We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega ,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$ -game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

In 2015, Guglielmi and Badia discussed optimal strategies in a particular type of service system with two strategic servers. In their setup, each server can be either active or inactive and an active server can be requested to transmit a sequence of packets. The servers have varying probabilities of successfully transmitting when they are active, and both servers receive a unit reward if the sequence of packets is transmitted successfully. Guglielmi and Badia provided an analysis of optimal strategies in four scenarios: where each server does not know the other’s successful transmission probability; one of the two servers is always inactive; each server knows the other’s successful transmission probability and they are willing to cooperate.

Unfortunately, the analysis by Guglielmi and Badia contained some errors. In this paper we correct these errors. We discuss three cases where both servers (I) communicate and cooperate; (II) neither communicate nor cooperate; (III) communicate but do not cooperate. In particular, we obtain the unique Nash equilibrium strategy in Case II through a Bayesian game formulation, and demonstrate that there is a region in the parameter space where there are multiple Nash equilibria in Case III. We also quantify the value of communication or cooperation by comparing the social welfare in the three cases, and propose possible regulations to make the Nash equilibrium strategy the socially optimal strategy for both Cases II and III.