Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.     

Sensitivity of the inverse wave crest problem

In a previous paper [Physica D 141 (3–4) (2000) 316], the inverse problem for wave crests was introduced and a solution strategy for two-wave interactions was given. Here these solutions are actually constructed, in particular for the cases with small interaction angle, moderate phase shifts, and/or symmetric interactions. Two detailed examples are presented and analyzed. The sensitivity of the method is investigated, and conclusions about the practical applicability are given.     

Absolute measurement of hadronic branching fractions of the Ds+ meson

The branching fractions of D(s)(+/-) meson decays serve to normalize many measurements of processes involving charm quarks. Using 298 pb(-1) of e(+)e(-) collisions recorded at a center of mass energy of 4.17 GeV, we determine absolute branching fractions for eight D(s)(+/-) decays with a double tag technique. In particular we determine the branching fraction B(D(s)(+)-->K(-)K(+}pi(+))=(5.50+/-0.23+/-0.16)%, where the uncertainties are statistical and systematic, respectively. We also provide partial branching fractions for kinematic subsets of the K(-)K(+)pi(+) decay mode.     

Comparison of D--> KS0 pi and D--> KL0 pi decay rates

We present measurements of D--> KS0 pi and D--> KL0 pi branching fractions using 281 pb(-1) of psi(3770) data at the CLEO-c experiment. We find that B(D0--> KS0 pi 0) is larger than B(D0--> KL0 pi 0), with an asymmetry of R(D0)=0.108+/-0.025+/-0.024. For B(D+--> KS0 pi+) and B(D+--> KL0 pi+), we observe no measurable difference; the asymmetry is R(D+)=0.022+/-0.016+/-0.018. The D0 asymmetry is consistent with the value based on the U-spin prediction A(D0--> K0 pi 0)/A(D0--> K0 pi 0)=-tan2 theta C, where theta C is the Cabibbo angle.     

Strong and ductile colossal carbon tubes with walls of rectangular macropores

We report a new type of carbon material-porous colossal carbon tubes. Compared with carbon nanotubes, colossal carbon tubes have a much bigger size, with a diameter of between 40 and 100 mum and a length in the range of centimeters. Significantly, the walls of the colossal tubes are composed of macroscopic rectangular columnar pores and exhibit an ultralow density comparable to that of carbon nanofoams. The porous walls of colossal tubes also show a highly ordered lamellar structure similar to that of graphite. Furthermore, colossal tubes possess excellent mechanical and electrical properties.     

Stable,Reactive, and Orthogonal Tetrazines: Dispersion Forces Promote the Cycloaddition with Isonitriles

The isocyano group is a structurally compact bioorthogonal functional group that reacts with tetrazines under physiological conditions. Now it is shown that bulky tetrazine substituents accelerate this cycloaddition. Computational studies suggest that dispersion forces between the isocyano group and the tetrazine substituents in the transition state contribute to the atypical structure–activity relationship. Stable asymmetric tetrazines that react with isonitriles at rate constants as high as 57 L mol^?1 s^?1 were accessible by combining bulky and electron‐withdrawing substituents. Sterically encumbered tetrazines react selectively with isonitriles in the presence of strained alkenes/alkynes, which allows for the orthogonal labeling of three proteins. The established principles will open new opportunities for developing tetrazine reactants with improved characteristics for diverse labeling and release applications with isonitriles.