Power Sequence Certification of Sequential Equilibrium in a Supply Chain Game: A Farkas Lemma Approach

Abstract

This paper investigates the consistency of off-path beliefs in the ordinal equilibrium of a supply chain game with imperfect information. In this game, the supplier chooses an unobservable quality level, and the retailer makes an order decision based on its beliefs about quality. Although the standard Nash equilibrium leads to the low-quality solution, the main issue is to examine the consistency of optimistic beliefs about high quality. For the consistency analysis, the framework of power sequences, based on the Farkas lemma and Reny (2026) representation, is used. It is shown that off-path beliefs are verifiable only if they correspond to a consistent divergence rate in the linear system. Consequently, the only belief supportable in the limit of power sequences is the definite belief in high quality. In the following, a structural bound on the power of attesting sequences in supply chain games is presented that depends on the graph structure of the decision paths. For the two-state game under study, this bound is computed explicitly. These results show that the marginal rate-based analysis provides a complementary perspective to the classical definition of consistency in ordinal equilibrium.