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1
Sautmann, Anja. "Age-Dependent Payoffs and Assortative Matching by Age in a Market with Search". American Economic Journal: Microeconomics 9, nr 2 (1.05.2017): 263–94. http://dx.doi.org/10.1257/mic.20150237.
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This paper considers a matching market with two-sided search and transferable utility where match payoffs depend on age at marriage (time until match) and search is finite. We define and prove existence of equilibrium, and provide sufficient conditions for positive assortative matching that build on restricting the slope and curvature of the marriage payoff function to generate single-peaked preferences in age and therefore convex matching sets. Payoff functions that are incompatible with positive sorting by age include all strictly increasing functions and constant flow payoffs enjoyed for some finite period. (JEL C78, D83, J12)
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Kufel, Tadeusz, Sławomir Plaskacz i Joanna Zwierzchowska. "Strong and Safe Nash Equilibrium in Some Repeated 3-Player Games". Przegląd Statystyczny 65, nr 3 (30.01.2019): 271–95. http://dx.doi.org/10.5604/01.3001.0014.0540.
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The paper examines an infinitely repeated 3-player extension of the Prisoner’s Dilemma game. We consider a 3-player game in the normal form with incomplete information, in which each player has two actions. We assume that the game is symmetric and repeated infinitely many times. At each stage, players make their choices knowing only the average payoffs from previous stages of all the players. A strategy of a player in the repeated game is a function defined on the convex hull of the set of payoffs. Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a strategy profile being resistant to deviations by coalitions. Constructed equilibrium strategies are safe, i.e. the non-deviating player payoff is not smaller than the equilibrium payoff in the stage game, and deviating players’ payoffs do not exceed the nondeviating player payoff more than by a positive constant which can be arbitrary small and chosen by the non-deviating player. Our construction is inspired by Smale’s good strategies described in Smale’s paper (1980), where the repeated Prisoner’s Dilemma was considered. In proofs we use arguments based on approachability and strong approachability type results.
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Nagel, Stefan, i Amiyatosh Purnanandam. "Banks’ Risk Dynamics and Distance to Default". Review of Financial Studies 33, nr 6 (17.10.2019): 2421–67. http://dx.doi.org/10.1093/rfs/hhz125.
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Abstract We adapt structural models of default risk to take into account the special nature of bank assets. The usual assumption of lognormally distributed asset values is not appropriate for banks. Typical bank assets are risky debt claims with concave payoffs. Because of the payoff nonlinearity, bank asset volatility rises following negative shocks to borrower asset values. As a result, standard structural models with constant asset volatility can severely understate banks’ default risk in good times when asset values are high. Additionally, bank equity return volatility is much more sensitive to negative shocks to asset values than in standard structural models.
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Maestri, Lucas. "Bonus Payments versus Efficiency Wages in the Repeated Principal-Agent Model with Subjective Evaluations". American Economic Journal: Microeconomics 4, nr 3 (1.08.2012): 34–56. http://dx.doi.org/10.1257/mic.4.3.34.
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We study an infinitely repeated principal-agent model with subjective evaluations. We compare the surplus in efficiency-wage equilibria and in bonus-payments equilibria. The agent receives a constant wage and is motivated by the threat of dismissal in efficiency-wage equilibria. The agent receives a bonus and quits the relationship after disagreements between his self-evaluation and the principal's performance appraisal in bonus-payments equilibria. We construct a class of equilibria with bonus payments that approach efficiency as patience increases. In contrast, payoffs from efficiency-wage equilibria are bounded away from the Pareto-payoff frontier for any discount factor. (JEL D82, J33, J41)
5
KÖHLER, WOLFGANG R. "UNIQUE EQUILIBRIA IN THE RUBINSTEIN BARGAINING MODEL WHEN THE PAYOFF SET IS NON-CONVEX". International Game Theory Review 08, nr 03 (wrzesień 2006): 469–82. http://dx.doi.org/10.1142/s0219198906001028.
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I give necessary and sufficient conditions on the payoff set that guarantee uniqueness of the equilibrium in the Rubinstein bargaining model. The conditions encompass a class of non-convex or disconnected payoff sets with discontinuous Pareto frontiers. Roughly speaking, the equilibrium is unique if the objective function of the corresponding Nash-bargaining game has a unique maximum. I extend the analysis to games where the time between offers is not constant.
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Lutsenko, Mikhail M. "Generalized Integral Equations for Timing Games". Contributions to Game Theory and Management 16 (2023): 182–91. http://dx.doi.org/10.21638/11701/spbu31.2023.11.
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We consider timing games, the payoff functions of which have additional break lines outside the unit square diagonal. A special case of such games are games with piecewise constant payoff functions. Solving these games is reduced to solving a pair of integral equations for the distribution functions of equalizing strategies. The proposed solution methods can be used in the study of random walks on a segment in a variable environment.
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Braithwaite, R. Scott. "Can Life Expectancy and QALYs Be Improved by a Framework for Deciding Whether to Apply Clinical Guidelines to Patients With Severe Comorbid Disease?" Medical Decision Making 31, nr 4 (10.02.2011): 582–95. http://dx.doi.org/10.1177/0272989x10386117.
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Background: Guidelines with short-term harms and long-term benefits are often applied to chronically ill patients who may not benefit. The payoff time framework has been proposed (i.e., do not apply a guideline if a patient’s life expectancy (LE) is shorter than when a guideline’s cumulative incremental benefits first exceed its cumulative incremental harms), but its health impact is unclear. Objective: To investigate whether the payoff time framework improves LE and/or quality-adjusted life-years (QALY) for chronically ill patients. Methods: I evaluate impact of the payoff time framework on LE and QALYs, assuming (1) high and constant background mortality rate from chronic illness (≥ 10% per year), (2) immediate guideline-related harm with probability < 1, and (3) constant guideline-related benefit that occurs over an extended time. I apply the framework to questions of whether to screen chronically ill 50-year-old women for colorectal cancer using colonoscopy, and whether to advocate intensive glucose control for chronically ill diabetics. Results: If a guideline’s payoff time is greater than a patient’s LE, then withholding that guideline will increase LE and QALYs for that patient. For a 50-year-old chronically ill woman with background mortality > 0.15 per year (corresponding to LE < 6.5 years), withholding CR screening will increase LE. For a diabetic with background mortality > 0.11 per year (corresponding to LE < 9.4 years), withholding CR screening will increase QALYs. Conclusion: The payoff time framework may indicate when withholding a guideline with short-term harms and long-term benefits may increase LE and/or QALY.
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CHATTERJEE, KRISHNENDU, i RUPAK MAJUMDAR. "DISCOUNTING AND AVERAGING IN GAMES ACROSS TIME SCALES". International Journal of Foundations of Computer Science 23, nr 03 (kwiecień 2012): 609–25. http://dx.doi.org/10.1142/s0129054112400308.
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We introduce two-level discounted and mean-payoff games played by two players on a perfect-information stochastic game graph. The upper level game is a discounted or mean-payoff game and the lower level game is a (undiscounted) reachability game. Two-level games model hierarchical and sequential decision making under uncertainty across different time scales. For both discounted and mean-payoff two-level games, we show the existence of pure memoryless optimal strategies for both players and an ordered field property. We show that if there is only one player (Markov decision processes), then the values can be computed in polynomial time. It follows that whether the value of a player is equal to a given rational constant in two-level discounted or mean-payoff games can be decided in NP ∩ coNP . We also give an alternate strategy improvement algorithm to compute the value.
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Szilagyi, Miklos N., i Iren Somogyi. "Agent-Based Simulation of N-Person Games with Crossing Payoff Functions". Complex Systems 17, nr 4 (15.12.2008): 427–39. http://dx.doi.org/10.25088/complexsystems.17.4.427.
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We report on computer simulation experiments using our agent-based simulation tool to model uniform N-person games with crossing payoff functions. We study the case when agents are greedy simpletons who imitate the action of their neighbor that received the highest payoff for its previous action. The individual agents may cooperate with each other for the collective interest or may defect, that is, pursue their selfish interests only. After a certain number of iterations the proportion of cooperators stabilizes to either a constant value or oscillates around such a value. The payoff (reward/penalty) functions are given as two straight lines: one for the cooperators and another for the defectors. The payoff curves are functions of the ratio of cooperators to the total number of agents. Even if the payoff functions are linear, four free parameters determine them. In this investigation only crossing payoff functions are considered. We have investigated the behavior of the agents systematically. The results show that the solutions are nontrivial and in some cases quite irregular. They show drastic changes for the Leader Game in the narrow parameter range of 1.72 ≤ P ≤ 1.75. This behavior is similar to that observed in [1] for the N-Person Chicken Game. Irregular solutions were also found for the Reversed Stag Hunt Game.
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Selten, Reinhard, i Thorsten Chmura. "Stationary Concepts for Experimental 2x2-Games". American Economic Review 98, nr 3 (1.05.2008): 938–66. http://dx.doi.org/10.1257/aer.98.3.938.
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Five stationary concepts for completely mixed 2x2-games are experimentally compared: Nash equilibrium, quantal response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium (Martin J. Osborne and Ariel Rubinstein 1998), and impulse balance equilibrium. Experiments on 12 games, 6 constant sum games, and 6 nonconstant sum games were run with 12 independent subject groups for each constant sum game and 6 independent subject groups for each nonconstant sum game. Each independent subject group consisted of four players 1 and four players 2, interacting anonymously over 200 periods with random matching. The comparison of the five theories shows that the order of performance from best to worst is as follows: impulse balance equilibrium, payoff-sampling equilibrium, action-sampling equilibrium, quantal response equilibrium, Nash equilibrium. (JEL C70, C91)
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Lee, Youngrok, Yehun Kim i Jaesung Lee. "Pricing Various Types of Power Options under Stochastic Volatility". Symmetry 12, nr 11 (20.11.2020): 1911. http://dx.doi.org/10.3390/sym12111911.
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The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in markets in order to provide high leverage strategy. In pricing power options, the classical Black–Scholes model which assumes a constant volatility is simple and easy to handle, but it has a limit in reflecting movements of real financial markets. As the alternatives of constant volatility, we focus on the stochastic volatility, finding more exact prices for power options. In this paper, we use the stochastic volatility model introduced by Schöbel and Zhu to drive the closed-form expressions for the prices of various power options including soft strike options. We also show the sensitivity of power option prices under changes in the values of each parameter by calculating the resulting values obtained from the formulas.
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Romanuke, Vadim. "Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values". Electrical, Control and Communication Engineering 8, nr 1 (1.07.2015): 13–19. http://dx.doi.org/10.1515/ecce-2015-0002.
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Abstract The paper suggests a method of obtaining an approximate solution of the infinite noncooperative game on the unit hypercube. The method is based on sampling uniformly the players’ payoff functions with the constant step along each of the hypercube dimensions. The author states the conditions for a sufficiently accurate sampling and suggests the method of reshaping the multidimensional matrix of the player’s payoff values, being the former player’s payoff function before its sampling, into a matrix with minimally possible number of dimensions, where also maintenance of one-to-one indexing has been provided. Requirements for finite NE-strategy from NE (Nash equilibrium) solution of the finite game as the initial infinite game approximation are given as definitions of the approximate solution consistency. The approximate solution consistency ensures its relative independence upon the sampling step within its minimal neighborhood or the minimally decreased sampling step. The ultimate reshaping of multidimensional matrices of players’ payoff values to the minimal number of dimensions, being equal to the number of players, stimulates shortened computations.
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Bereby-Meyer, Yoella, i Alvin E. Roth. "The Speed of Learning in Noisy Games: Partial Reinforcement and the Sustainability of Cooperation". American Economic Review 96, nr 4 (1.08.2006): 1029–42. http://dx.doi.org/10.1257/aer.96.4.1029.
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In an experiment, players' ability to learn to cooperate in the repeated prisoner's dilemma was substantially diminished when the payoffs were noisy, even though players could monitor one another's past actions perfectly. In contrast, in one-time play against a succession of opponents, noisy payoffs increased cooperation, by slowing the rate at which cooperation decays. These observations are consistent with the robust observation from the psychology literature that partial reinforcement (adding randomness to the link between an action and its consequences while holding expected payoffs constant) slows learning. This effect is magnified in the repeated game: when others are slow to learn to cooperate, the benefits of cooperation are reduced, which further hampers cooperation. These results show that a small change in the payoff environment, which changes the speed of individual learning, can have a large effect on collective behavior. And they show that there may be interesting comparative dynamics that can be derived from careful attention to the fact that at least some economic behavior is learned from experience.
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Romanuke, Vadim. "Finite uniform approximation of zero-sum games defined on a product of staircase-function continuous spaces". Annals of the University of Craiova - Mathematics and Computer Science Series 49, nr 2 (24.12.2022): 270–90. http://dx.doi.org/10.52846/ami.v49i2.1554.
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A method of finite approximation of zero-sum games defined on a product of staircase-function continuous spaces is presented. The method consists in uniformly sampling the player’s pure strategy value set, solving “smaller” matrix games, each defined on a subinterval where the pure strategy value is constant, and stacking their solutions if they are consistent. The stack of the “smaller” matrix game solutions is an approximate solution to the initial staircase game. The (weak) consistency, equivalent to the approxi-mate solution acceptability, is studied by how much the payoff and optimal situation change as the sampling density minimally increases. The consistency is decomposed into the payoff, optimal strategy support cardinality, optimal strategy sampling density, and support probability consistency. The most important parts are the payoff consistency and optimal strategy support cardinality (weak) consistency. However, it is practically reasonable to consider a relaxed payoff consistency, by which the game optimal value change in an appropriate approximation may grow at most by epsilon as the sampling density minimally increases. The weak consistency itself is a relaxation to the consistency, where the minimal decrement of the sampling density is ignored.
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Vadim Romanuke. "Equilibrium stacks for a non-cooperative game defined on a product of staircase-function continuous and finite strategy spaces". Statistics, Optimization & Information Computing 12, nr 1 (27.10.2023): 45–74. http://dx.doi.org/10.19139/soic-2310-5070-1356.
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A method of finite uniform approximation of 3-person games played with staircase-function strategies is presented. A continuous staircase 3-person game is approximated to a staircase trimatrix game by sampling the player’s pure strategy value set. The set is sampled uniformly so that the resulting staircase trimatrix game is cubic. An equilibrium of the staircase trimatrix game is obtained by stacking the equilibria of the subinterval trimatrix games, each defined on an interval where the pure strategy value is constant. The stack is an approximate solution to the initial staircase game. The (weak) consistency, equivalent to the approximate solution acceptability, is studied by how much the players’ payoff and equilibrium strategy change as the sampling density minimally increases. The consistency includes the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. The most important parts are the payoff consistency and equilibrium strategy support cardinality (weak) consistency, which are checked in the quickest and easiest way. However, it is practically reasonable to consider a relaxed payoff consistency, by which the player’s payoff change in an appropriate approximation may grow at most by epsilon as the sampling density minimally increases. The weak consistency itself is a relaxation to the consistency, where the minimal decrement of the sampling density is ignored. An example is presented to show how the approximation is fulfilled for a case of when every subinterval trimatrix game has pure strategy equilibria.
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Sumalpong, Felipe Jr Raypan, Michael Frondoza i Noel Lito Sayson. "British Put Option On Stocks Under Regime-Switching Model". European Journal of Pure and Applied Mathematics 16, nr 3 (30.07.2023): 1830–47. http://dx.doi.org/10.29020/nybg.ejpam.v16i3.4830.
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In a plain vanilla option, its holder is given the right, but not the obligation, to buy or sell the underlying stock at a specified price (strike price) at a predetermined date. If the exercise date is at maturity, the option is called a European; if the option is exercised anytime prior to maturity, it is called an American. In a British option, the holder can enjoy the early exercise feature of American option whereupon his payoff is the ‘best prediction’ of the European payoff given all the information up to exercise date under the hypothesis that the true drift of the stock equals a specified contract drift. In this paper, in contrast to the constant interest rate and constant volatility assumptions, we consider the British option by assuming that the economic state of the world is described by a finite state continuous-time Markov chain. Also, we provide a solution to a free boundary problem by using PDE arguments. However, closed form expression for the arbitrage-free price are not available in our setting.
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Истомин, Александр Владимирович, i Елена Витальевна Кузьмина. "GAME THEORETICAL RISK AND PAYOFF MANAGEMENT IN RAILWAY TRANSPORT". Транспорт: Наука, техника, управление, nr 5 (2.07.2024): 9–21. http://dx.doi.org/10.36535/0236-1914-2023-05-2.
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Управление рисками безопасности, связанными с железнодорожной инфраструктурой, представляют собой постоянный процесс действий и мер по регулированию изменяющейся транспортной ситуации. Игровой процесс в соревновании по контролю рисков критической железнодорожной инфраструктуры носит случайный характер, в интерпретации действий с неопределенными последствиями. Рассматривается новый тип игры, в которой неточность касается выигрышных ситуаций, а не действий игрока. Поэтапно изложено, как исправить отсутствие (канонического) порядка в контексте распределений вероятностей. Предложена реализация теоретико-игрового подхода с помощью гиперреальных чисел и нестандартного анализа. Security risk management related to the railway infrastructure, represent a constant process of actions and measures to regulate the changing transport situation. The gameplay in the competition to control the risks of critical railway infrastructure is random, in the interpretation of actions with uncertain consequences. A new type of game is being considered, in which the inaccuracy concerns winning situations, and not the actions of the player. It is described step by step how to correct the lack of (canonical) order in the context of probability distributions. The implementation of a game-theoretic approach using hyperreal numbers and non-standard analysis is proposed.
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Brown, Meta, Christopher J. Flinn i Andrew Schotter. "Real-Time Search in the Laboratory and the Market". American Economic Review 101, nr 2 (1.04.2011): 948–74. http://dx.doi.org/10.1257/aer.101.2.948.
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While widely accepted labor market search models imply a constant reservation wage policy, empirical evidence strongly suggests that reservation wages decline in search duration. This paper reports the results of the first real-time-search laboratory experiment. The controlled environment subjects face is stationary, and the payoff-maximizing reservation wage is constant. Nevertheless, subjects' reservation wages decline sharply over time. We investigate two hypotheses to explain this decline: 1. Searchers respond to the stock of accruing search costs. 2. Searchers experience non-stationary subjective costs of time spent searching. Our data support the latter hypothesis, and we substantiate this conclusion both experimentally and econometrically. (JEL C91, D83, J64)
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Romanuke, Vadim V., i Vladimir V. Kamburg. "Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game". Applied Computer Systems 20, nr 1 (1.12.2016): 5–14. http://dx.doi.org/10.1515/acss-2016-0009.
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Abstract Approximation in solving the infinite two-person non-cooperative games is studied in the paper. An approximation approach with conversion of infinite game into finite one is suggested. The conversion is fulfilled in three stages. Primarily the players’ payoff functions are sampled variously according to the stated requirements to the sampling. These functions are defined on unit hypercube of the appropriate Euclidean finite-dimensional space. The sampling step along each of hypercube dimensions is constant. At the second stage, the players’ payoff multidimensional matrices are reshaped into ordinary two-dimensional matrices, using the reversible index-to-index reshaping. Thus, a bimatrix game as an initial infinite game approximation is obtained. At the third stage of the conversion, the player’s finite equilibrium strategy support is checked out for its weak consistency, defined by five types of inequalities within minimal neighbourhood of every specified sampling step. If necessary, the weakly consistent solution of the bimatrix game is checked out for its consistency, strengthened in that the cardinality of every player’s equilibrium strategy support and their densities shall be non-decreasing within minimal neighbourhood of the sampling steps. Eventually, the consistent solution certifies the game approximation acceptability, letting solve even games without any equilibrium situations, including isomorphic ones to the unit hypercube game. A case of the consistency light check is stated for the completely mixed Nash equilibrium situation.
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BUCHMANN, BORIS, i STEFAN WEBER. "A CONTINUOUS TIME APPROXIMATION OF AN EVOLUTIONARY STOCK MARKET MODEL". International Journal of Theoretical and Applied Finance 10, nr 07 (listopad 2007): 1229–53. http://dx.doi.org/10.1142/s0219024907004627.
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We derive a continuous time approximation of the evolutionary market selection model of Blume and Easley (1992). Conditions on the payoff structure of the assets are identified that guarantee convergence. We show that the continuous time approximation equals the solution of an integral equation in a random environment. For constant asset returns, the integral equation reduces to an autonomous ordinary differential equation. We analyze its long-run asymptotic behavior using techniques related to Lyapunov functions, and compare our results to the benchmark of profit-maximizing investors.
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Bruss, F. Thomas, i Thomas S. Ferguson. "Multiple buying or selling with vector offers". Journal of Applied Probability 34, nr 04 (grudzień 1997): 959–73. http://dx.doi.org/10.1017/s0021900200101652.
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We consider a generalization of the house-selling problem to sellingkhouses. Let the offers,X1,X2, · ··,be independent, identically distributedk-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneouslykstopping rules,N1, · ··,Nk,one for each component. The payoff is the sum overjof thejth component ofminus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.
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Hauert, Christoph, Miranda Holmes i Michael Doebeli. "Evolutionary games and population dynamics: maintenance of cooperation in public goods games". Proceedings of the Royal Society B: Biological Sciences 273, nr 1600 (5.07.2006): 2565–71. http://dx.doi.org/10.1098/rspb.2006.3600.
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The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behaviour seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the Prisoner's Dilemma and public goods games. Traditional approaches to studying the problem of cooperation assume constant population sizes and thus neglect the ecology of the interacting individuals. Here, we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation. In public goods games, cooperation can gain a foothold if the population density depends on the average population payoff. Decreasing population densities, due to defection leading to small payoffs, results in smaller interaction group sizes in which cooperation can be favoured. This feedback between ecological dynamics and game dynamics can generate stable coexistence of cooperators and defectors in public goods games. However, this mechanism fails for pairwise Prisoner's Dilemma interactions and the population is driven to extinction. Our model represents natural extension of replicator dynamics to populations of varying densities.
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Bruss, F. Thomas, i Thomas S. Ferguson. "Multiple buying or selling with vector offers". Journal of Applied Probability 34, nr 4 (grudzień 1997): 959–73. http://dx.doi.org/10.2307/3215010.
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We consider a generalization of the house-selling problem to selling k houses. Let the offers, X1, X2, · ··, be independent, identically distributed k-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneously k stopping rules, N1, · ··, Nk, one for each component. The payoff is the sum over j of the jth component of minus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.
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Ohlendorf, Susanne. "Expectation Damages, Divisible Contracts, and Bilateral Investment". American Economic Review 99, nr 4 (1.08.2009): 1608–18. http://dx.doi.org/10.1257/aer.99.4.1608.
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This paper examines the efficiency of expectation damages as a breach remedy in a bilateral trade setting with renegotiation and relationship-specific investment by the buyer and the seller. As demonstrated by Edlin and Reichelstein (1996), no contract that specifies only a fixed quantity and a fixed per-unit price can induce efficient investment if marginal cost is constant and deterministic. We show that this result does not extend to more general payoff functions. If both parties face the risk of breaching, the first best becomes attainable with a simple price-quantity contract. (JEL D86, K12)
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Huang, Wanying, Philipp Strack i Omer Tamuz. "Learning in Repeated Interactions on Networks". Econometrica 92, nr 1 (2024): 1–27. http://dx.doi.org/10.3982/ecta20806.
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We study how long‐lived, rational agents learn in a social network. In every period, after observing the past actions of his neighbors, each agent receives a private signal, and chooses an action whose payoff depends only on the state. Since equilibrium actions depend on higher‐order beliefs, it is difficult to characterize behavior. Nevertheless, we show that regardless of the size and shape of the network, the utility function, and the patience of the agents, the speed of learning in any equilibrium is bounded from above by a constant that only depends on the private signal distribution.
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LEUNG, TIM, KAZUTOSHI YAMAZAKI i HONGZHONG ZHANG. "AN ANALYTIC RECURSIVE METHOD FOR OPTIMAL MULTIPLE STOPPING: CANADIZATION AND PHASE-TYPE FITTING". International Journal of Theoretical and Applied Finance 18, nr 05 (28.07.2015): 1550032. http://dx.doi.org/10.1142/s0219024915500326.
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We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Lévy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Lévy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Lévy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.
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KUMKOV, SERGEY S., VALERY S. PATSKO i JOSEF SHINAR. "ON LEVEL SETS WITH "NARROW THROATS" IN LINEAR DIFFERENTIAL GAMES". International Game Theory Review 07, nr 03 (wrzesień 2005): 285–311. http://dx.doi.org/10.1142/s0219198905000533.
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Examples with zero-sum linear differential games of fixed terminal time and a convex terminal payoff function depending on two components of the phase vector are considered. Such games can have an indifferent zone with constant value function. The level set of the value function associated with the indifferent zone is called the "critical" tube. In the selected examples, the critical tube and the neighboring level sets exhibit "narrow throats". Presence of such throats requires extremely precise computations for constructing the level sets. The paper presents different forms of critical tubes with narrow throats and indicates the combinations of problem parameters that can produce them.
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Brophy, Chris, i Gabriel Roy. "Benefits and Challenges of Pressure-Gain Combustion Systems for Gas Turbines". Mechanical Engineering 131, nr 03 (1.03.2009): 54–55. http://dx.doi.org/10.1115/1.2009-mar-8.
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This article discusses benefits and challenges of Pressure-Gain Combustion Systems for Gas Turbines. The article also highlights that one approach to substantially improve gas turbine thermal efficiency is to replace the nearly constant pressure combustion process with some form of pressure-gain heat release such as either a constant volume or detonative mode of combustion. These systems commonly possess some form of rotating inlet valve design to control the filling process for an annular array of combustors and maintain the appropriate amount of inlet isolation. Although evaluation of turbine life and performance needs to continue, turbine efficiencies approaching values comparable to those of steady-state operation have been reported. The article concludes that the collaborative efforts, such as listed in the article, are ultimately required in times of reduced funding for continued technology development. Even with the risks and challenges associated with this technology, a high payoff potential exists with hybrid gas turbine architectures.
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Романюк, Вадим Васильович. "Finite approximation of zero-sum games played in staircase-function continuous spaces". KPI Science News, nr 4 (14.02.2022): 19–38. http://dx.doi.org/10.20535/kpisn.2021.4.242769.
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Background. There is a known method of approximating continuous zero-sum games, wherein an approximate solutionis considered acceptable if it changes minimally by changing the sampling step minimally. However, the method cannotbe applied straightforwardly to a zero-sum game played with staircase-function strategies. Besides, the independence ofthe player’s sampling step selection should be taken into account.Objective. The objective is to develop a method of finite approximation of zero-sum games played in staircase-functioncontinuous spaces by taking into account that the players are likely to independently sample their pure strategy sets.Methods. To achieve the said objective, a zero-sum game, in which the players’ strategies are staircase functions of time,is formalized. In such a game, the set of the player’s pure strategies is a continuum of staircase functions of time, andthe time is thought of as it is discrete. The conditions of sampling the set of possible values of the player’s pure strategyare stated so that the game becomes defined on a product of staircase-function finite spaces. In general, the samplingstep is different at each player and the distribution of the sampled points (function-strategy values) is non-uniform.Results. A method of finite approximation of zero-sum games played in staircase-function continuous spaces is pre-sented. The method consists in irregularly sampling the player’s pure strategy value set, solving smaller-sized matrixgames, each defined on a subinterval where the pure strategy value is constant, and stacking their solutions if they areconsistent. The stack of the smaller-sized matrix game solutions is an approximate solution to the initial staircase game.The (weak) consistency of the approximate solution is studied by how much the payoff and optimal situation change asthe sampling density minimally increases by the three ways of the sampling increment: only the first player’s increment,only the second player’s increment, both the players’ increment. The consistency is decomposed into the payoff, opti-mal strategy support cardinality, optimal strategy sampling density, and support probability consistency. It is practicallyreasonable to consider a relaxed payoff consistency.Conclusions. The suggested method of finite approximation of staircase zero-sum games consists in the independentsamplings, solving smaller-sized matrix games in a reasonable time span, and stacking their solutions if they are con-sistent. The finite approximation is regarded appropriate if at least the respective approximate (stacked) solution ise-payoff consistent.
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Galashin, Mikhail, i Sergey V. Popov. "Teamwork Efficiency and Company Size". B.E. Journal of Theoretical Economics 16, nr 1 (1.01.2016): 337–66. http://dx.doi.org/10.1515/bejte-2014-0040.
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AbstractWe study how ownership structure and management objectives interact in determining the company size without assuming information constraints or any explicit costs of management. In symmetric agent economies, the optimal company size balances the returns to scale of the production function and the returns to collaboration efficiency. For a general class of payoff functions, we characterize the optimal company size, and we compare the optimal company size across different managerial objectives. We demonstrate the restrictiveness of common assumptions on effort aggregation (e.g., constant elasticity of effort substitution), and we show that common intuition (e.g., that corporate companies are more efficient and therefore will be larger than equal-share partnerships) might not hold in general.
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Grigorieva, Xeniya. "Multicriteria coalitional model of decision-making over the set of projects with constant payoff matrix in the noncooperative game". Applied Mathematical Sciences 8 (2014): 8473–79. http://dx.doi.org/10.12988/ams.2014.410894.
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HELBING, DIRK, i WENJIAN YU. "MIGRATION AS A MECHANISM TO PROMOTE COOPERATION". Advances in Complex Systems 11, nr 04 (sierpień 2008): 641–52. http://dx.doi.org/10.1142/s0219525908001866.
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A unified approach is proposed, which integrates game-theoretical models with models of directed motion. Specifically, strategic interactions in space and imitation of more successful neighboring strategies, as studied by spatial games, are combined with success-driven migration based on "test interactions" or wealth-related "neighborhood tagging." It turns out that such directed migration allows cooperators to evade defectors and (in constrast to purely diffusive motion) also to find other cooperators in order to form clusters ("islands of cooperation"). This can increase the cooperation to a great extent. Moreover, success-driven motion leads to interesting spatiotemporal pattern-formation phenomena, which are clearly different from those produced by previously studied spatial games. Despite the simplicity of the model, the forming patterns appear to realistically represent many stylized facts of social interactions, particularly phenomena such as social networking, urban aggregation, social segregation or turn-taking. Furthermore, migration games offer a game-theoretical explanation of social attraction and repulsion, based on payoff-related mobility, and they display a larger variety than conventional games, as their behavior is not invariant to shifting all payoffs by a constant amount. In summary, the combination of strategic interactions and imitation with models of directed motion has high potential for understanding empirical observations in the social, biological and economic sciences from a game-theoretical point of view. Finally, the breakdown of cooperation in the prisoner's dilemma at high values of temptation is explained by configurational analysis.
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D'Ascenzo, Andrea, Mattia D'Emidio, Michele Flammini i Gianpiero Monaco. "Digraph k-Coloring Games: New Algorithms and Experiments". Journal of Artificial Intelligence Research 81 (23.09.2024): 163–202. http://dx.doi.org/10.1613/jair.1.16174.
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We study digraph k-coloring games where strategic agents are vertices of a digraph and arcs represent agents' mutual unidirectional conflicts/idiosyncrasies. Each agent can select, as strategy, one of k different colors, and her payoff in a given state (a k-coloring) is given by the number of outgoing neighbors with a color different from her one. Such games model lots of strategic real-world scenarios and are related to several fundamental classes of anti-coordination games. Unfortunately, the problem of understanding whether an instance of the game admits a pure Nash equilibrium (NE), i.e., a state where no agent can improve her payoff by changing strategy, is NP-complete. Thus, in this paper, we focus on algorithms to compute an approximate NE: informally, a coloring is an approximate γ-NE, for some γ ≥ 1, if no agent can improve her payoff, by changing strategy, by a multiplicative factor of γ. Our contribution is manifold and of both theoretical and experimental nature. First, we characterize the hardness of finding pure and approximate equilibria in both general and special classes of digraphs. Second, we design and analyze three approximation algorithms with different theoretical guarantees on the approximation ratio, under different conditions; (i) algorithm APPROX-1 which computes, for any k ≥ 3, a Δo-NE for any n vertex graph having a maximum outdegree of Δo, in polynomial time; (ii) algorithm LLL-SPE, a randomized algorithm that, for any constant k ≥ 2, determines a γ-NE for some constant γ but only in digraphs whose minimum outdegree is sufficiently large, in polynomial time in expectation; (iii) algorithm APPROX-3 which, for any ε, computes a (1+ε)-NE by using O(log(n)/ε) colors, for any n-vertex digraph. Note that, the latter shows that a (1+ε)-NE exists and can be computed in polynomial time for k = O(log(n)). Finally, to assess how proposed algorithms behave in the typical case, we complete our study with an extensive experimental evaluation showing that, while newly introduced algorithms achieve bounded worst case behavior, they generally perform poorly in practice. Motivated by such unsatisfactory performance, we shift our attention to the best-response paradigm, successfully applied to other classes of games, and design and experimentally evaluate it a heuristic based on such paradigm. Our experiments provide strong evidences of such approach outperforming, in terms of approximation and computational time, all other methods and hence identify it as the most suited candidate for practical usage. More remarkably, it is also able to compute exact, pure NE in the great majority of cases. This suggests that, while these games are known to not always possess a pure NE, such an equilibrium often exists and can be efficiently computed, even by a distributed uncoordinated interaction of the agents.
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STACE, ANTONY WILLIAM. "A MOMENT MATCHING APPROACH TO THE VALUATION OF A VOLUME WEIGHTED AVERAGE PRICE OPTION". International Journal of Theoretical and Applied Finance 10, nr 01 (luty 2007): 95–110. http://dx.doi.org/10.1142/s0219024907004068.
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In this paper we develop a method to find the price of several options whose payoff depends on a volume weighted average price (VWAP). Fixed and floating strike VWAP, together with digital VWAP contracts are considered. Throughout we assume that the stock follows a geometric Brownian motion and the rate of trades evolves as a mean reverting process. It is assumed that the VWAP at the final time has a lognormal distribution. The parameters of the approximating lognormal distribution are obtained by matching the first two moments of the volume weighted average price with a lognormal process. A price is then obtained for the fixed strike and digital options when the market price of risk is a constant. We concentrate on the price for calls, prices for puts can be obtained in an analogous manner.
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Liu, Yang, Changfu Zong, Xiaojian Han, Dong Zhang, Hongyu Zheng i Chunmei Shi. "Spacing Allocation Method for Vehicular Platoon: A Cooperative Game Theory Approach". Applied Sciences 10, nr 16 (12.08.2020): 5589. http://dx.doi.org/10.3390/app10165589.
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Recently, spacing policies of the vehicular platoon have been widely developed to enhance safety, traffic efficiency, and fuel consumption. However, the integrated spacing policies aim to maximum overall benefit, and the distributed spacing policies intense to get optimal monomer benefit. Ignoring the fairness of the benefit allocation of each vehicle will reduce the motivation to constitute the platoon. To fill this critical gap, this study proposes a spacing allocation method by treating spacing decisions as cooperative games. A flock’s model which is used to be the payoff function is introduced based on bionic motion principles. We present a characteristic function of the platoon for the cooperative game model considering the specific structure of the platoon. The τ value, Shapley value, and average lexicographic value are introduced and applied to allocate the spacing fairly. Proposed methods are compared with constant distance policy in some typical situations. The simulation results demonstrate that the spacing policy based on cooperative game theory improved the stable time for consistency control and the convergence of longitudinal following error.
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Alzubaidi, Hasan. "Efficient exponential timestepping algorithm using control variate technique for simulating a functional of exit time of one-dimensional Brownian diffusion with applications in finance". Arabian Journal of Mathematics 9, nr 3 (13.07.2020): 495–511. http://dx.doi.org/10.1007/s40065-020-00287-w.
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Abstract The exponential timestepping Euler algorithm with a boundary test is adapted to simulate an expected of a function of exit time, such as the expected payoff of barrier options under the constant elasticity of variance (CEV) model. However, this method suffers from a high Monte Carlo (MC) statistical error due to its exponentially large exit times with unbounded samples. To reduce this kind of error efficiently and to speed up the MC simulation, we combine such an algorithm with an effective variance reduction technique called the control variate method. We call the resulting algorithm the improved Exp algorithm for abbreviation. In regard to the examples we consider in this paper for the restricted CEV process, we found that the variance of the improved Exp algorithm is about six times smaller than that of the Jansons and Lythe original method for the down-and-out call barrier option. It is also about eight times smaller for the up-and-out put barrier option, indicating that the gain in efficiency is significant without significant increase in simulation time.
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Dehghani, Sina, Hamed Saleh, Saeed Seddighin i Shang-Hua Teng. "Computational Analyses of the Electoral College: Campaigning Is Hard But Approximately Manageable". Proceedings of the AAAI Conference on Artificial Intelligence 35, nr 6 (18.05.2021): 5294–302. http://dx.doi.org/10.1609/aaai.v35i6.16668.
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In the classical discrete Colonel Blotto game—introduced by Borel in 1921—two colonels simultaneously distribute their troops across multiple battlefields. The winner of each battlefield is determined by a winner-take-all rule, independently of other battlefields. In the original formulation, each colonel’s goal is to win as many battlefields as possible. The Blotto game and its extensions have been used in a wide range of applications from political campaign—exemplified by the U.S presidential election—to marketing campaign, from (innovative) technology competition to sports competition. Despite persistent efforts, efficient methods for finding the optimal strategies in Blotto games have been elusive for almost a century—due to exponential explosion in the organic solution space—until Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin developed the first polynomial-time algorithm for this fundamental gametheoretical problem in 2016. However, that breakthrough polynomial-time solution has some structural limitation. It applies only to the case where troops are homogeneous with respect to battlegruounds, as in Borel’s original formulation: For each battleground, the only factor that matters to the winner’s payoff is how many troops as opposed to which sets of troops are opposing one another in that battleground. In this paper, we consider a more general setting of the two-player-multi-battleground game, in which multifaceted resources (troops) may have different contributions to different battlegrounds. In the case of U.S presidential campaign, for example, one may interpret this as different types of resources—human, financial, political—that teams can invest in each state. We provide a complexity-theoretical evidence that, in contrast to Borel’s homogeneous setting, finding optimal strategies in multifaceted Colonel Blotto games is intractable. We complement this complexity result with a polynomial-time algorithm that finds approximately optimal strategies with provable guarantees. We also study a further generalization when two competitors do not have zerosum/ constant-sum payoffs. We show that optimal strategies in these two-player-multi-battleground games are as hard to compute and approximate as Nash equilibria in general noncooperative games and economic equilibria in exchange markets.
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PAPIN, TIMOTHEE, i GABRIEL TURINICI. "PREPAYMENT OPTION OF A PERPETUAL CORPORATE LOAN: THE IMPACT OF THE FUNDING COSTS". International Journal of Theoretical and Applied Finance 17, nr 04 (czerwiec 2014): 1450028. http://dx.doi.org/10.1142/s0219024914500289.
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We investigate in this paper a perpetual prepayment option related to a corporate loan. The short interest rate and default intensity of the firm are supposed to follow Cox–Ingersoll–Ross (CIR) processes. A liquidity term that represents the funding costs of the bank is introduced and modeled as a continuous time discrete state Markov chain. The prepayment option needs specific attention as the payoff itself is a derivative product and thus an implicit function of the parameters of the problem and of the dynamics. We prove verification results that allows to certify the geometry of the exercise region and compute the price of the option. We show moreover that the price is the solution of a constrained minimization problem and propose a numerical algorithm building on this result. The algorithm is implemented in a two-dimensional code and several examples are considered. It is found that the impact of the prepayment option on the loan value is not to be neglected and should be used to assess the risks related to client prepayment. Moreover, the Markov chain liquidity model is seen to describe more accurately clients' prepayment behavior than a model with constant liquidity.
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Huang, Weini, Christoph Hauert i Arne Traulsen. "Stochastic game dynamics under demographic fluctuations". Proceedings of the National Academy of Sciences 112, nr 29 (6.07.2015): 9064–69. http://dx.doi.org/10.1073/pnas.1418745112.
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Frequency-dependent selection and demographic fluctuations play important roles in evolutionary and ecological processes. Under frequency-dependent selection, the average fitness of the population may increase or decrease based on interactions between individuals within the population. This should be reflected in fluctuations of the population size even in constant environments. Here, we propose a stochastic model that naturally combines these two evolutionary ingredients by assuming frequency-dependent competition between different types in an individual-based model. In contrast to previous game theoretic models, the carrying capacity of the population, and thus the population size, is determined by pairwise competition of individuals mediated by evolutionary games and demographic stochasticity. In the limit of infinite population size, the averaged stochastic dynamics is captured by deterministic competitive Lotka–Volterra equations. In small populations, demographic stochasticity may instead lead to the extinction of the entire population. Because the population size is driven by fitness in evolutionary games, a population of cooperators is less prone to go extinct than a population of defectors, whereas in the usual systems of fixed size the population would thrive regardless of its average payoff.
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Swartzman, Gordon L., Wayne M. Getz i Robert C. Francis. "Binational Management of Pacific Hake (Merluccius productus): A Stochastic Modeling Approach". Canadian Journal of Fisheries and Aquatic Sciences 44, nr 5 (1.05.1987): 1053–63. http://dx.doi.org/10.1139/f87-124.
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We use an age-structured stochastic recruitment model to examine a stock managed by two nations: the Pacific Hake (Merluccius productus) fishery. The model computes the mean and variance in stock and yield and the probability (risk) that the stock falls into a critically low condition. An optimization algorithm finds effort quotas that maximize expected yield over a short-term planning horizon while constraining the risk of critically low stock conditions. Yields produced exceed those of constant effort alternatives, but with year-to-year fluctuations in fishing effort including fishery closures. These occur in different years for the United States and Canadian fisheries depending on the condition of the stock and the incoming year class. Binational management of fishing effort is also presented as a two-person game with a weighted difference between fishery catch and effort as payoff. Nash (rational players), maxi–min (self-protecting), and Pareto (cooperative optimizing) strategies are examined. There are striking shifts in relative effort between United States and Canadian fisheries as the cost of effort changes relative to catch. This approach shows potential benefits to both players in playing a long-term cooperative strategy.
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Liu, Jie, Pengyi Wang, Jiayang Zhao i Yu Dong. "Estimation of peer pressure in dynamic homogeneous social networks". JUSTC 53 (2023): 1. http://dx.doi.org/10.52396/justc-2023-0035.
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Social interaction with peer pressure is widely studied in social network analysis. Game theory can be utilized to model dynamic social interaction and one class of game network models assumes that peopleos decision payoff functions hinge on individual covariates and the choices of their friends. However, peer pressure would be misidentified and induce a non-negligible bias when incomplete covariates are involved in the game model. For this reason, we develop a generalized constant peer effects model based on homogeneity structure in dynamic social networks. The new model can effectively avoid bias through homogeneity pursuit and can be applied to a wider range of scenarios. To estimate peer pressure in the model, we first present two algorithms based on the initialize expand merge method and the polynomial-time two-stage method to estimate homogeneity parameters. Then we apply the nested pseudo-likelihood method and obtain consistent estimators of peer pressure. Simulation evaluations show that our proposed methodology can achieve desirable and effective results in terms of the community misclassification rate and parameter estimation error. We also illustrate the advantages of our model in the empirical analysis when compared with a benchmark model.
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Atabayeva, Assiya, Anar Kurmanalina, Gaukhar Kalkabayeva, Aigerim Lambekova, Ainur Myrzhykbayeva i Yerbolsyn Akbayev. "Utilizing Investment in Fixed Assets and R&D as a Catalyst for Boosting Productivity to Stimulate Economic Growth". Economies 12, nr 10 (30.09.2024): 266. http://dx.doi.org/10.3390/economies12100266.
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Investments form the basis for high-quality economic growth by ensuring renewal of production capacities and improvement of technologies and processes, thereby increasing labor productivity. Investment research is key to understanding its impact on the country’s economic advancement and more effective government policymaking to stimulate investment activity. The purpose of this paper is to study the relationship between fixed capital and R&D investments and labor productivity growth, as well as to determine the optimal level of these investments in determining the greatest effect on labor productivity growth. We use a regression model and the least squares method to empirically analyze data for seven countries for the period between 1997 and 2022. We calculate estimated values using SPSS and Python. The results show a certain impact of fixed assets and R&D investments on labor productivity growth. However, the payoff varies across the countries under study. Furthermore, despite its relatively small volumes, investment in R&D brings a greater effect on productivity growth than investment in fixed capital. With other factors remaining constant, the calculated optimums for investments in fixed assets and R&D show maximum points of growth in labor productivity.
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Ananyev, Boris. "About control of guaranteed estimation". Cybernetics and Physics, Volume 7, 2018, Number 1 (18.06.2018): 18–25. http://dx.doi.org/10.35470/2226-4116-2018-7-1-18-25.
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The control problem by parameters in the course of the guaranteed state estimation of linear non-stationary systems is considered. It is supposed that unknown disturbances in the system and the observation channel are limited by norm in the space of square integrable functions and the initial state of the system is also unknown. The process of guaranteed state estimation includes the solution of a matrix Riccati equation that contains some parameters, which may be chosen at any instant of time by the first player (an observer) and the second player (an opponent of the observer). The purposes of players are diametrically opposite: the observer aims to minimize diameter of information set at the end of observation process, and the second player on the contrary aims to maximize it. This problem is interpreted as a differential game with two players for the Riccati equation. All the choosing parameters are limited to compact sets in appropriate spaces of matrices. The payoff of the game is interpreted through the Euclidean norm of the inverse Riccati matrix at the end of the process. A specific case of the problem with constant matrices is considered. Methods of minimax optimization, the theory of optimal control, and the theory of differential games are used. Examples are also given.
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ANÉ, THIERRY, i VINCENT LACOSTE. "UNDERSTANDING BID-ASK SPREADS OF DERIVATIVES UNDER UNCERTAIN VOLATILITY AND TRANSACTION COSTS". International Journal of Theoretical and Applied Finance 04, nr 03 (czerwiec 2001): 467–89. http://dx.doi.org/10.1142/s0219024901001073.
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The classical option valuation models assume that the option payoff can be replicated by continuously adjusting a portfolio consisting of the underlying asset and a risk-free bond. This strategy implies a constant volatility for the underlying asset and perfect markets. However, the existence of non-zero transaction costs, the consequence of trading only at discrete points in time and the random nature of volatility prevent any portfolio from being perfectly hedged continuously and hence suppress any hope of completely eliminating all risks associated with derivatives. Building upon the uncertain parameters framework we present a model for pricing and hedging derivatives where the volatility is simply assumed to lie between two bounds and in the presence of transaction costs. It is shown that the non-arbitrageable prices for the derivatives, which arise in this framework, can be derived by a non-linear PDE related to the convexity of the derivatives. We use Monte Carlo simulations to investigate the error in the hedging strategy. We show that the standard arbitrage is exposed to such large risks and transaction costs that it can only establish very wide bounds on equilibrium prices, obviously in contradiction with the very tight bid-ask spreads of derivatives observed on the market. We explain how the market spreads can be compatible with our model through portfolio diversification. This has important implications for price determination in options markets as well as for testing of valuation models.
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Escobedo-Trujillo, Beatris Adriana, José Daniel López-Barrientos, Carmen Geraldi Higuera-Chan i Francisco Alejandro Alaffita-Hernández. "Robust Statistic Estimation of Constrained Optimal Control Problems of Pollution Accumulation (Part I)". Mathematics 11, nr 4 (11.02.2023): 923. http://dx.doi.org/10.3390/math11040923.
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In this paper, we study a constrained optimal control on pollution accumulation where the dynamic system was governed by a diffusion process that depends on unknown parameters, which need to be estimated. As the true values are unknown, we intended to determine (adaptive) policies that maximize a discounted reward criterion with constraints, that is, we used Lagrange multipliers to find optimal (adaptive) policies for the unconstrained version of the optimal control problem. In the present context, the dynamic system evolves as a diffusion process, and the cost function is to be minimized by another function (typically a constant), which plays the role of a constraint in the control model. We offer solutions to this problem using standard dynamic programming tools under the constrained discounted payoff criterion on an infinite horizon and the so-called principle of estimation and control. We used maximum likelihood estimators by means of a minimum least square error approximation in a pollution accumulation model to illustrate our results. One of the advantages of our approach compared to others is the intuition behind it: find optimal policies for an estimated version of the problem and let this estimation tend toward the real version of the problem. However, most risk analysts will not be as used to our methods as they are to, for instance, the model predictive control, MATLAB’s robust control toolbox, or the polynomial chaos expansion method, which have been used in the literature to address similar issues.
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Kyrychko, Y. N., i K. B. Blyuss. "Vaccination games and imitation dynamics with memory". Chaos: An Interdisciplinary Journal of Nonlinear Science 33, nr 3 (marzec 2023): 033134. http://dx.doi.org/10.1063/5.0143184.
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In this paper, we model dynamics of pediatric vaccination as an imitation game, in which the rate of switching of vaccination strategies is proportional to perceived payoff gain that consists of the difference between perceived risk of infection and perceived risk of vaccine side effects. To account for the fact that vaccine side effects may affect people’s perceptions of vaccine safety for some period of time, we use a delay distribution to represent how memory of past side effects influences current perception of risk. We find disease-free, pure vaccinator, and endemic equilibria and obtain conditions for their stability in terms of system parameters and characteristics of a delay distribution. Numerical bifurcation analysis illustrates how stability of the endemic steady state varies with the imitation rate and the mean time delay, and this shows that it is not just the mean duration of memory of past side effects, but also the actual distribution that determines whether disease will be maintained in the population at some steady level, or if sustained periodic oscillations around this steady state will be observed. Numerical simulations illustrate a comparison of the dynamics for different mean delays and different distributions, and they show that even when periodic solutions are observed, there are differences in their amplitude and period for different distributions. We also investigate the effect of constant public health information campaigns on vaccination dynamics. The analysis suggests that the introduction of such campaigns acts as a stabilizing factor for endemic equilibrium, allowing it to remain stable for larger values of mean time delays.
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Guo, Xin. "Some risk management problems for firms with internal competition and debt". Journal of Applied Probability 39, nr 1 (marzec 2002): 55–69. http://dx.doi.org/10.1239/jap/1019737987.
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Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (Ut, Zt), where Ut is the size of the business unit and Zt is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process Xt and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for Ut and a singular control for Zt. The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.
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Guo, Xin. "Some risk management problems for firms with internal competition and debt". Journal of Applied Probability 39, nr 01 (marzec 2002): 55–69. http://dx.doi.org/10.1017/s0021900200021501.
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Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (U t , Z t ), where U t is the size of the business unit and Z t is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process X t and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for U t and a singular control for Z t . The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X 0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.
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Browne, Sid. "Reaching goals by a deadline: digital options and continuous-time active portfolio management". Advances in Applied Probability 31, nr 2 (czerwiec 1999): 551–77. http://dx.doi.org/10.1239/aap/1029955147.
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We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy that maximizes the probability of reaching a given wealth level by a given fixed terminal time, for the case where an investor can allocate his wealth at any time between n + 1 investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investment opportunity. We then use this to solve related problems for cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a correspondence between the probability maximizing results and the pricing and hedging of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to maximize the probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff. A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.
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Browne, Sid. "Reaching goals by a deadline: digital options and continuous-time active portfolio management". Advances in Applied Probability 31, nr 02 (czerwiec 1999): 551–77. http://dx.doi.org/10.1017/s000186780000923x.
Pełny tekst źródłaStreszczenie:
We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy that maximizes the probability of reaching a given wealth level by a given fixed terminal time, for the case where an investor can allocate his wealth at any time between n + 1 investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investment opportunity. We then use this to solve related problems for cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a correspondence between the probability maximizing results and the pricing and hedging of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to maximize the probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff. A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.