printacmref=false ifaamas [AAMAS ’24]Proc. of the 23rd International
Conference on Autonomous Agents and Multiagent Systems (AAMAS 2024)May 6 – 10,
2024 Auckland, New ZealandN. Alechina, V. Dignum, M. Dastani, J.S. Sichman
(eds.) 2024 2024 printacmref=false IIIT, Hyderabad Hyderbad India IIIT,
Hyderabad Hyderbad India

# Analyzing Crowdfunding of Public Projects Under Dynamic Beliefs

Sankarshan Damle<EMAIL_ADDRESS>and  Sujit Gujar
<EMAIL_ADDRESS>
###### 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.

###### Key words and phrases:

Civic Crowdfunding, Martingale Theory

15(0.35,1) In the Proceedings of the 23rd International Conference on
Autonomous Agents and Multiagent Systems (AAMAS), 2024, as an Extended
Abstract.

##  1\. Introduction

The process of raising funds for _public or private projects_ through
voluntary contributions is known as _crowdfunding_. As the contributors may be
strategic agents, researchers analyze crowdfunding game-theoretically Alaei et
al. (2016); Strausz (2017); Soundy et al. (2021); Damle et al. (2023). This
work focuses on the crowdfunding of public projects such as parks, libraries,
and community services.

##### Provision Point mechanism for Public projects (PPP)

. Bagnoli and Lipman (1989) present the seminal approach for crowdfunding of
public projects, which we refer to as PPP. In PPP, a project issuer (PI) sets
up the project’s crowdfunding by announcing a target threshold, known as the
_provision point_. PI seeks voluntary contributions from interested agents
towards this project before a known _deadline_. If the net contribution
crosses the provision point by the deadline, PI funds the public project
through them. If the target is not met, PI returns the contributions.

##### PPP & Free-riding.

As public projects are non-excludable, strategic agents in PPP may choose not
to contribute and _free-ride_. Moreover, PPP also admits several inefficient
equilibria Bagnoli and Lipman (1989); Healy (2006). The primary challenge in
crowdfunding of public projects is thus the lack of incentives for strategic
agents to contribute. _Provision Point mechanism with Refunds_ (PPR) Zubrickas
(2014) addresses this challenge with the introduction of _refund bonus
schemes_.

##### PPR.

With PPR, if the project is not funded, the agents receive their contribution
and an additional refund proportional to their contribution. Significantly,
the incentive structure avoids free-riding by incentivizing the agents to
contribute. PPR also overcomes inefficient equilibria as Zubrickas (2014)
proves that the total contribution equals the provision point at equilibrium.
Subsequent works Chandra et al. (2016); Damle et al. (2021) build on PPR by
introducing other refund schemes.

##### Modes of Crowdfunding.

The following two settings are possible for a project’s crowdfunding. (i)
_Offline_ : in which the participating agents are not aware of the history of
the contributions and the net contribution at any epoch. (ii) _Online_ : where
the net and the history of contributions are visible to each participating
agent (e.g., online platforms like kickstarter.com and spacehive.com). We
refer to crowdfunding over online settings as _sequential crowdfunding_.

Particularly for sequential crowdfunding, _blockchain_ -based online platforms
are becoming popular. Blockchain is an immutable, decentralized, and public
ledger Nakamoto (2008). These ledgers allow for pseudo-anonymous, transparent,
and verifiable payments while eliminating the middle person. In practice,
crowdfunding is now being deployed as _smart contracts_ over publicly
distributed ledgers such as the _Ethereum blockchain_ (e.g., weifund.io and
starbase.co). Carrying out transactions through incurs _gas_ (a form of
payment), and thus, there is a need to design efficient mechanisms for
sequential crowdfunding. Damle et al. (2021) present several refund schemes
and show these schemes consume fewer gas units, and therefore, the
corresponding crowdfunding mechanisms are efficient to deploy as smart
contracts over blockchains.

For an offline setting, PPR is an excellent choice. However, PPR induces a
simultaneous game Chandra et al. (2016). In sequential crowdfunding, such a
game result in the agents deferring their contribution until the deadline,
which in turn may result in the project not getting funded Chandra et al.
(2016); Cason et al. (2021), i.e., a “race” condition (RC). Chandra et al.
(2016) introduce _Provision Point mechanism with Securities_ (PPS), which
employs a temporal refund scheme to avoid the race condition. Furthermore,
Damle et al. (2021) study various aspects of refund schemes to avoid the race
condition and for efficient deployment in blockchain-based online settings.

##### Information Structure Damle et al. (2019).

In addition to the above, crowdfunding of public projects also depends on the
information available to the participating agents. To capture this, we define
the tuple consisting of each agent’s (i) valuation and (ii) belief as its
_information structure_. The existing literature majorly assumes that each
agent is interested in the funding of the public project, i.e., its
_valuation_ towards the project’s funding is non-negative. Additionally, the
literature also assumes that each agent has _symmetric_ belief, i.e., each
agent believes that the public project will be funded with probability $1/2$
and not with $1/2$. Note that in the real world, the beliefs may be
asymmetric. Damle et al. (2019) present PPRx (which leverages PPR) for public
projects when information structure allows positive valuation with asymmetric,
yet _static_ , beliefs.

With this background, we highlight a few observations that need to be
addressed in crowdfunding mechanism design.

######  Observation 1 (Kickstarter (2011)).

Empirically, the probability of funding a project _decreases_ with an
_increase_ in its duration.

######  Observation 2 (Cason et al. (2021)).

Empirically, agents prefer to contribute even in the absence of refunds (refer
to Figure 1 (left), Section 4).

##### Belief Evolution.

The above observations point to a change in the agent’s belief regarding
funding the public project. For instance, with Observation 1, we see that
agents become reluctant to fund projects whose target deadlines are greater.
Moreover, from Observation 2, it is natural to assume that the availability of
critical information, such as net contribution and the remaining time, will
also impact the agent’s belief.

We refer to such evolving beliefs as _dynamic_ beliefs. We model this belief
evolution as a _random walk_. We argue that each agent’s step size, at any
epoch, will be a posterior update depending on its prior belief and other
auxiliary information (e.g., net contribution or the time remaining).

##### PPRx-DB.

This work primarily incorporates dynamic beliefs to analyze incentive-based
civic crowdfunding mechanisms. To the best of our knowledge, PPRx is the only
mechanism that incorporates asymmetric but static beliefs. We study PPRx under
dynamic beliefs, and to distinguish our setting, we refer to it as _Provision
Point Mechanism for agents With Dynamic Belief_ (PPRx-DB). We argue that
agents’ belief evolution will be a random walk. We identify conditions on the
random walk under which we can characterize the sub-game perfect equilibria of
the sequential game induced by PPRx-DB. In particular, by utilizing the
evolution of each agent’s random walk as a martingale, super/sub-martingale,
we identify conditions wherein agents are naturally incentivized to contribute
as soon as they arrive (i.e., avoid the race condition). Thus, though
theoretically sound, complex mechanisms such as PPS may not be warranted in
practice (see Table 1).

Mechanism | Information Structure | Setting | On Blockchain | RC
---|---|---|---|---
Valuation | Belief
PPR Zubrickas (2014) | Positive | Static & Symmetric | Offline | - | ✓
PPS Chandra et al. (2016) | Positive | Static & Symmetric | Online | Inefficient | ✗
PPRG Damle et al. (2021) | Positive | Static & Symmetric | Online | Efficient | ✗
PPRN Damle et al. (2019) | Positive & Negative | Static & Symmetric | Offline | - | ✗
PPRx Damle et al. (2019) | Positive |  Static & Asymmetric | Offline | - | ✓
Chen et al. (2021) |  Binary ($\\{0,1\\}$) | Static & Symmetric | Offline | - | ✓
PPRx-DB | Positive |  Dynamic & Asymmetric | Online | Efficient |  ✗/✓‡
$\ddagger$: Depends on certain conditions on Agent Beliefs (see Table 2 for
details)
Table 1. Comparing existing literature with PPRx-DB. Here, “✗” denotes that
the mechanism avoids the Race Condition (RC), and “✓” denotes otherwise.

##  2\. Related Work

Designing mechanisms for crowdfunding of public projects with provision points
is an active research area Bagnoli and Lipman (1989); Marx and Matthews
(2000); Morgan (2000); Zubrickas (2014); Chandra et al. (2016, 2017); Damle et
al. (2019, 2023); Padala et al. (2022). Provision Point mechanism for Public
projects (PPP) Bagnoli and Lipman (1989) is the first mechanism in this class.
In PPP, agents contribute voluntarily to the public project until a deadline.
If the project is funded, the agents receive a positive payoff; if not, their
contributions are returned. However, PPP suffers from free-riding and has
several inefficient equilibria Healy (2006).

To overcome free-riding, Zubrickas (2014) proposes Provision Point mechanisms
with Refunds (PPR). PPR introduces the concept of a _refund bonus scheme_ paid
out to the contributing agents if the public project is not funded – along
with their contributions. More significantly, the equilibrium contributions in
PPR are such that the project is funded at equilibrium Zubrickas (2014).
Subsequently, researchers propose several mechanisms based on different agent
models and refund schemes. We summarize the existing literature for refund
mechanisms for crowdfunding of public projects with Table 1; where RC refers
to the race condition. PPRx Damle et al. (2019) is the closest work to ours;
however, PPRx assumes static beliefs.

##  3\. Preliminaries

We now (i) state our crowdfunding model, (ii) provide relevant game-theoretic
definitions, (iii) describe different information structures, (iv) summarize
PPRx and (v) define martingales.

##### Crowdfunding Model.

The sequential crowdfunding of a public project $P$ set up by a _Project
Issuer_ (PI) involves the following steps.

  1. (1)

PI announces a public project $P$, seeking voluntary contributions towards it.
The announcement also comprises of the _target threshold_ ($H_{0}$), the
_project deadline_ ($T$), and the _refund bonus scheme_ ($R$) (if deployed).
Let, $\mathbf{T}=\\{1,\dots,T\\}$ denote the set of discrete epochs.

  2. (2)

After step (1), each interested agent $j$ arrives to the crowdfunding platform
at $a_{j}\in\mathbf{T}$. Upon arrival, each agent $j$ observes its valuation
($\theta_{j}$) towards $P$.

  3. (3)

An interested agent $i$ may contribute $x_{i}$ to the crowdfunding platform at
an epoch of time $t_{i}\in\mathbf{T}$. Let $\mathbf{A}=\\{1,\dots,n\\}$ denote
the set of all contributing agents, with $C_{0}=\sum_{i\in\mathbf{A}}x_{i}$ as
the total contribution towards the public project; and
$\vartheta=\sum_{i\in\mathbf{A}}\theta_{i}$ as the total valuation of all the
contributing agents. Observe that, as we consider sequential crowdfunding and
as the agents are strategic, their contributions may depend on $\theta_{i}$
_as well as_ the total contribution up till $t_{i}$, i.e., $C_{t_{i}}$.

  4. (4)

PI states that $P$ is funded only if the total contribution equals $H_{0}$ at
any point before or at the deadline, i.e., $P$ is funded only if $\exists
t\in\mathbf{T}\mbox{\leavevmode\nobreak\ s.t.\leavevmode\nobreak\ }C_{t}\geq
H_{0}$. Otherwise, $P$ is not funded.

  5. (5)

In the event that $P$ is not funded, PI returns the contributions of all the
contributing agents. Agents may also be eligible for additional refunds
depending on the mechanism deployed.

Assumptions. As standard in crowdfunding literature, our work assumes that,

  * •

We assume that $\vartheta$ is public knowledge s.t. $\vartheta>H_{0}$
Zubrickas (2014); Chandra et al. (2016); Damle et al. (2019). It implies that
there is sufficient interest in the funding of the public project.

  * •

Agents arrive at the crowdfunding platform sequentially over time and _not_
simultaneously and can contribute only once to the project Chandra et al.
(2016); Damle et al. (2021).

  * •

Agents do not have _any_ information regarding the funding of the project
other than knowing the total contribution at any epoch of time Chandra et al.
(2016); Damle et al. (2021). Moreover, agents are also not aware of the number
of (i) contributions made before their arrival; and (ii) agents that are yet
to arrive at any epoch of time.

Steps 1-5 induce a game among the interested agents. To analyze this induced
game, we next define Sub-game Perfect Equilibrium. Towards these, let
$\psi_{i}=(x_{i},t_{i})$ denote the strategy of each agent $i\in\mathbf{A}$
with $\psi=(\psi_{1},\dots,\psi_{n})$ as the vector of their strategies. As
standard in the game theory literature, we use the subscript $-i$ to denote
vectors without agent $i$. Further, let the payoff derived by an agent $i$
with valuation $\theta_{i}$ and with the strategy profile, $\psi$ be
$\pi_{i}(\theta_{i};\psi)$.

##### Sub-game Perfect Equilibrium (SPE).

In a sequential setting, the agents may observe the actions of other agents
over time. Thus, we must characterize the strategy profile such that it is the
_best response_ for every agent to follow it at any time during the project,
i.e., for every sub-game induced. We refer to such a strategy profile as _Sub-
game Perfect Equilibrium_ defined as follows,

######  Definition 0 (Sub-game Perfect Equilibrium (SPE)).

A strategy profile $\psi^{\star}=(\psi_{1}^{\star},\dots,\psi_{n}^{\star})$ is
said to be a sub-game perfect equilibrium if for every agent $i$, if it
maximizes the payoff
$\pi_{i}(\psi^{\star}_{i},\psi^{\star}_{-i|h^{a_{i}}};\theta_{i}),$ i.e.
$\forall i\in\mathbf{A}$,

$\pi_{i}(\psi^{\star}_{i},\psi^{\star}_{-i|h^{a_{i}}};\theta_{i})\geq\pi_{i}(\psi_{i},\psi^{\star}_{-i|h^{a_{i}}};\theta_{i})\
\forall\psi_{i},\forall h^{t},\forall\theta_{i}.$

Here, $h^{t}$ is the history of the game till time $t$, constituting the net
contribution and $\psi^{\star}_{-i|h^{a_{i}}}$ indicates that the agents who
arrive after $a_{i}$ follow the strategy specified by $\psi^{\star}_{-i}$.
Informally it means that, at every stage of the game, irrespective of what has
happened, it is _Nash Equilibrium_ to follow the SPE strategy. That is,
dropping the dependence on the history, we note that
$\psi^{\star}=(\psi_{1}^{\star},\dots,\psi_{n}^{\star})$ satisfies Pure-
strategy Nash equilibrium when $\forall i\in\mathbf{A}$,

$\pi_{i}(\psi^{\star}_{i},\psi^{\star}_{-i};\theta_{i})\geq\pi_{i}(\psi_{i},\psi^{\star}_{-i};\theta_{i})\
\forall\psi_{i},\forall\theta_{i}.$

##### Agent Information Structure.

Similar to Damle et al. (2019), we consider agents with positive valuation and
asymmetric beliefs towards the funding of the public project. However, unlike
Damle et al. (2019), we _also_ consider that an agent’s belief may dynamically
evolve over time. To the best of our knowledge, our work is the first to
analyze such an information structure.

##### PPRx.

To incorporate asymmetric beliefs, Damle et al. (2019) introduce PPRx. PPRx
consists of two phases, (i) _Belief Phase_ (BP) with budget $B_{B}$ and
deadline $T_{B}$ wherein each agent $i$ arrives at $a_{i,1}$ to the
crowdfunding platform and submits its (static) belief $b_{i}$ at $t_{i,1}$
towards the funding of the project, based on which it gets a reward $m_{i}$
(known as Belief Based Reward); followed by (ii) _Contribution Phase_ (CP)
with budget $B_{C}$ and deadline $T_{C}$ $(T=T_{C}+T_{B})$ wherein each agent
$i$ arrives at $a_{i,2}$ to the crowdfunding platform and contributes $x_{i}$
at $t_{i,2}$ to the project. Thus, $\psi_{i}=(b_{i},t_{i,1},x_{i},t_{i,2})$ is
each agent $i$’s strategy in PPRx.

We create the following subsets for agents with “high” and “low” belief:
$\mathbf{A}_{H}=\\{i|\forall i\in\mathbf{A}\mbox{\leavevmode\nobreak\
s.t.\leavevmode\nobreak\ }b_{i}\geq 1/2\\}$ and $\mathbf{A}_{L}=\\{i|\forall
i\in\mathbf{A}\mbox{\leavevmode\nobreak\ s.t.\leavevmode\nobreak\
}b_{i}<1/2\\}$. Trivially, $\mathbf{A}=\mathbf{A}_{H}\sqcup\mathbf{A}_{L}$.
Let $\mathbbm{1}_{X}$ be an indicator random variable s.t. $\mathbbm{1}_{X}=1$
if $X$ is true and $\mathbbm{1}_{X}=0$ otherwise. Then, the payoff structure
in PPRx $\forall i\in\mathbf{A}_{H},$

$\pi_{i}=\mathbbm{1}_{{C_{0}}\geq
H_{0}}\cdot(\theta_{i}-x_{i}+m_{i})+\mathbbm{1}_{{C_{0}}<H_{0}}\cdot\left(\frac{x_{i}}{C_{0}}\cdot
B_{C}\right)$ (1)

Likewise, $\forall i\in\mathbf{A}_{L}$,

$\pi_{i}=\mathbbm{1}_{{C_{0}}\geq
H_{0}}\cdot(\theta_{i}-x_{i})+\mathbbm{1}_{{C_{0}}<H_{0}}\cdot\left(\frac{x_{i}}{C_{0}}\cdot
B_{C}+m_{i}\right)$ (2)

Funded and Unfunded Payoffs. We denote the payoff for agent $i$ when
$C_{0}\geq{H}_{0}$ as $\pi_{i}^{F}$ (i.e., funded) and when $C_{0}<{H}_{0}$ as
$\pi_{i}^{UF}$ (i.e., unfunded). In PPRx,

  * •

$\forall i\in\mathbf{A}_{H}$:

$\displaystyle\mathbb{E}[\pi_{i}^{F}(x_{i},\cdot)]$
$\displaystyle=b_{i}\cdot\left(\theta_{i}-x_{i}+m_{i}\right)$
$\displaystyle\mathbb{E}[\pi_{i}^{UF}(x_{i},\cdot)]$
$\displaystyle=(1-b_{i})\cdot\left(\frac{x_{i}}{C_{0}}\cdot B_{C}\right)$

  * •

$\forall i\in\mathbf{A}_{L}$:

$\displaystyle\mathbb{E}[\pi_{i}^{F}(x_{i},\cdot)]$
$\displaystyle=b_{i}\cdot\left(\theta_{i}-x_{i}\right)$
$\displaystyle\mathbb{E}[\pi_{i}^{UF}(x_{i},\cdot)]$
$\displaystyle=(1-b_{i})\cdot\left(\frac{x_{i}}{C_{0}}\cdot
B_{C}+m_{i}\right)$

Position of BBR. We now describe the intuition behind the different positions
of $m_{i}$ in Eq. 1 and Eq. 2. We first note that, in PPRx, only one set of
agents, either $\mathbf{A}_{H}$ or $\mathbf{A}_{L}$, get the BBR reward. The
agents in $\mathbf{A}_{H}$ get the reward when the project is funded, implying
their belief regarding the project’s funding was ‘correct.’ Likewise, agents
in $\mathbf{A}_{L}$ get the reward when the project is not funded in
accordance with their lower belief. The impact of such a utility structure is
that agents with a higher belief have a greater equilibrium contribution than
agents with lower belief – which is a desirable outcome.

##### Belief Based Reward (BBR).

Based on the belief submitted by each agent in the Belief Phase, PPRx gives
each agent $i$ a reward $m_{i}$, as follows Damle et al. (2019),

$m_{i}=\begin{cases}\frac{w_{i}}{\sum_{j\in\mathbf{A}_{H}}w_{j}}\times
B_{B}&\forall i\in\mathbf{A}_{H}\\\
\frac{w_{i}}{\sum_{j\in\mathbf{A}_{L}}w_{j}}\times B_{B}&\forall
i\in\mathbf{A}_{L}\\\ \end{cases}$ (3)

for $w_{i}=\frac{y_{i}}{\sum_{j}y_{j}}\ \forall j\in S_{t_{i}}$ where $y_{i}$
is the score calculated by the RBTS mechanism Witkowski and Parkes (2012)
depending on the belief $b_{i}$, while $S_{t_{i}}$ is the set consisting of
all the agents that have reported their belief till $t_{i}$. BBR is (i)
_incentive compatible_ and (ii) is a decreasing function of time. However,
PPRx does not incorporate the evolution of agents’ beliefs over time and its
dependence on the total contribution, i.e., does not incorporate dynamic
beliefs.

##### Martingale Theory.

A martingale is a sequence of random variables such that the next value is
equal to the current value in expectation, conditioned over all prior values.
However, for several applications, one cannot always guarantee this equality.
To analyze such scenarios, we interest ourselves in _bounding_ the expected
values. Such a sequence corresponds to a _super-martingale_ or a _sub-
martingale_. Formally, consider a discrete sequence of random variables
$X_{0},X_{1},\dots$ evolving over time. Such a collection of random variables
is referred to as a _stochastic process_ , denoted by
$\\{X_{t}\\}_{t\in\mathbf{T}}$.

######  Definition 0 (Martingales Williams (1991)).

A stochastic process $\\{X_{t}\\}_{t\in\mathbf{T}}$ such that
$\mathbb{E}[X_{t}]<\infty$, is a

  * •

Martingale if $\mathbb{E}[X_{t+1}|X_{0},\dots,X_{t}]=X_{t}$

  * •

Sub-martingale if $\mathbb{E}[X_{t+1}|X_{0},\dots,X_{t}]\geq X_{t}$; and

  * •

Super-martingale if $\mathbb{E}[X_{t+1}|X_{0},\dots,X_{t}]\leq X_{t}$.

In mechanism design literature, martingale theory is popularly used to model
the dynamic evolution of an agent’s private information. For e.g., Chawla et
al. (2016) model agent’s dynamic valuation for a product (such as Netflix
subscription) over time as a Martingale. Balseiro et al. (2018) model agent’s
expected utility as a Martingale to design a dynamic auction.

Figure 1. Plotting $\approx x/\theta$ for two randomly sampled agents using
the dataset available with Cason et al. (2021). 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.

##  4\. PPRx-DB Mechanism

This section analyzes PPRx with dynamic beliefs. We first present the agent’s
dynamic belief model. We then introduce PPRx-DB and provide agents’
equilibrium contribution and the equilibrium time of contribution. We begin by
presenting empirical evidence that agents’ beliefs evolve during the
crowdfunding process.

##### Observing Agent’s Belief Evolution.

Cason et al. (2021) conduct real-world experiments to primarily test the
impact of early refund bonus on a crowdfunding project’s success. We use their
data to provide the following insight regarding an agent’s evolving belief.

Figure 1 plots $x/\theta$, with varying $t$ and $C_{t}$, for two random agents
from the dataset. Post $t>60$ seconds; the agents do not get refunds for their
contributions. Yet, we observe that agents contribute post $t>60$ (Figure
1(left)). Agents’ contribution pattern also evolves with $C_{t}$ (Figure
1(right)).

###  4.1. Agent Dynamic Belief Model

We model the evolution of each agent’s belief as a stochastic process over
discrete epochs. Observe that this belief evolution may depend on available
information at an epoch (e.g., net contribution). After each epoch, as an
agent’s belief can increase or decrease, we model it as a random walk.

For each agent $i\in\mathbf{A}$, let $\\{b_{i,t}\\}_{t\in\mathbf{T}}$ denote
the random walk with $X_{i,t}$ as the random variable for the step size at an
epoch $t$. More formally, let each agent $i$’s prior belief regarding the
project’s funding be $b_{i,0}\in[0,1]$. At each epoch $t$, the agent’s belief
evolves in accordance with the available information, e.g., $C_{t}$ or
remaining epochs $T-t$.

Now, at each epoch $t\geq 1$, we denote agent $i$’s posterior belief regarding
the project’s funding as: $b_{i,t}=b_{i,t-1}+X_{i,t}$. The sizes of the
positive ($s_{i,+}()$) and negative ($s_{i,-}()$)steps (with “$\circ$” as
auxiliary information) are:

$X_{i,t}=\begin{cases}s_{i,+}(C_{t},T-t,\circ),&\text{with probability
}p\in[0,1]\\\ s_{i,-}(C_{t},T-t,\circ),&\text{with probability
}1-p\end{cases}$

Note 1. $X_{i,t}$ captures the agent’s belief evolution through the size of
the step sizes, dependent on the available information. That is, an agent
$i$’s belief evolves as $b_{i,t}\leftarrow b_{i,t-1}+X_{i,t}$ where $b_{i,0}$
is the agent’s prior belief111Future work can explore the exact
characterization of this random walk.. We now have a model for the random
walk, $\\{b_{i,t}\\}_{t\in\mathbf{T}}$. Our goal is to analytically derive
equilibrium strategies for the agents conditioned on the behavior of
$\\{b_{i,t}\\}_{t\in\mathbf{T}}$.

###  4.2. PPRx-DB: Formal Description

We refer to PPRx in such a setting as _Provision Point Mechanisms for agents
with Dynamic Belief_ (PPRx-DB) to enhance readability. Protocol 1 formally
describes PPRx-DB.

We denote $\mathbf{T}_{B}=\\{0,1,\dots,T_{B}\\}$ and
$\mathbf{T}_{C}=\\{T_{B+1},\dots,T_{B}+T_{C}\\}$, where $T=T_{B}+T_{C}$, as
the sets constituting the discrete epochs of time for the two phases. For the
analysis of the mechanism, let $\bar{\mathbf{T}}_{C}=\\{1,\dots,T_{C}\\}$ as
the set of epochs of time for the CP (w.l.o.g). Consequently, the strategy for
each agent $i\in\mathbf{A}$ becomes
$\psi_{i}=(\hat{b}_{i},t_{i,1},x_{i},t_{i,2})$. The payoff structure of PPRx-
DB is the same as in PPRx, i.e., Eqs. 1 and 2. We next game-theoretically
analyze PPRx-DB.

1 • Belief Phase (BP): (1) PI announces the start of the phase (2) Agent $i$
enters at $a_{i,1}$ and submits $\hat{b}_{i}$ at $t_{i,1}\in\mathbb{T}_{B}$
(3) Each participating agent $i$ is told their BBR reward $m_{i}$ •
Contribution Phase (CP): (1) PI announces the start of the phase (2) Agent $i$
enters at $a_{i,2}$ and submits $x_{i}$ at $t_{i,2}$ (3) The protocol
continues until the target is reached (4) PI announces end of phase if target
is not reached at $T_{C}$ • Refunds are distributed as per the outcome by PI
according to Eqs. 1 and 2 for set of agents $\mathbf{A}_{H}$ and
$\mathbf{A}_{L}$, respectively

Protocol 1 _PPRx-DB Mechanism_

###  4.3. PPRx-DB: Theoretical Analysis

In this subsection, we first discuss the funding of the public project at
equilibrium. Second, we provide the upper bound of the agents’ equilibrium
contribution. Last, we present the equilibrium time of contribution for agents
based on the underlying condition of the agent’s belief evolution.

####  4.3.1. Project Status at Equilibrium

In PPRx-DB, the public project is funded at equilibrium. That is, at
equilibrium, the total contribution _equals_ the provision point, i.e.,
$C_{0}=H_{0}$, when $\vartheta>H_{0}$. Consider the following lemma.

###### Lemma 3

In PPRx-DB, the public project is funded at equilibrium, i.e., $C_{0}=H_{0}$
if $\vartheta>H_{0}$ with $B_{B},B_{C}>0$.

###### Proof.

From Eq. 1 and Eq. 2, at equilibrium, $C_{0}<H_{0}$ cannot hold, since
$\exists i\in\mathbf{A}_{H}$ with $x_{i}<\theta_{i}+m_{i}$ or $\exists
i\in{\mathbf{A}}_{L}$ with $x_{i}<\theta_{i}$, at least, as $\vartheta>H_{0}$.
Such an agent $i$ can obtain a greater refund bonus by marginally increasing
its contribution $x_{i}$ as $B_{C}>0$. Moreover, if $C_{0}>H_{0}$, any
contributing agent can increase its payoff by marginally decreasing its
contribution. Thus, at equilibrium $C^{0}=H^{0}$ holds, i.e., the project is
funded at equilibrium. ∎

####  4.3.2. Equilibrium Contribution: Upper Bound

We now analyze the equilibrium contribution of each agent $i\in\mathbf{A}$ in
PPRx-DB. As each agent $i\in\mathbf{A}$ submits its belief $\hat{b}_{i}$ in
the BP, PI can categorize each agent $i$ to the sets $\mathbf{A}_{H}$ or
$\mathbf{A}_{L}$. We next independently compute equilibrium contributions for
the agents in $\mathbf{A}_{H}$ and $\mathbf{A}_{L}$, respectively.

1

##### For Agents with High Belief.

Lemma 4 presents the equilibrium contribution, $x_{i}^{\star}$, analysis for
each agent $i\in\mathbf{A}_{H}$. For the proof, we solve for $x_{i}^{\star}$
such that the (expected) funded payoff is greater than equal to the (expected)
unfunded payoff. This is because from Lemma 3 we know that at equilibrium the
contributions are such that $C_{0}=H_{0}$.

###### Lemma 4

In PPRx-DB, for each $i\in\mathbf{A}_{H}$, its equilibrium contribution is

$x_{i}^{\star}\leq\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}(\theta_{i}+m_{i})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}},$
(4)

where $t_{i,{2^{\star}}}\in\bar{\mathbf{T}}_{C}$ is its time of contribution
at equilibrium.

###### Proof.

Since at equilibrium $C_{0}=H_{0}$, each agent $i$ will contribute such that
its funded payoff is no less than the highest possible unfunded payoff. That
is, we solve for $x_{i}^{\star}$ such that
$\mathbb{E}[\pi_{i}^{F}]\geq\mathbb{E}[\pi_{i}^{UF}]$, for each
$i\in\mathbf{A}_{H}$. That is,

$\displaystyle
b_{i,t_{i,2^{\star}}}\cdot(\theta_{i}-x_{i}^{\star}+m_{i})\geq(1-b_{i,t_{i,2^{\star}}})\cdot\frac{x_{i}^{\star}}{H_{0}}\cdot
B_{C}$ $\displaystyle\implies
x_{i}^{\star}\leq\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}(\theta_{i}+m_{i})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}}$

This proves the lemma. ∎

2

##### For Agents with Low Belief.

Similar to our analysis for the set of agents in $\mathbf{A}_{H}$, Lemma 5
presents the equilibrium contribution analysis of each agent
$i\in\mathbf{A}_{L}$.

###### Lemma 5

In PPRx-DB, for each $i\in\mathbf{A}_{L}$, its equilibrium contribution is

$x_{i}^{\star}\leq\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}\theta_{i}+{H}_{0}m_{i}(1-b_{i,{t_{i,{2^{\star}}}}})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}},$
(5)

where $t_{i,{2^{\star}}}\in\bar{\mathbf{T}}_{C}$ is its time of contribution
at equilibrium.

###### Proof.

Similar to Lemma 4, we again solve for $x_{i}^{\star}$ such that
$\mathbb{E}[\pi_{i}^{F}]\geq\mathbb{E}[\pi_{i}^{UF}]$, for each
$i\in\mathbf{A}_{H}$. That is,

$\displaystyle
b_{i,t_{i,2^{\star}}}\cdot(\theta_{i}-x_{i})\geq(1-b_{i,t_{i,2^{\star}}})\cdot\left(\frac{x_{i}^{\star}}{H_{0}}\cdot
B_{C}+m_{i}\right)$ $\displaystyle\implies
x_{i}^{\star}\leq\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}\theta_{i}+{H}_{0}m_{i}(1-b_{i,{t_{i,{2^{\star}}}}})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}}$

This proves the lemma. ∎

####  4.3.3. Time of Equilibrium Contribution

Firstly, note that the refund bonus scheme in PPR (or PPRx) is independent of
time. This induces a simultaneous-move game in PPR Chandra et al. (2016) or
PPRx Damle et al. (2019). However, in PPRx-DB, the dynamic evolution of
agents’ belief towards the public project results in _variable_ expected
payoff for each agent – dependent on their belief at each epoch. Thus, unlike
PPR and PPRx, PPRx-DB does not induce a simultaneous-move game and can be
deployed in sequential settings.

The challenge remains to identify the time at which an agent will contribute
to the public project. Recall that we denote the funded payoff for agent $i$
as $\pi_{i}^{F}$ and the unfunded payoff as $\pi_{i}^{UF}$. Now, the complete
payoff structure for agent $i$ is,

$\pi_{i}(\cdot)=\mathbbm{1}_{C_{0}\geq{H}_{0}}\cdot\pi_{i}^{F}(\cdot)+\mathbbm{1}_{C_{0}<{H}_{0}}\cdot\pi_{i}^{UF}(\cdot).$

At equilibrium, the expected funded payoff is equal to the expected not funded
payoff (Lemmas 4 and 5). Thus, we have
$\mathbb{E}[\pi_{i}]=\mathbb{E}[\pi_{i}^{UF}]$, $\forall i$ at equilibrium. In
PPRx-DB, from Eq. 1 and Eq. 2, we also have

$\pi_{i}^{UF}(x_{i})=\frac{x_{i}}{C_{0}}\cdot B_{C}+c,$

where $c=0,\forall i\in\mathbf{A}_{H}$ and $c=m_{i},\forall
i\in\mathbf{A}_{L}$.

Now, the equilibrium time of contribution
$t_{i,2^{\star}},\leavevmode\nobreak\ \forall i\in\mathbf{A}$, can be
calculated as:

$t_{i,2^{\star}}=\operatorname*{arg\,max}_{t_{i,2}\in\mathbf{\bar{T}}_{C}}\mathbb{E}[\pi_{i}^{UF}(x_{i}^{\star})].$

The subsequent results indeed derive $t_{i,2^{\star}}$ for the set of agents
in $\mathbf{A}_{H}$ and $\mathbf{A}_{L}$. For these, we remark that when an
agent $i$ arrives at the Contribution Phase (CP), its belief at that epoch is
the same as its prior belief, i.e., $b_{i,a_{i,2}}=b_{i,0}$. This stems from
the fact that the agent has yet to observe the available information for any
meaningful belief update.

(a) When $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Super-martingale

(b) When $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Sub-martingale

Figure 2. Proof Intuition for Lemma 6: Deriving Time of Equilibrium
Contribution for Agent $i\in\mathbf{A}_{H}$

1

##### For Agents with High Belief.

Consider the following lemma for each agent $i\in\mathbf{A}_{L}$.

###### Lemma 6

In PPRx-DB, with $b^{\star}=\frac{\sqrt{B_{C}/H_{0}}}{1+\sqrt{B_{C}/H_{0}}}$,
for each $i\in\mathbf{A}_{H}$, if:

  1. (1)

$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Martingale, then
$t_{i,2^{\star}}=T_{C}.$

  2. (2)

$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Super-martingale, then

$t_{i,2^{\star}}=\begin{cases}a_{i,2}&\mbox{\leavevmode\nobreak\
if\leavevmode\nobreak\ }b_{i,0}\leq b^{\star}\\\ t\mbox{\leavevmode\nobreak\
s.t.\leavevmode\nobreak\ }b_{i,t}=b^{\star}&\mbox{\leavevmode\nobreak\
if\leavevmode\nobreak\ }b_{i,0}>b^{\star}\end{cases}$

  3. (3)

$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Sub-martingale, then

$t_{i,2^{\star}}=\begin{cases}a_{i,2}&\mbox{\leavevmode\nobreak\
if\leavevmode\nobreak\ }b_{i,0}\geq b^{\star}\\\ t\mbox{\leavevmode\nobreak\
s.t.}b_{i,t}=b^{\star}&\mbox{\leavevmode\nobreak\ if\leavevmode\nobreak\
}b_{i,0}<b^{\star}\end{cases}$

###### Proof.

We broadly divide the proof into the following two steps: (i) Firstly, we show
that $\mathbb{E}[\pi_{i}^{UF}(x_{i}^{\star})]$ is increasing in $b_{i,t}$ only
if $b_{i,t}\leq b^{\star}$. That is, $\mathbb{E}[\pi_{i}^{UF}(x_{i}^{\star})]$
is maximized at $b^{\star}$. In (ii), to decide on $t_{i,2^{\star}}$, we
condition on the underlying evolution of agent belief. Consider the following.

  * (i)

Deriving $b^{\star}$. We have,

$\displaystyle\mathbb{E}[\pi_{i}^{UF}]=(1-b_{i,t})\cdot\frac{B_{C}}{H_{0}}\cdot\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}(\theta_{i}+m_{i})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}}$

Now,

$\frac{\partial\mathbb{E}[\pi_{i}^{UF}]}{\partial
b_{i,t}}>0\iff\frac{b_{i,t}}{1-b_{i,t}}\leq\sqrt{\frac{B_{c}}{H_{0}}}.$

That is,
$b_{i,t}\leq\frac{\sqrt{B_{C}/H_{0}}}{1+\sqrt{B_{C}/H_{0}}}=b^{\star}.$

  * (ii)

Deriving $t_{i,2^{\star}}$. First, if
$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Martingale,
$\mathbb{E}[b_{i,t}]=b_{i,{t-1}}$. Thus, on expectation the value
$\mathbb{E}[\pi_{i}^{UF}]$ does not change. As such, agent $i$ has no
incentive to contribute early, resulting in the race condition, i.e.,
$t_{i,2^{\star}}=T_{C}$.

Second, if $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Super-martingale,
we have $\mathbb{E}[b_{i,t}]\leq b_{i,{t-1}}$. Since
$\mathbb{E}[\pi_{i}^{UF}]$ increases till $b^{\star}$, if agents initial
belief is less than $b^{\star}$, $b_{i,t}$ will not reach $b^{\star}$ (in
expectation) implying agent $i$ must contribute as soon as it arrives, i.e.,
$t_{i,2^{\star}}=a_{i,2}$. However, if agent $i$’s initial belief is greater
than $b^{\star}$ and since $\mathbb{E}[b_{i,t}]\leq b_{i,{t-1}}$, the agent
waits till an epoch $t^{\prime}$ s.t. $b_{i,t^{\prime}}=b^{\star}$. Figure
2(a) provides the proof intuition.

Last, if $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Sub-martingale, we
have $\mathbb{E}[b_{i,t}]\geq b_{i,{t-1}}$. Now, if agents initial belief is
greater than $b^{\star}$, it is incentivized to contribute as soon as it
arrives as its belief increases in expectation resulting in lesser
$\mathbb{E}[\pi_{i}^{UF}]$. Likewise, if its initial belief is less than
$b^{\star}$, than in expectation its belief will increase. That is, agent $i$
waits till an epoch $t^{\prime}$ s.t. $b_{i,t^{\prime}}=b^{\star}$. Figure
2(b) provides the proof intuition.

This proves the lemma. ∎

Agent Set | $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ | $x_{i,t_{i,2^{\star}}}$ | $t_{i,2^{\star}}$ | Race Condition
---|---|---|---|---
$\forall i\in\mathbf{A}_{H}$ | Martingale | $\leq\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}(\theta_{i}+m_{i})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}}$ | $T_{C}$ | ✓
Super-martingale | $a_{i,2}\mbox{\leavevmode\nobreak\ if\leavevmode\nobreak\ }b_{i,0}\leq b^{\star}$ | ✗
$t\mbox{\leavevmode\nobreak\ s.t.}b_{i,t}=b^{\star}\mbox{\leavevmode\nobreak\ if\leavevmode\nobreak\ }b_{i,0}>b^{\star}$ | ✗
Sub-martingale | $a_{i,2}\mbox{\leavevmode\nobreak\ if\leavevmode\nobreak\ }b_{i,0}\geq b^{\star}$ | ✗
$t\mbox{\leavevmode\nobreak\ s.t.}b_{i,t}=b^{\star}\mbox{\leavevmode\nobreak\ if\leavevmode\nobreak\ }b_{i,0}<b^{\star}$ | ✗
$\forall i\in\mathbf{A}_{L}$ | Martingale | $\leq\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}\theta_{i}+{H}_{0}m_{i}(1-b_{i,{t_{i,{2^{\star}}}}})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}}$ | $T_{C}$ | ✓
Super-martingale | $a_{i,2}$ | ✗
Sub-martingale | $T_{C}$ | ✓
Table 2. Summary of Our Results for PPRx-DB. Here, “✗” denotes that the
mechanism avoids race condition.

2

##### For Agents with Low Belief.

Similar to Lemma 6, we now analytically present time of equilibrium
contribution for agents in $\mathbf{A}_{L}$.

###### Lemma 7

In PPRx-DB, with $\theta_{i}>m_{i}$ and $\theta_{i}<\frac{m_{i}\cdot
H_{0}}{B_{C}}$ for each $i\in\mathbf{A}_{L}$, if

  1. (1)

$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Martingale, then
$t_{i,2^{\star}}=T_{C}$.

  2. (2)

$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Super-martingale, then
$t_{i,2^{\star}}=a_{i,2}$.

  3. (3)

$\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Sub-martingale, then
$t_{i,2^{\star}}=T_{C}$.

###### Proof.

Similar to the proof for Lemma 6, we broadly divide the proof in the following
two steps: (i) Firstly, we show that $\mathbb{E}[\pi_{i}^{UF}(x_{i}^{\star})]$
is increasing in $b_{i,t}$ if $\theta_{i}>m_{i}$ and
$\theta_{i}<\frac{m_{i}\cdot H_{0}}{B_{C}}$. In (ii), to decide on
$t_{i,2^{\star}}$, we condition on the underlying evolution of agent belief.
Consider the following.

  * (i)

$\mathbb{E}[\pi_{i}^{UF}]$ as an Increasing Function. We first derive the
condition in which $\mathbb{E}[\pi_{i}^{UF}]$ is an increasing function. We
have,

$\mathbb{E}[\pi_{i}^{UF}]=(1-b_{i,t})\cdot\frac{B_{C}}{H_{0}}\cdot\frac{{H}_{0}b_{i,{t_{i,{2^{\star}}}}}\theta_{i}+{H}_{0}m_{i}(1-b_{i,{t_{i,{2^{\star}}}}})}{B_{C}(1-b_{i,{t_{i,{2^{\star}}}}})+{H}_{0}b_{i,{t_{i,{2^{\star}}}}}}$

Now,

$\displaystyle\frac{\partial\mathbb{E}[\pi_{i}^{UF}]}{\partial
b_{i,t}}>0\iff(H_{0}-B_{C})(m_{i}-\theta_{i})b_{i,t}^{2}+$ $\displaystyle
2B_{C}(m_{i}-\theta_{i})b_{i,t}-(H_{0}+B_{C})m_{i}+B_{C}\theta_{i}>0$ (6)

For $\mathbb{E}[\pi_{i}^{UF}]$ to be increasing, the quadratic in Eq. 6 must
be increasing. That is, its $\Delta<0$ and the first term must be positive.
Through algebraic manipulations, we can show that these conditions will hold
$\forall i\in\mathbf{A}_{L}$ iff $\theta_{i}>m_{i}$ and
$\theta_{i}<\frac{m_{i}\cdot H_{0}}{B_{C}}$.

  * (ii)

Deriving $t_{i,2^{\star}}$. As $\mathbb{E}[\pi_{i}^{UF}]$ is increasing in
$b_{i,t}$ under $\theta_{i}>m_{i}$ and $\theta_{i}<\frac{m_{i}\cdot
H_{0}}{B_{C}}$, we now derive $t_{i,2^{\star}}$ conditioned on the nature of
the belief evolution. First, if $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is
a Martingale, then $\mathbb{E}[b_{i,t}]=b_{i,t}$. Trivially, the value
$\mathbb{E}[\pi_{i}^{UF}]$ will not change in expectation for such a case.
Thus, in practice, agent $i$ will defer its contribution to the deadline,
i.e., $t_{i,2^{\star}}=T_{C}$.

Second, if $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Super-martingale,
then $\mathbb{E}[b_{i,t}]\leq b_{i,t}$. In this case, a decrease in $b_{i,t}$
(in expectation) will imply a decrease in $\mathbb{E}[b_{i,t}]$. As such,
agent $i$ will contribute as soon as it arrives, i.e.,
$t_{i,2^{\star}}=a_{i,2}$.

Last, if $\\{b_{i,t}\\}_{t\in\bar{\mathbf{T}}_{C}}$ is a Sub-martingale, then
$\mathbb{E}[b_{i,t}]\geq b_{i,t}$. In this case, an increase in $b_{i,t}$ (in
expectation) will imply an increase in $\mathbb{E}[b_{i,t}]$. As such, agent
$i$ will defer its contribution till the deadline, i.e.,
$t_{i,2^{\star}}=T_{C}$.

This proves the lemma. ∎

Note 2. Table 2 summarizes the results presented in this section. We
analytically provide the equilibrium contribution and time of contribution
based on the underlying property of agents’ dynamic belief evolution. The
equilibrium time of contribution also implies if the mechanism avoids the race
condition or not. That is, when the equilibrium time of contribution equals
the deadline, the race condition persists.

###  4.4. PPRx-DB: SPE Strategy

What remains to be shown is that the strategy, for each $i\in\mathbf{A}$,
$\psi_{i}^{\star}=(b_{i}^{\star},t_{i,1^{\star}},x_{i}^{\star},t_{i,2^{\star}})$
where $b_{i}^{\star}=b_{i,0},t_{i,1^{\star}}=a_{i,1}$, $x_{i}^{\star}$ as
defined in Lemma 4 and Lemma 5 and $t_{i,2^{\star}}$ as defined in Lemma 6 and
Lemma 7 satisfies sub-game perfect equilibrium (SPE). To this end, consider
the following theorem.

######  Theorem 8.

For PPRx-DB, with the payoff structure as given by Eq. 1 and Eq. 2,
$\vartheta>H_{0}$ and $B_{B},B_{C}>0$, we have $C_{0}=H_{0}$ and the set of
strategies
$\psi_{i}^{\star}=(b_{i}^{\star},t_{i,1^{\star}},x_{i}^{\star},t_{i,2^{\star}})$
where $b_{i}^{\star}=b_{i,0},t_{i,1^{\star}}=a_{i,1}$ and
$(x_{i}^{\star},t_{i,2^{\star}})$ as defined in Lemma 4 and Lemma 6 $\forall
i\in\mathbf{A}_{H}$ and in Lemma 5 and Lemma 7 $\forall i\in\mathbf{A}_{L}$.

###### Proof.

Firstly, from Lemma 3, we know that $C_{0}=H_{0}$ if $\vartheta>H_{0}$. Next,
$b_{i}^{\star}=b_{i,0}$ and $t_{i,1^{\star}}=a_{i,1}$ follows from the
properties of BBR. More concretely, since BBR is incentive compatibility and
decreasing with time, each agent $i\in\mathbf{A}$ reports its prior belief as
soon as it arrive to the Belief Phase. Further, Lemmas 4,5 derive
$x_{i}^{\star}$ and Lemmas 6,7 derive $t_{i,2^{\star}},\forall
i\in\mathbf{A}_{H}$ and $\forall i\in\mathbf{A}_{L}$, respectively.

The equilibrium strategy $\psi_{i}^{\star}$ depending on the aggregate
contribution and current belief is also SPE. W.l.o.g., let agent $j$ arrive to
the Contribution Phase (CP) last. If $C_{a_{j,2}}=H_{0}$, then its best
response is contributing $x_{j,2^{\star}}=0$. If $H_{0}-C_{a_{j,2}}>0$, then
irrespective of $H_{0}$ and $C_{a_{j,2}}$ its best strategy is $x_{i}^{\star}$
(defined in Lemmas 4,5) and $t_{i,2^{\star}}$ (defined in 6,7). Using backward
induction, we argue that it is the best response for every agent $i$ to follow
its strategy $\psi_{i}^{\star}$, irrespective of history. That is,
$\psi_{i}^{\star}$ satisfy SPE, $\forall i\in\mathbf{A}$. ∎

Theorem 8 present the SPE strategy for an agent. Without additional
information/assumption regarding the belief evolution or future agents’
contribution, we believe that these are a good starting point for mechanism
design for crowdfunding of public projects with dynamic beliefs.

##  5\. Conclusion & Future Work

To the best of our knowledge, this paper is the first attempt at addressing
the persistent issue of static beliefs in the existing literature on
crowdfunding of public projects. Towards this, we model the dynamic belief
update for each agent as a random walk. Empirical evidence available justifies
this argument. Next, we analyzed PPRx with dynamic beliefs as PPRx-DB. We
first derived the agent’s equilibrium contribution as a function of their
dynamic beliefs. In order to derive the time of equilibrium contribution, we
condition the dynamic belief as a (i) Martingale, (ii) Super-martingale, and
(iii) Sub-martingale. Based on these underlying conditions, we provide the
time of equilibrium contribution. Consequently, we also showed the conditions
at which PPRx-DB avoids the race condition.

Discussion & Future Work. Significantly, our results highlight that simpler
mechanisms may also avoid the race condition, allowing a practitioner to save
on-chain deployment costs. Future work can build on these results by (i)
exploring other conditions that provide an analytical characterization of the
agent’s equilibrium time and contribution and (ii) empirically validating the
evolution of the agent’s dynamic belief as a martingale. In parallel, one can
even attempt to learn an ML model for an agent’s belief update.

The authors would like to thank Prof. Timothy Cason for providing access to
their dataset introduced in Cason et al. (2021).

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