Measuring productivity in networks: A game-theoretic approach

TL;DR

The paper tackles measuring and fairly distributing productivity in networks with peer effects, focusing on complete bipartite structures in logistics. It introduces Finite Attenuation Network (FAN) games where a coalition's value $v_{\delta}^{t}(S)$ aggregates intrinsic productivity and attenuated peer contributions within distance $t$, and shows FANs are convex and totally balanced, placing the Shapley value in the core. As distance grows, FAN games converge to Attenuation Network (AN) games under $\delta \in [0,1/\lambda_{\max}(N))$, with an explicit limit $v_{\delta}(S)=\frac{k_S+m_S+2k_Sm_S\delta}{1-k_Sm_S\delta^{2}}$ and a core-allocating $p^{N}(\delta)$. The paper further introduces the Link Ratio Productivity (LRP) distribution $\omega(\delta)$, the unique core-stable allocation satisfying EF, EB, and LBP, which also offers computational advantages over the Shapley value and emphasizes connectivity-based contributions. These results have practical implications for tailoring productivity sharing in logistics networks and suggest natural extensions to more general network topologies.

Abstract

Measuring individual productivity (or equivalently distributing the overall productivity) in a network structure of workers displaying peer effects has been a subject of ongoing interest in many areas ranging from academia to industry. In this paper, we propose a novel approach based on cooperative game theory that takes into account the peer effects of worker productivity represented by a complete bipartite network of interactions. More specifically, we construct a series of cooperative games where the characteristic function of each coalition of workers is equal to the sum of each worker intrinsic productivity as well as the productivity of other workers within a distance discounted by an attenuation factor. We show that these (truncated) games are balanced and converge to a balanced game when the distance of influence grows large. We then provide an explicit formula for the Shapley value and propose an alternative coalitionally stable distribution of productivity which is computationally much more tractable than the Shapley value. Lastly, we characterize this alternative distribution based on three sensible properties of a logistic network. This analysis enhances our understanding of game-theoretic analysis within logistics networks, offering valuable insights into the peer effects' impact when assessing the overall productivity and its distribution among workers.

Theorems & Definitions (23)

• Proposition 3.1
• Theorem 3.2
• Example 3.3
• Lemma 4.1
• Theorem 4.2
• Example 4.3
• Proposition 4.4
• Lemma 4.5
• Theorem 4.6
• Theorem 4.7