Mixed strategy may outperform pure strategy: An initial study

In pure strategy meta-heuristics, only one search strategy is applied for all time. In mixed strategy meta-heuristics, each time one search strategy is chosen from a strategy pool with a probability and then is applied. An example is classical genetic algorithms, where either a mutation or crossover operator is chosen with a probability each time. The aim of this paper is to compare the performance between mixed strategy and pure strategy meta-heuristic algorithms. First an experimental study is implemented and results demonstrate that mixed strategy evolutionary algorithms may outperform pure strategy evolutionary algorithms on the 0-1 knapsack problem in up to 77.8% instances. Then Complementary Strategy Theorem is rigorously proven for applying mixed strategy at the population level. The theorem asserts that given two meta-heuristic algorithms where one uses pure strategy 1 and another uses pure strategy 2, the condition of pure strategy 2 being complementary to pure strategy 1 is sufficient and necessary if there exists a mixed strategy meta-heuristics derived from these two pure strategies and its expected number of generations to find an optimal solution is no more than that of using pure strategy 1 for any initial population, and less than that of using pure strategy 1 for some initial population

Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps)

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Logit dynamics for strategic games mixing time and metastability

2010 - 2011A complex system is generally de_ned as a system emerging from the interaction of several and di_erent components, each one with their properties and their goals, usually subject to external inuences. Nowadays, complex systems are ubiquitous and they are found in many research areas: examples can be found in Economy (e.g., markets), Physics (e.g., ideal gases, spin systems), Biology (e.g., evolution of life) and Computer Science (e.g., Internet and social networks). Modeling complex systems, understanding how they evolve and predicting the future status of a complex system are major research endeavors. Historically, physicists, economists, sociologists and biologists have separately studied complex systems, developing their own tools that, however, often are not suitable for being adopted in di_erent areas. Recently, the close relation between phenomena in di_erent research areas has been highlighted. Hence, the aim is to have a powerful tool that is able to give us insight both about Nature and about Society, an universal language spoken both in natural and in social sciences, a modern code of nature. In a recent book [16], Tom Siegfried pointed out game theory as such a powerful tool, able to embrace complex systems in Economics [3, 4, 5], Biology [13], Physics [8], Computer Science [10, 11], Sociology [12] and many other disciplines. Game theory deals with sel_sh agents or players, each with a set of possible actions or strategies. An agent chooses a strategy evaluating her utility or payo_ that does not depend only on agent's own strategy, but also on the strategies played by the other players. The way players update their strategies in response to changes generated by other players de_nes the dynamics of the game and describes how the game evolves. If the game eventually reaches a _xed point, i.e., a state stable under the dynamics considered, then it is said that the game is in an equilibrium, through which we can make predictions about the future status of a game. The classical game theory approach assumes that players have complete knowledge about the game and they are always able to select the strategy that maximizes their utility: in this rational setting, the evolution of a system is modeled by best response dynamics and predictions can be done by looking at well-known Nash equilibrium. Another approach is followed by learning dynamics: here, players are supposed to \learn" how to play in the next rounds by analyzing the history of previous plays. By examining the features and the drawbacks of these dynamics, we can detect the basic requirements to model the evolution of complex systems and to predict their future status. Usually, in these systems, environmental factors can inuence the way each agent selects her own strategy: for example, the temperature and the pressure play a fundamental role in the dynamics of particle systems, whereas the limited computational power is the main inuence in computer and social settings. Moreover, as already pointed by Harsanyi and Selten [9], the complete knowledge assumption can fail due to limited information about external factors that could inuence the game (e.g., if it will rain tomorrow), or about the attitude of other players (if they are risk taking), or about the amount of knowledge available to other players. Equilibria are usually used to make predictions about the future status of a game: for this reason, we like that an equilibrium always exists and that the game converges to it. Moreover, in case that multiple equilibria exist, we like to know which equilibrium will be selected, otherwise we could make wrong predictions. Finally, if the dynamics takes too long time to reach its _xed point, then this equilibrium cannot be taken to describe the state of the players, unless we are willing to wait super-polynomially long transient time. Thus we would like to have dynamics that models bounded rationality and induces an equilibrium that always exists, it is unique and is quickly reached. Logit dynamics, introduced by Blume [6], models a noisy-rational behavior in a clean and tractable way. In the logit dynamics for a game, at each time step, a player is randomly selected for strategy update and the update is performed with respect to an inverse noise parameter _ (that represents the degree of rationality or knowledge) and of the state of the system, that is the strategies currently played by the players. Intuitively, a low value of _ represents the situation where players choose their strategies \nearly at random" because they are subject to strong noise or they have very limited knowledge of the game; instead, an high value of _ represents the situation where players \almost surely" play the best response, that is, they pick the strategies yielding high payo_ with higher probability. This model is similar to the one used by physicists to describe particle systems, where the behavior of each particle is inuenced by temperature: here, low temperature means high rationality and high temperature means low rationality. It is well known [6] that this dynamics de_nes an ergodic _nite Markov chain over the set of strategy pro_les of the game, and thus it is known that a stationary distribution always exists, it is unique and the chain converges to such distribution, independently of the starting pro_le. Since the logit dynamics models bounded rationality in a clean and tractable way, several works have been devoted to this subject. Early works about this dynamics have focused about long-term behavior of the dynamics: Blume [6] showed that, for 2 _ 2 coordination games and potential games, the long-term behavior of the system is concentrated in a speci_c Nash equilibrium; Al_os-Ferrer and Netzer [1] gave a general characterization of long term behavior of logit dynamics for wider classes of games. A lot of works have been devoted to evaluating the time that the dynamics takes to reach speci_c Nash equilibria of a game, called hitting time: Ellison [7] considered logit dynamics for graphical coordination games on cliques and rings; Peyton Young [15] extended this work for more general families of graphs; Montanari and Saberi [14] gave the exact graph theoretic property of the underlying interaction network that characterizes the hitting time in graphical coordination games; Asadpour and Saberi [2] studied the hitting time for a class of congestion games. Our approach is di_erent: indeed, our _rst contribution is to propose the stationary distribution of the logit dynamics Markov chain as a new equilibrium concept in game theory. Our new solution concept, sometimes called logit equilibrium, always exists, it is unique and the game converges to it from any starting point. Instead, previous works only take in account the classical equilibrium concept of Nash equilibrium, that it is known to not satisfying all the requested properties. Moreover, the approach of previous works forces to consider only speci_c values of the rationality parameter, whereas we are interested to analyze the behavior of the system for each value of _. In order to validate the logit equilibrium concept we follow two di_erent lines of research: from one hand we evaluate the performance of a system when it reaches this equilibrium; on the other hand we look for bounds to the time that the dynamics takes to reach this equi- librium, namely the mixing time. This approach is trained on some simple but interesting games, such as 2_2 coordination games, congestion games and two team games (i.e., games where every player has the same utility). Then, we give bounds to the convergence time of the logit dynamics for very interesting classes of games, such as potential games, games with dominant strategies and graphical coordination games. Speci_cally, we prove a twofold behavior of the mixing time: there are games for which it exponentially depends on _, whereas for other games there exists a function independent of _ such that the mixing time is always bounded by this function. Unfortunately, we show also that there are games where the mixing time can be exponential in the number of players. When the mixing is slow, in order to describe the future status of the system through the logit equilibrium, we need to wait a long transient phase. But in this case, it is natural to ask if we can make predictions about the future status of the game even if the equilibrium has not been reached yet. In order to answer this question we introduce the concept of metastable distribution, a probability distribution such that the dynamics quickly reaches it and spends a lot of time therein: we show that there are graphical coordination games where there are some distributions such that for almost every starting pro_le the logit dynamics rapidly converges to one of these distributions and remains close to it for an huge number of steps. In this way, even if the logit equilibrium is no longer a meaningful description of the future status of a game, the metastable distributions resort the predictive power of the logit dynamics. References [1] Carlos Al_os-Ferrer and Nick Netzer. The logit-response dynamics. Games and Economic Behavior, 68(2):413 { 427, 2010. [2] Arash Asadpour and Amin Saberi. On the ine_ciency ratio of stable equilibria in congestion games. In Proc. of the 5th International Workshop on Internet and Network Economics (WINE'09), volume 5929 of Lecture Notes in Computer Science, pages 545{ 552. Springer, 2009. [3] Robert J. Aumann and S. Hart, editors. Handbook of Game Theory with Economic Applications, volume 1. Elsevier, 1992. [4] Robert J. Aumann and S. Hart, editors. Handbook of Game Theory with Economic Applications, volume 2. Elsevier, 1994. [5] Robert J. Aumann and S. Hart, editors. Handbook of Game Theory with Economic Applications, volume 3. Elsevier, 2002. [6] Lawrence E. Blume. The statistical mechanics of strategic interaction. Games and Economic Behavior, 5:387{424, 1993. [7] Glenn Ellison. Learning, local interaction, and coordination. Econometrica, 61(5):1047{ 1071, 1993. [8] Serge Galam and Bernard Walliser. Ising model versus normal form game. Physica A: Statistical Mechanics and its Applications, 389(3):481 { 489, 2010. [9] John C. Harsanyi and Reinhard Selten. A General Theory of Equilibrium Selection in Games. MIT Press, 1988. [10] Elias Koutsoupias and Christos H. Papadimitriou. Worst-case equilibria. Computer Science Review, 3(2):65{69, 2009. Preliminary version in STACS 1999. [11] Hagay Levin, Michael Schapira, and Aviv Zohar. Interdomain routing and games. In STOC, pages 57{66, 2008. [12] Jan Lorenz, Heiko Rauhut, Frank Schweitzer, and Dirk Helbing. How social inuence can undermine the wisdom of crowd e_ect. Proceedings of the National Academy of Sciences, 108(22):9020{9025, 2011. [13] John Maynard Smith. Evolution and the theory of games. Cambridge University Press, 1982. [14] Andrea Montanari and Amin Saberi. Convergence to equilibrium in local interaction games. In Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09). IEEE, 2009. [15] Hobart Peyton Young. The di_usion of innovations in social networks, chapter in \The Economy as a Complex Evolving System", vol. III, Lawrence E. Blume and Steven N. Durlauf, eds. Oxford University Press, 2003. [16] Tom Siegfried. A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature. Joseph Henry Press, 1st ed edition, 2006. [edited by author]X n.s

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Selektionsintensität und Fixierungszeiten in evolutionären Systemen

Processes of Darwinian evolution are dynamic, nonlinear, and underly fluctuations. A way to analyze systems of Darwinian evolution is by using methods well established in statistical physics. The main mechanisms that are responsible for evolutionary changes are reproduction, mutation, and selection. Individuals reproduce and inherit genes and traits, such that a population evolves. Mutations occur spontaneously, e.g., by errors in reproduction, whereby different new types of genes or traits can emerge. Selection acts on different types. This thesis focuses on selection in systems that underlie the principles of Darwinian evolution, as well as fluctuations. Once there are different types, their interactions with each other can influence their reproductive rates. One important framework to look at such interactions is game theory. In evolutionary game theory, different types are identified with different strategies, and the payoff of a strategy affects the reproductive success. An important property of evolutionary games is that, in general, the evolutionary success of a strategy varies with the composition of the population. The event of a mutation taking over a population is called fixation. The quantities mainly considered in this thesis are the fixation times of a mutant strategy. They are a measure for the time a population spends reaching the state of only mutants, when starting from a few. The role of selection is to control the payoff differences between types, which gives rise to several regimes of selection. In the absence of selection evolution is neutral and fluctuations dominate. An important limit case is weak selection, which introduces a small bias to the random evolutionary changes. In this thesis, weak selection analysis plays an important part in the classification of different evolutionary processes. This allows to simplify the nonlinear dynamical system and thus an analytical description. Here, approximative formulations of the fixation times under weak selection are presented, and the universality of the weak selection regime is addressed. On intermediate scales, one can observe that the average fixation time of an advantageous mutation increases with selection, although the probability of fixation also increases. One can then move on to strong selection, such that selection dominates the dynamics even in small systems. Here, one can observe segregation effects, where the initial condition determines the fate of the finite population in a deterministic way. Another important evolutionary mechanism is gene flow, e.g., caused by migration between populations of the same species. In this context, migration can counterbalance selection. In systems with bi-stable evolutionary dynamics, the migration-selection equilibrium can lead to coexistence that is stable for a long time. This thesis gives a quantitative analysis of the dynamical and statistical properties of such a system. To this end, the extinction (fixation) times are analyzed also in the nonlinearly coupled population system.Prozesse Darwinscher Evolution sind dynamisch, nichtlinear, und unterliegen Fluktuationen. Systeme Darwinscher Evolution können mit wohl etablierten Methoden der statistischen Physik analysiert werden. Die für evolutionäre Veränderungen wesentlichen Mechanismen sind Reproduktion, Mutation und Selektion. Individuen reproduzieren sich und vererben Gene und Merkmale, so dass die Population evolviert. Mutationen treten spontan auf, z.B. durch Fehler in der Reproduktion, wodurch verschiedene neue Typen von Genen oder Merkmalen entstehen können. Selektion wirkt auf verschiedene Typen. Diese Arbeit konzentriert sich auf Selektion in Systemen, welche den Prinzipien Darwinscher Evolution, sowie Fluktuationen unterliegen. Die Wechselwirkungen verschiedener Typen untereinander können die jeweiligen reproduktiven Raten beeinflussen. Eine wichtige Disziplin, welche solche Wechselwirkungen betrachtet, ist die Spieltheorie. In der evolutionären Spieltheorie identifiziert man verschiedene Typen mit verschiedenen Strategien. Der (spieltheoretische) Erfolg einer Strategie beeinflusst deren reproduktiven Erfolg. Eine wichtige Eigenschaft evolutionärer Spiele ist, dass der evolutionäre Erfolg einer Strategie im Allgemeinen mit der Zusammensetzung der Population variiert. Der Begriff Fixierung bezeichnet das Ereignis der Übernahme einer Population durch eine Mutation. Hauptsächlich werden in dieser Arbeit die Fixierungszeiten einer mutierten Strategie betrachtet. Sie sind ein Maß für die Zeit, die eine Population benötigt, um von einem Zustand mit nur wenigen zu einem Zustand mit ausschließlich Mutanten zu gelangen. Selektion kontrolliert die Erfolgsdifferenz zwischen Typen. Dies ermöglicht die Definition verschiedener Regime der Selektion. Ohne Selektion ist Evolution neutral und Fluktuationen dominieren. Ein wichtiger Grenzfall ist schwache Selektion, welche eine gerichtete Veränderung zu diesen zufälligen evolutionären Veränderungen hinzufügt. In dieser Arbeit spielt die Analyse der schwachen Selektion eine bedeutende Rolle in der Klassifikation verschiedener evolutionärer Prozesse. Sie erlaubt eine Vereinfachung der nichtlinearen Systeme und damit eine analytische Beschreibung. Es werden approximative Formulierungen der Fixierungszeiten unter schwacher Selektion präsentiert und die Universalität dieses Grenzfalls betrachtet. Auf Zwischenskalen kann man beobachten, dass die Fixierungszeit einer vorteilhaften Mutation mit der Selektion ansteigt, obwohl die entsprechende Fixierungswahrscheinlichkeit ebenso größer wird. Davon ausgehend kann man zur Betrachtung starker Selektion übergehen, so dass Selektion die Dynamik auch in kleinen Systemen dominiert. Hierbei lassen sich Segregationseffekte beobachten: Das Schicksal der Population ist deterministisch durch die Anfangsbedingung bestimmt. Ein weiterer wichtiger Mechanismus der Evolution ist der Genfluss, welcher z.B.~durch Migration zwischen Population der selben Art erzeugt wird. In diesem Zusammenhang kann Migration der Selektion entgegenwirken. In Systemen bistabiler evolutionärer Dynamik kann solch ein Migrations-Selektionsgleichgewicht zu lang stabiler Koexistenz führen. Die vorliegenden Arbeit gibt hier eine quantitative Analyse der dynamischen und statistischen Eigenschaften. Zu diesem Zweck werden die Austerbe- oder Fixierungszeiten des nichtlinear gekoppelten Populationssystems analysiert

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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A random journey through dynamics and finance: pullback attractors, price impact, nonlinear valuation and FX market.

The main objective of this thesis is to explore several areas of Random Dynamical Systems and Mathematical Finance. We start by considering random dynamical systems with two different sources of noise, which we call common and intrinsic. We study the interplay between these two sources of randomness from a novel point of view, going beyond the usual statistical approach. We determine the stochastic Fokker-Planck equation describing the system and prove that such equation has a pullback attractor for almost all realizations of the common noise. On the mathematical finance side, we start by discussing consistency properties of jump-diffusion models with respect to inversion, with applications to the Foreign Exchange market. We first solve the constant jump size case, and then analyze the more involved case of the compound Poisson process. We determine a fairly general class of admissible densities for the jump size in the domestic measure. Then, we delve into the nonlinear valuation framework under credit risk, collateral and funding costs, generalizing the mathematical framework of \cite{brigo2019nonlinear} for what concerns in particular the filtrations and the default times. Finally, we propose a first theory of price impact in presence of an interest-rates term structure. We formulate an instantaneous and transient price impact model for zero-coupon bond, defining a cross price impact that is endogenous to the term structure. We extend this setup to coupon-bearing bonds, HJM framework and conclude by solving an optimal execution problem in interest rates market.Open Acces