Computing optimal coalition structures in polynomial time

The optimal coalition structure determination problem is in general computationally hard. In this article, we identify some problem instances for which the space of possible coalition structures has a certain form and constructively prove that the problem is polynomial time solvable. Specifically, we consider games with an ordering over the players and introduce a distance metric for measuring the distance between any two structures. In terms of this metric, we define the property of monotonicity, meaning that coalition structures closer to the optimal, as measured by the metric, have higher value than those further away. Similarly, quasi-monotonicity means that part of the space of coalition structures is monotonic, while part of it is non-monotonic. (Quasi)-monotonicity is a property that can be satisfied by coalition games in characteristic function form and also those in partition function form. For a setting with a monotonic value function and a known player ordering, we prove that the optimal coalition structure determination problem is polynomial time solvable and devise such an algorithm using a greedy approach. We extend this algorithm to quasi-monotonic value functions and demonstrate how its time complexity improves from exponential to polynomial as the degree of monotonicity of the value function increases. We go further and consider a setting in which the value function is monotonic and an ordering over the players is known to exist but ordering itself is unknown. For this setting too, we prove that the coalition structure determination problem is polynomial time solvable and devise such an algorithm

Pareto-optimal matching allocation mechanisms for boundedly rational agents

Is the Pareto optimality of matching mechanisms robust to the introduction of boundedly rational behavior? To address this question I define a restrictive and a permissive notion of Pareto optimality and consider the large set of hierarchical exchange mechanisms which contains serial dictatorship as well as Gale’s top trading cycles. Fix a housing problem with boundedly rational agents and a hierarchical exchange mechanism. Consider the set of matchings that arise with all possible assignments of agents to initial endowments in the given mechanism. I show that this set is nested between the sets of Pareto optima according to the restrictive and the permissive notion. These containment relations are generally strict, even when deviations from rationality are minimal. In a similar vein, minimal deviations from rationality suffice for the set of outcomes of Gale’s top trading cycles with all possible initial endowments to differ from the set of outcomes of serial dictatorship with all possible orders of agents as dictators. © 2016 The Author(s

Sn-Beta zeolites with borate salts catalyse the epimerization of carbohydrates via an intramolecular carbon shift

Carbohydrate epimerization is an essential technology for the widespread production of rare sugars. In contrast to other enzymes, most epimerases are only active on sugars substituted with phosphate or nucleotide groups, thus drastically restricting their use. Here we show that Sn-Beta zeolite in the presence of sodium tetraborate catalyses the selective epimerization of aldoses in aqueous media. Specifically, a 5 wt% aldose (for example, glucose, xylose or arabinose) solution with a 4:1 aldose:sodium tetraborate molar ratio reacted with catalytic amounts of Sn-Beta yields near-equilibrium epimerization product distributions. The reaction proceeds by way of a 1,2 carbon shift wherein the bond between C-2 and C-3 is cleaved and a new bond between C-1 and C-3 is formed, with C-1 moving to the C-2 position with an inverted configuration. This work provides a general method of performing carbohydrate epimerizations that surmounts the main disadvantages of current enzymatic and inorganic processes.National Science Foundation (U.S.). Materials Research Science and Engineering Centers (Program) (Award DMR-0819762)DuPont MIT Alliance (Graduate Research Fellowship)National Institutes of Health (U.S.) (Grant EB-001960)National Institutes of Health (U.S.) (Grant EB-002026)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374