  We study random joint choice rules, allowing for interdependence of choice
across agents. These capture random choice by multiple agents, or a single
agent across goods or time periods. Our interest is in separable choice rules,
where each agent can be thought of as acting independently of the other. A
random joint choice rule satisfies marginality if for every individual choice
set, we can determine the individual's choice probabilities over alternatives
independently of the other individual's choice set. We offer two
characterizations of random joint choice rules satisfying marginality in terms
of separable choice rules. While marginality is a necessary condition for
separability, we show that it fails to be sufficient. We provide an additional
condition on the marginal choice rules which, along with marginality, is
sufficient for separability.
