Online Contract Design

TL;DR

This work introduces the Online Multi-Agent Contract (OMAC) model, which jointly models sequential, adversarial arrivals and rational, contract-driven participation. The authors develop a randomized, balance-point based algorithm that achieves a tight $1/2$-competitive ratio for additive rewards, demonstrating that deterministic online strategies cannot achieve a bounded constant, and showing that the inclusion of XOS rewards breaks even randomized guarantees. The balance-point technique provides a new analytical tool to align online hiring with incentive-compatible contracts under adversarial arrivals, and the paper situates OMAC within online budgeted maximization and preemption literature. The results illuminate the boundary between tractable and intractable reward structures in online contracting and point to rich avenues for stochastic variants and model enhancements with practical relevance to dynamic, contract-driven teams.

Abstract

We initiate the study of online contracts, which integrate the game-theoretic considerations of economic contract theory, with the algorithmic and informational challenges of online algorithm design. Our starting point is the classic online setting with preemption of Buchbinder et al. [SODA'15], in which a hiring principal faces a sequence of adversarial agent arrivals. Upon arrival, the principal must decide whether to tentatively accept the agent to their team, and whether to dismiss previous tentative choices. Dismissal is irrevocable, giving the setting its online decision-making flavor. In our setting, the agents are rational players: once the team is finalized, a game is played where the principal offers contracts (performance-based payment schemes), and each agent decides whether or not to work. Working agents reward the principal, and the goal is to choose a team that maximizes the principal's utility. Our main positive result is a 1/2-competitive algorithm when agent rewards are additive, which matches the best-possible competitive ratio. Our algorithm is randomized and this is necessary, as we show that no deterministic algorithm can attain a bounded competitive ratio. Moreover, if agent rewards are allowed to exhibit combinatorial structure known as XOS, even randomized algorithms might fail. En route to our competitive algorithm, we develop the technique of balance points, which can be useful for further exploration of online contracts in the adversarial model.

Figures (1)

• Figure 1: Geometric intuition for balance points. For a team $S$ and quality level $x$, the balance point $b(S,x)$ is marked on the $x$-axis, which represents allocated share. Recall that $S^{x-1}$ denotes the subset of agents of quality at least $q^{x-1}$. Consider two subsets $S', S" \subseteq Q_x$ of quality $q^x$. The horizontal segments show the shares allocated to $S^{x-1} \cup S'$ (red and blue) and $S^{x-1} \cup S"$ (red and green). Since the share of $S^{x-1} \cup S'$ is closer to the balance point than that of $S^{x-1} \cup S"$, by Definition \ref{['def: bp']} this implies $g(S^{x-1} \cup S') \geq g(S^{x-1} \cup S")$. The $y$-axis represents the principal's utility, and the dashed parabola depicts the utility as a function of shares allocated to agents of quality $q^x$. The principal's utility is maximized at the balance point, and indeed $g(S^{x-1} \cup S') \geq g(S^{x-1} \cup S")$.

Theorems & Definitions (39)

• Definition 3.1: Quality
• Definition 3.2: Balance point
• Lemma 3.4
• Corollary 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Lemma 3.9
• Theorem 4.1
• Corollary 4.2