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Published in: Dynamic Games and Applications 4/2021

01-03-2021

Games Without Winners: Catching-up with Asymmetric Spillovers

Author: Anton Bondarev

Published in: Dynamic Games and Applications | Issue 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.

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Appendix
Available only for authorised users
Footnotes
1
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.
 
2
Both \(u_{k}(t), g_{k}(i,t)\) are arbitrary functions called optimal strategies, defined as optimizers of players’ payoffs via Maximum Principle.
 
3
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.
 
4
Observe that this is not a standard Stackelberg leadership, the notion of the leader is to be defined further.
 
5
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.
 
6
It will be demonstrated that this minor extension of asymmetry leads to qualitatively different results and substantially more complicated dynamics, see Remark 1.
 
7
To be defined later.
 
8
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.
 
9
Note that \(i_{k}\) is not just one index, but a range of technologies.
 
10
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).
 
11
Observe that regimes refer to the dynamical system while modes to the global behaviour of the overall game.
 
12
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).
 
13
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.
 
14
defined further on.
 
15
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.
 
16
provided assumption on both steady states are (saddle-type) stable, arguments are the same as for the cooperative game case.
 
17
I omit all standard types of dynamics possible once some of Lemmas 79 do not hold, since these can be described via the same mechanics as in Gromov and Gromova [24].
 
18
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.
 
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Metadata
Title
Games Without Winners: Catching-up with Asymmetric Spillovers
Author
Anton Bondarev
Publication date
01-03-2021
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 4/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00379-y

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