on repeated games with complete information we consider n person repeated games with complete information and standard signalling we first prove several properties of the sets of feasible payoffs and nash equilibrium payoffs for the n stage game and for the discounted game in the second part we determine the set of equilibrium payoffs for the prisoner s dilemma corresponding to the critical value of the discount factor

multi peak solutions for nonlinear choquard equation in the plane in this paper we consider the nonlinear choquard equation in the plane with general nonlinearity satisfying berestycki lions condition by means of global penalization method we obtain the existence of multi peak solutions which concentrate near local minimal points of potential in this paper we consider the nonlinear choquard equation in the plane with general nonlinearity satisfying berestycki lions condition by means of global penalization method we obtain the existence of multi peak solutions which concentrate near local minimal points of potential

cooperation in repeated games when the number of stages is not commonly known it is shown that an exponentially small departure from the common knowledge assumption on the number t of repetitions of the prisoners dilemma already enables cooperation more generally with such a departure any feasible individually rational outcome of any one shot game can be approximated by a subgame perfect equilibrium of a finitely repeated version of that game the sense in which the departure from common knowledge is small is as follows i with probability one the players know t with precision k ii with probability 1 e the players know t precisely moreover this knowledge is mutual of order et iii the deviation of t from its finite expectation is exponentially small

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

gibbs sampler based coordination of autonomous swarms in this paper a novel gibbs sampler based algorithm is proposed for coordination of autonomous swarms the swarm is modeled as a markov random field mrf on a graph with a time varying neighborhood system determined by local interaction links the gibbs potential is designed to reflect global objectives and constraints it is established that with primarily local sensing communications the swarm configuration converges to the global minimizer s of the potential function the impact of the gibbs potential on the convergence speed is investigated finally a hybrid algorithm is developed to improve the efficiency of the stochastic scheme by integrating the gibbs sampler based method with the deterministic gradient flow method simulation results are presented to illustrate the proposed approach and verify the analyses

multi peak solution for nonlinear magnetic choquard type equation in this paper we study a class of nonlinear magnetic choquard type equation involving a magnetic potential and nonlocal nonlinearities by using the method of penalization argument we show that there exists a family of solutions having multiple concentration regions which are concentrate at the minimum points of potential v

strongly interacting bumps for the schrodinger newton equations we study concentrated bound states of the schrodinger newton equations h2 e x u 0 0 x r3 u 12 2 0 x r3 x 0 u x 0 as x moroz et al an analytical approach to the schrodinger newton equations nonlinearity 12 201 1999 proved the existence and uniqueness of ground states of u 0 0 x r3 u 12 2 0 x r3 x 0 u x 0 as x we first prove that the linearized operator around the unique ground state radial solution 0 u0 with 0 r ae r r 1 o 1 as r x u0 r b r 1 o 1 as r x for some a b 0 has a kernel whose dimension is exactly 3 corresponding to the translational modes using this result we further show that if for some positive integer k the points pi r3 i 1 2 k with pi pj for i j are all local minimum or local maximum or nondegenerate critical points of e p then for h small enough there exist solutions of the schrodinger newton equations with k bumps which concentrate at pi we also prove that given a local maximum point p0 of e p the

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

existence and concentration of ground state solutions for a critical nonlocal schrodinger equation in r2 we study the following singularly perturbed nonlocal schrodinger equation e2 u v x u e 2 1 x f u f u inr2 where v x is a continuous real function on r2 f s is the primitive of f s 0 2 and e is a positive parameter assuming that the nonlinearity f s has critical exponential growth in the sense of trudinger moser we establish the existence and concentration of solutions by variational methods

finitely repeated games etude systematique des equilibres parfaits de sous jeux pour des jeux repetes de facon finie avec information complete

the folk theorem for repeated games a neu condition we are concerned here with perfect folk theorems for infinitely repeated games with complete information folk theorems assert that any feasible and individually rational payoff vector of the stage game is a subgame perfect equilibrium payoff in the associated infinitely repeated game with little or no discounting where payoff streams are evaluated as average discounted or average values respectively it is obvious that feasibility and individual rationality are necessary conditions for a payoff vector to be an equilibrium payoff the surprising content of the folk theorems is that these conditions are also almost sufficient perhaps the first folk theorem type result is due to friedman 1971 who showed that any feasible payoff which pareto dominates a nash equilibrium payoff of the stage game will be an equilibrium payoff in the associated repeated game with sufficiently patient players this kind of result is sometimes termed a nash threats folk theorem a reference to its method of proof for the more permissive kinds of folk theorems considered here the seminal results are those of aumann and shapley 1976 and rubinstein 1977 1979 these authors assume that payoff streams are undiscounted 2 fudenberg and maskin 1986 establish an analogous result for discounted repeated games as the discount factor goes to 1 their result uses techniques of proof rather different from those used by aumann shapley and rubinstein respectively see their paper for an insightful discussion of this point and quite generally for more by way of background it is a key reference for subsequent work in this area including our own for the two player case the result of fudenberg and maskin 1986 is a complete if and only if characterization modulo the requirement of strict rather than weak individual rationality which we retain in this note and does not employ additional conditions for three or more players fudenberg and maskin introduced a full dimensionality condition the convex hull f of the set of feasible payoff vectors of the stage game must have dimension n where n is the number of players or equivalently a nonempty interior this condition has been widely adopted in proving folk theorems for related environments such as finitely repeated games benoit and krishna 1985 and overlapping generations games kandori 1992 smith 1992 full dimensionality is a sufficient condition fudenberg and maskin present an example of a three player stage game in which the conclusion of the folk theorem is false in this example all players receive the same payoffs in all contingencies the convex hull of the set of feasible payoffs is one dimensional this example violates full dimensionality in a rather extreme way less extreme violations may also lead to

cooperation and effective computability a common interest game is a game in which there exists a unique pair of payoffs which strictly pareto dominates all other payoffs we consider the undiscounted repeated game obtained by the infinite repetition of such a two player stage game we show that if supergame strategies are restricted to be computable within church s thesis the only pair of payoffs which survives any computable tremble with sufficiently large support is the pareto efficient pair the result is driven by the ability of the players to use the early stages of the game to communicate their intention to play cooperatively in the future

finitely repeated games with finite automata the paper studies the implications of bounding the complexity of the strategies players may select on the set of equilibrium payoffs in repeated games the complexity of a strategy is measured by the size of the minimal automation that can implement it a finite automation has a finite number of states and an initial state it prescribes the action to be taken as a function of the current state and a transition function changing the state of the automaton as a function of its current state and the present actions of the other players the size of an automaton is its number of states the main results imply in particular that in two person repeated games the set of equilibrium payoffs of a sequence of such games g n n 1 2 converges as n goes to infinity to the individual rational and feasible payoffs of the one shot game whenever the bound on one of the two automata sizes is polynomial or subexponential in n and both the length of the game and the bounds of the automata sizes are at least n a special case of such result justifies cooperation in the finitely repeated prisoner s dilemma without departure from strict utility maximization or complete information but under the assumption that there are bounds possibly very large to the complexity of the strategies that the players may use

the logarithmic choquard equation sharp asymptotics and nondegeneracy of the groundstate we derive the asymptotic decay of the unique positive radially symmetric solution to the logarithmic choquard equation u au 12 ln 1 x u 2 uin r2 and we establish its nondegeneracy for the corresponding three dimensional problem the nondegeneracy property of the positive ground state to the choquard equation was proved by e lenzmann 2009 13

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

erratum standing waves for nonlinear schrodinger equations with a general nonlinearity erratum arch rational mech anal doi 10 1007 s00205 006 0019 3through answering to some questions from joao marcos bezzerra do oon the paper above we found some mistakes in the paper we thank him forhis interest and careful reading here we correct these mistakes first at the beginning of the proof of proposition 4 14 17 line on page193 the correct version is by compactness of s

existence of semiclassical ground state solutions for a generalized choquard equation abstract in this paper we study a generalized quasilinear choquard equation e p p u v x u p 2 u e n r n q y f u y x y q x f u in r n where p is the p laplacian operator 1 p n v and q are two continuous real functions on r n 0 n f s is the primate function of f s and e is a positive parameter under suitable assumptions on p and f we establish a new concentration behavior of solutions for the quasilinear choquard equation by variational methods the results are also new for the semilinear case p 2

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

standing waves for nonlinear schrodinger equations with a general nonlinearity for elliptic equations e2 u v x u f u 0 x rn n 3 we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of v as e 0 under conditions on f which we believe to be almost optimal