Technical Note—Analysis of Scrip Systems: On an Open Question in Johnson et al. (2014)
回应了Johnson等人（2014）提出的开放问题，在不依赖惩罚假设的情况下，证明当玩家数量足够大且贴现因子接近1时，存在一个ε-纳什均衡，其中每个玩家都采用始终交易策略。
In a recent paper, Johnson et al. (2014) [Johnson K, Simchi-Levi D, Sun P (2014) Analyzing scrip systems. Oper. Res. 62(3):524–534.]use an infinitely repeated game with discounting, among a set of homogeneous players, to model a scrip system. In each period, a randomly chosen player requests service; all the other players have a choice of whether or not to volunteer to provide service. Among the players who volunteer, the service provider is chosen using the minimum-scrip rule: a player with the minimum number of scrips is chosen as the service provider. The authors study the always-trade strategy for a player, that is, the strategy of a player always willing to provide service, regardless of the distribution of scrips among the players. A key result of their work is that, under the minimum-scrip rule, for any number of players and for every discount factor close enough to one, there exists a Nash equilibrium in which each player plays the always-trade strategy. This result, however, is established under an assumption of severe punishment: If a player selected to provide service refuses to do so even once, then that player will be forever banned from participating in the system, thus losing all potential future benefit. Johnson et al. (2014) explain that this assumption is unsatisfactory, particularly for large scrip systems, because of difficulties in detecting players who refuse to provide service and verifying the reason(s) for their refusal, and the consequent possibility of players being unfairly banned. Motivated by this concern, the authors suggest an important direction of future research: investigate whether or not the always-trade strategy is an equilibrium without the punishment assumption. In this note, we address the question posed in Johnson et al. (2014) for a generalization of their model. Our generalization allows for the possibility that players might genuinely not be available to provide service, say because of sickness. Without using the assumption of punishment, we show that when the number of players is large enough and the discount factor is close enough to one, there exists an ε-Nash equilibrium in which each player plays the always-trade strategy.