Construction of Nash Equilibrium in a Game Version of Elfving’s Multiple Stopping Problem

Abstract

Multi-person stopping games with players' priorities are considered. Players observe sequentially offers Y 1 , Y 2 , . . . at jump times T 1 , T 2 , . . . of a Poisson process. Y 1 , Y 2 , . . . are independent identically distributed random variables. Each accepted offer Y n results in a reward G n = Y n r(T n ), where r is a non-increasing discount function. If more than one player wants to accept an offer, then the player with the highest priority (the lowest ordering) gets the reward. We construct Nash equilibrium in the multi-person stopping game using the solution of a multiple optimal stopping time problem with structure of rewards {G n }. We compare rewards and stopping times of the players in Nash equilibrium in the game with the optimal rewards and optimal stopping times in the multiple stopping time problem. It is also proved that presented Nash equilibrium is a Pareto optimum of the game. The game is a generalization of the Elfving stopping time problem to multi-person stopping games with priorities.Keywords Stopping game · Nash equilibrium · Pareto-optimality · Multiple stopping Suppose that a company is going to open m new departments that will be ordered (ranked) according to their importance. Rank 1 denotes the most important department, and consequently rank m refers to the least important one. So, the firm offers m secretary positions. The applications will be considered up to a fixed deadline U . There is a selection committee that interviews candidates arriving one by one at jump times of a Poisson process. We assume that candidates' "skills" form a sequence of i.i.d. random variables with known probability distribution. The selection process is organized similarly to the classical secretary problem but taking into account the departments' rankings. So, a candidate who is being interviewed may be accepted for department 1; if she is rejected, she may be accepted for department 2, and so on until the first acceptance. Candidates rejected for department i cannot be considered in the future. The aim is to select candidates with maximal expected "skills". So, one may say that each department acts as an independent player in a stopping game with priorities.We will formulate the problem as an m-person stopping game with priorities in which random offers are presented at jump times of a homogeneous Poisson process. Such a game has been considered in Ferenstein and Krasnosielska [8]. In this paper, we propose a new solution and we prove that a proposed strategy is a Nash equilibrium, which allows removing some assumption made in Ferenstein and Krasnosielska [8]. The difference between the solution proposed in this paper and those in [8] will be more thoroughly discussed at the end of the paper.The game considered is a generalization, to the case of several players, of the optimal stopping time problem formulated and solved first by Elfving [6] Assume that the set of points of discontinuity of r is finite. Let G n = Y n r(T n ), n ∈ N 0 , whereNote that without loss of generality we can assu...

Construction of Nash equilibrium based on multiple stopping problem in multi-person game

“…We will also present its properties. The game is a generalization of the one presented in Krasnosielska (2011) and Krasnosielska-Kobos and Ferenstein (2013). Suppose that there are m > 1 ordered players.…”

“…The numerous examples of multiple stopping problems in which functions γ i have been obtained can be found in Sakaguchi (1972), Stadje (1985Stadje ( , 1987) (see also Krasnosielska-Kobos and Ferenstein 2013) and Krasnosielska-Kobos (2015).…”

“…A special case of the considered game has been presented in Krasnosielska (2009), Krasnosielska (2011), and Krasnosielska-Kobos and Ferenstein (2013). In mentioned papers, in opposite to this paper, authors considered a game with the same rewards' structure as in the Elfving problem [see Elfving (1967) and Siegmund (1967)], under the assump-tion that each player has only one commodity for sale.…”

“…The idea of using the solution of a multiple stopping problem to construct a Nash equilibrium is based on Krasnosielska (2011) and Krasnosielska-Kobos and Ferenstein (2013).…”

On Multilateral Hierarchical Dynamic Decisions

“…Earlier studies of such issues (cf. Ferenstein(1992), , Ramsey and Cierpiał(2009), Dorobantu et al(2009), Krasnosielska-Kobos and Ferenstein(2013), Ferguson(2016)) showed their complexity, and detailed models of the analyzed cases a way to overcome difficulties in modeling and setting goals with the help of created models. The basic difficulty, except for cases when the decision is made by one agent, consists in determining the goals of the team, which can not always be determined so that the task can be reduced to the optimization of the objective function as the result of scalarisation.…”