Nonatomic games on Loeb spaces

In the setting of noncooperative game theory, strategic negligibility of individual agents, or diffuseness of information, has been modeled as a nonatomic measure space, typically the unit interval endowed with Lebesgue measure. However, recent work has shown that with uncountable action sets, for example the unit interval, there do not exist pure-strategy Nash equilibria in such nonatomic games. In this brief announcement, we show that there is a perfectly satisfactory existence theory for nonatomic games provided this nonatomicity is formulated on the basis of a particular class of measure spaces, hyperfinite Loeb spaces. We also emphasize other desirable properties of games on hyperfinite Loeb spaces, and present a synthetic treatment, embracing both large games as well as those with incomplete information.

(+)-Germacrene A Biosynthesis : The Committed Step in the Biosynthesis of Bitter Sesquiterpene Lactones in Chicory

The leaves and especially the roots of chicory (Cichorium intybus L.) contain high concentrations of bitter sesquiterpene lactones such as the guianolides lactupicrin, lactucin, and 8-deoxylactucin. Eudesmanolides and germacranolides are present in smaller amounts. Their postulated biosynthesis through the mevalonate-farnesyl diphosphate-germacradiene pathway has now been confirmed by the isolation of a (+)-germacrene A synthase from chicory roots. This sesquiterpene cyclase was purified 200-fold using a combination of anion-exchange and dye-ligand chromatography. It has a Km value of 6.6 μm, an estimated molecular mass of 54 kD, and a (broad) pH optimum around 6.7. Germacrene A, the enzymatic product, proved to be much more stable than reported in literature. Its heat-induced Cope rearrangement into (−)-β-elemene was utilized to determine its absolute configuration on an enantioselective gas chromatography column. To our knowledge, until now in sesquiterpene biosynthesis, germacrene A has only been reported as an (postulated) enzyme-bound intermediate, which, instead of being released, is subjected to additional cyclization(s) by the same enzyme that generated it from farnesyl diphosphate. However, in chicory germacrene A is released from the sesquiterpene cyclase. Apparently, subsequent oxidations and/or glucosylation of the germacrane skeleton, together with a germacrene cyclase, determine whether guaiane- or eudesmane-type sesquiterpene lactones are produced.