Decentralized pricing in multi-product firms and marketplaces can lose revenue because each decision maker ignores how its price affects the demand for other products. We quantify this loss through the price of anarchy of a game in which the price of each product is set to maximize that product's own revenue. Demand is linear, and cross-price effects may be substitutes or complements and need not be symmetric. Under a two-sided diagonal dominance condition with parameter $μ\in[0,1)$, which bounds the aggregate absolute cross-price effects in each row and column relative to the own-price effect, we show that the game has a unique Nash equilibrium and that equilibrium revenue is at least a fraction $4(1-μ)/(2-μ)^2$ of the centralized optimum. This bound was previously known for symmetric gross substitutes; we show that it remains valid for signed and asymmetric interactions, and that it is tight. Because asymmetry breaks the simultaneous diagonalization underlying existing analyses, we instead evaluate the revenue gap at the equilibrium itself. This yields an exact formula for the worst-case price of anarchy of each demand matrix in terms of a normalized interaction matrix, which can be much sharper than the scalar bound in heterogeneous interaction networks. The same argument extends the guarantee to jointly concave nonlinear demand under pointwise dominance of the demand Jacobian, for every pure-strategy equilibrium.

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