Linear Contracts for Supermodular Functions Based on Graphs

Kanstantsin Pashkovich; Jacob Skitsko

Home/Archive/arXiv

arXiv · 2412.13290

Linear Contracts for Supermodular Functions Based on Graphs

Kanstantsin Pashkovich·Jacob Skitsko

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study linear contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a linear contract with several agents, each of whom can decide to take a costly action or not. The principal observes only the outcome of the agents' collective actions, not the actions themselves, and obtains a reward from this outcome. Agents that take an action incur a cost, and so naturally agents require a fraction of the principal's reward as an incentive for taking their action. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized. Our focus is on the case when the agents are vertices in a graph and the principal's reward corresponds to the number of edges between agents who take their costly action. This case represents the natural scenario when an action of each agent complements actions of other agents though collaborations. Recently, Deo-Campo Vuong et.al. showed that for this problem it is impossible to provide any finite multiplicative approximation or additive FPTAS unless $\mathcal{P} = \mathcal{NP}$. On a positive note, the authors provided an additive PTAS for the case when all agents have the same cost. They asked whether an additive PTAS can be obtained for the general case, i.e when agents potentially have different costs. We answer this open question in positive.

Explore related subjects

cs.GT

Keep this discovery

BibTeX RIS

Kanstantsin Pashkovich, Jacob Skitsko. 2024-12-17. Linear Contracts for Supermodular Functions Based on Graphs. https://arxiv.org/abs/2412.13290

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗

arXiv2026

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗

arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗