  The airport game is a classical and well-known model of fair cost-sharing for
a single facility among multiple agents. This paper extends it to the so-called
assignment setting, that is, for multiple facilities and agents, each agent
chooses a facility to use and shares the cost with the other agents. Such a
situation can be often seen in sharing economy, such as sharing fees for office
desks among workers, taxis among customers of possibly different destinations
on a line, and so on. Our model is regarded as a coalition formation game based
on the fair cost-sharing of the airport game; we call our model \emph{a fair
ride allocation on a line}. As criteria of solution concepts, we incorporate
Nash stability and envy-freeness into our setting. We show that a Nash-stable
feasible allocation that minimizes the social cost of agents can be computed
efficiently if a feasible allocation exists. For envy-freeness, we provide
several structural properties of envy-free allocations. Based on these, we
design efficient algorithms for finding an envy-free allocation when at least
one of (1) the number of facilities, (2) the capacity of facilities, and (3)
the number of agent types, is small. Moreover, we show that a consecutive
envy-free allocation can be computed in polynomial time. On the negative front,
we show the NP-hardness of determining the existence of an allocation under two
relaxed envy-free concepts.
