Optimal allocations with incomplete record-keeping and no commitment

We study a random-matching, absence-of-double-coincidence environment in which people cannot precommit and in which there are two imperfect ways of keeping track of what other people have done in the past: money and a public record of all past actions that is updated with an average lag. We study how the magnitude of that lag affects the allocations that are optimal from among allocations that are stationary and feasible and that satisfy incentive constraints which arise from the absence of commitment and the imperfect ways of keeping track of what others have done in the past.