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Dynamic Games and Applications

Erschienen in:

01.03.2021

Games Without Winners: Catching-up with Asymmetric Spillovers

verfasst von: Anton Bondarev

Erschienen in: Dynamic Games and Applications | Ausgabe 4/2021

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Abstract

Multi-modal differential R&D game with asymmetric players is studied. It is demonstrated that under sufficiently asymmetric players there is no long-run ‘winner’ in this game in terms of developed technologies and all players try to imitate each other. Moreover, this outcome may be the only equilibrium in the cooperative game. In decentralized setting, additional complex types of dynamics are observed: permanent fluctuations around symmetric (pseudo)equilibrium and chaotic dynamics. This last is possible only once strategies of players are interdependent. These new emergent dynamics types call for additional regulation tools which are shortly discussed. It is shown that cooperative solution is qualitatively similar for any number of players, while non-cooperative solution is progressively complex given players are asymmetric. Results are extended to an arbitrary linear-quadratic multi-modal differential game with spillovers, and the structure necessary for the onset of non-deterministic chaos is discussed.

Vorheriger Artikel An Impulse-Regime Switching Game Model of Vertical Competition

Nächster Artikel Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

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although the revenue and profits are not explicitly introduced, it is assumed that the higher the quality of a given product, the higher is the profit from its sales on the market.

Both \(u_{k}(t), g_{k}(i,t)\) are arbitrary functions called optimal strategies, defined as optimizers of players’ payoffs via Maximum Principle.

This is not an essential restriction. If controls are allowed to jump one uses a suitable jump condition as e.g. in Boltyanski [4]. However, this significantly complicates exposition without affecting the main geometric insights of dynamics.

Observe that this is not a standard Stackelberg leadership, the notion of the leader is to be defined further.

All results below remain valid for the case \(\alpha ^{-}_{k}<\alpha ^{+}_{k}\). The only important thing is that coefficients are not equal across regimes player-wise, since otherwise only standard regime-switching as in Long et al. [31], Gromov and Gromova [24], Bondarev and Greiner [7], etc., may occur.

It will be demonstrated that this minor extension of asymmetry leads to qualitatively different results and substantially more complicated dynamics, see Remark 1.

To be defined later.

This is the case, since all dynamics in \(q_{k}(i,t)\) are governed by n(t) state-costate dynamics which in turn is linear-quadratic after substituting for \(\psi _{k}(i)\) as shown in Appendix A.

Note that \(i_{k}\) is not just one index, but a range of technologies.

By the equilibrium here the Nash equilibrium (either open-loop or Markov-perfect) is understood. We do not distinguish those two types for now since it is not important for subsequent analysis).

Observe that regimes refer to the dynamical system while modes to the global behaviour of the overall game.

if this is not the case, the problem either does not have a long-run solution (both steady states are unstable), or this solution is uniquely defined (one stable steady state).

the procedure to obtain the sliding dynamics is described in e.g. Colombo and Jeffrey [12] to which the reader is referred for technical details.

defined further on.

Since the problem is linear-quadratic in each of the regimes, Markov perfect solution via Hamilton–Jacobi–Bellman approach would yield topologically equivalent results (number of steady states and geometry of the state space), so we resort to a simpler concept of the open-loop equilibrium. This does not mean equilibria will be the same as OLNE is not a Markov perfect equilibrium.

provided assumption on both steady states are (saddle-type) stable, arguments are the same as for the cooperative game case.

I omit all standard types of dynamics possible once some of Lemmas 7–9 do not hold, since these can be described via the same mechanics as in Gromov and Gromova [24].

Roughly speaking, non-deterministic chaos is a set-valued flow, while standard chaotic dynamics is deterministic extremely sensitive to initial conditions. See Colombo and Jeffrey [12] for definition.

Akcigit U, Kerr WR (2018) Growth through heterogeneous innovations. J Political Econ 126(4):1374–1443CrossRef

Antonelli C (2016) Technological congruence and the economic complexity of technological change. Struct Change Econ Dyn 38(C):15–24CrossRef

Aseev SM, Veliov VM (2015) Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions. Proce Steklov Inst Math 291(1):22–39MathSciNetCrossRef

Boltyanski V (2004) The maximum principle for variable structure systems. Int J Control 77(17):1445–1451MathSciNetCrossRef

Bondarev A (2014) Endogenous specialization of heterogeneous innovative activities of firms under the technological spillovers. J Econ Dyn Control 38:235–249MathSciNetCrossRef

Bondarev A (2016) Intensity of R&D competition and the generation of innovations in heterogeneous setting. J Evolut Econ 26(3):621–653CrossRef

Bondarev A, Greiner A (2018) Catching-up and falling behind: Effects of learning in an R&D differential game with spillovers. J Econ Dyn Control 91:134–156MathSciNetCrossRef

Bondarev A, Greiner A (2019) Endogenous growth and structural change through vertical and horizontal innovations. Macroecon Dyn 23(1):52–79CrossRef

Brito PB, Costa LF, Dixon H (2013) Non-smooth dynamics and multiple equilibria in a Cournot–Ramsey model with endogenous markups. J Econ Dyn Control 37(11):2287–2306MathSciNetCrossRef

10.

Brito PB, Costa LF, Dixon HD (2017) From sunspots to black holes: singular dynamics in macroeconomic models. In: Nishimura K, Venditti A, Yannelis NC (eds) Sunspots and non-linear dynamics: essays in honor of Jean–Michel grandmont. Springer International Publishing, Cham, pp 41–70CrossRef

11.

Cellini R, Lambertini L (2009) Dynamic R&D with spillovers: competition versus cooperation. J Econ Dyn Control 33(3):568–582CrossRef

12.

Colombo A, Jeffrey MR (2011) Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows. SIAM J Appl Dyn Syst 10(2):423–451MathSciNetCrossRef

13.

D’Aspremont C, Jacquemin A (1988a) Cooperative and noncooperative R&D in duopoly with spillovers. Am Econ Rev 78(5):1133–1137

14.

D’Aspremont C, Jacquemin A (1988b) Cooperative and noncooperative R&D in duopoly with spillovers. Am Econ Rev 78(5):1133–1137

15.

Dawid H, Kopel M, Kort P (2013) R&D competition versus R&D cooperation in oligopolistic markets with evolving structure. Int J Ind Organ 31(5):527–537CrossRef

16.

Deal KR, Sethi SP, Thompson GL (1979) A bilinear-quadratic game in advertising. In: Liu PT, Sutinen JG (eds) Control Theory Math Econ. Marcel Dekker, New York, pp 91–109

17.

Di Bernardo M, Budd C, Champneys A, Kowalczyk P, Nordmark A, Olivar G, Piiroinen P (2008) Bifurcations in nonsmooth dynamical systems. SIAM Rev 50(4):629–701MathSciNetCrossRef

18.

Dixon DD (1995) Piecewise deterministic dynamics from the application of noise to singular equations of motion. J Phys A Math General 28(19):5539MathSciNetCrossRef

19.

Dockner E, Jorgensen S, Long N, Sorger G (2000) Differential games in economics and management sciences. Cambridge University Press, CambridgeCrossRef

20.

Giachetti C (2013) Changes in industrial leadership: technological discontinuities and firms’ aggressive actions. Palgrave Macmillan, London, pp 45–78

21.

Gnyawali DR, Park B-J (2009) Co-opetition and technological innovation in small and medium-sized enterprises: a multilevel conceptual model. J Small Bus Manag 47(3):308–330CrossRef

22.

Gnyawali DR, Park B-J (2011) Co-opetition between giants: collaboration with competitors for technological innovation. Res Policy 40:650–663CrossRef

23.

Greiner A, Bondarev A (2017) Optimal R&D investment with learning-by-doing: multiple steady states and thresholds. Optim Control Appl Methods 38(6):956–962MathSciNetCrossRef

24.

Gromov D, Gromova E (2017) On a class of hybrid differential games. Dyn Games Appl 7(2):266–288MathSciNetCrossRef

25.

Hamano M, Zanetti F (2017) Endogenous product turnover and macroeconomic dynamics. Rev Econ Dyn 26:263–279CrossRef

26.

He X, Prasad A, Sethi SP, Gutierrez GJ (2007) A survey of Stackelberg differential game models in supply and marketing channels. J Syst Sci Syst Eng 16(4):385–413CrossRef

27.

Hinloopen J, Smrkolj G, Wagener F (2013) From mind to market: a global, dynamic analysis of R&D. J Econ Dyn Control 37(12):2729–2754MathSciNetCrossRef

28.

Jeffrey M, Colombo A (2009) The two-fold singularity of discontinuous vector fields. SIAM J Appl Dyn Syst 8:624–640MathSciNetCrossRef

29.

Kline D (2001) Positive feedback, lock-in, and environmental policy. Policy Sci 34(1):95–107CrossRef

30.

Lamperti F, Napoletano M, Roventini A (2020) Green transitions and the prevention of environmental disasters: market-based versus command-and-control policies. Macroecon Dyn 24(7):1861–1880CrossRef

31.

Long NV, Prieur F, Tidball M, Puzon K (2017) Piecewise closed-loop equilibria in differential games with regime switching strategies. J Econ Dyn Control 76:264–284MathSciNetCrossRef

32.

Motta M, Thisse J-F, Cabrales A (1997) On the persistence of leadership or leapfrogging in international trade. Int Econ Rev 38(4):809–24CrossRef

33.

Peretto P, Connolly M (2007) The Manhattan metaphor. J Econ Growth 12(4):329–350CrossRef

34.

Petrosjan L, Zaccour G (2003) Time-consistent Shapley value allocation of pollution cost reduction. J Econ Dyn Control 27(3):381–398MathSciNetCrossRef

35.

Ritala P, Tidström A (2014) Untangling the value-creation and value-appropriation elements of coopetition strategy: a longitudinal analysis on the firm and relational levels. Scand J Manag 30(4):498–515CrossRef

36.

Samano M, Santugini M, Zaccour G (2017) Dynamics in research joint ventures and R&D collaborations. J Econ Dyn Control 77:70–92MathSciNetCrossRef

37.

Skritek B, Tsachev T, Veliov V (2014) Optimality conditions and the Hamiltonian for a distributed optimal control problem on controlled domain. Appl Math Optim 70:141–164MathSciNetCrossRef

38.

Teixeira MA (1990) Stability conditions for discontinuous vector fields. J Differ Equ 88(1):15–29MathSciNetCrossRef

39.

Wagener FOO (2003) Skiba points and heteroclinic bifurcations, with applications to the shallow lake system. J Econ Dyn Control 27(9):1533–1561MathSciNetCrossRef

Titel: Games Without Winners: Catching-up with Asymmetric Spillovers
verfasst von: Anton Bondarev
Publikationsdatum: 01.03.2021
Verlag: Springer US
Erschienen in: Dynamic Games and Applications / Ausgabe 4/2021
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-021-00379-y

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