By Frank Thuijsman, Florian Wagener
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Extra info for Advances in Dynamic and Evolutionary Games: Theory, Applications, and Numerical Methods (Annals of the International Society of Dynamic Games)
To the best of our knowledge, this is the first problem to be studied in optimal control of queues, going back to the seminal paper of Naor (1969). Naor considered an M/M/1 queue, in which a controller has to decide whether arrivals should enter a queue or not. The objective was to minimize a cost with the form of a weighted difference between the average expected waiting time of those that enter, and the acceptance rate of customers. The strategy that minimizes the above cost was shown to be of a threshold type where arrivals are accepted as long as the queue size does not exceed some threshold L and are otherwise denied access.
References Basar T, Olsder GJ (1982) Dynamic noncooperative game theory. Academic, London Berkovitz LD (1964) A variational approach to differential games. In: Dresher M, Shapley LS, Tucker AW (eds) Advances in game theory. Annals of mathematics studies, vol 52. Princeton University Press, Princeton, pp 127–174 Dutkevich YG, Petrosyan LA (1972) Games with a “life-line”. The case of l-capture. SIAM J Control 10(1):40–47 Fleming WH (1961) The convergence problem for differential games. J Math Anal Appl 3:102– 116 Haurie A (1976) A note on nonzero-sum differential games with bargaining solutions.
18) t0 / is time-consistent. x0 ; T t0 / in various classes of games. 1 Dynamic Games with Perfect Information 23 In the remainder of this section, we discuss the new concept of strong timeconsistency, and we define time-consistent solutions for cooperative games with terminal payoffs. t/; ˇ/ is the vector of total payoffs to the players up to time t. t/; T t/ that fail to satisfy this equation. t/; T t/ provides a “good” continuation of the original agreement. x0 ; T t0 /; is satisfied. By slightly strengthening this requirement, we obtain the concept of strong timeconsistency.