Random assignments and outside options

Most environments where assignment mechanisms (possibly random) are used are such that participants have outside options. For instance private schools and private housing are options that participants in a public choice or public housing assignment problems may have. We postulate that, in cardinal mechanisms, chances inside the assignment process could favor agents with better outside options. By imposing a robustness to outside options condition, we conclude that, on the universal domain of cardinal preferences, any mechanism must be (interim) ordinal.