In the last decade, social planners have used crowdfunding to raise funds for public projects. As these public projects are non-excludable, the beneficiaries may free-ride. Thus, there is a need to design incentive mechanisms for such strategic agents to contribute to the project. The existing mechanisms, like PPR or PPRx, assume that the agent's beliefs about the project getting funded do not change over time, i.e., their beliefs are static. Researchers highlight that unless appropriately incentivized, the agents defer their contributions in static settings, leading to a ``race'' to contribute at the deadline. In this work, we model the evolution of agents' beliefs as a random walk. We study PPRx -- an existing mechanism for the static belief setting -- in this dynamic belief setting and refer to it as PPRx-DB for readability. We prove that in PPRx-DB, the project is funded at equilibrium. More significantly, we prove that under certain conditions on agent's belief evolution, agents will contribute as soon as they arrive at the mechanism. Thus, we believe that by incorporating dynamic belief evolution in analysis, the social planner can mitigate the concern of race conditions in many mechanisms.

翻译：在过去十年中，社会规划者已利用众筹为公共项目筹集资金。由于这些公共项目具有非排他性，受益者可能会"搭便车"。因此，需要为这类策略性参与者设计激励机制以促使他们为项目做出贡献。现有机制（如PPR或PPRx）假设参与者对项目能否获得资金支持的信念不会随时间改变，即其信念是静态的。研究者指出，在静态信念环境中，若缺乏适当激励，参与者会推迟其贡献，导致在截止日期前出现"竞相贡献"的现象。本工作将参与者信念的演化建模为随机游走过程。我们针对动态信念环境研究PPRx（一种现有静态信念设定机制），为便于表述将其称为PPRx-DB。我们证明，在PPRx-DB中，项目将在均衡状态下获得资金支持。更重要的是，我们证明在参与者信念演化的特定条件下，参与者将在抵达机制后立即做出贡献。因此，我们认为通过将动态信念演化纳入分析，社会规划者可以缓解多种机制中的竞相贡献问题。