uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

probability inequalities for decomposition integrals recently in mathematical economics even and lehrer introduced the decomposition integral even and lehrer 2014 in this paper general versions of some well known probabilistic inequalities for the decomposition integrals and the superdecomposition integrals are discussed that are still open for research the main results of this paper generalize some previous results for particular integral inequalities obtained by several researchers in generalized probability theory

optimal investments for robust utility functionals in complete market models this paper introduces a systematic approach to the problem of maximizing the robust utility of the terminal wealth of an admissible strategy in a general complete market model where the robust utility functional is defined by a set q of probability measures our main result shows that this problem can often be reduced to determining a least favorable measure q0 q which is universal in the sense that it does not depend on the particular utility function the robust problem is thus equivalent to a standard utility maximization problem with respect to the subjective probability measure q0 by using the huber strassen theorem from robust statistics it is shown that q0 always exists if q is the i core of a 2 alternating capacity besides other examples we also discuss the problem of robust utility maximization with uncertain drift in a black scholes market and the case of weak information

optimal stopping under uncertainty in drift and jump intensity this paper studies the optimal stopping problem in the presence of model uncertainty ambiguity we develop a numerically implementable method to solve this problem in a general setting allowing

record statistics for biased random walks with an application to financial data we consider the occurrence of record breaking events in random walks with asymmetric jump distributions the statistics of records in symmetric random walks was previously analyzed by majumdar and ziff phys rev lett 101 050601 2008 and is well understood unlike the case of symmetric jump distributions in the asymmetric case the statistics of records depends on the choice of the jump distribution we compute the record rate p n c defined as the probability for the nth value to be larger than all previous values for a gaussian jump distribution with standard deviation that is shifted by a constant drift c for small drift in the sense of c n 1 2 the correction to p n c grows proportional to arctan n and saturates at the value c 2 for large n the record rate approaches a constant which is approximately given by 1 2 c exp c 2 2 2 for c 1 these asymptotic results carry over to other continuous jump distributions with finite variance as an application we compare our analytical results to the record statistics of 366 daily stock prices from the standard poor s 500 index the biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices and after detrending the number of upper records is in good agreement with the symmetric random walk however the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified

stochastic entropy production in diffusive systems computing the stochastic entropy production associated with the evolution of a stochastic dynamical system is a well established problem in a small number of cases such as the ornstein uhlenbeck process of which we give a complete exposition the distribution of entropy production can be obtained analytically for a general potential it is much harder a recent development in solving the fokker planck equation in which the solution is written as a product of positive functions addresses any system governed by the condition of detailed balance thereby permitting nonlinear potentials using examples in one and higher dimension we demonstrate how such a framework is very convenient for the computation of stochastic entropy production in diffusion processes

vector expected utility and attitudes toward variation this paper proposes a model of decision under ambiguity deemed vector expected utility or veu in this model an uncertain prospect or savage act is assessed according to a a baseline expected utility evaluation a nd b anadjustment that reflects the individual s perception of ambiguity and her attitudes toward it the adjustment is itself a function of the act s exposure to distinct sources of ambiguity as well as its variability the key elements of the veu model are a baseline probability and a collection of random variables or adjustment factors which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events the adjustment to the baseline expected utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor which reflects exposure to the corresponding ambiguity source a behavioral characterization of the veu model is provided furthermore an updating rule for veu preferences is proposed and characterized the suggested updating rule facilitates the analysis of sophisticated dynamic choice with veu preferences

record statistics and persistence for a random walk with a drift we study the statistics of records of a one dimensional random walk of n steps starting from the origin and in presence of a constant bias c at each time step the walker makes a random jump of length eta drawn from a continuous distribution f eta which is symmetric around a constant drift c we focus in particular on the case were f eta is a symmetric stable law with a levy index 0 mu leq 2 the record statistics depends crucially on the persistence probability which as we show here exhibits different behaviors depending on the sign of c and the value of the parameter mu hence in the limit of a large number of steps n the record statistics is sensitive to these parameters c and mu of the jump distribution we compute the asymptotic mean record number after n steps as well as its full distribution p r n we also compute the statistics of the ages of the longest and the shortest lasting record our exact computations show the existence of five distinct regions in the c 0 mu leq 2 strip where these quantities display qualitatively different behaviors we also present numerical simulation results that verify our analytical predictions

nonlinear functionals on c x and choquet integrals abstract we establish a retraction from monotone functionals defined on c x where x is a compact hausdorff space with pointwise convergence topology to functionals represented by choquet integrals with respect to monotone measures we prove an analogue of the riesz representation theorem in the case of choquet integrals

on the determination of strength of belief for decision support under uncertainty part ii fusing strengths of belief the problem of decision making under uncertainty is described and the difficulty of comparing actions with uncertain outcomes is noted we discuss the role of valuation functions which convert an uncertain outcome into a single representative value and hence aid in the comparison of uncertain alternatives the need for information about the strength of belief that a given outcome will occur is pointed out we then look at various methods for obtaining this information we consider in particular detail the situation in which the strength of belief is dependent on the outcome payoff

decision making on the sole basis of statistical likelihood this paper presents a new axiomatic decision theory for choice under uncertainty unlike bayesian decision theory where uncertainty is represented by a probability function in our theory uncertainty is given in the form of a likelihood function extracted from statistical evidence the likelihood principle in statistics stipulates that likelihood functions encode all relevant information obtainable from experimental data in particular we do not assume any knowledge of prior probabilities consequently a bayesian conversion of likelihoods to posterior probabilities is not possible in our setting we make an assumption that defines the likelihood of a set of hypotheses as the maximum likelihood over the elements of the set we justify an axiomatic system similar to that used by von neumann and morgenstern for choice under risk our main result is a representation theorem using the new concept of binary utility we also discuss how ambiguity attitudes are handled applied to the statistical inference problem our theory suggests a novel solution the results in this paper could be useful for probabilistic model selection

fuzzy integral representation abstract in this paper we give representation results for nonlinear functionals in arbitrary spaces as fuzzy integrals the main assumption is that of comonotonic maxitivity i e i f g if ig for comonotonic f g under the more restrictive condition of stochastic dominance we prove a more specific representation as a fuzzy integral with respect to a distorted probability measure we discuss the particular cases of possibility measures and upper fuzzy expectations some partial randon nikodym results are proved for both the choquet and the fuzzy integrals finally an example is shown where the fuzzy integral representation is useful

choquet integration on riesz spaces and dual comonotonicity we give a general integral representation theorem theorem 6 for nonadditive functionals de ned on an archimedean riesz space x with order unit additivity is replaced by a weak form of modularity or equivalently dual comonotonic additivity and integrals are choquet integrals those integrals are de ned through the kakutani 8 isometric identi cation of x with a c k space we further show that our novel notion of dual comonotonicity naturally generalizes and characterizes the notions of comonotonicity found in the literature when x is assumed to be a space of functions

sugeno integral based on absolutely monotone real set functions the class of all absolutely monotone and sign stable set functions with m a 0 denoted by amss is introduced necessary and sufficient conditions are obtained for a set function m m a 0 to be a member of amss a sugeno type integral of an a measurable real valued function is defined with respect to an absolutely monotone and sign stable set function and some of its properties are shown a representation of a comonotone cosigned additive functional by sugeno integral based on m amss is obtained

pari mutuel probabilities as an uncertainty model abstract the pari mutuel model is a betting scheme that has its origins in horse racing and that has been applied in a number of contexts mostly economics in this paper we consider the set of probability measures compatible with a pari mutuel model characterize its extreme points and investigate the properties of the associated lower and upper probabilities we show that the pari mutuel model can be embedded within the theory of probability intervals and prove necessary and sufficient conditions for it to be a belief function or a minitive measure in addition we also investigate the combination of different pari mutuel models and their definition on product spaces

reformulating decision theory using fuzzy set theory and shafer s theory of evidence abstract utilities and probabilities in decision theory are usually assessed by asking individuals to indicate their preferences between various uncertain choices in this paper we argue that 1 the utility of a consequence can be assessed as the membership function of the consequence in the fuzzy set satisfactory 2 the probability of an event instead of being directly assessed should be inferred from the evidence associated with that event the degree of evidence is quantified using shaferian basic probability assignments in addition we use the heisenberg uncertainty principle to argue for a change in one of the technical assumptions underlying decision theory as a result of this change some kinds of evidence will be observable in certain experiments but unobservable in others since probabilities are defined over the potential outcomes of an experiment they will only be defined over some but not all the evidence as a result the probabilities associated with different experiments could be inconsistent this formulation emphasizes the importance of new distinctions and not just new information in updating probabilities we argue that this formulation addresses many of the observed empirical deviations between decision theory and experiment it also addresses the anomalies of quantum physics we close with a brief discussion of directions for further research

on the continuity of the concave integral whenever a functional is concave it is natural to ask whether its sendograph is a closed convex set if so the hahn banach theory implies that the functional can be represented as the infimum of all continuous linear functionals greater than or equal to it we refer to such representation as a dual representation dominated convergence of the concave integral for capacities is characterized in terms of dual representation whenever sequences of functions converge pointwise outside a set of zero capacity

tight approximations of dynamic risk measures this paper compares two frameworks for measuring risk in a multiperiod setting the first corresponds to applying a single coherent risk measure to the cumulative future costs and the second involves applying a composition of one step coherent risk mappings we characterize several necessary and sufficient conditions under which one measurement always dominates the other and introduce a metric to quantify how close the two measures are using this notion we address the question of how tightly a given coherent measure can be approximated by lower or upper bounding compositional measures we exhibit an interesting asymmetry between the two cases the tightest upper bound can be exactly characterized and corresponds to a popular construction in the literature whereas the tightest lower bound is not readily available we show that testing domination and computing the approximation factors are generally np hard even when the risk measures are comonotonic and law invariant however we characterize conditio

a way to choquet calculus in this paper we deal with the choquet integral and derivative with respect to fuzzy measures on the nonnegative real line and present a way to choquet calculus as a new research paradigm in choquet calculus a representation for calculating the continuous choquet integral is first given by restricting the integrand to a class of nondecreasing and continuous functions and the fuzzy measure to a class of distorted lebesgue measures next the derivative of functions with respect to distorted lebesgue measures is defined as the inverse operation of the choquet integral then elementary properties in choquet calculus are explored in addition we clarify the relation of choquet calculus with fractional calculus where the fractional choquet integral and derivative are newly defined in addition we consider differential equations with respect to distorted lebesgue measures and give their solutions finally we introduce conditional distorted lebesgue measures and explore their properties

qualitative decision theory from savage s axioms to nonmonotonic reasoning this paper investigates to what extent a purely symbolic approach to decision making under uncertainty is possible in the scope of artificial intelligence contrary to classical approaches to decision theory we try to rank acts without resorting to any numerical representation of utility or uncertainty and without using any scale on which both uncertainty and preference could be mapped our approach is a variant of savage s where the setting is finite and the strict preference on acts is a partial order it is shown that although many axioms of savage theory are preserved and despite the intuitive appeal of the ordinal method for constructing a preference over acts the approach is inconsistent with a probabilistic representation of uncertainty the latter leads to the kind of paradoxes encountered in the theory of voting it is shown that the assumption of ordinal invariance enforces a qualitative decision procedure that presupposes a comparative possibility representation of uncertainty originally due to lewis and usual in nonmonotonic reasoning our axiomatic investigation thus provides decision theoretic foundations to the preferential inference of lehmann and colleagues however the obtained decision rules are sometimes either not very decisive or may lead to overconfident decisions although their basic principles look sound this paper points out some limitations of purely ordinal approaches to savage like decision making under uncertainty in perfect analogy with similar difficulties in voting theory

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

on the integral representation of g expectations with terminal constraints abstract in this paper we study the integral representation of g expectations with two kinds of terminal constraints and obtain the corresponding necessary and sufficient conditions

what common ground exists for descriptive prescriptive and normative utility theories descriptive and normative modeling of decision making under risk and uncertainty have grown apart over the past decade psychological models attempt to accommodate the numerous violations of rationality axioms including independence and transitivity meanwhile normatively oriented decision analysts continue to insist on the applied usefulness of the subjective expected utility seu model as this gap has widened two facts have remained largely unobserved first most people in real situations attempt to behave in accord with the most basic rationality principles even though they are likely to fail in more complex situations second the seu model is likely to provide consistent and rational answers to decision problems within a given problem structure but may not be invariant across structures thus people may be more rational than the psychological literature gives them credit for and applications of the seu model may be susceptible to some violations of invariance principles this paper attempts

statistical mechanics of topological phase transitions in networks we provide a phenomenological theory for topological transitions in restructuring networks in this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges the associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge dual graph of a fully connected network in our studies based on an exact enumeration method monte carlo simulations and theoretical considerations we find a rich variety of topological phase transitions when the temperature is varied these transitions signal singular changes in the essential features of the global structure of the network depending on the energy function chosen the observed transitions can be best monitored using the order parameters f s5s max m i e the size of the largest connected component divided by the number of edges or f k5k max m the largest degree in the network divided by the number of edges if for example the energy is chosen to be e52s max the observed transition is analogous to the percolation phase transition of random graphs for this choice of the energy the phase diagram in the k t plane is constructed single vertex energies of the form e5 i f k i where k i is the degree of vertex i are also studied depending on the form of f ki first order and continuous phase transitions can be observed in case of f k i 52 k i1a ln ki the transition is continuous and at the critical temperature scale free graphs can be recovered finally by abruptly decreasing the temperature nonequilibrium processes e g nucleation and growth of particular topological phases can also be interpreted by the present approach

time since maximum of brownian motion and asymmetric levy processes motivated by recent studies of record statistics in relation to strongly correlated time series we consider explicitly the drawdown time of a levy process which is defined as the time since it last achieved its running maximum when observed over a fixed time period we show that the density function of this drawdown time in the case of a completely asymmetric jump process may be factored as a function of t multiplied by a function of t t this extends a known result for the case of pure brownian motion we state the factors explicitly for the cases of exponential down jumps with drift and for the downward inverse gaussian levy process with drift

a note on cc integral abstract recently choquet like copula based aggregation functions were introduced and discussed by lucca et al afterwards cmin integral i e a choquet like minimum based operator was examined by dimuro et al we show the link of these aggregation functions with universal integrals of klement et al in particular cmin integral is shown to be an alternative approach to the sugeno integral assuming symmetry an axiomatic characterization of the discussed aggregation functions is given and their coincidence with ordered modular averages in short oma operators is shown

two axiomatic approaches to decision making using possibility theory this paper proposes a utility theory for decision making under uncertainty that is described by possibility theory we show that our approach is a natural generalization of the two axiomatic systems that correspond to pessimistic and optimistic decision criteria proposed by dubois et al the generalization is achieved by removing axioms that are supposed to reflect attitudes toward uncertainty namely pessimism and optimism in their place we adopt an axiom that imposes an order on a class of canonical lotteries that realize either in the best or in the worst prize we prove an expected utility theorem for the generalized axiomatic system based on the newly introduced concept of binary utility

risk aversion in cumulative prospect theory this paper characterizes the conditions for strong risk aversion and second order stochastic dominance for cumulative prospect theory strong risk aversion implies a convex weighting function for gains and a concave one for losses it does not necessarily imply a concave utility function the latter does follow if the weighting functions are continuous by investigating the exact relationship between loss aversion and strong risk aversion a natural index for the degree of loss aversion is derived

testing theories of financial decision making we describe the observable content of some of the most widely used models of decision under uncertainty models of translation invariant preferences in particular we characterize the models of variational maxmin constant absolute risk aversion and constant relative risk aversion utilities in each case we present a revealed preference axiom that is satisfied by a dataset if and only if the dataset is consistent with the corresponding utility representation we test our axioms using data from an experiment on financial decisions

long and short time asymptotics of the first passage time of the ornstein uhlenbeck and other mean reverting processes the first passage problem of the ornstein uhlenbeck ou process to a boundary is a long standing problem with no known closed form solution except in specifc cases taking this as a starting point and extending to a general mean reverting process we investigate the long and short time asymptotics using a combination of hopf cole and laplace transform techniques as a result we are able to give a single formula that is correct in both limits as well as being exact in certain special cases we demonstrate the results using a variety of other models

on the smooth ambiguity model a reply epstein 2009 describes three ellsberg style thought experiments and argues that they pose difficulties for the smooth ambiguity model of decision making under uncertainty developed by klibanoff marinacci and mukerji 2005 we revisit these thought exeperiments and find to the contrary that they either point to strengths of the smooth ambiguity model compared to other models such as the maximum expected utility model gilboa and schmeidler 1989 or in the case of one thought experiment raise criticisms that apply equally to a broad range of current ambiguity models

the midweight method to measure attitudes toward risk and ambiguity this paper introduces a parameter free method for measuring the weighting functions of prospect theory and rank dependent utility these weighting functions capture risk attitudes subjective beliefs and ambiguity attitudes our method called the midweight method is based on a convenient way to obtain midpoints in the weighting function scale it can be used both for risk known probabilities and for uncertainty unknown probabilities the resulting integrated treatment of risk and uncertainty is particularly useful for measuring ambiguity i e the difference between uncertainty and risk compared to existing methods to measure weighting functions and attitudes toward uncertainty and ambiguity our method is more efficient and can accommodate violations of expected utility under risk an experiment demonstrates the tractability of our method yielding plausible results such as ambiguity aversion for moderate and high likelihoods but ambiguity seeking for low likelihoods as predicted by ellsberg this paper was accepted by george wu decision analysis

uncertainty aversion with second order utilities and probabilities subjective expected utility theory does not distinguish between attitudes toward uncertainty ambiguous probabilities and attitudes toward risk unambiguous probabilities both are explained in terms of nonlinear utility for money rather than properties of events per se hence the decision maker displays the same attitude toward all sources of risk and uncertainty there is ample evidence that real decision makers do not always behave or even wish to behave in this way and instead they often distinguish between risk and uncertainty as in ellsbergs 1961 paradox this paper presents a simple axiomatic model of nonneutral attitudes toward uncertainty and a behavioral test for uncertainty aversion that is applicable even if utility is state dependent the decision maker may display different degrees of aversion toward gambles on different kinds of events e g being systematically more averse toward gambles on events whose probabilities are more ambiguous for such a decision maker the elicitation of preferences among objective gambles may not yield the correct measure of risk aversion for modeling real world decisions

universal statistics of longest lasting records of random walks and levy flights we study the record statistics of random walks after n steps x0 x1 xn with arbitrary symmetric and continuous distribution p of the jumps i xi xi 1 we consider the age of the records i e the time up to which a record survives depending on how the age of the current last record is defined we propose three distinct sequences of ages indexed by i ii iii associated to a given sequence of records we then focus on the longest lasting record which is the longest element among this sequence of ages to characterize the statistics of these longest lasting records we compute i the probability that the record of the longest age is broken at step n denoted by q n which we call the probability of record breaking and ii the duration of the longest lasting record we show that both q n and the full statistics of are universal i e independent of the jump distribution p we compute exactly the large n asymptotic behaviors of q n as well as when it exists and show that each case gives rise to a different universal constant associated to random walks including levy flights while two of them appeared before in the excursion theory of brownian motion for which we provide here a simpler derivation the third case gives rise to a non trivial new constant ciii 0 241 749 associated to the records of random walks other observables characterizing the ages of the records exhibiting an interesting universal behavior are also discussed

non additive interval valued f transform this article proposes a new interval valued fuzzy transform its construction is based on a possibilistic interpretation of the partition on which the fuzzy transform is built the main advantage of this approach is that it provides specific interval valued functions whose interpretation is straightforward this interpretation relates to a traditional sampling reconstruction framework where little is known about the sampling and or reconstructing kernels numerous properties of the proposed approach are proved that could be useful for function analysis and comparison in the experimental section we illustrate some properties of the proposed transform while highlighting interesting features of the obtained framework

choquet based optimisation in multiobjective shortest path and spanning tree problems this paper is devoted to the search of choquet optimal solutions in finite graph problems with multiple objectives the choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives we first present a condition on preferences name hereafter preference for interior points that characterizes preferences favouring compromise solutions a natural attitude in various contexts such as multicriteria optimisation robust optimisation and optimisation with multiple agents within choquet expected utility theory this condition amounts to using a submodular capacity and a convex utility function under these assumptions we focus on the fast determination of choquet optimal paths and spanning trees after investigating the complexity of these problems we introduce a lower bound for the choquet integral computable in polynomial time then we propose different algorithms using this bound either based on a controlled enumeration of solutions ranking approach or an implicit enumeration scheme branch and bound finally we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes

a strong law of large numbers for capacities we consider a totally monotone capacity on a polish space and a sequence of bounded p i i d random variables we show that on a full set any cluster point of empirical averages lies between the lower and the upper choquet integrals of the random variables provided either the random variables or the capacity are continuous

a characterization of monotone unidimensional latent variable models recently the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by ellis and van den wollenberg and by junker we generalize their work with a de finetti like characterization of the distribution of repeated measures x x 1 x 2 that can be represented with mixtures of likelihoods of independent but not identically distributed random variables where the data satisfy a stochastic ordering property with respect to the mixing variable the random variables x j may be arbitrary real valued random variables we show that the distribution of x can be given a monotone unidimensional latent variable representation that is useful in the sense of junker if and only if this distribution satisfies conditional association ca and a vanishing conditional dependence vcd condition which asserts that finite subsets of the variables in x become independent as we condition on a larger and larger segment of the remaining variables in x it is also interesting that the mixture representation is in a certain ordinal sense unique when ca and vcd hold the characterization theorem extends and simplifies the main result of junker and generalizes methods of ellis and van den wollenberg to a much broader class of models exchangeable sequences of binary random variables also satisfy both ca and vcd as do exchangeable sequences arising as location mixtures in the same way that de finetti s theorem provides a path toward justifying standard i i d mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations this theorem justifies one dimensional latent variable components in hierarchical models in terms of our intuitions about positive association and redundancy between observations because these conditions are on the joint distribution of the observable data x they may also be used to construct asymptotically power 1 tests for unidimensional latent variable models

beliefs about beliefs without probabilities this paper constructs a space of states of the world representing the exhaustive uncertainty facing each player in a strategic situation the innovation is that preferences are restricted primarily by regularity conditions and need not conform with subjective expected utility theory the construction employs a hierarchy of preferences rather than of beliefs as in the standard bayesian model the framework is sufficiently general to accommodate uncertainty averse preferences such as exhibited in the ellsberg paradox and to allow common knowledge of expected utility or choquet expected utility to be well defined formally applications include the provision of i foundations for a harsanyi style game of incomplete information and ii a rich framework for the axiomatization of solution concepts for complete information normal form games

measurement analogies comparisons of behavioral and physical measures two examples of behavioral measurement are explored utility theory and a global psychophysical theory of intensity that closely parallel the foundations of classical physical measurement in several ways first the qualitative attribute in question can be manipulated in two independent ways second each method of manipulation is axiomatized and each leads to a measure of the attribute that because they are order preserving must be strictly monotonically related third a law like constraint somewhat akin to the distribution property underlying e g mass measurement links the two types of manipulation fourth given the numerical measures that result from each manipulation the linking law between them can be recast as a functional equation that establishes the connection between the two measures of the same attribute fifth a major difference from most physical measurement is that the resulting measures are themselves mathematical functions of underlying physical variables of money and probability in the utility case and of physical intensity and numerical proportions in the psychophysical case axiomatizing these functions although still problematic appears to lead to interesting results and to limit the degrees of freedom in the representations

entropy of stochastic blockmodel ensembles stochastic blockmodels are generative network models where the vertices are separated into discrete groups and the probability of an edge existing between two vertices is determined solely by their group membership in this paper we derive expressions for the entropy of stochastic blockmodel ensembles we consider several ensemble variants including the traditional model as well as the newly introduced degree corrected version karrer et al phys rev e 83 016107 2011 which imposes a degree sequence on the vertices in addition to the block structure the imposed degree sequence is implemented both as soft constraints where only the expected degrees are imposed and as hard constraints where they are required to be the same on all samples of the ensemble we also consider generalizations to multigraphs and directed graphs we illustrate one of many applications of this measure by directly deriving a log likelihood function from the entropy expression and using it to infer latent block structure in observed data due to the general nature of the ensembles considered the method works well for ensembles with intrinsic degree correlations i e with entropic origin as well as extrinsic degree correlations which go beyond the block structure

characterizing and reasoning about probabilistic and non probabilistic expectation expectation is a central notion in probability theory the notion of expectation also makes sense for other notions of uncertainty we introduce a propositional logic for reasoning about expectation where the semantics depends on the underlying representation of uncertainty we give sound and complete axiomatizations for the logic in the case that the underlying representation is a probability b sets of probability measures c belief functions and d possibility measures we show that this logic is more expressive than the corresponding logic for reasoning about likelihood in the case of sets of probability measures but equi expressive in the case of probability belief and possibility finally we show that satisfiability for these logics is np complete no harder than satisfiability for propositional logic

self organization of collaboration networks we study collaboration networks in terms of evolving self organizing bipartite graph models we propose a model of a growing network which combines preferential edge attachment with the bipartite structure generic for collaboration networks the model depends exclusively on basic properties of the network such as the total number of collaborators and acts of collaboration the mean size of collaborations etc the simplest model defined within this framework already allows us to describe many of the main topological characteristics degree distribution clustering coefficient etc of one mode projections of several real collaboration networks without parameter fitting we explain the observed dependence of the local clustering on degree and the degree degree correlations in terms of the aging of collaborators and their physical impossibility to participate in an unlimited number of collaborations

a soft robust model for optimization under ambiguity in this paper we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set we apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant our primary objective is to develop this framework and relate it to the standard notion of robustness which deals with only a single guarantee across one uncertainty set first we show that our approach connects closely to the theory of convex risk measures we show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems we then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach finally we illustrate the methodology on an asset allocation example consisting of historical market data over a 25 year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk return trade off each case results in a higher out of sample expected return for a relatively minor degradation of out of sample downside performance

local homogeneity in latent trait models a characterization of the homogeneous monotone irt model the stochastic subject formulation of latent trait models contends that within a given subject the event of obtaining a certain response pattern may be probabilistic ordinary latent trait models do not imply that these within subject probabilities are identical to the conditional probabilities specified by the model the latter condition is called local homogeneity it is shown that local homgeneity is equivalent to subpopulation invariance of the model in case of the monotone irt model local homogeneity implies absence of item bias absence of item specific traits and the possibility to join overlapping subtests the following characterization theorem is proved the homogeneous monotone irt model holds for a finite or countable item pool if and only if the pool is experimentally independent and pairwise nonnegative association holds in every positive subpopulation

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

integrals based on monotone set functions an overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set called here monotone measures our survey includes not only the choquet integral 1954 10 the shilkret integral 1971 66 and the sugeno integral 1974 71 and some of their properties but also some more general and more recent concepts as universal integrals klement et al 2010 27 and decomposition integrals even 2014 13 together with some of their properties such as integral inequalities and convergence theorems

symmetric upper probabilities as a first step toward developing statistical models based on upper and lower probabilities we study upper probabilities and upper expectations on the unit interval that are symmetric by which we mean invariant with respect to equimeasurability these upper probabilities are generalizations of uniform probability measures we give some characterizations of these upper probabilities specifically we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization we also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star shaped with respect to the origin and to the point 1 1 we derive inner and outer approximations to symmetric classes of probabilities based on the upper probability the class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized we provide a geometric characterization of a hierarchy of upper probabilities including fine s generalized upper probabilities and 2 alternating choquet capacities in particular we establish a 1 1 correspondence between symmetric 2 alternating capacities and nonincreasing density functions we prove that undominated generalized upper probabilities do not exist in the symmetric case examples from robust statistics are considered an example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations a corollary is that symmetry of the choquet integral does not imply symmetry of the upper expectation functional

derivation of a cardinal utility through a weak trade off consistency requirement an axiomatic model is presented in which a utility function over consequences unique up to location and unit is derived the assumptions apply to a binary relation over purely subjective acts namely no exogenous probabilities are assumed the main assumption used is a weak trade off consistency condition the model generalizes the biseparable model of ghirardato and marinacci ghirardato p marinacci m 2001 risk ambiguity and the separation of utility and beliefs math oper res 26 4 864 890

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

probabilistic sophistication and multiple priors we show that under fairly mild conditions a maximin expected utility preference relation is probabilistically sophisticated if and only if it is subjective expected utility this abstract was borrowed from another version of this item

choquet representability of submodular functions let omega be an arbitrary set equipped with an algebra mathcal a subseteq 2 omega and let f b mathcal a rightarrow mathbb r be a functional defined on the set b mathcal a of bounded measurable functions x omega rightarrow mathbb r we provide necessary and sufficient conditions for a submodular functional f to be representable as a choquet integral from standard properties of the choquet integral the functional f should be positively homogeneous and constant additive our first result shows that these two properties together with submodularity characterize a subadditive choquet integral when omega is finite in the general case f is a subadditive choquet integral if and only if it satisfies the three previous properties together with sup norm continuity this result provides another characterization of subadditive choquet integrals different from the seminal paper by schmeidler proc am math soc 97 2 255 261 1986 that relies on comonotonic additivity

some characterizations of strongly subadditive discounting functions the aim of this paper is to characterize some parameters which measure the force of discount in a discounting function the instantaneous discount rate and the geometric mean of a discounting function and relate them with the strong subadditivity and the so called subadditivity of second order we distinguish two intertemporal choice frameworks depending on the consideration of time as an interval or both as an interval and a delay and we establish some necessary and sufficient conditions to identify strongly subadditive and subadditive of second order discounting functions

record statistics of financial time series and geometric random walks the study of record statistics of correlated series in physics such as random walks is gaining momentum and several analytical results have been obtained in the past few years in this work we study the record statistics of correlated empirical data for which random walk models have relevance we obtain results for the records statistics of select stock market data and the geometric random walk primarily through simulations we show that the distribution of the age of records is a power law with the exponent lying in the range 1 5 1 8 further the longest record ages follow the frechet distribution of extreme value theory the records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data

additivizations of nonadditive measures the concept of an additivization of a nonadditive probability measure is introduced and studied the relation to cooperative game theory is explored and it is shown that in many cases of interest additivizations may be considered as a special type of semivalues

signed integral representations of comonotonic additive functionals we establish integral representation results for suitably pointwise continuous and comonotonic additive functionals of bounded variation de ned on stone lattices as an application we prove a comonotonic version of the daniell stone theorem 2000 mathematics subject classi cation primary 28a12 28a25 46g12 secondary 91b06

unbounded utility for savage s foundations of statistics and other models a general procedure for extending finite dimensional additive like representations for binary relations to infinite dimensional integral like representations is developed by means of a condition called truncation continuity the restriction of boundedness of utility met throughout the literature can now be dispensed with and for instance normal distributions or any other distribution with finite first moment can be incorporated classical representation results of expected utility such as savage 1954 von neumann and morgenstern 1944 anscombe and aumann 1963 de finetti 1937 and many others can now be extended the results are generalized to schmeidler s 1989 approach with nonadditive measures and choquet integrals and quiggin s 1982 rank dependent utility the different approaches have been brought together in this paper to bring to the fore the unity in the extension process

common mathematical foundations of expected utility and dual utility theories we show that the main results of the expected utility and dual utility theories can be derived in a unified way from two fundamental mathematical ideas the separation principle of convex analysis and integral representations of continuous linear functionals from functional analysis our analysis reveals the dual character of utility functions we also derive new integral representations of dual utility models

a discrete choquet integral for ordered systems a model for a choquet integral for arbitrary finite set systems is presented the model includes in particular the classical model on the system of all subsets of a finite set the general model associates canonical non negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system which are then extended to arbitrary valuations on the set system it is shown that the general choquet integral can be computed by a simple monge type algorithm for so called intersection systems which include as a special case weakly union closed families generalizing lovasz classical characterization we give a characterization of the superadditivity of the choquet integral relative to a capacity on a union closed system in terms of an appropriate model of supermodularity of such capacities

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

unified model for network dynamics exhibiting nonextensive statistics we introduce a dynamical network model which unifies a number of network families which are individually known to exhibit q exponential degree distributions the present model dynamics incorporates static nongrowing self organizing networks preferentially growing networks and preferentially rewiring networks further it exhibits a natural random graph limit the proposed model generalizes network dynamics to rewiring and growth modes which depend on internal topology as well as on a metric imposed by the space they are embedded in in all of the networks emerging from the presented model we find q exponential degree distributions over a large parameter space we comment on the parameter dependence of the corresponding entropic index q for the degree distributions and on the behavior of the clustering coefficients and neighboring connectivity distributions

on the confidence preferences model in this paper we study the model of decision under uncertainty consistent with confidence preferences in that model a decision maker held beliefs represented by a fuzzy set of priors and tastes captured by a standard affine utility index on consequences first we find some interesting properties concerning the well known maxmin expected utility model taking into account the point of view of the confidence preferences model further we provide new examples of preferences that capture ambiguity averse attitudes weaker than ambiguity attitudes featured by maxmin expected utility theory finally we discuss the axiomatic foundations for the confidence preferences model with optimistic behavior

what determines the shape of the probability weighting function under uncertainty decision weights are an important component in recent theories of decision making under uncertainty to better explain these decision weights a two stage approach has been proposed first the probability of an event is judged and then this probability is transformed by the probability weighting function known from decision making under risk we extend the two stage approach by allowing the probability weighting function to depend on the type of uncertainty using this more general approach properties of decision weights can be attributed to properties of probability judgments and or to properties of probability weighting we present an empirical study that shows that it is indeed necessary to allow the probability weighting function to be source dependent the analysis includes an examination of properties of the probability weighting function under uncertainty that have not been considered yet

decision making with monotone lower probabilities of infinite order fabrice properties of convex and monotone capacities of infinite order in polish spaces are studied and used to justify the representation of certain situations of imprecise risk imprecisely known probabilities by lower probabilities which are monotone of infinite order decision making with imprecise risk is then modeled and linear utility theory is shown to be generalizable to the case of imprecise risk

equimeasurable rearrangements with capacities in the classical theory of monotone equimeasurable rearrangements of functions equimeasurability i e that two functions have the same distribution is defined relative to a given additive probability measure these rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired however in all of these problems uncertainty refers to the classical bayesian understanding of the term where the idea of ambiguity is absent arguably knightian uncertainty or ambiguity is one of the cornerstones of modern decision theory it is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to the non bayesian or neo bayesian context this paper introduces the idea of a monotone equimeasurable rearrangement in the context of nonadditive probabilities or capacities that satisfy a property that i call strong diffuseness the latter is a strengthening of the usual notion of diffusen

ambiguity in the small and in the large this paper considers local and global multiple prior representations of ambiguity for preferences that are i monotonic ii bernoullian that is admit an affine utility representation when restricted to constant acts and iii locally lipschitz continuous we do not require either certainty independence or uncertainty aversion we show that the set of priors identified by ghirardato maccheroni and marinacci s 2004 unambiguous preference relation can be characterized as a union of clarke differentials we then introduce a behavioral notion of locally better deviation at an act and show that it characterizes the clarke differential of the preference representation at that act these results suggest that the priors identified by these preference statements are directly related to local optimizing behavior

scale free networks from a hamiltonian dynamics contrary to many recent models of growing networks we present a model with fixed number of nodes and links where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced by applying a local rewiring move the network reaches equilibrium states assuming broad degree distributions which have a power law form in an intermediate range of the parameters used interestingly in the same range we find nontrivial hierarchical clustering

complex networks emerging from fluctuating random graphs analytic formula for the hidden variable distribution in analogy to superstatistics which connects boltzmann gibbs statistical mechanics to its generalizations through temperature fluctuations complex networks are constructed from fluctuating erd os r enyi random graphs using a quantum mechanical method the exact analytic formula for the hidden variable distribution is presented which describes the nature of the fluctuations and generates a generic degree distribution through the poisson transformation as an example a static scale free network is discussed and the corresponding hidden variable distribution is found to decay as a power law and to diverge at the origin

choice based elicitation and decomposition of decision weights for gains and losses under uncertainty this paper reports the results of an experimental parameter free elicitation and decomposition of decision weights under uncertainty assuming cumulative prospect theory utility functions were elicited for gains and losses at an individual level using the tradeoff method subsequently decision weights were elicited through certainty equivalents of uncertain two outcome prospects furthermore decision weights were decomposed using observable choice instead of invoking other empirical primitives as in previous experimental studies the choice based elicitation of decision weights allows for a quantitative study of their characteristics and also allows among other things for the examination of the sign dependence hypothesis for observed choice under uncertainty our results confirm concavity of the utility function in the gain domain and bounded subadditivity of decision weights and choice based subjective probabilities we also find evidence for sign dependence of decision weights

some properties and convergence theorems of set valued choquet integrals the paper aims at discussing the set valued choquet integral which is the integral of set valued random variables with respect to capacities we mainly present representation theorems of the set valued random variable by using a sequence of choquet integrable selections and then we investigate some properties of set valued choquet integrals especially subadditive property and inequality of the metric of set valued choquet integrals we also prove fatou s lemmas lebesgue dominated convergence theorem and monotone convergence theorem of set valued choquet integrals under the weaker conditions than that in previous works

statistical mechanics of scale free networks at a critical point complexity without irreversibility based on a rigorous extension of classical statistical mechanics to networks we study a specific microscopic network hamiltonian the form of this hamiltonian is derived from the assumption that individual nodes increase or decrease their utility by linking to nodes with a higher or lower degree than their own we interpret utility as an equivalent to energy in physical systems and discuss the temperature dependence of the emerging networks we observe the existence of a critical temperature tc where total energy utility and network architecture undergo radical changes along this topological transition we obtain ensemble averages of scale free networks with complex hierarchical topology the scale free nature emerges strictly within equilibrium with a clearly defined microcanonical ensemble and the principle of detailed balance fulfilled this provides evidence that complex networks may arise without irreversibility the utility approach establishes a link between classical statistical physics and a wide variety of applications in socioeconomic statistical systems

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

interacting epidemics on overlay networks the interaction between multiple pathogens spreading on networks connecting a given set of nodes presents an ongoing theoretical challenge here we aim to understand such interactions by studying bond percolation of two different processes on overlay networks of arbitrary joint degree distribution we find that an outbreak of a first pathogen providing immunity to another one spreading subsequently on a second network connecting the same set of nodes does so most effectively if the degrees on the two networks are positively correlated in that case the protection is stronger the more heterogeneous the degree distributions of the two networks are if on the other hand the degrees are uncorrelated or negatively correlated increasing heterogeneity reduces the potential of the first process to prevent the second one from reaching epidemic proportions we generalize these results to cases where the edges of the two networks overlap to arbitrary amount or where the immunity granted is only partial if both processes grant immunity to each other we find a wide range of possible situations of coexistence or mutual exclusion depending on the joint degree distribution of the underlying networks and the amount of immunity granted mutually these results generalize the concept of a coexistence threshold and illustrate the impact of large scale network structure on the interaction between multiple spreading agents

choquet calculus using operational matrices abstract in generalized probability theory choquet calculus on a real line including choquet integral equations and differential equations with respect to non additive measures has been recently discussed by sugeno torra and narukawa these studies were based on laplace transformation as a basic tool however in many problems it is not possible to provide laplace transformation on the other hand in many problems differential equations with respect to non additive measures often do not have exact solutions these reasons permit us to propose an alternative approach for numerical choquet calculus based on operational matrices which also readily gives the approximate numerical solution of choquet integral equations and differential equations with respect to non additive measures finally we present some illustrative examples for numerical choquet calculus based on operational matrices which the laplace transformation does not seem to be applicable in these cases