Bounded incentives in manipulating the probabilistic serial rule

The Probabilistic Serial mechanism is valued for its fairness and efficiency in addressing the random assignment problem. However, it lacks truthfulness, meaning it works well only when agents' stated preferences match their true ones. Significant utility gains from strategic actions may lead self-interested agents to manipulate the mechanism, undermining its practical adoption. To gauge the potential for manipulation, we explore an extreme scenario where a manipulator has complete knowledge of other agents' reports and unlimited computational resources to find their best strategy. We establish tight incentive ratio bounds of the mechanism. Furthermore, we complement these worst-case guarantees by conducting experiments to assess an agent's average utility gain through manipulation. The findings reveal that the incentive for manipulation is very small. These results offer insights into the mechanism's resilience against strategic manipulation, moving beyond the recognition of its lack of incentive compatibility

Equilibria Under the Probabilistic Serial Rule

The probabilistic serial (PS) rule is a prominent randomized rule for assigning indivisible goods to agents. Although it is well known for its good fairness and welfare properties, it is not strategyproof. In view of this, we address several fundamental questions regarding equilibria under PS. Firstly, we show that Nash deviations under the PS rule can cycle. Despite the possibilities of cycles, we prove that a pure Nash equilibrium is guaranteed to exist under the PS rule. We then show that verifying whether a given profile is a pure Nash equilibrium is coNP-complete, and computing a pure Nash equilibrium is NP-hard. For two agents, we present a linear-time algorithm to compute a pure Nash equilibrium which yields the same assignment as the truthful profile. Finally, we conduct experiments to evaluate the quality of the equilibria that exist under the PS rule, finding that the vast majority of pure Nash equilibria yield social welfare that is at least that of the truthful profile.Comment: arXiv admin note: text overlap with arXiv:1401.6523, this paper supersedes the equilibria section in our previous report arXiv:1401.652

Complexity of Manipulating Sequential Allocation

Sequential allocation is a simple allocation mechanism in which agents are given pre-specified turns and each agents gets the most preferred item that is still available. It has long been known that sequential allocation is not strategyproof. Bouveret and Lang (2014) presented a polynomial-time algorithm to compute a best response of an agent with respect to additively separable utilities and claimed that (1) their algorithm correctly finds a best response, and (2) each best response results in the same allocation for the manipulator. We show that both claims are false via an example. We then show that in fact the problem of computing a best response is NP-complete. On the other hand, the insights and results of Bouveret and Lang (2014) for the case of two agents still hold

Social Welfare in One-Sided Matching Mechanisms

We study the Price of Anarchy of mechanisms for the well-known problem of one-sided matching, or house allocation, with respect to the social welfare objective. We consider both ordinal mechanisms, where agents submit preference lists over the items, and cardinal mechanisms, where agents may submit numerical values for the items being allocated. We present a general lower bound of $\Omega(\sqrt{n})$ on the Price of Anarchy, which applies to all mechanisms. We show that two well-known mechanisms, Probabilistic Serial, and Random Priority, achieve a matching upper bound. We extend our lower bound to the Price of Stability of a large class of mechanisms that satisfy a common proportionality property, and show stronger bounds on the Price of Anarchy of all deterministic mechanisms

Algorithms for Manipulating Sequential Allocation

Sequential allocation is a simple and widely studied mechanism to allocate indivisible items in turns to agents according to a pre-specified picking sequence of agents. At each turn, the current agent in the picking sequence picks its most preferred item among all items having not been allocated yet. This problem is well-known to be not strategyproof, i.e., an agent may get more utility by reporting an untruthful preference ranking of items. It arises the problem: how to find the best response of an agent? It is known that this problem is polynomially solvable for only two agents and NP-complete for arbitrary number of agents. The computational complexity of this problem with three agents was left as an open problem. In this paper, we give a novel algorithm that solves the problem in polynomial time for each fixed number of agents. We also show that an agent can always get at least half of its optimal utility by simply using its truthful preference as the response

Practical algorithms and experimentally validated incentives for equilibrium-based fair division (A-CEEI)

Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is an equilibrium-based solution concept for fair division of discrete items to agents with combinatorial demands. In theory, it is known that in asymptotically large markets: 1. For incentives, the A-CEEI mechanism is Envy-Free-but-for-Tie-Breaking (EF-TB), which implies that it is Strategyproof-in-the-Large (SP-L). 2. From a computational perspective, computing the equilibrium solution is unfortunately a computationally intractable problem (in the worst-case, assuming $\textsf{PPAD}\ne \textsf{FP}$). We develop a new heuristic algorithm that outperforms the previous state-of-the-art by multiple orders of magnitude. This new, faster algorithm lets us perform experiments on real-world inputs for the first time. We discover that with real-world preferences, even in a realistic implementation that satisfies the EF-TB and SP-L properties, agents may have surprisingly simple and plausible deviations from truthful reporting of preferences. To this end, we propose a novel strengthening of EF-TB, which dramatically reduces the potential for strategic deviations from truthful reporting in our experiments. A (variant of) our algorithm is now in production: on real course allocation problems it is much faster, has zero clearing error, and has stronger incentive properties than the prior state-of-the-art implementation.Comment: To appear in EC 202