integral equation theory for fluids ordered by an external field separable interactions the structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories a general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles this approach is then illustrated for the case of two off lattice models the ferromagnetic heisenberg model and a simple liquid crystal model for which the numerical solution of integral equations such as the percus yevick the hypernetted chain and the reference hypernetted chain closure equations are presented and compared with monte carlo simulation results and the analytical solution of the mean spherical approximation the zero field case is also examined and the spontaneous ordering is analyzed in detail mainly in what concerns the appearance of infinite wavelength singularity in the ornstein zernike equation and the relation with the one body closure equations and the long range orientational ordering that occurs in particular it is shown that the wertheim one body closure equation appears as a sum rule compatible with the ornstein zernike equation the relation between the elastic constant and the long range tail of the pair correlation function is made explicit in particular the long range behavior of the various terms in the expansion of the pair correlation function is depicted the numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations as to which one would be the most accurate in all cases the general trends in the thermodynamical and structural properties seem to indicate that the percus yevick approximation is generally better in the strong ordering case whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes

maximum marginal likelihood estimation for semiparametric item analysis the item characteristic curve icc defining the relation between ability and the probability of choosing a particular option for a test item can be estimated by using polynomial regression splines these provide a more flexible family of functions than is given by the three parameter logistic family the estimation of spline iccs is described by maximizing the marginal likelihood formed by integrating ability over a beta prior distribution some simulation results compare this approach with the joint estimation of ability and item parameters

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

phase diagram in a continuum model of the classical heisenberg ferromagnet mean spherical approximation abstract the continuum heisenberg model of spin fluid is considered the pair correlation functions and the single particle distribution function of ferromagnetic state are obtained by the self consistent solution of the ornstein zernike and the lovett equations in the mean spherical approximation on the basis of the analytical solution the phase diagram is calculated and compared with the available monte carlo results for the same model the agreement between theory and simulation is quite satisfactory especially for the liquid gas coexistence line the msa suggests that only ordered states are involved in the gas liquid transition near the gas liquid critical point this result confirms the monte carlo prediction about the existence of a magnetic critical point

identification of the 1pl model with guessing parameter parametric and semi parametric results in this paper we study the identification of a particular case of the 3pl model namely when the discrimination parameters are all constant and equal to 1 we term this model 1pl g model the identification analysis is performed under three different specifications the first specification considers the abilities as unknown parameters it is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero the second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter it is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero the third specification corresponds to a semi parametric 1pl g model where the distribution g generating the abilities is a parameter of interest it is not only shown that after fixing a difficulty parameter and a guessing parameter at zero the item parameters are identified but also that under those restrictions the distribution g is not identified it is finally shown that after introducing two identification restrictions either on the distribution g or on the item parameters the distribution g and the item parameters are identified provided an infinite quantity of items is available

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

dealing with label switching in mixture models summary in a bayesian analysis of finite mixture models parameter estimation and clustering are sometimes less straightforward than might be expected in particular the common practice of estimating parameters by their posterior mean and summarizing joint posterior distributions by marginal distributions often leads to nonsensical answers this is due to the so called label switching problem which is caused by symmetry in the likelihood of the model parameters a frequent response to this problem is to remove the symmetry by using artificial identifiability constraints we demonstrate that this fails in general to solve the problem and we describe an alternative class of approaches relabelling algorithms which arise from attempting to minimize the posterior expected loss under a class of loss functions we describe in detail one particularly simple and general relabelling algorithm and illustrate its success in dealing with the label switching problem on two examples

a note on approximate nash equilibria in view of the intractability of finding a nash equilibrium it is important to understand the limits of approximation in this context a subexponential approximation scheme is known richard j lipton evangelos markakis aranyak mehta playing large games using simple strategies in ec 2003 and no approximation better than 14 is possible by any algorithm that examines equilibria involving fewer than logn strategies ingo althofer on sparse approximations to randomized strategies and convex combinations linear algebra and its applications 1994 199 we give a simple linear time algorithm examining just two strategies per player and resulting in a 12 approximate nash equilibrium in any 2 player game for the more demanding notion of approximately well supported nash equilibrium due to constantinos daskalakis paul w goldberg christos h papadimitriou the complexity of computing a nash equilibrium siam journal on computing in press preliminary version appeared in stoc 2006 no nontrivial bound is known we show that the problem can be reduced to the case of win lose games games with all utilities 0 or 1 and that an approximation of 56 is possible contingent upon a graph theoretic conjecture subsequent work extends the 14 impossibility result of ingo althofer s paper as mentioned above to 12 tomas feder hamid nazerzadeh amin saberi approximating nash equilibria using small support strategies in ec 2007 making our 12 approximate nash equilibrium algorithm optimal among the algorithms that only consider mixed strategies of sublogarithmic size support moreover techniques similar to our techniques for approximately well supported nash equilibria are used in spyros kontogiannis paul g spirakis efficient algorithms for constant well supported approximate equilibria in bimatrix games in icalp 2007 for obtaining an efficient algorithm for 0 658 approximately well supported nash equilibria unconditionally

discrete fixed points models complexities and applications we study three discrete fixed point concept sperner dpzp brouwer under two different models the polynomial time function model and the oracle function model we fully characterize the computational complexities of these three problems the computational complexity unification of the above problems gives us more choices in the study of different applications as an example by a reduction from dpzp we derive asymptotically equal lower and upper bound for tucker in the oracle model the same reduction also allows us to derive a single proof for the ppad completeness of tucker in any constant dimension which is significantly simpler than the recent proofs

intrinsic robustness of the price of anarchy the price of anarchy defined as the ratio of the worst case objective function value of a nash equilibrium of a game and that of an optimal outcome quantifies the inefficiency of selfish behavior remarkably good bounds on this measure are known for a wide range of application domains however such bounds are meaningful only if a game s participants successfully reach a nash equilibrium this drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria such as mixed nash equilibria and correlated equilibria and to sequences of outcomes generated by natural experimentation strategies such as successive best responses and simultaneous regret minimization we establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives first we identify a canonical sufficient condition for an upper bound on the price of anarchy of pure nash equilibria which we call a smoothness argument second we prove an extension theorem every bound on the price of anarchy that is derived via a smoothness argument extends automatically with no quantitative degradation in the bound to mixed nash equilibria correlated equilibria and the average objective function value of every outcome sequence generated by no regret learners smoothness arguments also have automatic implications for the inefficiency of approximate equilibria for bicriteria bounds and under additional assumptions for polynomial length best response sequences third we prove that in congestion games smoothness arguments are complete in a proof theoretic sense despite their automatic generality they are guaranteed to produce optimal worst case upper bounds on the price of anarchy

mcmc estimation and some model fit analysis of multidimensional irt models a bayesian procedure to estimate the three parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented the procedure is a generalization of a procedure by albert 1992 for estimating the two parameter normal ogive model the procedure supports analyzing data from multiple populations and incomplete designs it is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure the technique is illustrated using simulated and real data

market equilibrium under separable piecewise linear concave utilities we consider fisher and arrow debreu markets under additively separable piecewise linear concave utility functions and obtain the following results for both market models if an equilibrium exists there is one that is rational and can be written using polynomially many bits there is no simple necessary and sufficient condition for the existence of an equilibrium the problem of checking for existence of an equilibrium is np complete for both market models the same holds for existence of an e approximate equilibrium for e e o n 5 under standard mild sufficient conditions the problem of finding an exact equilibrium is in ppad for both market models finally building on the techniques of chen et al 2009a we prove that under these sufficient conditions finding an equilibrium for fisher markets is ppad hard

query complexity of approximate nash equilibria we study the query complexity of approximate notions of nash equilibrium in games with a large number of players n our main result states that for n player binary action games and for constant e the query complexity of an e well supported nash equilibrium is exponential in n as a consequence of this result we get an exponential lower bound on the rate of convergence of adaptive dynamics to approximate nash equilibria

learning equilibria of games via payoff queries a recent body of experimental literature has studied empirical game theoretical analysis in which we have partial knowledge of a game consisting of observations of a subset of the pure strategy profiles and their associated payoffs to players the aim is to find an exact or approximate nash equilibrium of the game based on these observations it is usually assumed that the strategy profiles may be chosen in an on line manner by the algorithm we study a corresponding computational learning model and the query complexity of learning equilibria for various classes of games we give basic results for exact equilibria of bimatrix and graphical games we then study the query complexity of approximate equilibria in bimatrix games finally we study the query complexity of exact equilibria in symmetric network congestion games for directed acyclic networks we can learn the cost functions and hence compute an equilibrium while querying just a small fraction of pure strategy profiles for the special case of parallel links we have the stronger result that an equilibrium can be identified while only learning a small fraction of the cost values

a complementary pivot algorithm for market equilibrium under separable piecewise linear concave utilities using lemke s scheme we give a complementary pivot algorithm for computing an equilibrium for arrow debreu markets under separable piecewise linear concave splc utilities despite the polynomial parity argument on directed graphs ppad completeness of this case experiments indicate that our algorithm is practical on randomly generated instances the number of iterations it needs is linear in the total number of segments i e pieces in all the utility functions specified in the input our paper settles a number of open problems 1 eaves 1976 gave an lcp formulation and a lemke type algorithm for the linear arrow debreu model we generalize both to the splc case hence settling the relevant part of his open problem 2 our path following algorithm for splc markets together with a result of todd 1976 gives a direct proof of membership of such markets in ppad and settles a question of vazirani and yannakakis 2011 3 we settle a question of devanur and kannan 2008 of obtaining a sy

on the complexity of an expanded tarski s fixed point problem under the componentwise ordering abstract let be a finite lattice of integer points in a box of r n and f an increasing mapping in terms of the componentwise ordering from to itself the well known tarski s fixed point theorem asserts that f has a fixed point in a simple expansion of f from to a larger lattice c of integer points in a box of r n yields that the smallest point in c is always a fixed point of f an expanded tarski s fixed point problem by introducing an integer labeling rule and applying a cubic triangulation of the euclidean space we prove in this paper that the expanded tarski s fixed point problem is in the class ppa when f is given as an oracle it is shown in this paper that nash equilibria of a bimatrix game can be reformulated as fixed points different from the smallest point in c of an increasing mapping from c to itself this implies that the expanded tarski s fixed point problem has at least the same complexity as that of the nash equilibrium problem as a byproduct we also present a homotopy like simplicial method to compute a tarski fixed point of f the method starts from an arbitrary lattice point and follows a finite simplicial path to a fixed point of f

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

a semiparametric bayesian model for randomised block designs summary a model is proposed for a bayesian semiparametric analysis of randomised block experiments the model is a hierarchical model in which a dirichlet process is inserted at the middle stage for the distribution of the block effects this model allows an arbitrary distribution of block effects and it results in effective estimates of treatment contrasts block effects and the distribution of block effects an effective computational strategy is presented for describing the posterior distribution

semidefinite programming for min max problems and games we consider two min max problems 1 minimizing the supremum of finitely many rational functions over a compact basic semi algebraic set and 2 solving a 2 player zero sum polynomial game in randomized strategies with compact basic semi algebraic sets of pure strategies in both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations in the spirit of the moment approach developed in lasserre siam j optim 11 796 817 2001 math program b 112 65 92 2008 this provides a unified approach and a class of algorithms to compute nash equilibria and min max strategies of several static and dynamic games each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation and sometimes even for finite convergence a behavior similar to what was observed in polynomial optimization

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

maximum marginal likelihood estimation of a monotonic polynomial generalized partial credit model with applications to multiple group analysis we present a semi parametric approach to estimating item response functions irf useful when the true irf does not strictly follow commonly used functions our approach replaces the linear predictor of the generalized partial credit model with a monotonic polynomial the model includes the regular generalized partial credit model at the lowest order polynomial our approach extends liang s a semi parametric approach to estimate irfs unpublished doctoral dissertation 2007 method for dichotomous item responses to the case of polytomous data furthermore item parameter estimation is implemented with maximum marginal likelihood using the bock aitkin em algorithm thereby facilitating multiple group analyses useful in operational settings our approach is demonstrated on both educational and psychological data we present simulation results comparing our approach to more standard irf estimation approaches and other non parametric and semi parametric alternatives

on total functions existence theorems and computational complexity abstract nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between p and np we show that this class which we call tfnp contains a host of important problems whose membership in p is currently not known these include besides factoring local optimization brouwer s fixed points a computational version of sperner s lemma bimatrix equilibria in games and linear complementarity for p matrices

on the complexity of nash equilibria and other fixed points we reexamine what it means to compute nash equilibria and more generally what it means to compute a fixed point of a given brouwer function and we investigate the complexity of the associated problems specifically we study the complexity of the following problem given a finite game gamma with 3 or more players and given epsilon 0 compute an approximation within epsilon of some actual nash equilibrium we show that approximation of an actual nash equilibrium even to within any nontrivial constant additive factor epsilon 1 2 in just one desired coordinate is at least as hard as the long standing square root sum problem as well as a more general arithmetic circuit decision problem that characterizes p time in a unit cost model of computation with arbitrary precision rational arithmetic thus placing the approximation problem in p or even np would resolve major open problems in the complexity of numerical computation we show similar results for market equilibria it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium where the economy is given by explicit algebraic formulas for the excess demand functions we define a class fixp which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits straight line programs over basis max min with rational constants we show that the exact or approximate computation of nash equilibria for 3 or more players is complete for fixp the price equilibrium problem for exchange economies with algebraic demand functions is another fixp complete problem we show that the piecewise linear fragment of fixp equals ppad many other problems in game theory economics and probability theory can be cast as fixed point problems for such algebraic functions we discuss several important such problems computing the value of shapley s stochastic games and the simpler games of condon extinction probabilities of branching processes probabilities of stochastic context free grammars and termination probabilities of recursive markov chains we show that for some of them the approximation or even exact computation problem can be placed in ppad while for others they are at least as hard as the square root sum and arithmetic circuit decision problems

market equilibria with hybrid linear leontief utilities we introduce a new family of utility functions for exchange markets this family provides a natural and continuous hybridization of the traditional linear and leontief utilities and might be useful in understanding the complexity of computing approximating market equilibria although computing an equilibrium in a market with this family of utility functions this is ppad hard in general in this paper we present an algorithm for finding an approximate arrow debreu equilibrium when the leontief components of the market are grouped finite and well conditioned

markov chain monte carlo in approximate dirichlet and beta two parameter process hierarchical models summary we present some easy to construct random probability measures which approximate the dirichlet process and an extension which we will call the beta two parameter process the nature of these constructions makes it simple to implement markov chain monte carlo algorithms for fitting nonparametric hierarchical models and mixtures of nonparametric hierarchical models for the dirichlet process we consider a truncation approximation as well as a weak limit approximation based on a mixture of dirichlet processes the same type of truncation approximation can also be applied to the beta two parameter process both methods lead to posteriors which can be fitted using markov chain monte carlo algorithms that take advantage of blocked coordinate updates these algorithms promote rapid mixing of the markov chain and can be readily applied to normal mean mixture models and to density estimation problems we prefer the truncation approximations since a simple device for monitoring the adequacy of the approximation can be easily computed from the output of the gibbs sampler furthermore for the dirichlet process the truncation approximation offers an exponentially higher degree of accuracy over the weak limit approximation for the same computational effort we also find that a certain beta two parameter process may be suitable for finite mixture modelling because the distinct number of sampled values from this process tends to match closely the number of components of the underlying mixture distribution

bayesian density regression summary the paper considers bayesian methods for density regression allowing a random probability distribution to change flexibly with multiple predictors the conditional response distribution is expressed as a non parametric mixture of regression models with the mixture distribution changing with predictors a class of weighted mixture of dirichlet process priors is proposed for the uncountable collection of mixture distributions it is shown that this specification results in a generalized polya urn scheme which incorporates weights that are dependent on the distance between subjects predictor values to allow local dependence in the mixture distributions we propose a kernel based weighting scheme a gibbs sampling algorithm is developed for posterior computation the methods are illustrated by using simulated data examples and an epidemiologic application

fast algorithms for rank 1 bimatrix games the rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices this paper comprehensively analyzes games of rank one and shows the following 1 for a game of rank r the set of its nash equilibria is the intersection of a generically one dimensional set of equilibria of parameterized games of rank r 1 with a hyperplane 2 one equilibrium of a rank 1 game can be found in polynomial time 3 all equilibria of a rank 1 game can be found by following a piecewise linear path in contrast such a path following method finds only one equilibrium of a bimatrix game 4 the number of equilibria of a rank 1 game may be exponential 5 there is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank it is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component

inapproximability results for constrained approximate nash equilibria abstract we study the problem of finding approximate nash equilibria that satisfy certain conditions such as providing good social welfare in particular we study the problem ne sw find an approximate nash equilibrium ne that is within of the best social welfare achievable by an ne our main result is that if the exponential time hypothesis eth is true then solving 1 8 o ne o sw for an n n bimatrix game requires n log n time building on this result we show similar conditional running time lower bounds for a number of other decision problems for ne where for example the payoffs or supports of players are constrained we show quasi polynomial lower bounds for these problems assuming eth where these lower bounds apply to nash equilibria for all 1 8 the hardness of these other decision problems has so far only been studied in the context of exact equilibria

constant rank two player games are ppad hard finding a nash equilibrium in a two player normal form game 2 nash is one of the most extensively studied problems within mathematical economics as well as theoretical computer science such a ga

the complexity of uniform nash equilibria and related regular subgraph problems we investigate the complexity of finding nash equilibria in which the strategy of each player is uniform on its support set we show that even for a restricted class of win lose bimatrix games deciding the existence of such uniform equilibria is an np complete problem our proof is graph theoretical motivated by this result we also give np completeness results for the problems of finding regular induced subgraphs of large size or regularity which can be of independent interest

the complexity of non monotone markets we introduce the notion of non monotone utilities which covers a wide variety of utility functions in economic theory we then prove that it is ppad hard to compute an approximate arrow debreu market equilibrium in markets with linear and non monotone utilities building on this result we settle the long standing open problem regarding the computation of an approximate arrow debreu market equilibrium in markets with ces utility functions by proving that it is ppad complete when the constant elasticity of substitution parameter is any constant less than 1

semiparametric thurstonian models for recurrent choices a bayesian analysis we develop semiparametric bayesian thurstonian models for analyzing repeated choice decisions involving multinomial multivariate binary or multivariate ordinal data our modeling framework has multiple components that together yield considerable flexibility in modeling preference utilities cross sectional heterogeneity and parameter driven dynamics each component of our model is specified semiparametrically using dirichlet process dp priors the utility latent variable component of our model allows the alternative specific utility errors to semiparametrically deviate from a normal distribution this generates a robust alternative to popular thurstonian specifications that are based on underlying normally distributed latent variables our second component focuses on flexibly modeling cross sectional heterogeneity the semiparametric specification allows the heterogeneity distribution to mimic either a finite mixture distribution or a continuous distribution such as the normal whichever is supported by the data thus special features such as multimodality can be readily incorporated without the need to overtly search for the best heterogeneity specification across a series of models finally we allow for parameter driven dynamics using a semiparametric state space approach this specification adds to the literature on robust kalman filters the resulting framework is very general and integrates divergent strands of the literatures on flexible choice models bayesian nonparametrics and robust time series specifications given this generality we show how several existing thurstonian models can be obtained as special forms of our model we describe markov chain monte carlo methods for the inference of model parameters report results from two simulation studies and apply the model to consumer choice data from a frequently purchased product category the results from our simulations and application highlight the benefits of using our semiparametric approach

oriented euler complexes and signed perfect matchings this paper presents oriented pivoting systems as an abstract framework for complementary pivoting it gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign a special case are the nash equilibria of a bimatrix game at the ends of lemke howson paths which have opposite index for euler complexes or oiks an orientation is defined which extends the known concept of oriented abstract simplicial manifolds ordered room partitions for a family of oriented oiks come in pairs of opposite sign for an oriented oik of even dimension this sign property holds also for unordered room partitions in the case of a two dimensional oik these are perfect matchings of an euler graph with the sign as defined for pfaffian orientations of graphs a near linear time algorithm is given for the following problem given a graph with an eulerian orientation with a perfect matching find another perfect matching of opposite sign in contrast the complementary pivoting algorithm for this problem may be exponential

bayesian nonparametric inference for random distributions and related functions in recent years bayesian nonparametric inference both theoretical and computational has witnessed considerable advances however these advances have not received a full critical and comparative analysis of their scope impact and limitations in statistical modelling many aspects of the theory and methods remain a mystery to practitioners and many open guestions remain in this paper we discuss and illustrate the rich modelling and analytic possibilities that are available to the statistician within the bayesian nonparametric and or semiparametric framework

bridging the gap between phenomenology and microscopic theory asymptotes in nematic colloids the ornstein zernike equation is applied to nematic colloids with up down symmetry to determine how the electrostatic analogy and other phenomenological results appear in molecular theory in contrast to phenomenological approaches the molecular theory does not assume particular boundary conditions anchoring at colloidal surfaces for our molecular parameters the resulting anchoring appears to be realistic neither rigid nor infinitely weak for this case the effective force between a colloidal pair at large separation remains essentially constant over the entire region of nematic stability we show that a simple van der waals approximation gives a potential of mean force that in some important aspects is similar to the phenomenological results obtained in the limit of weak anchoring at large separations the potential varies as sigma8 where sigma is the colloidal diameter in contrast the more sophisticated mean spherical approximation yields a sigma6 dependence consistent with phenomenological calculations employing rigid boundary conditions we show that taking proper account of the correlation or magnetic coherence length xi inherent in the nematic sample is essential in an analysis of the sigma dependence at infinite xi the leading sigma dependence is sigma6 but this shifts to sigma8 when xi is finite the correlation length also influences the orientational behavior of the effective interaction the so called quadrupole interaction that determines the long range behavior at infinite xi transforms into a superposition of screened multipoles when xi is finite the basic approach employed in this paper can be readily applied to a broad range of physically interesting systems these include patterned and nonspherical colloids colloids trapped at interfaces and nematic fluids in confined geometries such as droplets

constructive epsilon nash equilibria for nonzero sum differential games in this paper a class of infinite horizon nonzero sum differential games and their nash equilibria are studied and the notion of epsilon alpha nash equilibrium strategies is introduced dynamic strategies satisfying partial differential inequalities in place of the hamilton jacobi isaacs partial differential equations associated with the differential games are constructed these strategies constitute local epsilon alpha nash equilibrium strategies for the differential game the proposed methods are illustrated on a differential game for which the nash equilibrium strategies are known and on a lotka volterra model with two competing species simulations indicate that both dynamic strategies yield better performance than the strategies resulting from the solution of the linear quadratic approximation of the problem

kernel smoothing approaches to nonparametric item characteristic curve estimation the option characteristic curve the relation between ability and probability of choosing a particular option for a test item can be estimated by nonparametric smoothing techniques what is smoothed is the relation between some function of estimated examinee ability rankings and the binary variable indicating whether or not the option was chosen this paper explores the use of kernel smoothing which is particularly well suited to this application examples show that with some help from the fast fourier transform estimates can be computed about 500 times as rapidly as when using commonly used parametric approaches such as maximum marginal likelihood estimation using the three parameter logistic distribution simulations suggest that there is no loss of efficiency even when the population curves are three parameter logistic the approach lends itself to several interesting extensions

the linear complementarity problems with a few variables per constraint in this paper we consider the sparse linear complementarity problem denoted by k lcp the coefficient matrices are restricted to have at most k nonzero entries per row it is known that the 1 lcp is solvable in linear time and the 3 lcp is strongly np hard we show that the 2 lcp is strongly np hard and it can be solved in polynomial time if it is sign balanced i e each row of the matrix has at most one positive and one negative entry our second result matches the currently best known complexity bound for the corresponding sparse linear feasibility problem in addition we show that an integer variant of the sign balanced 2 lcp is weakly np hard and pseudo polynomially solvable and the generalized 1 lcp is strongly np hard

settling the complexity of computing two player nash equilibria we prove that bimatrix the problem of finding a nash equilibrium in a two player game is complete for the complexity class ppad polynomial parity argument directed version introduced by papadimitriou in 1991 our result building upon the work of daskalakis et al 2006a on the complexity of four player nash equilibria settles a long standing open problem in algorithmic game theory it also serves as a starting point for a series of results concerning the complexity of two player nash equilibria in particular we prove the following theorems bimatrix does not have a fully polynomial time approximation scheme unless every problem in ppad is solvable in polynomial time the smoothed complexity of the classic lemke howson algorithm and in fact of any algorithm for bimatrix is not polynomial unless every problem in ppad is solvable in randomized polynomial time our results also have a complexity implication in mathematical economics arrow debreu market equilibria are ppad hard to compute

the complexity of computing a nash equilibrium in 1951 john f nash proved that every game has a nash equilibrium ann of math 2 54 1951 pp 286 295 his proof is nonconstructive relying on brouwer s fixed point theorem thus leaving open the questions is there a polynomial time algorithm for computing nash equilibria and is this reliance on brouwer inherent many algorithms have since been proposed for finding nash equilibria but none known to run in polynomial time in 1991 the complexity class ppad polynomial parity arguments on directed graphs for which brouwer s problem is complete was introduced c papadimitriou j comput system sci 48 1994 pp 489 532 motivated largely by the classification problem for nash equilibria but whether the nash problem is complete for this class remained open in this paper we resolve these questions we show that finding a nash equilibrium in three player games is indeed ppad complete and we do so by a reduction from brouwer s problem thus establishing that the two problems are computationally equivalent our reduction simulates a stylized brouwer function by a graphical game m kearns m littman and s singh graphical model for game theory in 17th conference in uncertainty in artificial intelligence uai 2001 relying on gadgets graphical games performing various arithmetic and logical operations we then show how to simulate this graphical game by a three player game where each of the three players is essentially a color class in a coloring of the underlying graph subsequent work x chen and x deng setting the complexity of 2 player nash equilibrium in 47th annual ieee symposium on foundations of computer science focs 2006 established by improving our construction that even two player games are ppad complete here we show that this result follows easily from our proof

faster algorithms for extensive form game solving via improved smoothing functions sparse iterative methods in particular first order methods are known to be among the most effective in solving large scale two player zero sum extensive form games the convergence rates of these methods depend heavily on the properties of the distance generating function that they are based on we investigate both the theoretical and practical performance improvement of first order methods foms for solving extensive form games through better design of the dilated entropy function a class of distance generating functions related to the domains associated with the extensive form games by introducing a new weighting scheme for the dilated entropy function we develop the first distance generating function for the strategy spaces of sequential games that has only a logarithmic dependence on the branching factor of the player this result improves the overall convergence rate of several foms working with dilated entropy function by a factor of omega b dd where b is the branching factor of the player and d is the depth of the game tree thus far counterfactual regret minimization methods have been faster in practice and more popular than foms despite their theoretically inferior convergence rates using our new weighting scheme and a practical parameter tuning procedure we show that for the first time the excessive gap technique a classical fom can be made faster than the counterfactual regret minimization algorithm in practice for large games and that the aggressive stepsize scheme of cfr is the only reason that the algorithm is faster in practice

ranking games that have competitiveness based strategies an extensive literature in economics and social science addresses contests in which players compete to outperform each other on some measurable criterion often referred to as a player s score or output players incur costs that are an increasing function of score but receive prizes for obtaining higher score than their competitors in this paper we study finite games that are discretized contests and the problems of computing exact and approximate nash equilibria our motivation is the worst case hardness of nash equilibrium computation and the resulting interest in important classes of games that admit polynomial time algorithms for games that have a tie breaking rule for players scores we present a polynomial time algorithm for computing an exact equilibrium in the 2 player case and for multiple players a characterization of nash equilibria that shows an interesting parallel between these games and unrestricted 2 player games in normal form when ties are allowed via a reduction from these games to a subclass of anonymous games we give approximation schemes for two special cases constant sized set of strategies and constant number of players

computing equilibria in discounted dynamic games game theory gt is an essential formal tool for interacting entities however computing equilibria in gt is a hard problem when the same game can be played repeatedly over time the problem becomes even more complicated the existence of multiple game states makes the problem of computing equilibria in such games extremely difficult in this paper we approach this problem by first proposing a method to compute a nonempty subset of approximate up to any precision subgame perfect equilibria in repeated games we then demonstrate how to extend this method to approximate all subgame perfect equilibria in a repeated game and also to solve more complex games such as markov chain games and stochastic games we observe that in stochastic games our algorithm requires additional strong assumptions to become tractable while in repeated and markov chain games it allows approximating all subgame perfect equilibria reasonably fast and under considerably weaker assumptions than previous methods

xy spin fluids in an external magnetic field an integral equation approach the phase behavior of ising spin fluids is studied in the presence of an external magnetic field with the integral equation method the calculations are performed on the basis of a soft mean spherical approximation using an efficient algorithm for solving the coupled set of the ornstein zernike equations the closure relations and the external field constraint the phase diagrams are obtained in the whole thermodynamic space including the magnetic field h for a wide class of ising fluid models with various ratios r of the strengths of magnetic to nonmagnetic yukawa like interactions the influence of varying the inverse screening lengths z 1 and z 2 corresponding to the magnetic and nonmagnetic yukawa parts of the potential is investigated too it is shown that changes in r as well as in z 1 and z 2 can lead to different topologies of the phase diagrams in particular depending on the value of r the critical temperature of the liquid gas transition either decreases monotonically behaves nonmonotonically or increases monotonically with increasing h the para ferro magnetic transition is also affected by changes in r and the screening lengths at h 0 the ising fluid maps onto a simple model of a symmetric nonmagnetic binary mixture for h infinity it reduces to a pure nonmagnetic fluid the results are compared with available simulations and the predictions of other theoretical methods it is demonstrated that the mean spherical approximation appears to be more accurate compared with mean field theory especially for systems with short ranged attraction potentials when z 1 and z 2 are large in the kac limit z 1 z 2 0 both approaches tend to nearly the same results

integral equation procedure based on tailored orthogonal functions for the xy spin fluid in an external magnetic field the classical xy model describes particles in three dimensional space that carry magnetic moments or spins whose motion is restricted to rotations in a plane introduction of an external magnetic field lying in the same plane then generates a system that is anisotropic in the azimuthal angle phi we use numerical simulations and integral equation techniques to study this system producing in the latter case a formalism that is identical to that of the simpler isotropic version having no external field the basis for this simplification is a generalization em phi of the ordinary exponential basis set eimphi that restores orthogonality in the presence of the external field we display results of sample calculations obtained with two integral equation closures reference hypernetted chain and soft mean spherical approximation both coupled to the lovett mou buff wertheim relation along with results from the numerical simulations for comparison construction of the em phi is described in an appendix

on mutual concavity and strategically zero sum bimatrix games we study the fundamental problem 2nash of computing a nash equilibrium ne point in bimatrix games we start by proposing a novel characterization of the ne set via a bijective map to the solution set of a parameterized quadratic program neqp whose feasible space is the highly structured set of correlated equilibria ce this is to our knowledge the first characterization of the subset of ce points that are in 1 1 correspondence with the ne set of the game and contributes to the quite lively discussion on the relation between the spaces of ce and ne points in a bimatrix game e g 15 26 33 we proceed with studying a property of bimatrix games which we call mutually concavity mc that assures polynomial time tractability of 2nash due to the convexity of a proper parameterized quadratic program either neqp or a parameterized variant of the mangasarian stone formulation 23 for a particular value of the parameter we prove various characterizations of the mc games which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero sum szs games of moulin vial 25 this gives an alternative explanation of the polynomial time tractability of 2nash for these games not depending on the solvability of zero sum games moreover the recognition of the mc property for an arbitrary game is much faster than the recognition szs property this along with the comparable time complexity of linear programs and convex quadratic programs leads us to a much faster algorithm for 2nash in mc games we conclude our discussion with a comparison of mc games or szs games to k rank games which are known to admit for 2nash a fptas when k is fixed 18 and a polynomial time algorithm for k 1 2 we finally explore some closeness properties under well known ne set preserving transformations of bimatrix games

on the bayesian nonparametric generalization of irt type models we study the identification and consistency of bayesian semiparametric irt type models where the uncertainty on the abilities distribution is modeled using a prior distribution on the space of probability measures we show that for the semiparametric rasch poisson counts model simple restrictions ensure the identification of a general distribution generating the abilities even for a finite number of probes for the semiparametric rasch model only a finite number of properties of the general abilities distribution can be identified by a finite number of items which are completely characterized the full identification of the semiparametric rasch model can be only achieved when an infinite number of items is available the results are illustrated using simulated data

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

estimation of finite mixture distributions through bayesian sampling summary a formal bayesian analysis of a mixture model usually leads to intractable calculations since the posterior distribution takes into account all the partitions of the sample we present approximation methods which evaluate the posterior distribution and bayes estimators by gibbs sampling relying on the missing data structure of the mixture model the data augmentation method is shown to converge geometrically since a duality principle transfers properties from the discrete missing data chain to the parameters the fully conditional gibbs alternative is shown to be ergodic and geometric convergence is established in the normal case we also consider non informative approximations associated with improper priors assuming that the sample corresponds exactly to a k component mixture

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

a bayesian semiparametric item response model with dirichlet process priors in item response theory irt item characteristic curves iccs are illustrated through logistic models or normal ogive models and the probability that examinees give the correct answer is usually a monotonically increasing function of their ability parameters however since only limited patterns of shapes can be obtained from logistic models or normal ogive models there is a possibility that the model applied does not fit the data as a result the existing method can be rejected because it cannot deal with various item response patterns to overcome these problems we propose a new semiparametric irt model using a dirichlet process mixture logistic distribution our method does not rely on assumptions but only requires that the iccs be a monotonically nondecreasing function that is our method can deal with more types of item response patterns than the existing methods such as the one parameter normal ogive models or the two or three parameter logistic models we conducted two simulation studies whose results indicate that the proposed method can express more patterns of shapes for iccs and can estimate the ability parameters more accurately than the existing parametric and nonparametric methods the proposed method has also been applied to facial expression recognition data with noteworthy results