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Analyzing Crowdfunding of Public Projects Under Dynamic Beliefs

Sankarshan Damle, Sujit Gujar

TL;DR

This work tackles the race condition in sequential crowdfunding of public projects by modeling agents’ beliefs as dynamic and evolving via a random walk. It extends the PPRx mechanism to a dynamic-belief setting (PPRx-DB), deriving equilibrium contributions and times of contribution under different belief evolutions, including Martingale, Super-, and Sub-martingale regimes. The authors prove that, with positive refund budgets and total valuation exceeding the target, the project is funded at equilibrium and identify conditions under which agents contribute upon arrival, thereby mitigating the race. The results offer a practical theoretical foundation suggesting that dynamic-belief considerations can simplify mechanism design and reduce on-chain costs, with future work aimed at empirical validation and learning belief-update models.

Abstract

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.

Analyzing Crowdfunding of Public Projects Under Dynamic Beliefs

TL;DR

This work tackles the race condition in sequential crowdfunding of public projects by modeling agents’ beliefs as dynamic and evolving via a random walk. It extends the PPRx mechanism to a dynamic-belief setting (PPRx-DB), deriving equilibrium contributions and times of contribution under different belief evolutions, including Martingale, Super-, and Sub-martingale regimes. The authors prove that, with positive refund budgets and total valuation exceeding the target, the project is funded at equilibrium and identify conditions under which agents contribute upon arrival, thereby mitigating the race. The results offer a practical theoretical foundation suggesting that dynamic-belief considerations can simplify mechanism design and reduce on-chain costs, with future work aimed at empirical validation and learning belief-update models.

Abstract

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.
Paper Structure (30 sections, 6 theorems, 11 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 30 sections, 6 theorems, 11 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Provision Point mechanism for Public projects (PPP)
  3. PPP & Free-riding.
  4. PPR.
  5. Modes of Crowdfunding.
  6. Information Structure damle2019ijcai.
  7. Belief Evolution.
  8. PPRx-DB.
  9. Related Work
  10. Preliminaries
  11. Crowdfunding Model.
  12. Sub-game Perfect Equilibrium (SPE).
  13. Agent Information Structure.
  14. PPRx.
  15. Belief Based Reward (BBR).
  16. ...and 15 more sections

Key Result

lemma 1

In PPRx-DB, the public project is funded at equilibrium, i.e., $C_0=H_0$ if $\vartheta>H_0$ with $B_B,B_C>0$.

Figures (4)

  • Figure 1: Plotting $\approx x/\theta$ for two randomly sampled agents using the dataset available with cason2021early. The black vertical line in the left plots represents the end of the refund period. We observe that the agents contribute even post the refund stage, possibly implying a change in their beliefs.
  • Figure : (a) When $\{b_{i,t}\}_{t\in\bar{\mathbf{T}}_C}$ is a Super-martingale
  • Figure : (a) When $\{b_{i,t}\}_{t\in\bar{\mathbf{T}}_C}$ is a Super-martingale
  • Figure : (b) When $\{b_{i,t}\}_{t\in\bar{\mathbf{T}}_C}$ is a Sub-martingale

Theorems & Definitions (8)

  • Definition 1: Sub-game Perfect Equilibrium (SPE)
  • Definition 2: Martingales martingalepdf
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • Theorem 1