Kalai–Smorodinsky Bargaining Solution¶
Select the Pareto-efficient feasible agreement that equalizes each bargainer's gain above disagreement as a proportion of that bargainer's maximum feasible gain.
Core Idea¶
The Kalai–Smorodinsky bargaining solution assigns a cooperative two-person bargain by equalizing proportional gains. Given a feasible utility set \(F\subseteq\mathbb R^2\), disagreement point (d), and ideal point (b) whose coordinates are the players' separately attainable maxima, it selects the Pareto-efficient feasible point on the ray from (d) toward (b). Equivalently, it maximizes a common (t) such that (d+t(b-d)) remains feasible.
It replaces Nash's independence of irrelevant alternatives with a monotonicity principle: expanding opportunities for one player without harming the other's opportunity frontier should not reduce the first player's selected utility.
Scope of Application¶
The rule is used to compare normative bargaining solutions, allocate jointly produced surplus, and study how changes in feasible opportunities should affect negotiated outcomes. Thomson situates it among axiomatic cooperative bargaining models and their extensions. Moulin uses the geometry to contrast solution concepts and distributive principles.
Clarity¶
State the feasible-set assumptions, disagreement point, utility normalization, ideal-point construction, and whether the frontier intersects the ideal ray uniquely. Distinguish weak and strong Pareto efficiency and specify any extension used for nonconvex sets, more than two players, or asymmetric claims.
Manages Complexity¶
The method compresses a continuum of feasible agreements into three geometric objects: disagreement, ideal point, and Pareto frontier. It makes the distributive commitment visible: each player receives the same fraction of the best gain they could individually hope to achieve.
Abstract Reasoning¶
- Represent feasible agreements in utility space.
- Fix the no-agreement payoff (d).
- Restrict attention to individually rational outcomes.
- Compute each player's maximum feasible utility and form (b).
- Parameterize proportional gains as (d+t(b-d)).
- Increase (t) until feasibility ends.
- Select the maximal feasible point and verify Pareto efficiency.
- Test how the selection moves when resources or feasible opportunities expand.
Knowledge Transfer¶
The portable pattern is normalize each claimant's gain by its own opportunity ceiling, then advance all claimants at the same normalized rate until feasibility binds. It transfers to fair resource allocation. The proposed immediate parent is Allocation.
Relationships to Other Abstractions¶
Current abstraction Kalai–Smorodinsky Bargaining Solution Domain-specific
Parents (1) — more general patterns this builds on
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Kalai–Smorodinsky Bargaining Solution is a kind of Allocation Prime
Allocation is the proposed immediate parent.
Hierarchy path (1) — routes to 1 parentless root
- Kalai–Smorodinsky Bargaining Solution → Allocation → Scarcity → Constraint
Neighborhood in Abstraction Space¶
Kalai–Smorodinsky Bargaining Solution sits in a sparse region of the domain-specific corpus (89th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Coase Theorem — 0.81
- Fourier–Motzkin Elimination — 0.79
- Median Voter Theorem — 0.79
- Pirate game — 0.79
- Arrow–Debreu Model — 0.78
Computed from structural-signature embeddings · 2026-09-08