One of the central economic paradigms in multi-agent systems is that agents should not be better off by acting dishonestly. In the context of collective decision-making, this axiom is known as strategyproofness and turns out to be rather prohibitive, even when allowing for randomization. In particular, Gibbard's random dictatorship theorem shows that only rather unattractive social decision schemes (SDSs) satisfy strategyproofness on the full domain of preferences. In this paper, we obtain more positive results by investigating strategyproof SDSs on the Condorcet domain, which consists of all preference profiles that admit a Condorcet winner. In more detail, we show that, if the number of voters $n$ is odd, every strategyproof and non-imposing SDS on the Condorcet domain can be represented as a mixture of dictatorial SDSs and the Condorcet rule (which chooses the Condorcet winner with probability $1$). Moreover, we prove that the Condorcet domain is a maximal connected domain that allows for attractive strategyproof SDSs if $n$ is odd as only random dictatorships are strategyproof and non-imposing on any sufficiently connected superset of it. We also derive analogous results for even $n$ by slightly extending the Condorcet domain. Finally, we also characterize the set of group-strategyproof and non-imposing SDSs on the Condorcet domain and its supersets. These characterizations strengthen Gibbard's random dictatorship theorem and establish that the Condorcet domain is essentially a maximal domain that allows for attractive strategyproof SDSs.

翻译：多智能体系统中的核心经济范式之一是，智能体不应通过不诚实行为获得更好结果。在集体决策语境下，这一公理被称为防策略性，事实证明即使允许随机化，该约束也相当严格。特别地，吉巴德随机独裁定理表明，在完整偏好域上，只有相当不具吸引力的社会决策方案（SDS）满足防策略性。本文通过研究康多塞特域上的防策略SDS取得了更积极的结果。更具体地说，我们证明：若选民数$n$为奇数，则康多塞特域上每个防策略且非强制的SDS都可表示为独裁SDS与康多塞特规则（以概率$1$选择康多塞特胜者）的混合。此外，我们证明当$n$为奇数时，康多塞特域是允许存在有吸引力防策略SDS的最大连通域——因为在其任何充分连通超集上，只有随机独裁是防策略且非强制的。通过轻微扩展康多塞特域，我们也推导出$n$为偶数时的类似结论。最后，我们刻画了康多塞特域及其超集上群体防策略且非强制SDS的集合。这些刻画强化了吉巴德随机独裁定理，并确立了康多塞特域本质上是允许存在有吸引力防策略SDS的最大域。