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

Anna Krasnosielska-Kobos,Elżbieta Ferenstein

doi:10.1007/s13235-012-0070-7

Anna Krasnosielska-Kobos, Elżbieta Ferenstein

Open Access

https://doi.org/10.1007/s13235-012-0070-7

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Abstract

Multi-person stopping games with players’ priorities are considered. Players observe sequentially offers Y1,Y2,… at jump times T1,T2,… of a Poisson process. Y1,Y2,… are independent identically distributed random variables. Each accepted offer Yn results in a reward Gn=Ynr(Tn), 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 {Gn}. 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.

Highlights

Suppose that a company is going to open m new departments that will be ordered according to their importance
Summary In Theorem 4, we have proved that the strategy profile ψm is a Nash equilibrium
The expected reward of Player i in the m-person game in Nash equilibrium is equal to the expected reward from selling the ith good in the future, if there are i instead of i − 1 goods for sale (Lemma 8)

Summary

Introduction

Suppose that a company is going to open m new departments that will be ordered (ranked) according to their importance. 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]. Stadje [19] considered an optimal multi-stopping time problem in Elfving setting, in which the final reward is the sum of selected discounted random variables. Various stopping games with rewards observed at jump times of a Poisson process were considered in Dixon [4], Enns and Ferenstein [7], Saario and Sakaguchi [17], Ferenstein and Krasnosielska [9]. An extensive bibliography on stochastic games can be found in Nowak and Szajowski [15]

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