Uncertain Price Competition in a Duopoly with Heterogeneous Availability

We study the price competition in a duopoly with an arbitrary number of buyers. Each seller can offer multiple units of a commodity depending on the availability of the commodity which is random and may be different for different sellers. Sellers seek to select a price that will be attractive to the buyers and also fetch adequate profits. The selection will in general depend on the number of units available with the seller and also that of its competitor - the seller may only know the statistics of the latter. The setting captures a secondary spectrum access network, a non-neutral Internet, or a microgrid network in which unused spectrum bands, resources of ISPs, and excess power units constitute the respective commodities of sale. We analyze this price competition as a game, and identify a set of necessary and sufficient properties for the Nash Equilibrium (NE). The properties reveal that sellers randomize their price using probability distributions whose support sets are mutually disjoint and in decreasing order of the number of availability. We prove the uniqueness of a symmetric NE in a symmetric market, and explicitly compute the price distribution in the symmetric NE.Comment: 45 pages, Accepted for publication in IEEE Transaction on Automatic Contro

Undrained stability of active and passive trapdoors

The recent growth in the number of sinkhole occurrences due to human activities has highlighted the need for better understanding and prediction of the problem. This paper investigates the use of Broms and Bennermark’s original stability number for trapdoor problems in cohesive soil. The shear-strength-reduction method built in a finite-difference method software program (FLAC) is used to obtain the factor of safety (FOS) under different combinations of pressures for collapse and blowout. Unlike previous research on the use of critical pressure ratios, the FOS results are now functions of the original stability number and depth ratio. The obtained numerical results are compared and validated by using rigorous upper- and lower-bound finite-element limit analysis, as well as other existing solutions available in the literature. Surface failure extents are also examined in the paper. The dimensionless ratios employed in this study are useful for preparing design charts with a broad range of trapdoor geometries and soil parameters