Sincere and sophisticated players in the envy-free allocation problem

Abstract

We study the problem of allocating a collective endowment of n objects among n agents when monetary compensation is available. We require budget balance and assume complete information. We also assume that money is desirable and that no object is infinitely better than another (in terms of money). Our preference domain contains, but is not restricted to, quasi-linear preferences. Examples are the dissolution of a partnership among partners who know each other well and the allocation of rooms and the division of the rent among housemates who collectively lease a house.An allocation is envy-free if no agent prefers the consumption of any other agent to her own consumption. We study the manipulation of the solutions that associate with each preference profile an envy-free allocation when some agents are sincere, i.e., unconditionally report their true preferences (and this is common knowledge), and the rest are strategic.(1) For each preference profile, the set of limit Nash equilibrium outcomes of the direct revelation game associated with each envy-free solution is the set of envy-free allocations, for the true preferences, that are not Pareto dominated for the strategic agents by another envy-free allocation. This implies that strategic agents take advantage, to some extent, of sincere agents. However, this has a limit. Strategic agents can only force a sincere agent to receive her worst envy-free allocation for the true preference profile. This allocation is generally better than the worst allocation she could receive if the reports of the strategic agents were not restricted by Nash behavior.(2) Independently of the envy-free solution that is operated, if there is at least another strategic agent, the worst case scenario equilibrium payoff for a strategic agent is equal to her worst case scenario equilibrium payoff if the agent were sincere. This suggests that given the opportunity to commit to unconditionally revealing her preferences, an agent may choose to do so even though her true preferences are not a dominant strategy. However, generically the best case scenario equilibrium payoff for a strategic agent is strictly better than the best case scenario equilibrium payoff if the agent were sincere. Thus, generically each agent may strictly prefer not to commit to unconditionally revealing her preferences, even when the agent's true preference profile is a dominant strategy.An allocation is an equal income competitive allocation if there is a vector of prices that sustains the allocation as a competitive outcome in which each agent is endowed with an equal share of the aggregate income at the given prices. It is well known that in our environment the set of equal income competitive allocations coincides with the set of envy-free allocations. Thus, our results can be equally interpreted as the study of market manipulation in an environment with sincere and sophisticated agents. Even though no market solution is strategy-proof, the manipulation of each market solution results only in market outcomes with respect to true preferences. Moreover, in the presence of sincere agents, strategic agents manipulate the market and extract from sincere agents the maximum surplus they could unanimously agree on within the limits of the true-profile market outcomes.