On the approximation performance of fictitious play in finite games

We study the performance of Fictitious Play (FP), when used as a heuristic for finding an approximate Nash equilibrium of a two-player game. We exhibit a class of two-player games having payoffs in the range $[0,1]$ that show that FP fails to find a solution having an additive approximation guarantee significantly better than $1/2$ . Our construction shows that for $n\times n$ games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity $1/2 - O(1/n^{1-\delta })$ for arbitrarily small $\delta $ . We also show an essentially matching upper bound of $1/2 - O(1/n)$ .     

SmI(2)-promoted radical addition of nitrones to alpha,beta-unsaturated amides and esters: synthesis of gamma-amino acids via a nitrogen equivalent to the ketyl radical

[reaction: see text] Alkyl nitrones undergo radical addition reactions to a series of alpha,beta-unsaturated amides and esters when subjected to samarium diiodide via a nitrogen equivalent to a ketyl radical anion. This reaction conveniently provides access to a variety of functionalized gamma-amino acids. The methodology was extended to the asymmetric synthesis of 4-substituted gamma-amino acids, via the nitrone radical addition reaction to acrylates/amides possessing a chiral auxiliary.     

Can decarbonylation of acyl radicals be overcome in radical addition reactions? En route to a solution employing N-acyl oxazolidinones and SmI2/H2O

The application of acyl radicals in radical addition reactions in the absence of a CO atmosphere is generally limited to aryl or alpha-unsubstituted alkyl acyl radicals due to competing decarbonylations where the rate constant for this degradation process surpasses 104 s-1. In this work, a potential solution to avoid the problem of decarbonylations is presented employing N-acyl oxazolidinones which are reduced to acyl radical equivalents in the presence of samarium diiodide and water. In the company of an acrylamide, acrylate, or acrylonitrile, the product from a formal acyl radical addition is obtained in yields up to 87%. Examples are given where the decarbonylation rate constants even exceed 108 s-1. It is proposed that the reaction proceeds via a ketyl-like intermediate.     

Regioselective Heck couplings of alpha,beta-unsaturated tosylates and mesylates with electron-rich olefins

[chemical reaction: see text]. Highly regioselective Heck couplings of alpha,beta-unsaturated tosylate and mesylate derivatives with N-acyl N-vinylamines and vinyl ethers were achieved. Several 2-alkoxy-1,3-dienes and 2-acylamino-1,3-butadienes were synthesized in good yields using 1.5 mol % of Pd2(dba)3, 3 mol % of DPPF, and diisopropylethylamine in dioxane. When working with alpha,beta-unsaturated ketones and esters, this method provides a less costly alternative to similar couplings using a triflate electrophile.     

Direct Access to α,α‐Difluoroacylated Arenes by Palladium‐Catalyzed Carbonylation of (Hetero)Aryl Boronic Acid Derivatives

A palladium‐catalyzed carbonylative coupling of (hetero)aryl boronates or boronic acid salts with carbon monoxide and α‐bromo‐α,α‐difluoroamides and bromo‐α,α‐difluoroesters is described herein. The method is useful for the synthesis of a diverse selection of (hetero)aryl α,α‐difluoro‐β‐ketoamides and α,α‐difluoro‐β‐ketoesters, which are useful building blocks for the generation of functionalized difluoroacylated and difluoroalkyl arenes. The method could be further extended to a one‐pot protocol for the formation of difluoroacetophenones.     

Direct vinylation and difluorovinylation of arylboronic acids using vinyl- and 2,2-difluorovinyl tosylates via the Suzuki-Miyaura cross coupling

General reaction conditions were developed for the Pd(0)-catalyzed Suzuki-Miyaura coupling reaction of aryl boronic acids with a simple electrophilic vinylation reagent, vinyl tosylate, providing access to styrene derivatives in good yields. The easily accessible vinyl tosylate represents a stable and less toxic alternative to the vinyl halides and the triflate/nonaflate derivatives. Furthermore, this methodology was expanded to provide a facile and straightforward approach for the introduction of a gem-difluorovinyl substituent onto an aromatic ring using the similar and also readily available 2,2-difluorovinyl tosylate as the electrophilic complement.     

Resolution enhancement in solid-state NMR of oriented membrane proteins by anisotropic differential linebroadening

We demonstrate that a significant improvement in the spectral resolution may be achieved in solid-state NMR experiments of proteins in inhomogeneously disordered oriented lipid bilayers. Using 1H homonuclear decoupling instead of standard 1H heteronuclear decoupling, the 15N line widths may be reduced by up to seven times for such samples. For large oriented membrane proteins, such resolution enhancements may be crucial for assignment and structural interpretation.     

An efficient method for the preparation of tertiary esters by palladium-catalyzed alkoxycarbonylation of aryl bromides

The palladium-catalyzed alkoxycarbonylation of aryl bromides is described for the efficient preparation of tertiary esters. The protocol proved compatible with a wide variety of functionalized (hetero)aromatic bromides, as well as several different sterically hindered tertiary alcohols, affording the alkoxycarbonylated products in high yields. Finally, the formation of aromatic trityl esters is discussed.