New Fisher-Market Model Solves Fractional Credit Assignment in Planning Systems

Personal and organizational planning systems often struggle to connect planned tasks with actual logged actions, relying on exclusive, all-or-nothing links that miss genuinely related effort and falsely flag stalled goals. In a new arXiv paper, researcher Salavat Ishbulatov introduces Attribution Markets, a framework that reframes this bridge as a quasi-linear Fisher market. Here, planned tasks act as budget-constrained buyers, performed actions as divisible goods, and a fused signal (text, structure, time) sets each buyer's valuation. Two market instruments—a seller reserve price and a buyer cash option—ensure conservation, a hard budget cap, and a provable junk filter, all proven as theorems.

The paper extends the market with a concave completion utility that discounts progress as a task nears its plan. Standard convergence theory doesn't transfer, so the author introduces a satiation-threshold fixed point with existence (via Brouwer) and local uniqueness under explicit diagonal-dominance. Empirical validation on random and adversarial instances highlights a weak spot: the sharp, zero-entropy equilibrium is more sensitive to affinity noise than entropy-regularized optimal transport. This is resolved with a one-parameter entropy-regularized generalization that unifies the two approaches, plus a noise-adaptive rule for regularization strength. The 35-page paper includes full reproducibility parameters and relates the work to multi-touch attribution and online Fisher-market algorithms.