficiency: the grand coalition's available worth is fully distributed. The second is individual rationality: no player receives less than it can guarantee by acting alone. An imputation is therefore not yet a prediction or a uniquely fair answer. It is the feasible negotiating set remaining after waste and individually unacceptable allocations are removed.
Scope of Application¶
Imputations are used in cooperative cost and surplus sharing, coalition formation, joint ventures, public projects, logistics, insurance, voting-power analysis, and dynamic cooperative games. The definition assumes transferable utility: one common payoff scale can be redistributed among players.
For non-transferable utility, externalities among coalitions, uncertain coalition values, or participation constraints richer than stand-alone worth, the baseline must be generalized. Cost games also require a consistent reversal of inequality and efficiency conventions.
Clarity¶
Publish (N), the characteristic function or at least (v(N)) and all singleton values, the payoff sign convention, and both tests. Calling a vector “an imputation” because it divides a total is incomplete. Likewise, individual rationality must be evaluated against each player's modeled outside option, not a vague fairness judgment.
Manages Complexity¶
The definition reduces an unconstrained continuum of payoff vectors to an affine hyperplane clipped by participation half-spaces. This geometrical object can be inspected for emptiness, dimension, extreme points, and intersections with the core. It cleanly separates the first question—what agreements are minimally acceptable?—from the harder question—which acceptable agreement should be chosen?
Abstract Reasoning¶
- Specify the coalitional game and transferable payoff unit.
- Compute the grand coalition's worth.
- compute each singleton's stand-alone worth.
- Form the efficiency hyperplane.
- Intersect it with every individual-rationality constraint.
- Test whether the imputation set is nonempty.
- If selection is required, apply a declared stronger solution concept.
- Check coalition stability separately rather than inferring it from individual rationality.
- For a dynamic game, verify that continuation allocations remain acceptable over time.
Knowledge Transfer¶
The portable pattern is define the minimally admissible allocation set by combining full-budget use with a floor at each participant's outside option. It transfers to negotiated budgets, consortium revenue sharing, and resource pooling. The proposed immediate parent is Allocation.
Relationships to Other Abstractions¶
Current abstraction Imputation (Game Theory) Domain-specific
Parents (1) — more general patterns this builds on
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Imputation (Game Theory) is a kind of Allocation Prime
Allocation is the proposed immediate parent.
Hierarchy path (1) — routes to 1 parentless root
- Imputation (Game Theory) → Allocation → Scarcity → Constraint
Neighborhood in Abstraction Space¶
Imputation (Game Theory) sits in a sparse region of the domain-specific corpus (91st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Fair Division & Cooperative Power (7 abstractions)
Nearest neighbors
- Shapley value — 0.81
- Mixed Strategy Equilibrium — 0.80
- Berge–Zhukovskii Equilibrium — 0.79
- Subgame Perfect Equilibrium — 0.78
- Traveler's Dilemma — 0.78
Computed from structural-signature embeddings · 2026-09-08