Cooperative Dynamic Games with Durable Controls: Theory and Application
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Cooperative Dynamic Games with Durable Controls: Theory and Application Leon A. Petrosyan1 · David W.K. Yeung2,3,4
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Durable controls that have effects lasting over a certain period of time are prevalent in real-life situations. Revenue-generating investments, toxic waste disposal, durable goods, emission of pollutants, regulatory measures, coalition agreements, diffusion of knowledge, advertisement and investments to build up physical capital are vivid examples of durable controls. Durable controls may affect both the decision-makers’ payoffs and the evolution of the state dynamics. This paper develops a new class of cooperative dynamic games with multiple durable controls of different lag durations affecting both the players’ payoffs and the state dynamics. A novel dynamic optimization theorem involving durable controls is derived. Dynamically consistent cooperative solutions are provided. An illustration of a dynamic game with durable controls causing lagged effects in capital formation, pollution accumulation and revenue generation from investment is presented. The analysis widens the application of dynamic game theory in practical problems. Further theoretical developments and relevant applications along this line would be expected. Keywords Cooperative dynamic games · Durable controls · Control lags · Dynamically consistent solution
1 Introduction Durable controls that have influences lasting over a certain period of time are frequently observed in real-life situations. In the presence of durable controls, modification of the dynamic optimization techniques has to be made to accommodate the lagged effects of * David W.K. Yeung [email protected] Leon A. Petrosyan [email protected] 1
Faculty of Applied Mathematics‑Control Processes, St Petersburg State University, St Petersburg, Russia 198904
2
Center of Game Theory, St Petersburg State University, St Petersburg, Russia 198904
3
SRS Consortium for Advanced Study in Cooperative Dynamic Games, Hong Kong Shue Yan University, Hong Kong, China
4
Department of Finance, Asia University, Taichung City, Taiwan, China
Vol.:(0123456789)
Dynamic Games and Applications
the controls. Most of the existing works on lagged control optimization involved lags in the state dynamics (see Bellman [4], Arthur [2], Burdet and Sethi [7] and Hartl and Sethi [9], Brandt-Pollmann et al. [6], Huschto et al. [10], Sethi and McGuire [19] and Winkler et al. [23]). Bokov [5] considered continuous-time optimal control with control lags in the payoff and time lags in the state dynamics. In reality, both the decision-maker’s payoffs and the evolution of the state dynamics can be subjected to effects from durable controls. For instance, durable goods, vehicles, factories, real estate, revenue-generating investments, toxic waste disposal, business contracts and regulations affect the decision-maker’s payoffs for more than one stage. At the same time, tariffs have delayed impacts on the economy’s g
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