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

Authors: F. Hosseini Shekarabi
Journal: Journal of Computational Engineering
 (2014)

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## 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.


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## Body

## 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 [1–8]. 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 [9–13]. 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 Section2, 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 areN pure strategies R1 to RN and that the players can use mixed strategies: this consists of playing the pure strategies R1 to RN with some probabilities q1 to qN with qi≥0 and ∑‍qi=1. A strategy corresponds to a point q in the simplex
(1)SN-1={q=(q1,q2,…,qN)∈RN:qi≥0,∑i=1N‍qi=1}.
The corners of the simplex are the standard unit vectors ei, where the ith component is 1 and all others are 0 and correspond to the N pure strategies Ri, i=1,…,N.Let us denote byaij the payoff for a player using the pure strategy Ri against a player using the pure strategy Rj. Here Matrix A=(aij) is called payoff matrix. An Ri-strategist obtain the expected payoff (Aq*)i=∑j‍aijqj* against a q*-strategist. The payoff for a q-startegist against a q* strategist is given by
(2)A(q,q*):=q·Aq*=∑i,j=1N‍aijqiqj*.We consider a population of individuals as a player of the game and denote byf(t,q) the density of population adopting the q strategy at time t; the evolution in time of f, due to dynamics of the game, is driven by
(3)∂tf(t,q)=f(t,q)(∫SN‍A(q,q*)f(t,q*)dq*-ϕ(f)),
where the term
(4)∫SN‍A(q,q*)f(t,q*)dq*,
represents the payoff of the strategy q against all the others strategies, A(q,q*) being the interacting kernel between the q-strategist and the q*-strategist. The last term of (3) is defined by
(5)ϕ(f):=∬SNf(t,q)A(q,q*)f(t,q*)dq*dq
and represents the average payoff of the population.Since∑i=1N‍qi=1, we can reduce the number of variables, considering
(6)qN=1-∑i=1N-1‍qi
and obtaining the (N-1)-dimensional model (3) on the simplex
(7)τN-1∶={∑i=1N-1‍pi≤1p=(p1,p2,…,pN-1)∈RN-1:pi≥0,∑i=1N-1‍pi≤1},
namely,
(8)∂tf(t,p)=f(t,p)(∫τN-1‍A(p,p*)f(t,p*)dp*-ϕ(f)),
with A(p,p*) defined by
(9)A(p,p*)∶=p·Ap*=∑i,j=1N-1‍aijpipj*,
and ϕ defined by
(10)ϕ(f):=∬τN-1f(t,p)A(p,p*)f(t,p*)dp*dp.Remark 1 (see [14]).
If we take an initial condition(11)f(0,p)=f0(p)≥0,
with ∫N-1‍f0(p)dp=1, then it is easy to see that f≥0 for all t>0 and if f0(p¯)=0 for some p¯, then f(t,p¯)=0 for all t>0. We also know that
(12)∫τN-1‍f(t,p)dp=1,∀t>0.
This follows from the mass conservation; by integrating (8) with respect to p and using (10) and (12) we have
(13)∂t∫τN-1‍f(t,p)dp=0.
Let us introduce the moments for f:
(14)Mk(f)∶=∫τN-1‍pkf(p)dp=∫τN-1‍p1k1p2k2…pN-1kN-1f(p)dp,
with k∶=(k1,k2,…,kN-1). Using Mk(f), the payoff and the average payoff (10)
(15)∫τN-1‍A(p,p*)f(t,p*)dp*=∑j=1N-1‍Mej(f)(∑i=1N-1‍ϑi,jpi+ςj)+aN,N+∑i=1N-1‍υipi,ϕ(f)=∑j=1N-1‍Mej(f)(∑i=1N-1‍ϑi,jMej(f)+ςj)+aN,N+∑i=1N-1‍υiMei(f),
where ei∈RN-1 is the standard unit vector with the ith component equal to 1 and all others equal to 0. Moreover, ϑi,j∶=ai,j-ai,N-aN,j+aN,N, ςj∶=aN,j-aN,N, υi∶=ai,N-aN,N.
In the final form of (8), that will be used later in this paper, the only integral terms are the first moments Mei:
(16)∂tf(t,p)=f(t,p)(∑i=1N-1(pi-Mei(f))(υi+∑i=1N-1ϑi,jMej(f)))Global Existence of the Solutions. We consider the Cauchy problem (11)–(16) for t≥0 and p∈τN-1; that is,
(17)∂tf(t,p)=f(t,p)((υi+∑i=1N-1‍ϑi,jMej(f))∑i=1N-1‍(pi-Mei(f))∂tf(t,p)×(υi+∑i=1N-1‍ϑi,jMej(f))(υi+∑i=1N-1‍ϑi,jMej(f))),f(0,p)=f0(p)
with f0(p)≥0 and ∫τN-1‍f(t,p)dp=1.Proposition 2 (Local existence see [14]).
For allM>0 there exists T(M)>0 such that if ∥f0(p)∥≤M;then there exists a unique solution f∈C([0,T~]×τN-1) for the problem (17), for all T~≤T(M).


### 2.1. Two Strategies Games

Assume there are two different strategies, whose interplay is ruled by the payoff matrix:(18)A=(abcd).
In this case the simplex τ1 is just the interval [0,1] and so we have a population where individuals are going to play the first strategy with probability p∈[0,1] and the second strategy with probability 1-p. The payoff (2) is given by
(19)A(p,p*)∶=(p1-p)(abcd)(p*1-p*)=(a+d-b-c)pp*+(b-d)p+(c-d)p*+d=αpp*+βp+γp*+δ
with
(20)α∶=(a+d-b-c),β∶=b-d,γ∶=c-d,δ∶=d.
The one dimensional Cauchy problem (17) reads
(21)∂tf(p)=f(p)(αM1(f)+β)(p-M1(f))fdsfdfdffddfffhhfdffft≥0,p∈[0,1]f(0,p)=f0(p)p∈[0,1]
with f0(p)≥0 and ∫01‍f0(p)dp=1.For more detail see [14].


## 2.1. Two Strategies Games

Assume there are two different strategies, whose interplay is ruled by the payoff matrix:(18)A=(abcd).
In this case the simplex τ1 is just the interval [0,1] and so we have a population where individuals are going to play the first strategy with probability p∈[0,1] and the second strategy with probability 1-p. The payoff (2) is given by
(19)A(p,p*)∶=(p1-p)(abcd)(p*1-p*)=(a+d-b-c)pp*+(b-d)p+(c-d)p*+d=αpp*+βp+γp*+δ
with
(20)α∶=(a+d-b-c),β∶=b-d,γ∶=c-d,δ∶=d.
The one dimensional Cauchy problem (17) reads
(21)∂tf(p)=f(p)(αM1(f)+β)(p-M1(f))fdsfdfdffddfffhhfdffft≥0,p∈[0,1]f(0,p)=f0(p)p∈[0,1]
with f0(p)≥0 and ∫01‍f0(p)dp=1.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 functionf(x) defined on the real line R is given by
(22)f(x)≈∑j=-NN‍f(jh)S(j,h)(x),
where S(j,h) is sinc function defined by
(23)S(j,h)(x)∶=sin[(π/h)(x-jh)][(π/h)(x-jh)].
And the step size h is suitably chosen for a given positive integer n=2N+1. Sinc for interpolation points xk=kh is given by
(24)S(j,h)(kh)=δjk(0)={1k=j,0k≠j.Assuming thatf(t) 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 N. The approximation may be extended to approximate f(t) on the interval [0,1] 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 τ=ϕ(t) and inverse transformation t=ψ(τ) such that ϕ(0)=-∞ and ϕ(1)=∞. We may write the sinc approximation employing the transformation for the function f(t) to be
(25)f(x)≈∑j=-NN‍f(ψ(jh))S(j,h)(x)(ϕ(t)),
where the mesh size h represents the separation between sinc points on the τ∈(-∞,∞) domain. In order to have the sinc approximation on a finite interval (0,1) conformal map is employed as follows:
(26)ϕ(z)=ln⁡(z1-z).
This map carries the eye-shaped complex domain
(27)DE={z=x+iy:|arg(z1-z)|<d≤π2},
onto the infinite strip
(28)Ds={w=u+iv:|v|<d≤π2}.
For the sinc method, the basic function on the interval (0,1) for z∈DE is derived from the composite translated sinc functions:
(29)Sj(z)=s(j,h)oϕ(z)=Sinc(ϕ(z)-jhh).
Exhibiting kroneckor delta behavior on the grid points
(30)xk=ϕ-1(kh)=ekh1+ekh,k=±1,±2,….
Thus we may define the inverse images of the real line and of the evenly space nodes {kh}k=-∞∞ as
(31)Γ={ϕ-1(x)∈DE:-∞<x<∞}=(0,1).
And quadrature formulas for f(t) over [0,1] are
(32)f(x)≈∑k=-NN‍f(xk)S(k,h)oϕ(x),∫01‍f(x)dx≈h∑k=-NN‍f(xk)ϕ′(xk).Definition 3.
LetB(DE) be the class of functions F which are analytic in DE satisfy
(33)∫ψ(t+L)‍|F(z)dz|⟶0,t⟶±∞,
where, L={iv:|v|<d≤π/2} and on the boundary of DE (denoted by ∂DE) satisfying
(34)N(F)=∫∂DE‍|F(z)dz|<∞.
Interpolation for function in B(DE) is defined in the following theorem whose proof can be found in [15].Theorem 4.
Ifϕ′F∈B(DE) then for all x∈Γ(35)|F(x)-∑k=-∞∞‍F(xk)S(k,h)oϕ(x)|≤N(Fϕ′)2πdsinh⁡(πd/h)≤2N(Fϕ′)πde-πd/h.
Moreover, if |F(x)|≤Ce-α|ϕ(x)|,x∈Γ for some positive constants C and α, and if the selection h=πd/αN≤2πd/ln⁡2 then
(36)|F(x)-∑k=-NN‍F(xk)S(k,h)oϕ(x)|≤C2Nexp⁡(-πdαN),x∈Γ,
where, C2 depends only on F, d, and α. The above expressions show sinc interpolation on B(DE) converge exponentially [17]. We also require derivatives of composite sinc functions evaluated at the nodes. The expressions required for the present discussion are [25]
(37)δk,j(0)=[S(k,h)oϕ(x)]x=xj={1k=j,0k≠j,δk,j(1)=ddϕ[S(k,h)oϕ(x)]x=xj=1h{0k=j,(-1)j-kj-kk≠j,δk,j(2)=d2dϕ2[S(k,h)oϕ(x)]x=xj=1h2{-π23k=j,-2(-1)j-k(j-k)2k≠j.


## 4. Collocation Method

LetIh={tn:0=t0<t1<⋯<tN=T} be a given mesh (not necessarily uniform) on I and set σn∶=(tn,tn+1], σ¯n∶=[tn,tn+1], with hn∶=tn+1-tn(n=0,1,…,N-1). The quantity h∶=max⁡{hn:0≤n≤N-1} will be called the diameter of the mesh Ih.Definition 5.
Suppose thatIh is a given partition on I. The piecewise polynomials space Sμ(d)(Ih) with μ≥0,-1≤d≤μ is defined by
(38)Sμ(d)(Ih)={q(t)∈Cd[I,R]:q∣σn∈πμ;0≤n≤N-1}.
Hereσn=(tn,tn+1] and πμ denote the space of polynomials of degree not exceeding μ, and it is easy to see that Sμ(d)(Ih)(39)Sμ(d)(Ih)=N(μ-d)+d+1.
The collocation solution is determined by uh that satisfies the given equation on a given suitable finite subset Xh of I, where Xh contains the collocation points:
(40)Xh={tn+cihn:0≤c1≤⋯≤cm≤1;0≤n≤N-1}
is determined by the points of the partition Ih and the given collocation parameters {ci}∈[0,1]. The collocation solution uh∈Sm(0)(Ih) for
(41)y´(t)=f(t,y(t))y(0)=y0
is defined by the collocation equation
(42)u´h(t)=f(t,uh(t)),t∈Xh,uh(0)=y0.
It will be convenient (and natural) to work with the local Lagrange basis representations of uh. These polynomials in σn can be written as
(43)Lj(z)=Πk≠jm(z-ck)(cj-ck),z∈[0,1],j=1,…,m,
where Lj(z) belong to πm-1. Also we have
(44)uh(tn+vhn)=∑j=1m‍Lj(v)Yn,j,v∈(0,1],dsdsdsdsdsdsdsdsdYn,j∶=uh(tn+cjhn).
From (44) we can obtain the local representation of uh∈Sm(0)(Ih) on σ¯n, hence we can achieve that
(45)uh′(tn+vhn)=1hn∑j=1m‍Lj′(v)Yn,j.
The unknown approximations Yn,i(i=1,…,m) in (44) are defined by the solution of a system of (generally nonlinear) algebraic equations obtained by setting t=tn,i∶=tn+cihn in the collocation equation (42) and employing the local representation (44). This system is
(46)1hn∑j=1m‍Lj′(ci)Yn,j=f(tn,i,Yn,i),i=1,…,m.
It corresponds to (44) with v=1,
(47)yn+1∶=uh(tn+hn)=∑j=1m‍Lj(1)Yn,j(n=0,1,…,N-1).
I present the result on global convergence for the linear initial-value problem(48)y´(t)=a(t)y(t)+g(t),t∈I,y(0)=y0.Theorem 6.
Assume that(a)
the given functions in (48) satisfy a,g∈Cm(I);(b)
h¯>0 is such that, for any h∈(0,h¯), each of the linear systems of method has a unique solution.Then the estimates
(49)∥y-uh∥∞∶=max⁡t∈I⁡|y(t)-uh(t)|≤C0∥y(m+1)∥∞hm,(50)∥y´-u´h∥∞∶=max⁡t∈I⁡|y´(t)-uh´(t)|≤C1∥y(m+1)∥∞hm.See [34].


## 5. Construction of the Method

Let{0=t0<t1<⋯<tN=T} is a partition of [0,T]. In every interval σn=(tn,tn+1] we assume that f(t,p), p∈[0,1], solution of one dimensional mixed strategy model is approximated by the finite expansion of sinc basis function and Lagrange polynomials:
(51)f(t,p)≃∑j=0m‍∑k=-NN‍cj,kS(k,h)oϕ(p)Ψj(t),
where, Ψj(t) is a polynomial of degree m. Also, initial value is according to
(52)f(tn,p)=ftn(p).
If we replace approximation (51) in (21) we have
(53)∑j=0m‍∑k=-NN‍cj,kS(k,h)oϕ(p)Ψj′(t)=∑j=0m‍∑k=-NN‍cj,kS(k,h)oϕ(p)Ψj(t)gdfgfdgfdgfg×(αM1(f(t,p))+β)(p-M1(f(t,p))),
where M1(f) is taken by
(54)M1(f(t,p))=∑j=0m‍∑k=-NN‍cj,k∫01‍p·S(k,h)oϕ(p)Ψj(t)dp.
By substituting collocation points for t and p and using quadrature rule (32), a nonlinear system is given.After solving this system we calculatecj,k and finally f(t,p) in σn, and also, f(t,p) at tn+1:
(55)f(tn+1,p)=∑j=0m‍∑k=-NN‍cj,kS(k,h)oϕ(p)Ψj(tn+1),
where f(tn+1,p) is used as an initial value for next interval σn+1. After N 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:
(56)A=(RSTP).
If both players cooperate both obtain R fitness units (reward payoff); if both defect, each receives P (punishment payoff); if one player cooperates and the other defects, the cooperator gets S (suckers payoff) while the defector gets T (temptation payoff). The payoff values are ranked T>R>P>S and 2R>T+S. We know that cooperators are always dominated by defectors.For the numerical tests we fix the following normalized payoff matrix:(57)A=(10bϵ),
with b=1.1 and ϵ=0.001. In this case we have α=1-b+ϵ<0 and β=-ϵ<0 and so β/α>0. This means that stationary solutions are expected to be given by concentrated Dirac masses. For general perturbation we have that p¯=0 is linearly stable.We In order to conform the results above, initial condition is considered as below:(1)
(58)f0(p)=1,∀p∈[0,1],(2)
(59)f0(p)=-p2+23p+1,∀p∈[0,1],(3)
(60)f0(p)={2p∈[14,12]∪[34,1],0elsewhere.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 f tends to concentrate at the point p=0, to what we expected.Figure 1
Plot of the evolution over time off(t,p) for the Prisoner’s Dilemma game with f0(p)=1.Figure 2
Plot of the evolution over time off(t,p) for the Prisoner’s Dilemma game with f0(p)=-p2+(2/3)p+1.Figure 3
Plot of the evolution over time off(t,p) for the Prisoner’s Dilemma game with f0(p)={2p∈[1/4,1/2]∪[3/4,1],0elsewhere..


## 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.



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