A Note on the Majority Dynamics in Inhomogeneous Random Graphs

In this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

Informational Robustness in Intertemporal Pricing

We introduce a robust approach to study dynamic monopoly pricing of a durable good in the face of buyer learning. A buyer receives information about her willingness-to-pay for the seller’s product over time, and decides when to make a one-time purchase. The seller does not know how the buyer learns but commits to a pricing strategy to maximize profits against the worst-case information arrival process. We show that a constant price path delivers the robustly optimal profit, with profit and price both lower than under known values. Thus, under the robust objective, intertemporal incentives do not arise at the optimum, despite the possibility for information arrival to influence the timing of purchases. We delineate whether constant prices remain optimal (or not) when the seller seeks robustness against a subset of information arrival processes. As part of the analysis, we develop new techniques to study dynamic Bayesian persuasion.

Knowledge Transfer, Transitional Dynamics and Optimal Research & Development Policy in a Dynamic Monopoly Setting

This paper focuses on the question of whether or not a reduction of the knowledge barrier is good for welfare. Based on a dynamic monopoly setting with simultaneous investment decisions in process as well as in product Research & Development (R&D), we show that a reduction of the knowledge barrier has ambiguous welfare consequences: due to a lower knowledge barrier, product quality and welfare increase in the short-run. However, this may not necessarily be the case in the long-run. One reason is that a positive long-lasting knowledge barrier shock triggers the monopolist sub-optimally to reduce its product R&D investments today and in the future at the cost of future product quality. This in turn may reduce welfare. Accordingly, to realize the first-best level of product quality, the long-run optimal R&D subsidy rate for product innovations increases with a reduction of the knowledge barrier.