Table of Contents
Journal of Computational Engineering
Volume 2014 (2014), Article ID 320420, 8 pages
http://dx.doi.org/10.1155/2014/320420
Research Article

Sinc Collocation Method for Finding Numerical Solution of Integrodifferential Model Arisen in Continuous Mixed Strategy

Department of Mathematics, Shahid Rajaee Teacher Training University, Lavizan, Tehran, Iran

Received 22 February 2014; Revised 9 August 2014; Accepted 21 August 2014; Published 17 September 2014

Academic Editor: Fu-Yun Zhao

Copyright © 2014 F. Hosseini Shekarabi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

One of the new techniques is used to solve numerical problems involving integral equations and ordinary differential equations known as Sinc collocation methods. This method has been shown to be an efficient numerical tool for finding solution. The construction mixed strategies evolutionary game can be transformed to an integrodifferential problem. Properties of the sinc procedure are utilized to reduce the computation of this integrodifferential to some algebraic equations. The method is applied to a few test examples to illustrate the accuracy and implementation of the method.

1. Introduction

Evolutionary game dynamics is a fast developing field, with applications in biology, economics, sociology, politics, interpersonal relationships, and anthropology. Background material and countless references can be found in [18]. In the present paper we consider a continuous mixed strategies model for population dynamics based on an integrodifferential representation. Analogous models for population dynamics based on the replicator equation with continuous strategy space were investigated in [913]. For the moment based model has proved global existence of solutions and studied the asymptotic behavior and stability of solutions in the case of two strategies [14].

In the last three decades a variety of numerical methods based on the sinc approximation have been developed. Sinc methods were developed by Stenger [15] and Lund and Bowers [16] and it is widely used for solving a wide range of linear and nonlinear problems arising from scientific and engineering applications including oceanographic problems with boundary layers [17], two-point boundary value problems [18], astrophysics equations [19], Blasius equation [20], Volterras population model [21], Hallens integral equation [22], third-order boundary value problems [23], system of second-order boundary value problems [24], fourth-order boundary value problems [25], heat distribution [26], elastoplastic problem [27], inverse problem [28, 29], integrodifferential equation [30], optimal control [15], nonlinear boundary-value problems [31], and multipoint boundary value problems [32]. Very recently authors of [33] used the sinc procedure to solve linear and nonlinear Volterra integral and integrodifferential equations.

The content of this paper is arranged in seven sections. In Section 2, I discuss the modeling of the problem in an integrodifferential form. Section 3, introduces some general concepts concerning the sinc approximation. Section 4, contains some preliminaries in collocation method. In Section 5, the method is applied for solving the problem. In Section 6, some numerical examples has been provided. Finally, Section 7 provides the conclusion of this work.

2. Mathematical Model

The model we consider here is an integrodifferential model for continuous mixed strategies. In game theory, a dominant strategy is the one that gives a player the most benefit no matter what the other players do. A player’s strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player’s behavior. A player’s strategy set defines what strategies are available for them to play. A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face. A player’s strategy set is the set of pure strategies available to that player. A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player, even if their strategy set is finite.

A payoff is a number, also called utility that reflects the desirability of an outcome to a player, for whatever reason. When the outcome is random, payoffs are usually weighted with their probabilities. The expected payoff incorporates the player’s attitude towards risk.

Assume that we have a game where there are pure strategies to and that the players can use mixed strategies: this consists of playing the pure strategies to with some probabilities to with and . A strategy corresponds to a point in the simplex The corners of the simplex are the standard unit vectors , where the th component is 1 and all others are 0 and correspond to the pure strategies , .

Let us denote by the payoff for a player using the pure strategy against a player using the pure strategy . Here Matrix is called payoff matrix. An -strategist obtain the expected payoff against a -strategist. The payoff for a -startegist against a strategist is given by

We consider a population of individuals as a player of the game and denote by the density of population adopting the strategy at time ; the evolution in time of , due to dynamics of the game, is driven by where the term represents the payoff of the strategy against all the others strategies, being the interacting kernel between the q-strategist and the -strategist. The last term of (3) is defined by and represents the average payoff of the population.

Since , we can reduce the number of variables, considering and obtaining the -dimensional model (3) on the simplex namely, with defined by and defined by

Remark 1 (see [14]). If we take an initial condition with , then it is easy to see that for all and if for some , then for all . We also know that
This follows from the mass conservation; by integrating (8) with respect to and using (10) and (12) we have Let us introduce the moments for : with . Using , the payoff and the average payoff (10) where is the standard unit vector with the th component equal to 1 and all others equal to 0. Moreover, , , .
In the final form of (8), that will be used later in this paper, the only integral terms are the first moments :

Global Existence of the Solutions. We consider the Cauchy problem (11)–(16) for and ; that is, with and .

Proposition 2 (Local existence see [14]). For all there exists such that if then there exists a unique solution for the problem (17), for all .

2.1. Two Strategies Games

Assume there are two different strategies, whose interplay is ruled by the payoff matrix: In this case the simplex is just the interval and so we have a population where individuals are going to play the first strategy with probability and the second strategy with probability . The payoff (2) is given by with The one dimensional Cauchy problem (17) reads with and .

For more detail see [14].

3. Sinc Interpolation

The goal of this section is to recall notations and definition of the sinc function that are used. The sinc approximation for a function defined on the real line is given by where is sinc function defined by And the step size is suitably chosen for a given positive integer . Sinc for interpolation points is given by

Assuming that is analytic on the real line and decays exponentially on the real line, it has been shown that the error of the approximation decays exponentially with increasing . The approximation may be extended to approximate on the interval by selection of an appropriate transfer function to transform the interval onto the real line and impose the exponential decay. We denote such variable transformation and inverse transformation such that and . We may write the sinc approximation employing the transformation for the function to be where the mesh size represents the separation between sinc points on the domain. In order to have the sinc approximation on a finite interval conformal map is employed as follows: This map carries the eye-shaped complex domain onto the infinite strip For the sinc method, the basic function on the interval for is derived from the composite translated sinc functions: Exhibiting kroneckor delta behavior on the grid points Thus we may define the inverse images of the real line and of the evenly space nodes as And quadrature formulas for over are

Definition 3. Let be the class of functions which are analytic in satisfy where, and on the boundary of (denoted by ) satisfying Interpolation for function in is defined in the following theorem whose proof can be found in [15].

Theorem 4. If then for all Moreover, if for some positive constants and , and if the selection then where, depends only on , , and . The above expressions show sinc interpolation on converge exponentially [17]. We also require derivatives of composite sinc functions evaluated at the nodes. The expressions required for the present discussion are [25]

4. Collocation Method

Let be a given mesh (not necessarily uniform) on and set , , with . The quantity will be called the diameter of the mesh .

Definition 5. Suppose that is a given partition on . The piecewise polynomials space with is defined by
Here and denote the space of polynomials of degree not exceeding , and it is easy to see that The collocation solution is determined by that satisfies the given equation on a given suitable finite subset of , where contains the collocation points: is determined by the points of the partition and the given collocation parameters . The collocation solution for is defined by the collocation equation It will be convenient (and natural) to work with the local Lagrange basis representations of . These polynomials in can be written as where belong to . Also we have
From (44) we can obtain the local representation of on , hence we can achieve that The unknown approximations in (44) are defined by the solution of a system of (generally nonlinear) algebraic equations obtained by setting in the collocation equation (42) and employing the local representation (44). This system is It corresponds to (44) with ,
I present the result on global convergence for the linear initial-value problem

Theorem 6. Assume that(a)the given functions in (48) satisfy ;(b) is such that, for any , each of the linear systems of method has a unique solution.Then the estimates

See [34].

5. Construction of the Method

Let is a partition of . In every interval we assume that , , solution of one dimensional mixed strategy model is approximated by the finite expansion of sinc basis function and Lagrange polynomials: where, is a polynomial of degree . Also, initial value is according to If we replace approximation (51) in (21) we have where is taken by By substituting collocation points for and and using quadrature rule (32), a nonlinear system is given.

After solving this system we calculate and finally in , and also, at : where is used as an initial value for next interval . After times, solution is achieved.

6. Numerical Examples

Prisoner’s Dilemma Game. One interesting example of a game is given by the so-called Prisoner’s Dilemma game in which there are two players and two possible strategies. The players have two options, cooperate or defect. The payoff matrix is the following: If both players cooperate both obtain fitness units (reward payoff); if both defect, each receives (punishment payoff); if one player cooperates and the other defects, the cooperator gets (suckers payoff) while the defector gets (temptation payoff). The payoff values are ranked and . We know that cooperators are always dominated by defectors.

For the numerical tests we fix the following normalized payoff matrix: with and . In this case we have and and so . This means that stationary solutions are expected to be given by concentrated Dirac masses. For general perturbation we have that is linearly stable.

We In order to conform the results above, initial condition is considered as below:(1)(2)(3)For implementation of proposed method, I used Maple15 and plotted the numerical results in Figures 1, 2, and 3. Figure 1 shows that the density tends to concentrate at the point , to what we expected.

320420.fig.001
Figure 1: Plot of the evolution over time of for the Prisoner’s Dilemma game with .
320420.fig.002
Figure 2: Plot of the evolution over time of for the Prisoner’s Dilemma game with .
320420.fig.003
Figure 3: Plot of the evolution over time of for the Prisoner’s Dilemma game with .

7. Conclusion

In this paper, the collocation method with sinc and Lagrange polynomials are employed to construct an approximation to the solution of continuous mixed strategy. It is found that the results of the present works agree well with trapezoidal rule. Properties of the sinc procedure are utilized to reduce the computation of this integrodifferential to some nonlinear equations. There are several advantages over classical methods to using approximations based on sinc numerical methods. First, unlike most numerical techniques, it is now well-established that they are characterized by exponentially decaying errors. Secondly, approximation by sinc functions handles singularities in the problem. Thirdly, due to their rapid convergence, sinc numerical methods do not suffer from the common instability problems associated with other numerical methods. Also, in this case the advantages of collocation method are used. The method is applied to test examples to illustrate the accuracy and implementation of the method.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

  1. J. W. Weibull, Evolutionary Game Theory, The MIT Press, 1995. View at MathSciNet
  2. D. Fudenberg and D. Levine, The Theory o f Learning in Games, MIT Press, 1998. View at MathSciNet
  3. L. Samuelson, Evolutionary Games and Equilibrium Selection, MIT Press, 1998. View at MathSciNet
  4. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  5. H. Gintis, Game Theory Evolving, Princeton University Press, 2000.
  6. R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press, 2003. View at MathSciNet
  7. Th. Vincent and J. Brown, Evolutionary Game Theory, Natural Selection and Darwinian Dynamics, Cambridge University Press, Cambridge, UK, 2005.
  8. J. Hofbauer and K. Sigmund, “Evolutionary game dynamics,” Bulletin of the American Mathematical Society, vol. 40, no. 4, pp. 479–519, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. I. M. Bomze, “Dynamical aspects of evolutionary stability,” Monatshefte für Mathematik, vol. 110, no. 3-4, pp. 189–206, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. R. Cressman, “Stability of the replicator equation with continuous strategy space,” Mathematical Social Sciences, vol. 50, no. 2, pp. 127–147, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. J. Hofbauer, J. Oechssler, and F. Riedel, “Brown-von Neumann-NASh dynamics: the continuous strategy case,” Games and Economic Behavior, vol. 65, no. 2, pp. 406–429, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. T. W. L. Norman, “Dynamically stable sets in infinite strategy spaces,” Games and Economic Behavior, vol. 62, no. 2, pp. 610–627, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. J. Oechssler and F. Riedel, “Evolutionary dynamics on infinite strategy spaces,” Economic Theory, vol. 17, no. 1, pp. 141–162, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. A. Boccabella, R. Natalini, and L. Pareschi, “On a continuous mixed strategies model for evolutionary game theory,” Kinetic and Related Models, vol. 4, no. 1, pp. 187–213, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, Pennsylvania, USA, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  17. D. F. Winter, K. L. Bowers, and J. Lund, “Wind-driven currents in a sea with a variable eddy viscosity calculated via a Sinc-Galerkin technique,” International Journal for Numerical Methods in Fluids, vol. 33, no. 7, pp. 1041–1073, 2000. View at Google Scholar · View at MathSciNet
  18. B. Bialecki, “Sinc-collocation methods for two-point boundary value problems,” IMA Journal of Numerical Analysis, vol. 11, no. 3, pp. 357–375, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. K. Parand and A. Pirkhedri, “Sinc-Collocation method for solving astrophysics equations,” New Astronomy, vol. 15, no. 6, pp. 533–537, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. K. Parand, M. Dehghan, and A. Pirkhedri, “Sinc-collocation method for solving the Blasius equation,” Physics Letters A, vol. 373, no. 44, pp. 4060–4065, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. K. Parand, Z. Delafkar, N. Pakniat, A. Pirkhedri, and M. K. Haji, “Collocation method using sinc and rational Legendre functions for solving Volterra's population model,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1811–1819, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. A. Saadatmandi, M. Razzaghi, and M. Dehghan, “Sinc-collocation methods for the solution of Hallen's integral equation,” Journal of Electromagnetic Waves and Applications, vol. 19, no. 2, pp. 245–256, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. A. Saadatmandi and M. Razzaghi, “The numerical solution of third-order boundary value problems using sinc-collocation method,” Communications in Numerical Methods in Engineering, vol. 23, no. 7, pp. 681–689, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. M. Dehghan and A. Saadatmandi, “The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method,” Mathematical and Computer Modelling, vol. 46, no. 11-12, pp. 1434–1441, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. M. El-Gamel, S. H. Behiry, and H. Hashish, “Numerical method for the solution of special nonlinear fourth-order boundary value problems,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 717–734, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. P. N. Dinh Alain, P. H. Quan, and D. D. Trong, “Sinc approximation of the heat distribution on the boundary of a two-dimensional finite slab,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 1103–1111, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. K. Abdella, X. Yu, and I. Kucuk, “Application of the Sinc method to a dynamic elasto-plastic problem,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 626–645, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. J. Lund and C. R. Vogel, “A fully-Galerkin method for the numerical solution of an inverse problem in a parabolic partial differential equation,” Inverse Problems, vol. 6, no. 2, pp. 205–217, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. A. Shidfar, R. Zolfaghari, and J. Damirchi, “Application of sinc-collocation method for solving an inverse problem,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 545–554, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. J. Rashidinia and M. Zarebnia, “The numerical solution of integro-differential equation by means of the sinc method,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1124–1130, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. M. El-Gamel and A. I. Zayed, “Sinc-Galerkin method for solving nonlinear boundary-value problems,” Computers & Mathematics with Applications, vol. 48, no. 9, pp. 1285–1298, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. A. Saadatmandi and M. Dehghan, “The use of sinc-collocation method for solving multi-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 593–601, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. A. Mohsen and M. El-Gamel, “On the numerical solution of linear and nonlinear Volterra integral and integro-differential equations,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3330–3337, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  34. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, 2004. View at Publisher · View at Google Scholar · View at MathSciNet