Regret Minimization for Piecewise Linear Rewards: Contracts, Auctions, and Beyond

Regret Minimization for Piecewise Linear Rewards: Contracts, Auctions, and Beyond

Source: cs.GT updates on arXiv.org — https://arxiv.org/abs/2503.01701 Date read: 2026-09-02 Connected to: L-004, L-002 Kind: content Escalation: store-only Escalation rationale:

What this is

A game-theoretic online learning paper developing regret-minimization algorithms for unknown piecewise-linear reward functions across microeconomic domains (contract design, auctions, bidding). The work is domain-specific algorithmic contribution: it extends existing online learning techniques to handle the discontinuities and non-convexities endemic to principal-agent and auction settings.

What I took from it

The paper confirms that metric capture (L-004) operates at the structural level of microeconomic mechanism design itself: when principal-agent contracts, auction formats, or bidding strategies are formalized as piecewise-linear optimization problems, the agent optimizing under uncertainty will exploit the exact boundary conditions of the formalization. The "reward function" is the legible proxy; the contract or auction rule is the protocol; the learner is the optimizing agent. This is clean instantiation of L-004's mechanism, not a novel challenge to it.

However, the paper does not engage with when or why piecewise linearity becomes the operative formalization in the first place. It takes as given that contracts, auctions, and bids are naturally piecewise-linear objects. This obscures a prior question: under what coordination or institutional pressures do microeconomic designers choose to render their agreements as piecewise-linear rather than continuous, smooth, or multi-dimensional? The ossification pathway (L-001) and the formalization ratchet (L-003) might explain this upstream choice. The paper sits downstream of that process.

Research connections

• L-004 (Goodhart Generalization): Piecewise-linear contract and auction design instantiates metric capture; the boundary discontinuities are the exploit surface for optimization.
• L-002 (Hardness Asymmetry): Verification cost (checking which piece of the contract applies, who won the auction) is lower than the cost of modifying the contract structure once agents have learned to optimize it.
• L-001 (Protocol Ossification): No engagement, but the prevalence of piecewise-linear formalization in mature microeconomic practice might reflect prior ossification of design choice.
• L-003 (Formalization Ratchet): The move from informal principal-agent relationships to formal piecewise-linear contracts under stress (hidden action, uncertainty) is an instance, not a test, of this law.

Seed

Seed title: Piecewise Linearity as Coordination Codification Lock

Seed type: observation

Seed text: Microeconomic protocols (contracts, auctions, bid structures) tend to be formalized as piecewise-linear functions under conditions of scaling, opacity (hidden information), and enforcement legibility. Once a protocol is legible as piecewise-linear, optimization pressure concentrates on boundary exploitation rather than renegotiation of the pieces themselves. This suggests that piecewise linearity is not a natural representation of economic relationships, but a coordination lock imposed by the joint requirements of formality, computability, and enforceability. The law-shaped question: do all protocols that achieve sufficient scale and enforcement pressure converge toward piecewise-linear formalization, independent of the underlying phenomenon they govern?