On a special two-person dynamic game
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Matsumoto, Akio; Szidarovszky, Ferenc; Hamidi, Maryam Article On a special two-person dynamic game Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Matsumoto, Akio; Szidarovszky, Ferenc; Hamidi, Maryam (2023) : On a special two-person dynamic game, Games, ISSN 2073-4336, MDPI, Basel, Vol. 14, Iss. 6, pp. 1-22, https://doi.org/10.3390/g14060067 This Version is available at: https://hdl.handle.net/10419/330060 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Citation: Matsumoto, A.; Szidarovszky, F.; Hamidi, M. On a Special Two-Person Dynamic Game. Games 2023,14, 67. https://doi.org/ 10.3390/g14060067 Academic Editors: Ulrich Berger, Heinrich H. Nax and Yuval Heller Received: 29 September 2023 Revised: 17 October 2023 Accepted: 19 October 2023 Published: 24 October 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). games Article On a Special Two-Person Dynamic Game Akio Matsumoto 1,*, Ferenc Szidarovszky 2and Maryam Hamidi 3 1Department of Economics, Chuo University, 742-1 Higashi-Nakano, Tokyo 192-0393, Japan 2 Department of Mathematics, Corvinus University, Föván tur 8, 1093 Budapest, Hungary; [email protected] 3Department of Industrial and Systems Engineering, Lamar University, 2210 Cherry Engineering Building, Beaumort, TX 77710, USA; [email protected] *Correspondence: [email protected] Abstract: The asymptotical properties of a special dynamic two-person game are examined under bestresponse dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper. Keywords: delay two-person game; stability switching; single and multiple delays; delay differential equation; best reply dynamics 1. Introduction Game theory is one of the most frequently studied fields in mathematical economics. Its foundation and main concepts are discussed in many textbooks and monographs, for example, Dresher (1961), Szép and Forgó (1985), Vorob’ev (1994), and Matsumoto and Szidarovszky (2016) [ 1 – 4 ]. These include both two-person and general n -person games. In the earliest decades, the existence and uniqueness of the Nash equilibrium were the central research issues (von Neumann and Morgenstern, 1944; Nash, 1951; Fudenberg and Tirole, 1991; Rosen, 1965) [ 5 – 8 ], and then the dynamic extensions of these static games received increasing attention (Okuguchi, 1976; Hahn, 1962) [ 9 , 10 ]. Models have been developed and examined in both discrete and continuous time scales. In most studies, the dynamic processes are driven by either gradient adjustments or best-response dynamics. In the first case, only interior equilibria are the steady states of the resulting dynamic systems. In the second case, this difficulty is avoided; however, the construction of the dynamic processes requires knowledge of the best responses of the players (Bischi et al., 2010) [ 11 ]. In both cases, the asymptotical stability of the equilibrium in examined by applying the Lyapunov theory (Cheban, 2013, Saeed, 2017) [ 12 , 13 ] or local linearization (Bischi et al., 2010) [ 11 ]. In the discrete case, the theory of noninvertable maps and critical sets is a commonly used approach for nonlinear discrete systems (Gumowski and Mira, 1980; Mira et al., 1996) [ 14 , 15 ], while for continuous systems Bellman (1969), LaSalle (1968), and Sánchez Games 2023,14, 67. https://doi.org/10.3390/g14060067 https://www.mdpi.com/journal/games
Games 2023,14, 67 2 of 22 (1968) [ 16 – 18 ] can be suggested as the main references. After developments in the stability theory of differential equations, time delays were added to dynamic models, in which it is usually assumed that the players make decisions based on delayed information about their own and others’ actions. This additional assumption makes the models more realistic, since data collection, determining the best decision alternatives, and their implementation need time (Bellman and Cooke, 1963) [ 19 ]. Delay systems have many applications in population dynamics (Kuang, 1993; Cushing, 1977) [ 20 , 21 ]; economics (Gori et al., 2014; Ösbay et al., 2017) [ 22 , 23 ]; and engineering (Berezowski, 2001; Wang et al., 1999) [ 24 , 25 ], among other disciplines. A two-person continuous game is considered, in which the strategy sets S1 and S2 are compact intervals and the payoff functions ϕk(x1 , x2) ( k= 1,2) are continuous and strictly concave in xk , which is the strategy of player k . Under these conditions, ϕk(x1 , x2) has a unique maximizer Rk(x`) for all x`∈S`(`6=k) . This function is called the best response of player k . At each time period, the players select strategies to move closer to their best responses. In the discrete case, this concept is realized by the difference equation system x1(t+1) = x1(t) + α1[R1(x2(t)) −x1(t)], (1) x2(t+1) = x2(t) + α2[R2(x1(t)) −x2(t)], (2) where α1 and α2 are positive adjustment coefficients. In order to avoid overshooting, it is assumed that both are less than unity. In the continuous case, the following differential equation system describes the dynamic evolution of the strategies: ˙ x1(t) = α1[R1(x2(t)) −x1(t)], (3) ˙ x2(t) = α2[R2(x1(t)) −x2(t)], (4) where α1,α2>0. We provide one example here. Consider a duopoly, when two firms produce the same product and sell their outputs in the same market. Let x1 and x2 denote the production levels of the firms. The production costs are C1(x1) = a1x1+b1and C2(x2) = a2x2+b2, and the selling unit price is p(x1+x2) = A−B(x1+x2). The profit of firm i(i=1, 2) is given as the difference between its revenue and cost: ϕi(x1,x2) = xiA−Bxi−Bxj−(aixi+bi) (i6=j). It is a natural assumption that firm i does not have instantaneous information on the output level of the competitor, so at time t , it believes that the output of the competitor is xj(t−τj). The marginal profit of the firm is clearly ∂ϕi(x1,x2) ∂x1=A−2Bxi−Bxj−ai. Hence, at time t, the best believed response of firm iis Ri(xj(t−τj)) = A−ai−Bxj(t−τj) 2Bif numerator is positive, 0 otherwise.
Games 2023,14, 67 3 of 22 The best response dynamics have the following forms: xi(t+1) = xi(t) + αiRi(xj(t−τj)) −xi(t)for i=1,2 in the discrete-time case, and ˙ xi(t) = αiRi(xj(t−τj)) −xi(t)for i=1, 2 in the continuous-time case, leading to a delay difference and differential equations. In this paper, a special two-person game is examined with best-response dynamics in discrete and continuous time scales. The main question we try to answer is the following: How does the presence of time delays affect the asymptotical properties of a dynamics extension of a special stable two-person game? In the case of one delay, the stability interval or intervals are determined when the stability of the equilibrium is still preserved. In the case of multiple delays, the stability regions in the delay space are determined, where the equilibrium is still stable. This paper introduces a mathematical methodology to find the stability intervals and regions. The paper is developed as follows. Section 2considers discrete time scales and is divided into two parts: models with and without time delays. The structure of Section 3is similar, with two subsections. Section 4introduces and examines alternative models, and Section 5offers concluding remarks and further research areas. 2. Discrete Systems 2.1. Stability without Time Delays Assuming the differentiability of the best-response functions, we can linearize Equations (1 ) and (2) about an equilibrium that is also a steady state of the system. Let x∗ 1 and x∗ 2 denote the equilibrium strategies and let ¯ x1(t) = x1(t)−x∗ 1and ¯ x2(t) = x2(t)−x∗ 2 be the discrepancies of the strategies from their equilibrium levels. Linearizing Equations (1) and (2) around (x∗ 1,x∗ 2)gives the following: ¯ x1(t+1) = (1−α1)¯ x1(t) + α1R0 1(x∗ 2)¯ x2(t)(5) and ¯ x2(t+1) = (1−α2)¯ x2(t) + α2R0 2(x∗ 1)¯ x1(t). (6) This is a linear system. The asymptotic behavior of the state trajectories depends on the locations of the eigenvalues of the system. For finding the eigenvalues, we consider exponential solutions ¯ x1(t) = λtu and ¯ x2(t) = λtv . Substituting them into Equations (5)and (6) , we see that λt+1u= (1−α1)λtu+α1R0 1(x∗ 2)λtv, λt+1v= (1−α2)λtv+α2R0 2(x∗ 1)λtu. After simplifying both equations by λt , a linear algebraic system is obtained for u and v, and nonzero solutions exist if and only if the determinant of the system is zero: 0=det 1−α1−λ α1R0 1(x∗ 2) α2R0 2(x∗ 1)1−α2−λ! =λ2−λ(2−α1−α2)+ (1−α1)(1−α2)−α1α2R0 1(x∗ 2)R0 2(x∗ 1). (7) This is a quadratic polynomial λ2+a1λ+a2with a1=−(2−α1−α2)and a2= (1−α1)(1−α2)−α1α2R0 1(x∗ 2)R0 2(x∗ 1).
Games 2023,14, 67 4 of 22 It is well known (see, for example, Appendix F of Bischi et al., 2010) [ 11 ] that its roots are inside the unit circle if and only if 1+a1+a2>0, 1−a1+a2<0, a2<1. In our case, these conditions can be written as R0 1(x∗ 2)R0 2(x∗ 1)<1, R0 1(x∗ 2)R0 2(x∗ 1)<1+4−2α1−2α2 α1α2, R0 1(x∗ 2)R0 2(x∗ 1)>1−α1+α2 α1α2. Since α1 and α2 are below unity, the first inequality is stronger than the second one, so we have the following: Proposition 1. The equilibrium is locally asymptotically stable if and only if 1−α1+α2 α1α2 <R0 1(x∗ 2)R0 2(x∗ 1)<1. (8) Notice that α1+α2 α1α2=1 α2+1 α1 >2, so the left-hand side of condition (8) is negative and below −1. 2.2. Stability with Delays It is now assumed that the players have access only to delayed information about the strategies of the others. We can select the time unit as the length of the common delay. Thus, in Equation (1), x2(t) is replaced by x2(t− 1 ) , and in (2), x1(t) is replaced by x1(t− 1 ) . Assuming again exponential solutions ¯ x1(t) = λtuand ¯ x2(t) = λtv, we have λt+1u= (1−α1)λtu+α1R0 1(x∗ 2)λt−1v, λt+1v= (1−α2)λtv+α2R0 2(x∗ 1)λt−1u. After simplifying by λt−1, the resulting algebraic system has the form λ2u= (1−α1)λu+α1R0 1(x∗ 2)v, λ2v= (1−α2)λv+α2R0 2(x∗ 1)u. A nonzero solution of uand vexists if and only if 0=det (1−α1)λ−λ2α1R0 1(x∗ 2) α2R0 2(x∗ 1)(1−α2)λ−λ2! =λ4−λ3(2−α1−α2)+ (1−α1)(1−α2)λ2−α1α2R0 1(x∗ 2)R0 2(x∗ 1). This is a quartic equation λ4+a1λ3+a2λ2+a3λ+a4with a1=−(2−α1−α2),a2= (1−α1)(1−α2),a3=0 and a4=−α1α2R0 1(x∗ 2)R0 2(x∗ 1).
Games 2023,14, 67 5 of 22 Farebrother (1973) [ 26 ] showed that the sufficient and necessary conditions to only have roots inside the unit circle are as follows: a4<1, 3+3a4>a2, 1+a1+a2+a3+a4>0, 1−a1+a2−a3+a4>0, (1−a4)(1−a2 4)−a2(1−a4)2+ (a1−a3)(a3−a1a4)>0. Notice that the first four inequalities are linear in a4 as well as in R0 1(x∗ 2)R0 2(x∗ 1) ; however, the last inequality is cubic in a4 , so it is difficult to find simple stability conditions. Nevertheless, the first four inequalities give necessary stability conditions. They can be written as R0 1(x∗ 2)R0 2(x∗ 1) >−1 α1α2, <α1+α2−α1α2+2 3α1α2, <1, <1+4−2α1−2α2 α1α2. Notice that −1 α1α2 <−1 α1+α2−α1α2+2 3α1α2=1 31 α2+1 α1−1 3+2 3 1 α1α2 >1 and 1+4−2α1−2α2 α1α2 >1. Hence, the necessary conditions are −1 α1α2 <R0 1(x∗ 2)R0 2(x∗ 1)<1. (9) In this case, −α1α2<a4<1. Let F(a4)denote the left-hand side of last stability condition. Notice that F(1) = −a2 1<0 and F(0) = 1−a2>0, so its sign is indeterminate in general. We will now examine the addition condition F(a4)> 0 in interval (−α1α2 ,1 ) . Clearly, F(a4)=(1−a4)(1−a2 4)−a2(1−a4)2−a2 1a4 =a3 4+a2 4(−1−a2) + a4(−1+2a2−a2 1) + 1−a2. Notice first that −1−a2≤ −1, 1 −a2≥0 and −1+2a2−a2 1=−(α1−1)2−(α2−1)2−1≤ −1. Since F0(a4) = 3a2 4+2a4(−1−a2) + (−1+2a2−a2 1),
Games 2023,14, 67 6 of 22 the stationary points of F(a4)are a± 4=1 6h2(1+a2)±√Di, where D=4(1+a2)2+12(1−2a2+a2 1)≥4(1+a2)2+12 ≥16, implying that a+ 4>0 and a− 4<0. Clearly, a+ 4>1 6[2+4]=1, being outside the range of the necessary condition. Additionally, F0(a4)>0 for a4<a− 4and a4>a+ 4, and F0(a4)<0 for a− 4<a4<a+ 4. Notice that F(0) = 1−a2≥0 and F(1) = −a2 1≤0, and so F(a4) has three real roots: one before a− 4 , one between 0 and 1, and one after 1. Consequently, if a∗ 4 denotes the negative root and a∗∗ 4 the positive root between 0 and 1, then F(a4)>0 if a∗ 4<a4<a∗∗ 4 or −a∗∗ 4 α1α2 <R0 1(x∗ 2)R0 2(x∗ 1)<min1, −a∗ 4 α1α2. (10) Proposition 2. A necessary condition for the local asymptotical stability of the equilibrium in the delayed model is condition (9). A sufficient and necessary condition is the simultaneous satisfaction of (9) and (10). 3. Continuous Systems 3.1. Stability without Time Delays We will now examine the stability of systems (3) and (4). Similarly to the discrete case, we linearize the system around the equilibrium (x∗ 1,x∗ 2)to have the following: ˙ x1(t) = −α1x1(t) + α1R0 1(x∗ 2)x2(t), (11) ˙ x2(t) = α1R0 2(x∗ 1)x1(t)−α2x2(t), (12) where overbar is avoided for simple notation. In order to find the eigenvalues, we look for the solutions in exponential forms as before: x1(t) = eλtuand x2(t) = eλtv. The substitution of these solutions into Equations (11) and (12) gives λeλtu=−α1eλtu+α1R0 1(x∗ 2)eλtv, λeλtv=α2R0 2(x∗ 1)eλtu−α2eλtv. After simplifying both equations by eλt , the resulting algebraic system for u and v has nonzero solutions if and only if
Games 2023,14, 67 7 of 22 0=det −α1−λ α1R0 1(x∗ 2) α2R0 2(x∗ 1)−α2−λ! =λ2+λ(α1+α2)+α1α2−α1α2R0 1(x∗ 2)R0 2(x∗ 1). This is a quadratic polynomial, λ2+a1λ+a2, with a1=α1+α2and a2=α1α21−R0 1(x∗ 2)R0 2(x∗ 1). It is well known (Bischi et al., 2010) [ 11 ] that the real parts of the roots of this quadratic equation are negative if and only if both a1 and a2 are positive, which implies the following: Proposition 3. The equilibrium with dynamics (3) and (4) is locally asymptotically stable if R0 1(x∗ 2)R0 2(x∗ 1)<1 (13) and unstable if R0 1(x∗ 2)R0 2(x∗ 1)>1. In comparing conditions (8), (9), (10), and (13), it is clear that the stability of the equilibrium with the continuous model is implied by its stability for the discrete cases with and without delays. In comparing the two discrete cases, notice that the sufficient and necessary stability condition of the no-delay case implies the necessary conditions for the delay case since −1 α1α2 <1−α1+α2 α1α2 or (1−α2)(α1−1)<0. 3.2. Stability with Delays Assume again that both players face delays, τ1 and τ2 , in the information about the strategies of the others. Therefore, in Equation (3), x2(t) is replaced by x2(t−τ1) , and in (4), x1(t) is replaced by x1(t−τ2) . The linearized equations can therefore be written as follows: ˙ x1(t) = −α1x1(t) + α1R0 1(x∗ 2)x2(t−τ2), (14) ˙ x2(t) = α1R0 2(x∗ 1)x1(t−τ1)−α2x2(t). (15) The eigenvalues clearly depend on the lengths of the delays. Searching for exponential solutions x1(t) = eλtuand x2(t) = eλtv and substituting them into these equations, we obtain λeλtu=−α1eλtu+α1R0 1(x∗ 2)eλ(t−τ2)v, λeλtv=α2R0 2(x∗ 1)eλ(t−τ1)u−α2eλtv. After simplifying by eλt , the resulting algebraic equations for u and v have nonzero solutions if and only if 0=det −α1−λ α1R0 1(x∗ 2)e−λτ2 α2R0 2(x∗ 1)e−λτ1−α2−λ! =λ2+λ(α1+α2)+α1α21−R0 1(x∗ 2)R0 2(x∗ 1)e−λ(τ1+τ2). (16) Notice first that the characteristic equation does not depend on the individual values of τ1 and τ2 —it depends on only τ=τ1+τ2 . We know that at τ1=τ2= 0 (no-delay case),
Games 2023,14, 67 8 of 22 the equilibrium is locally asymptotically stable if (13) holds. By increasing the value of τ from zero, the stability might be lost. In this case, an eigenvalue has a zero real part, λ=iω . Since the complex conjugate of an eigenvalue is also an eigenvalue, we can assume that ω> 0. Substituting this eigenvalue into the characteristic Equation (16), we have the following: −ω2+iω(α1+α2)+α1α21−R0 1(x∗ 2)R0 2(x∗ 1)(cos ωτ −isin ωτ)=0. The separation of the real and imaginary parts shows that α1α2R0 1(x∗ 2)R0 2(x∗ 1)cos ωτ =−ω2+α1α2, (17) α1α2R0 1(x∗ 2)R0 2(x∗ 1)sin ωτ =−ω(α1+α2). (18) By adding the squares of these equations, we have ω4+ω2α12+α2 2+α2 1α2 2h1−R0 1(x∗ 2)R0 2(x∗ 1)2i=0. (19) If R0 1(x∗ 2)R0 2(x∗ 1)≤ 1, then no stability switch occurs. If − 1 ≤R0 1(x∗ 2)R0 2(x∗ 1)< 1, then the equilibrium remains locally asymptotically stable for all τ> 0. If R0 1(x∗ 2)R0 2(x∗ 1) = 1, then the characteristic equation of the no-delay case has a negative and a zero eigenvalue. Thus, no conclusion can be drawn about stability, and the same is the case for all τ>0. Assume next that R0 1(x∗ 2)R0 2(x∗ 1)> 1. In this case, Equation (19) has two real roots for ω2, ω2 ±=−(α2 1+α2 2)±√D 2, where D= (α2 1+α2 2)2−4α2 1α2 2h1−R0 1(x∗ 2)R0 2(x∗ 1)2i>(α2 1+α2 2)2, implying that ω2 +> 0 and ω2 −< 0. Thus, we have a unique solution ω+ . From (17) and (18), we see that sin ωτ <0 and cos ωτ >0 if ω2 +<α1α2, =0 if ω2 +=α1α2, <0 if ω2 +>α1α2. Therefore, we have the critical values of the delay: τn=1 ω+"2nπ−cos−1 −ω2 ++α1α2 α1α2R0 1(x∗ 2)R0 2(x∗ 1)!#for n=1, 2, . . . (20) The direction of the stability switches can be determined by Hopf bifurcation. For this purpose, we select τ as the bifurcation parameter and consider the eigenvalues as functions of τ : λ=λ(τ) . Implicitly differentiating the characteristic Equation (16) with respect to τ , we have 2λλ0+λ0(α1+α2)−α1α2R0 1(x∗ 2)R0 2(x∗ 1)e−λτ(−λ−λ0τ) = 0, implying that λ0h2λ+α1+α2+α1α2τR0 1(x∗ 2)R0 2(x∗ 1)e−λτi+α1α2R0 1(x∗ 2)R0 2(x∗ 1)λe−λτ =0. From Equation (16), we know that α1α2R0 1(x∗ 2)R0 2(x∗ 1)e−λτ =λ2+λ(α1+α2) + α1α2,
Games 2023,14, 67 15 of 22 λv=α2R0 2(x∗ 1)u−v, with determinant equation 0=det λ+α1e−λτ1−α1R0 1(x∗ 2)e−λτ2 −α2R0 2(x∗ 1)λ+α2! =λ2+e−λτ1(α1λ+α1α2)+α2λ−α1α2R0 1(x∗ 2)R0 2(x∗ 1)e−λτ2, =P0(λ) + P1(λ)e−λτ1+P2(λ)e−λτ2, (46) where P0(λ) = λ2+α2λ, P1(λ) = α1λ+α1α2, P2(λ) = −α1α2R0 1(x∗ 2)R0 2(x∗ 1). In order to guarantee that Equation (46) is the characteristic equation of a delay system, we assume the following: (a) deg(P0(λ))≥max{deg(P1(λ)), deg(P2(λ))} , so there are finitely many eigenvalues on C+. This clearly holds. (b) P0( 0 ) + P1( 0 ) + P2( 0 )6= 0, which holds if R0 1(x∗ 2)R0 2(x∗ 1)6= 1, as is assumed in the following. (c) Polynomials P0(λ) , P1(λ) , and P2(λ) have no common roots. This is obvious if R0 1(x∗ 2) R0 2(x∗ 1)6= 0. If R0 1(x∗ 2)R0 2(x∗ 1) = 0, then λ=−α2 is a common root, which does not remove stability. (d) lim λ→∞ P1(λ) P0(λ) + P2(λ) P0(λ)<1, which is again obvious, since P0(λ) is quadratic, P1(λ) is linear, and P2(λ) is constant. We will now follow the suggestions given by Gu et al. (2005) [ 27 ]. Dividing both sides of (46) by P0(λ), we obtain 1+a1(λ)e−λτ1+a2(λ)e−λτ2=0, (47) where a1(λ) = P1(λ) P0(λ)=α1 λ and a2(λ) = P2(λ) P0(λ)=−α1α2R0 1(x∗ 2)R0 2(x∗ 1) λ2+α2λ. We are looking for pure complex solutions, λ=iω . If P0(iω) = 0, then from (46), we have P1(iω) + P2(iω)e−iω(τ2−τ1)=0, implying that τ2−τ1=1 ωarg−P2(iω) P1(iω)+2nπ. However, if P1(iω) = 0 as well, then P2(iω)6= 0 implies no solution, and P2(iω) = 0 implies that arbitrary positive τ1 and τ2 values are solutions. Notice that this case cannot occur, since P0(iω)has only two real roots, λ=0 and λ=−α2. Consider next the case of P0(iω)6= 0. Then, notice that the complex vectors 1, a1(iω)e−iωτ1, and a2(iω)e−iωτ2form a triangle in the complex plane, as shown in Figure 1.
Games 2023,14, 67 16 of 22 Figure 1. Triangle constructed by these vectors. In the special case of a1(iω) = 0, (47) implies that 1+a2(iω)e−iωτ2=0, so τ1is arbitrary and τn 2=1 ω[arg(−a2(iω))+2nπ]. If a2(iω) = 0, then, similarly, 1+a1(iω)e−iωτ1=0, implying that τ2is arbitrary and τm 1=1 ω[arg(−a1(iω))+2mπ]. If a1(iω)and a2(iω)are nonzero, then the three vectors form a triangle if and only if |a1(iω)|+|a2(iω)|≥1, (48) −1≤|a1(iω)|−|a2(iω)|≤1. (49) In our case, a1(iω) = −iα1 ωso |a1(iω)|=α1 ω, a2(iω) = −α1α2R0 1(x∗ 2)R0 2(x∗ 1) −ω2+iωα2so |a2(iω)|=α1α2R0 1(x∗ 2)R0 2(x∗ 1) ωqω2+α2 2 . Relations (48) and (49) are quartic inequalities. In general, they cannot be solved, but if the numerical values are known, then the roots of the quartic equations can be determined, and certain segments between the roots provide solutions. The law of cosine can be used to find angles θ1and θ2: θ1=cos−1 1+|a1(iω)|2−|a2(iω)|2 2|a1(iω)|!, and θ2=cos−1 1+|a2(iω)|2−|a1(iω)|2 2|a2(iω)|!.
Games 2023,14, 67 17 of 22 The triangle can be placed above and under the horizontal axis, so from Figure 1, we have arg[a1(iω)]−ωτ1±θ1=π and arg[a2(iω)]−ωτ2∓θ2=π, implying that the critical values are as follows: τn± 1(ω) = 1 ω[arg[a1(iω)]+ (2n−1)π±θ1](50) and τm± 2(ω) = 1 ω[arg[a2(iω)]+ (2m−1)π∓θ2]. (51) Let Ω denote the set of all solutions of conditions (48) and (49), which consists finitely of many intervals: Ωk(k=1, 2, . . . N). For k=1, 2, . . . N, we define Tk± n,m=τn± 1(ω),τm± 2(ω)ω∈Ωk and Tk=∪n∪m(Tk+ n,m∪Tk− n,m)∩R2 +. The set T of all stability-switching curves is the union of all sets Tk . By not restricting arg[a1(iω)] and arg[a2(iω)] to interval [ 0,2 π] , we make them continuous functions of ω in Ωk, so with fixed nand m,Tk+ n,mand Tk− n,malso become continuous curves. Notice that any left endpoint ω` k and right endpoint ωr k of Ω satisfy at least one of the conditions (48) and (49) by equality, so one of the following equalities must hold: |a1(iω)|+|a2(iω)|=1, (52) |a2(iω)|−|a1(iω)|=1, (53) |a1(iω)|−|a2(iω)|=1, (54) and the case of ω` k= 0 and ωr k=∞ are also possible. If (52) holds, then θ1=θ2= 0 and Tk+ n,m is connected with Tk− n,m at this endpoint. If (53) holds, then similarly, θ1=π and θ2= 0, so Tk+ n,m is connected with Tk− n+1,m at this endpoint, and if (54) is satisfied, then θ1=0 and θ2=π, showing that at this endpoint Tk+ n,mis connected with Tk− n,m−1. If ω` k=0, then as ω→ 0, both Tk+ n,m and Tk− n,m converge to infinity. This discussion shows that the stability-switching curve T is the intersection of R2 + and conforms to one of the following types: (i) A series of closed curves; (ii) A series of spiral-like curves with either horizontal, vertical, or diagonal axes; (iii) A series of open-ended curves with endpoints converging to infinity. If a point ( τ1 , τ2 ) crosses the stability-switching curve, then stability switching might occur. Before discussing the direction of stability switching (stability loss or gain), some comments are in order. The direction of a curve is positive if it corresponds to an increasing value of ω . The direction is reversed if a curve is passing through an endpoint. Moving along a curve in the positive direction, the region on the left-hand side is called the region on the left, and the region on the right right is similar. We define for any point on Tthe following: R`=Reha`(iω)e−iωτ`iand I`=Imha`(iω)e−iωτ`ifor `=1, 2 and ω∈Ω.
Games 2023,14, 67 18 of 22 The following result is given by Gu et al. (2005) [27]. Theorem 1. Let ω∈Ωk and (τ1 , τ2)∈Tk such that λ=iω is a simple pure complex eigenvalue. As a point (τ1 , τ2) moves from the right to the left of the corresponding curve of Tk , a pair of eigenvalues cross the imaginary axes to the right (stability loss) if R2I1−R1I2> 0. If the inequality is reversed, then the crossing is from right to left (stability gain). 4.3. Three-Delay Model Consider next the case in which the players have different delays in the data about their own strategies. In this case, the delay equations become ˙ x1(t) = α1[R1(x2(t)) −x1(t−τ1)], ˙ x2(t) = α2[R2(x1(t)) −x2(t−τ2)]. By linearizing these equations around the equilibrium, we obtain ˙ x1(t) = −α1x1(t−τ1) + α1R0 1(x∗ 2)x2(t), (55) ˙ x2(t) = α2R0 2(x∗ 1)x1(t)−α2x2(t−τ2). (56) Substituting the exponential solutions x1(t) = eλtuand x2(t) = eλtv into these equations, we have λeλtu=−α1eλ(t−τ1)u+α1R0 1(x∗ 2)eλtv, λeλtv=α2R0 2(x∗ 1)eλtu−α2eλ(t−τ2)v. After simplifying these equations by eλt , an algebraic equation system is obtained for uand v, and nonzero solutions exist if and only if 0=det λ+α1e−λτ1−α1R0 1(x∗ 2) −α2R0 2(x∗ 1)λ+α2e−λτ2! =λ2+α1λe−λτ1+α2λe−λτ2−α1α2R0 1(x∗ 2)R0 2(x∗ 1) + α1α2e−λ(τ1+τ2), which can be written as P0(λ) + P1(λ)e−λτ1+P2(λ)e−λτ2+P3(λ)e−λ(τ1+τ2)=0, (57) where P0(λ) = λ2−α1α2R0 1(x∗ 2)R0 2(x∗ 1), P1(λ) = α1λ,P2(λ) = α2λand P3(λ) = α1α2. The following assumptions are made to guarantee that (57) can be the characteristic equation of a delay system and to exclude some trivial cases: (a) There are a finite number of eigenvalues on C+when deg(P0(λ))≥max{deg(P1(λ)), deg(P2(λ)), deg(P3(λ))}. This holds, since P0(λ) is quadratic, while P1(λ) and P2(λ) are linear and P3(λ) is constant. (b) The zero frequency λ=0 is not an eigenvalue with any τ1and τ2, P0(0) + P1(0) + P2(0) + P3(0)6=0.
Games 2023,14, 67 19 of 22 This condition holds if R0 1(x∗ 2)R0 2(x∗ 1)6= 1, so we have to assume this relation in the following discussion. (c) Polynomials P0(λ) , P1(λ) , P2(λ) , and P3(λ) have no common roots, which is trivial in our case. (d) lim P1(λ) P0(λ) + P2(λ) P0(λ) + P3(λ) P0(λ)<1, which also holds, since P0(λ)is quadratic and the others have lower degrees. In examining the stability switches with Equation (57), we will follow the ideas of Lin and Wang (2012) [28]. We can rewrite Equation (57) at λ=iωas follows: P0(iω) + P1(iω)e−iωτ1+P2(iω) + P3(iω)e−iωτ1e−iωτ2=0. (58) Since e−iωτ2=1, P0(iω) + P1(iω)e−iωτ1=P2(iω) + P3(iω)e−iωτ1 or P0+P1e−iωτ1¯ P0+¯ P1eiωτ1=P2+P3e−iωτ1¯ P2+¯ P3eiωτ1, (59) where an overbar indicates a complex conjugate and the arguments of P0 , P1 , P2 , and P3 are omitted for simple notation. A simple calculation shows that (59) has the equivalent form, |P0|2+|P1|2−|P2|2−|P3|2=2A1(ω)cos ωτ1−2B1(ω)sin ωτ1, (60) where A1(ω) = Re(P2¯ P3−P0¯ P1)and B1(ω) = Im(P2¯ P3−P0¯ P1). With any value of ω, this is a trigonometric equation for τ1. In our case, P0(iω) = −ω2−α1α2R0 1(x∗ 2)R0 2(x∗ 1), P1(iω) = iωα1, P2(iω) = iωα2, P3(iω) = α1α2. Thus, A1(ω) = Rehiωα1α22+ω2+α1α2R10(x2∗)R20(x1∗)(−iωα1)i=0 and B1(ω) = ωα1α2 2−α1ω3−α2 1α2ωR0 1(x∗ 2)R0 2(x∗ 1). Similarly, |P0|2+|P1|2−|P2|2−|P3|2 =ω2+α1α2R0 1(x∗ 2)R0 2(x∗ 1)2+ (α1ω)2−(α2ω)2−(α1α2)2 =ω4+ω2h2α1α2R0 1(x∗ 2)R0 2(x∗ 1) + α2 1−α2 2i−(α1α2)2h1−R0 1(x∗ 2)R0 2(x∗ 1)2i.
Games 2023,14, 67 20 of 22 (A) We assume first that ω is a solution of equation P2¯ P3−P0¯ P1= 0. Then, A1(ω) = B1(ω) = 0, and an arbitrary value of τ1> 0 is a solution. The corresponding values of τ2 can be obtained from Equation (58) as τn 2=1 ωarg−P2+P3e−iωτ1 P0+P1e−iωτ1+2nπ. (61) If the denominator is zero, then (58) implies that the numerator is also zero, so an arbitrary τ2>0 is a solution. (B) We assume next that P2¯ P3−P0¯ P16= 0, so A1(ω)2+B1(ω)2> 0. Clearly, there is a φ1(ω)such that cos(φ1(ω))=A1(ω) q|A1(ω)|2+|B1(ω)|2=0, sin(φ1(ω))=sign(B1(ω)). Then, (60) can be rewritten as |P0|2+|P1|2−|P2|2−|P3|2=2q|A1(ω)|2+|B1(ω)|2cos(φ1(ω) + ωτ1), (62) and by defining ψ1(ω) = cos−1 |P0|2+|P1|2−|P2|2−|P3|2 2q|A1(ω)|2+|B1(ω)|2 ∈[0, π] it becomes clear that τ± 1n=1 ω(±ψ1(ω)−φ1(ω) + 2nπ). (63) From (62), we see that a solution for τ1exists if and only if |P0|2+|P1|2−|P2|2−|P3|2≤2q|A1(ω)|2+|B1(ω)|2, (64) or in our case |P0|2+|P1|2−|P2|2−|P3|2−2B1|P0|2+|P1|2−|P2|2−|P3|2+2B1≤0. The two factors must have different signs, or one of them has to be zero. This inequality and (64) cannot be solved in general, but if numerical values of the model parameters are available, then computer solutions can present finitely many segments Ωk as the set of all solutions for ω , which is denoted by Ω . For each ω∈Ω , the corresponding critical values are given by (63), and then from (58), we have τm n=1 ωarg−P2(iω) + P3(iω)e−iωτ1 P0(iω) + P1(iω)e−iωτ1+2mπ if the denominator is nonzero. If it is zero, then (58) implies that the numerator is also zero, so an arbitrary τ2is the solution. A more simple method can be suggested by interchanging τ1 and τ2 and repeating the above procedure for τ2 . It can be shown that for each segment Ωk , the critical values form the curves as follows: Tk± n,m=±ψ1(ω)−φ1(ω) + 2nπ ω,∓ψ2(ω)−φ2(ω) + 2mπ ω ω∈Ωk, which is continuous in R2.
Games 2023,14, 67 21 of 22 With any left and right endpoints τ` kand τr k, F(ω` k) = F(ωr k) = 0, where F(ω) = |P0|2+|P1|2−|P2|2−|P3|22−4A1(ω)2+B1(ω)2; therefore, coshψi(ω` k)i=δ` iπand cos[ψi(ωr k)]=δr iπ, where δ` k , δr k∈ { 0,1 } . The connection of the segments of the stability-switching curves can be given as follows: Tk+ n,mconnects Tk− n+δ` 1,m−δ` 2and Tk− n+δ` 1,m−δr 2at its two ends: (i) If δ` 1,δ` 2=δr 1,δr 2 , then Tk+ n,m and Tk+ n−δ` 1,m−δ` 2 form a loop, so the set of stabilityswitching curves is a collection of closed continuous curves. (ii) Otherwise, it is a collection of continuous curves with endpoints on the axes or extending to infinity in R2. The direction of stability switching now depends on R`=RehP`(iω)e−iωτ`+P3(iω)e−iω(τ1+τ2)i and I`=ImhP`(iω)e−iωτ`+P3(iω)e−iω(τ1+τ2)i, and Theorem 1 remains valid with these quantities. 5. Conclusions In engineering, population dynamics, and realistic economies, there are many examples where only delayed responses can be observed and instantaneous data are not available. In such cases, the design and decisions are based on delayed data and information. In this paper, we presented some important methods to deal with this situation. Under discrete time scales, it is usually assumed that the lengths of the delays are positive integers that only increase the order of the governing difference equations (Matsumoto and Szidarovszky, 2018) [ 29 ]. In the continuous case, different methods can be used based on the number of delays. It is always assumed that the equilibrium is locally asymptotically stable without delays, and stability is preserved if the delays are sufficiently small based on the fact that the matrix eigenvalues continuously depend on the matrix elements. However, by increasing the lengths of the delays, this stability might be lost. In the one-delay case, the critical value of the delay was determined when stability turns into instability ( Sections 3.2 and 4.1 ). In the two-delay case, the two-dimensional space of the delays was considered, and the stability-switching curves were determined (Sections 4.2 and 4.3). If a pair of delays crosses this curve from the region containing the origin, then stability is lost. The same is the situation in the three-delay case, when two delays and their sum affect the dynamic properties of the equilibrium. The mathematical methodology is different in these cases, which was illustrated in the case of a special two-person game wherein the dynamic evolution of the strategies was governed by best-response dynamics. Several cases were used for the presentation of the methodology, which could be very useful in solving practical problems including one, two, or even three delays. The material of this paper can be extended in several directions. One could include more players, different dynamic rules, and more variants of the delayed quantities. In addition, the Bayesian methodology is used in population dynamics to assess the survival probabilities of competing species (Dragicevic, 2015) [ 30 ] or, in general, to find the probability of the occurence of certain properties among the players. For the same issues, artificial neural networks could be an alternative approach (Poulton, 2001; Swingler, 1996) [31,32].
Games 2023,14, 67 22 of 22 Author Contributions: Conceptualization, all authors.; methodology, A.M. and F.S.; software, A.M.; validation, A.M., F.S. and M.H.; formal analysis, F.S.; investigation, M.H.; writing—original draft preparation, F.S.; writing—review and editing, M.H.; visualization, A.M.; project administration, F.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful to three referees; their comments and suggestions were very helpful in finalizing the paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. Dresher, M. Games of Strategy: Theory and Applications; Applied Mathematics Series; Prentice Hall: Englewood Cliffs, NJ, USA, 1961 . 2. Matsumoto, A.; Szidarovszky, F. Game Theory and Its Applications; Springer: Tokyo, Japan, 2016. 3. Szep, J.; Forgo, F. 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