Research Note: Continuous-State Dynamic Games with Quantal Response: Strategic Foundations of the Kyle-Ho-Stoll Parity Index in Dealer Markets

This paper reexamines the Parity Index framework—the ratio of inventory pressure to information incorporation in dealer markets—using strategic equilibrium rather than adaptive heuristics. We constructed a dynamic stochastic game in which dealers simultaneously choose spreads and depths to maximize profits while managing inventory risk, solved via continuous-state dynamic programming and Quantal Response Equilibrium. Despite fundamentally different behavioural foundations, the Parity Index's structural relationships proved robust: fragmentation decouples inventory from prices, non-linear costs decrease the index; and the interaction of fragmentation and non-linear costs determines the market regime boundaries. Although, the strategic behavioural model amplified the effects observed in the adaptive model. This convergence demonstrates that the index captures universal economic forces whose qualitative effects transcend model architecture. Policy implications become contingent on market characteristics: strategic depth management makes moderate competition welfare-improving in liquid markets while preserving consolidation's benefits in thin markets. The findings establish the Parity Index as a behaviourally robust framework and demonstrate that adaptive models provide reliable structural guidance when strategic analysis is computationally prohibitive.