The effect of quantization of prior probabilities in a collection of distributed Bayesian binary hypothesis testing problems over which the priors themselves vary is studied, with focus on conflicting agents. Conflict arises from differences in Bayes costs, even when all agents desire correct decisions and agree on the meaning of correct. In a setting with fusion of local binary decisions by majority rule, Nash equilibrium local decision strategies are found. Assuming that agents follow Nash equilibrium decision strategies, designing quantizers for prior probabilities becomes a strategic form game, we discuss its Nash equilibria. We also propose two different constrained quantizer design games, find Nash equilibrium quantizer designs, and compare performance. The system has deadweight loss: equilibrium decisions are not Pareto optimal.