We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and the competitive equilibrium, is known to be not implementable in a truthful fashion, which has been its main drawback. To alleviate this issue, we propose a new mechanism, which we call the Partial Allocation mechanism, that discards a carefully chosen fraction of the allocated resources in order to incentivize the agents to be truthful in reporting their valuations. This mechanism introduces a way to implement interesting truthful outcomes in settings where monetary payments are not an option. For a multi-dimensional domain with an arbitrary number of agents and items, and for the very large class of homogeneous valuation functions, we prove that our mechanism provides every agent with at least a 1 e � 0.368 fraction of her Proportionally Fair valuation. To the best of our knowledge, this is the first result that gives a constant factor approximation to every agent for the Proportionally Fair solution. To complement this result, we show that no truthful mechanism can guarantee more than 0.5 approximation, even for the restricted class of additive linear valuations. In addition to this, we uncover a connection between the Partial Allocation mechanism and VCG-based mechanism design. We also ask whether better approximation ratios are possible in more restricted settings. In particular, motivated by the massive privatization auction in the Czech republic in the early 90s we provide another mechanism for additive linear valuations that works really well when all the items are highly demanded.
Despite the extensive work on fair resource allocation, truthfulness considerations have not played a major role in this literature. Most results related to truthfulness were weakened by the assumption that each agent would be truthful in reporting her valuations unless this strategy was dominated. Very recent work @cite_14 @cite_16 @cite_12 @cite_24 studies truthful cake cutting variations using the standard notion of truthfulness according to which an agent need not be truthful unless doing so is a dominant strategy. study truthful cake-cutting with agents having piecewise uniform valuations and they provide a polynomial-time mechanism that is truthful, proportional, and envy-free. They also design randomized mechanisms for more general families of valuation functions, while prove the existence of truthful (in expectation) mechanisms satisfying proportionality in expectation for general valuations. aim to achieve envy-free Pareto optimal allocations of multiple divisible goods while reducing, but not eliminating, the agents' incentives to lie. The extent to which untruthfulness is reduced by their proposed mechanism is only evaluated empirically and depends critically on their assumption that the resource limitations are soft constraints. Very recent work by provides evidence that truthfulness comes at a significant cost in terms of efficiency.
@cite_24 @cite_14 @cite_12 @cite_16
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We characterize methods of dividing a cake between two bidders in a way that is incentive-compatible and Pareto-efficient. In our cake cutting model, each bidder desires a subset of the cake (with a uniform value over this subset), and is allocated some subset. Our characterization proceeds via reducing to a simple one-dimensional version of the problem, and yields, for example, a tight bound on the social welfare achievable. Cake cutting is a common metaphor for the division of a heterogeneous divisible good. There are numerous papers that study the problem of fairly dividing a cake; a small number of them also take into account self-interested agents and consequent strategic issues, but these papers focus on fairness and consider a strikingly weak notion of truthfulness. In this paper we investigate the problem of cutting a cake in a way that is truthful and fair, where for the first time our notion of dominant strategy truthfulness is the ubiquitous one in social choice and computer science. We design both deterministic and randomized cake cutting algorithms that are truthful and fair under different assumptions with respect to the valuation functions of the agents. A natural requirement of a resource allocation system is to guarantee fairness to its participants. Fair allocation can be achieved either by distributed protocols known as cake-cutting algorithms or by centralized approaches, which first collect the agents’ preferences and then decide on the allocation. Compared with cake-cutting algorithms, centralized approaches ageless restricted and can therefore achieve more favorable allocations. Our work uses as a starting point a recent centralized algorithm that achieves an envy-free (i.e., fair) and Pareto optimal (i.e., efficient) allocation of multiple divisible goods. In fair allocation algorithms, agents who do not follow the protocol cannot prevent other agents from being allocated a fair share, but in certain situations, agents can increase their own allocation by submitting untruthful preferences. A recent article has shown that the only mechanisms that do not allow such manipulations(i.e., the only incentive-compatible mechanisms) are dictatorial. Nevertheless, we present a method that reduces possible gains from untruthful manipulation. Our mechanism uses a heuristic to approximate optimal manipulations, and compensates agents who submitted suboptimal preferences by increasing their allocation. We empirically demonstrate that when our method is used, the additional benefit that agents can achieve by untruthful manipulation is insignificant, hence they have insignificant incentives to lie. We address the problem of fair division, or cake cutting, with the goal of finding truthful mechanisms. In the case of a general measure space ("cake") and non-atomic, additive individual preference measures - or utilities - we show that there exists a truthful "mechanism" which ensures that each of the k players gets at least 1 k of the cake. This mechanism also minimizes risk for truthful players. Furthermore, in the case where there exist at least two different measures we present a different truthful mechanism which ensures that each of the players gets more than 1 k of the cake. We then turn our attention to partitions of indivisible goods with bounded utilities and a large number of goods. Here we provide similar mechanisms, but with slightly weaker guarantees. These guarantees converge to those obtained in the non-atomic case as the number of goods goes to infinity.