Civic Crowdfunding for Agents with Negative Valuations and Agents with Asymmetric Beliefs

1 Introduction

Crowdfunding is a process of raising funds from a large pool of interested agents and is an active area of research [Alaei et al., 2016; Arieli et al., 2017; Chandra et al., 2017; Strausz, 2017; Shen et al., 2018]. The process, when applied for the provision of public projects, is called civic crowdfunding. In the last decade, civic crowdfunding has grown to be instrumental in providing a platform through which citizens can collectively finance social initiatives such as libraries, public parks, etc.

In the standard approach for civic crowdfunding, the social planner uses the voluntary contribution mechanism with a provision point, the provision point mechanism [Bagnoli and Lipman, 1989]. The social planner sets up a target amount, referred to as the provision point, to be raised. If the contributions by the agents exceed the provision point, social planner provisions the project; otherwise, returns the contributions. The mechanism, however, has been shown to have several inefficient equilibria [Bagnoli and Lipman, 1989; Brubaker, 1975; Schmidtz, 1991].

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Provision Point mechanism with Refund bonus (PPR) by Zubrickas [2014] introduces an additional refund bonus to be paid to all the contributing agents (along with their contribution) in case the project is not provisioned. Chandra et al. [2016] showed that in sequential setting, wherein the history of contributions is known to the agents, PPR collapses to a simultaneous-move game, among the contributing agents. Towards this, they proposed Provision Point mechanism with Securities (PPS) with refunds based on complex prediction markets, and showed that it induces a sequential game, in which the project is provisioned at equilibrium. Thus in this paper, for a sequential game, we focus on PPS while focusing on PPR for a simultaneous game.

Note that in all these mechanisms only those agents with positive valuations towards the project contribute to its provision. However, several agents may prefer the project to not be provisioned, i.e., their valuations may be negative for the project getting provisioned. For instance, consider the construction of a garbage dump yard in a locality. While the project may be welcomed by a number of agents, a certain set of agents may wish to relocate the project from its current location to another. In other words, these agents may not prefer the construction of the dump yard - in the locality proposed. In such a scenario, the construction of the dump yard (as well as the locality in which it is constructed) must depend on the majority's opinion of it. The civic crowdfunding literature does not address such negative valuations. If addressed, civic crowdfunding with agents with negative valuations can provide a natural way for preference aggregation.

We define an agent's information structure as consisting of its valuation and its belief towards the project's provision. Based on their valuation, we categorize these agents as follows: positive (negative) agents i.e., agents with positive (negative) valuations or positive (negative) preferences towards the project's provision. The mechanisms, mentioned above, also assume that apart from knowing the history of contributions, agents do not have any information regarding the provision of the project, i.e., every agent's belief is symmetric towards the project's provision.

Motivated to break these barriers on an agent's information structure in existing literature for civic crowdfunding, in this paper, we address these two limitations by (i) handling symmetric agents with negative preferences and (ii) handling positive agents with asymmetric belief towards the project's provision, independently. Relaxing both the assumptions in one mechanism is still illusive.

To incorporate civic crowdfunding for agents with negative valuations, we require mechanisms that integrate negative agents. For this, we set up two parallel markets, with two different targets - one for the provision, i.e., provision point and one against the provision, i.e., rejection point, for the project. The project is provisioned (not provisioned) if the provision (rejection) point is reached first. A strategic agent may choose to contribute in a market, against its preference. Thus, the challenge in such a setting remains to ingeniously design a refund scheme such that the agents are incentivized to contribute based on their preferences. For this, we propose a methodology through which existing mechanisms for positive preferences also allow for agents with negative preferences, such that agents contribute to the market based on their true preferences. In particular, we adapt existing PPR and PPS mechanisms to design PPRN and PPSN mechanisms. We prove that in these mechanisms at equilibrium, either the provision point or the rejection point holds.

Further, designing mechanisms for civic crowdfunding for agents with asymmetric beliefs is not trivial. For instance, a rational agent with significant belief towards the project's provision may choose to free-ride, as it believes that the project will be provisioned regardless of its contribution. Such asymmetric agents need to be further incentivized to contribute towards the project's provision. For this, we propose a novel reward scheme Belief Based Reward (BBR) that rewards an agent based on their belief towards the project's provision. We deploy a peer prediction mechanism for information aggregation of each agent's belief. With BBR, we propose a novel class of mechanisms for civic crowdfunding which incentivizes agents with asymmetric beliefs to contribute towards the provision, such that the project is provisioned at equilibrium. To the best of our knowledge, there is no prior work addressing negative valuations and asymmetric beliefs in civic crowdfunding literature.

2 Preliminaries

In this section, we present the required preliminaries. We begin by defining our crowdfunding model.

2.1 Model

For the civic crowdfunding of public project PP, the Project Maker (PM), sets up a market for its provision. The PM announces a provision point h0 as the target to be reached until a deadline T . Let A = {1, . . . , n} be the set of all contributing agents wherein each Agent i has valuation θi ≥ 0 if PP gets provisioned. The agents contribute x = (x1, . . . , xn) to the crowdfunding mechanism. Let ϑ = ∑i=n i=1 θi; the total valuation of all the agents and X = ∑i=n i=1 xi; the sum of the contributions. A project is provisioned if X ≥ h0 at the end of deadline T and not provisioned otherwise. Such mechanisms are referred to as provision point mechanisms. Note that θis are private to the agents and provision point mechanisms are indirect mechanisms to aggregate these. This setup induces a game among the agents.

Let σ = (σ1, . . . , σn) be the vector of strategy profile of every agent. Agent i's strategy consists of its contribution xi towards the project's provision along with other mechanism dependent parameters. We use the subscript -i to represent vectors without Agent i. The utility for Agent i with valuation θi for the project, when all the agents play the strategy profile σ is ui(σ; θi).

We now define some game-theoretic definitions necessary for the analysis of the mechanisms presented in this paper.

Definition 1 (Nash Equilibrium (NE)). A strategy profile σ∗ = (σ∗ 1 , . . . , σ∗n ) is said to be a Nash equilibrium (NE) if for every Agent i, it maximizes the utility ui(σ∗ ; θi) i.e., ∀i ∈ A,

For crowdfunding in sequential setting, i.e., when the contributing agents arrive over time, the strategy profile of every Agent i also constitutes the time ti at which it contributes to the mechanism. Let, ai be the time at which Agent i arrives to the mechanism. Further, let h0t denote the amount remaining for the project to be provisioned at time t. With this, we define,

Definition 2 (Sub-game Perfect Equilibrium (SPE)). A strategy profile σ∗ = (σ∗ 1 , . . . , σ∗n ) is said to be a sub-game perfect equilibrium if for every Agent i, it maximizes the utility ui(σ∗ i , σ∗-i|Hai ; θi) i.e., ∀i ∈ A,

Here, Ht is the history of the game till time t, constituting the agents' arrivals and their contributions and σ∗-i|Hai indicates that the players who arrive after ai follow the strategy specified by σ∗-i .

In the next subsection, we describe existing mechanisms in the literature for civic crowdfunding.

2.2 Provision Point Mechanisms

In this paper, we focus on the class of mechanisms for civic crowdfunding which require the project to aggregate a minimum level (Provision Point of the project) of funding before the PM can claim it. Several provision point mechanisms for civic crowdfunding have been proposed [Brubaker, 1975; Groves and Ledyard, 1977; Bagnoli and Lipman, 1989; Schmidtz, 1991; Tabarrok, 1998; Chen, 2008; Chandra et al., 2017; Damle et al., 2018]; but we focus on PPR [Zubrickas, 2014] and PPS [Chandra et al., 2016] mechanisms.

Provision Point Mechanism with Refund (PPR)

To counter the problem of free-riding, PPR offers a refund bonus to the agents in case the project does not get provisioned. The refund bonus scheme is directly proportional to agent's contribution. Let IX be an indicator random variable which takes the value 1 if X is true and 0 otherwise. Then the utility structure of PPR, for each agent i ∈ A is given as,

where B is the total budget kept aside by the PM, and is distributed to the agents who contributed along with their contribution, as a refund, in case the project is not provisioned.

The refund bonus in PPR is independent of time of contribution and therefore all agents delay their contributions as close to the deadline as possible and wait to free-ride till the end. This may lead to the project not getting provisioned in practice (for eg., because of server failure near the deadline or transactions not getting processed in time). Therefore, such strategies are undesirable.

Provision Point Mechanism with Securities (PPS)

PPS addresses the shortcomings of PPR by offering agents refunds based on the time of their contribution. An early contributor is paid higher refund than a late contributor for the same contribution. The utility structure of PPS, for each agent i ∈ A is given as,

where, ti and rti i are Agent i's time of contribution and the number of securities allocated to it, respectively. rti i depends on its contribution xi at ti, as well as the total number of securities issued in the market at time ti, denoted by qti , i.e. [Chandra et al., 2016, Eq. 6],

Here, C0 is the cost function governing the underlying prediction market in PPS obtained from the general cost function C, by fixing the number of positive outcome securities. A cost function must satisfy [Chandra et al., 2016, CONDITIONS 1-4,6] to be used in PPS. The properties of the cost function, C0, relevant to this paper are,

• Property 1. rti i is a monotonically increasing function of xi, i.e., ∂rti i xi > 1, ∀θi < h0 , ∀i [Chandra et al., 2016, CONDITION-7].
• Property 2. The refund rti - xi is a decreasing function of ti and qti as qti is non-decreasing function of ti, ∀i [Chandra et al., 2016, Step-2 (Theorem 3)].

The existing literature for civic crowdfunding is limited through its assumptions on the information structure of the contributing agents. We define an agent's information structure as consisting of its valuation and its belief towards the project's provision. Based on their valuations, we categorize agents as follows: positive (negative) agent i.e., Agent i with θi ≥ 0 (θi < 0) or with positive (negative) preference towards PP's provision. Let P (N) denote the set of all positive (negative) agents, such that A = P ∪ N. Further, let ϑ1 = ∑i θi ∀i ∈ P as the total valuation for PP getting provisioned and ϑ2 = ∑i (-θi) ∀i ∈ N as the total valuation for PP not getting provisioned, i.e., ϑ = ϑ1 -ϑ2 . For preference aggregation, the PM's goal is to determine whether the majority prefers PP to be provisioned or not, i.e., whether or not ϑ ≥ 0.

We also consider agents with asymmetric beliefs towards PP's provision i.e., agents may believe that the project may be provisioned with probability (belief) 1/2 ± ϵ or may not be provisioned with probability 1/2 ∓ ϵ for some ϵ ≥ 0. Let k1 i = (1/2 + ϵi) and k2 i = (1/2 - ϵi) for some ϵi ≥ 0 such that k1 i + k2 i = 1, ∀i ∈ A. Let A+ (A-) be the set of agents which believe that PP will (will not) be provisioned i.e., every agent i ∈ A+ (i ∈ A-) has belief k1 i (k2 i ) that PP will be provisioned, such that A = A+ ∪ A-.

In this paper, we require each agent to truthfully elicit its belief regarding the provision of the public project. Since an agent's opinion (belief) is its private information, we look for mechanisms which incentivize it to elicit its true opinion. In mechanism design theory, such mechanisms are called incentive compatible (IC). Further, these mechanisms must also be individually rational (IR) i.e., each Agent i must have nonnegative utility 1 . Towards this, we make use of peer prediction mechanisms.

2.3 Peer PredictionMechanisms

Peer prediction mechanisms (PPM) allow for elicitation and aggregation of subjective opinions from a set of agents. These are generally deployed in situations where there is no method of verifying an agent's honesty (of their opinion) or their ability. In the literature, there are a number of existing peer prediction mechanisms [Miller et al., 2005; Jurca and Faltings, 2007; Lambert and Shoham, 2008; Dasgupta and Ghosh, 2013; Radanovic and Faltings, 2013].

For illustrative purposes, in this paper, we focus on Robust Bayesian Truth Serum (RBTS) mechanism ([Witkowski and Parkes, 2012]).

Robust Bayesian Truth Serum

While RBTS mechanism properties hold for an arbitrary number of signals, we present the binary version of the mechanism, as we are interested in the elicitation of an agent's belief towards the provision of the project. In RBTS, each agent is required to submit, from [Witkowski and Parkes, 2012]:

1. Information Report: Let fi = {0, 1} be Agent i's reported signal.
2. Prediction Report: Let gi ∈ [0, 1] be Agent i's report about the frequency of high signals among the citizens.

Based on information report and prediction report, RBTS assigns a score to each agent. We describe the calculation of the scores in the complete version [Damle et al., 2019]. RBTS mechanism is IC and ex-post IR ([Witkowski and Parkes, 2012, Theorem 10]) for the elicitation of binary information for all n ≥ 3, without relying on knowledge of the common prior. Note that many other peer prediction mechanisms are IC only when n → ∞ and hence we chose RBTS.

1 For more details on mechanism design, refer to [Garg et al., 2008a; Garg et al., 2008b]

3 Civic Crowdfunding for Agents With Negative Valuations

In this section, we introduce a methodology through which civic crowdfunding mechanisms can incorporate symmetric agents with negative preferences (valuations) towards the public project's provision. For this, the PM sets up two separate markets, i.e., one for the provision and one against the provision of the project. Thus, agents now have a greater scope for manipulation. In such a setting, a strategic agent may choose to contribute in a market, against its preference, if its expected utility for contributing in that market is more than if it contributes in the market based on its preference. Therefore, to incorporate agents with negative preference, we must ingeniously construct the refund scheme in a way so that the agents are incentivized to contribute in the market based on their true preferences.

To illustrate this methodology, we provide two mechanisms for the same by adopting existing mechanisms in PPR and PPS; as namely, PPRN and PPSN. For these mechanisms, let h1 (h2 ) be the target for provision (rejection) of PP with X 1 (X 2 ) as the total funding received towards (against) its provision. Further, let h1t (h2t ) denote the amount remaining for the project to be provisioned (not provisioned) at time t.

We present Provision Point Mechanism with Refund for Negative Preference (PPRN) leveraging PPR in the complete version of this paper [Damle et al., 2019]. In the next subsection, we present Provision Point Mechanism for Securities with Negative Preference (PPSN) by leveraging PPS.

3.1 Provision Point Mechanism for Securities with Negative Preference (PPSN)

We now propose a mechanism for civic crowdfunding for agents with negative valuations by leveraging PPS, namely PPSN.

Protocol

In PPSN, we consider a mechanism with two independent PPS prediction markets - PPS+ and PPS-. In PPS+, agents contribute for the project to be provisioned (and buy negative securities) while in PPS-, agents contribute for the project to not be provisioned (and buy positive securities). Note that the markets being independent, the prices in both the markets are also independent of the other. Provision point for PPS+ is reached when the total contribution in it reaches h1 , and rejection point for PPS- when the total contribution in it reaches h2 . Let, X 1 be the total contribution received by the project in PPS+ and X 2 be the total contribution received by the project in PPS-. The goal of the mechanism is to provision the project or not based on whichever target is first reached.

Let si ∈ {1, 2}, be a private preference variable for Agent i, such that si = 1, ∀i ∈ P and si = 2, ∀i ∈ N.

Common Refund Scheme

An agent may not contribute in the market based on its preference if its expected refund is more in case it contributes in the other market. To prevent this, we present a common refund scheme that ensures that the agent obtains same refund in spite of which market it chooses to contribute. In this, Agent i contributes xi in any market based on a refund that depends on the minimum of the issued securities present in both the markets i.e., Qti = min(qti PPS+ , qti PPS-). Based on this, Agent i is issued securities (Rti i ) for a contribution xi given by

from [Chandra et al., 2016, Eq. 6]. Thus, Agent i's refund in this scheme is Rti i -xi.

However, the number of issued securities only changes for the market in which the agent contributes xi. For instance, if Agent i contributes xi at time t to PPS+, then the change in the issued securities in PPS+ will be given by

Proposition 1. The securities allotted to an agent with total issued securities as Qt is always greater than or equal to those it would have received with securities qtPPS+ or qtPPSfor the same contribution and the same cost function C0.

Proof. The statement follows from the fact that the number of securities allotted, for the same contribution, is a decreasing function of the total issued securities (Property 2).

Proposition 2. The refund given by Rti i -xi for Agent i, is a decreasing function with respect to time ti.

Proof. The securities allotted to Agent i, Rti i , decreases as Qti increases (Proposition 1). Further, since Qt = min(qtPPS+, qtPPS-) and qtPPS+ and qtPPS- are nondecreasing with respect to time t (Property 2); Qt is a nondecreasing function of time. Thus, Rti i -xi for Agent i, is a decreasing function with respect to time ti.

Let us call PPS+ as p1, and PPS- as p2. Thus, psi = p1, ∀i ∈ P and psi = p2, ∀i ∈ N. Further, let the market in which Agent i contributes be ˜ psi .

Agent Utility

The utility for Agent i ∈ A with ˜ psi = p1, in PPSN is as follows,

The utility for Agent i ∈ A with ˜ psi = p2, in PPSN is as follows,

Discussion.

1. Trivially, from Eqs. 1 and 2, PPSN is (ex-post) IR. PPSN, however, is not IC [Damle et al., 2019].
2. The utility structure in PPSN can be intuitively explained as follows: As θi is Agent i's valuation for the project getting provisioned, it is accounted for when the project is provisioned, i.e., when X 1 ≥ h1 and X 2 < h2 . Further, in PPS+, the refund share is distributed when the provision point is not reached, i.e., when X 1 < h1 . Likewise, in PPS-, the refund share is distributed when the rejection point is not reached, i.e., when X 2 < h2 .

Equilibrium Analysis

Theorem 3.1 provides the equilibrium analysis of PPSN.

Theorem 3.1. For PPSN, with the utility as given by Eq. 1 and Eq. 2 ∀i ∈ A, C : R2 → R as the cost function, C 1 0 : R → R as the cost function obtained from C by fixing the number positive outcome securities satisfying Property 1 and used in the market p1 satisfying (C 1 0 )-1 (h1 + C0(0)) < ϑ1 , and C 2 0 : R → R as the cost function obtained from C by fixing the number of negative outcome securities satisfying 2 Property 1 and used in the market p2 satisfying C -1 0 (h2 + C0(0)) < ϑ2 , with ϑ1 > h1 and ϑ2 > h2 , a set of strategies in the set σ∗ i = (x∗i , t∗i , ˜ psi ),

∀l ∈ {1, 2}; are sub-game perfect equilibria ∀i ∈ A, such that at equilibrium either X 1 = h1 or X 2 = h2 holds.

Proof. Since every agent i ∈ A is symmetric in its belief towards the project's provision, its expected utility is given by 1/2(θi +Ri) -xi irrespective of whether ˜ psi = p1 or p2. Therefore, no agent has any incentive to deviate from its true preference. Hence, ˜ psi = psi , ∀i ∈ A..

The rest of the proof follows similar to [Chandra et al., 2016] and is presented in [Damle et al., 2019].

Discussion. For PPSN, it can be seen that ϑ = ϑ1 - ϑ2 . The project is always provisioned if ϑ1 > h1 and ϑ ≥ 0 or is never provisioned if ϑ2 > h2 and ϑ < 0. Here it must be noted that it can happen that ϑ1 > h1 and ϑ2 > h2 are simultaneously satisfied. In that case, if ϑ1 > ϑ2 , the project attains provision point faster than rejection point and vice-versa.

The significance of this result is that, in PPSN (and PPRN), at equilibrium, the project is provisioned if the majority prefers it, i.e., only when ϑ ≥ 0. Thus, this methodology allows for truthful aggregation of private preference of each agent with respect to public projects.

4 Civic Crowdfunding for Agents with Asymmetric Beliefs

In this section, we present a General Mechanism which incentivizes agents with asymmetric beliefs towards the public project's provision, to contribute towards it. In this section, we restrict our attention to the case where every agent has a positive valuation towards project's provision.

The General Mechanism involves two phases: a Belief Phase (BP) and a Contribution Phase (CP). In BP, each Agent i submits its belief for the provision of the project for which it is allocated some share (denoted by bi) of the reward calculated through Belief Based Reward (BBR) scheme described in the next subsection. In CP, each Agent i submits its contribution (xi) to the project which is dependent on the refund obtained in the BP as well as on the provision point mechanism deployed for civic crowdfunding.

2 Trivially, both the cost functions are the same. Hence, from hereon we refer to both of themwithout the superscript i.e., as C0.

The mechanism requires two separate bonuses for both the phases, which the PM announces at the start of the project. Let BB (BC ) be the bonus allocated for the BP and the CP, respectively. Further, let a1i (a2i ) be the time at which Agent i arrives to the mechanism for the BP (CP) with t1i (t2i ) as the time at which it reports its belief (contribution). Let the deadline for the BP (CP) be T B (T C ) announced at the start.

Unlike in the case of civic crowdfunding for agents with symmetric beliefs, an asymmetric agent which has significant belief towards the project getting provisioned or not, may choose to free-ride and not contribute. Therefore, we introduce a reward scheme that further incentivizes such agent's to contribute towards the project.

4.1 Belief Based Reward (BBR)

To quantitatively measure the reward share to be distributed to every contributing agent in the BP, we use a PPM. We consider PPMs which incentivize truthful elicitation of an agent's belief i.e., PPMs which are IC.

Let the score of Agent i depending on its belief (1/2 ± ϵi) be mi. Further, let Sti be the set consisting of all the agents that have reported their belief including the Agent i, who reports its belief at time ti. For T B as the deadline, ST B consists of all the agents that have reported their belief. Let mi, ∀i ∈ ST B be the agent scores calculated after the deadline. For,

Agent i's reward in the scheme is,

We refer to the reward scheme given by Eq. 3 as Belief Based Reward (BBR). With this, we show the following proposition:

Proposition 3. BBR is a decreasing function of time.

Proof. From Eq. 3, BBR is inversely proportional to the order in which agents report their beliefs. As the arrival of agents is non-decreasing w.r.t. time, BBR is a decreasing function of time.

In addition, BBR is also strongly budget balanced, i.e., in BBR the entire budget is utilized.

RBTS Reward Scheme

In this paper, we use RBTS Mechanism to calculate the mechanism score mi for each Agent i. For this, every agent submits its prediction and information report as described earlier.

In this reward scheme, let fi = 0 denote that Agent i has belief that the project will be provisioned and fi = 1 denote that Agent i has belief that the project will not be provisioned. Thus, through each agent's prediction report, the PM knows whether an agent belongs to the set A+ or the set A-.

We present Provision Point Mechanism with Refunds for Agents with Asymmetric Beliefs (PPRx) by plugging PPR refund bonus scheme for the CP in [Damle et al., 2019]. In the next subsection, we present Provision Point Mechanism with Securities for Agents with Asymmetric Beliefs (PPSx) by plugging PPS refund bonus scheme for the CP.

4.2 Provision Point Mechanismwith Securities for Agents with Asymmetric Beliefs (PPSx)

In PPSx, we plugin PPS refund bonus scheme for the CP.

Agent Utility

The utility for Agent i ∈ A+, in PPSx is as follows,

Similarly, the utility for Agent i ∈ A-, in PPSx is as follows,

Discussion.

1. From Eqs. 4 and 5, PPSx is (ex-post) IR. PPSx, however, is not IC [Damle et al., 2019].
2. At the end of the mechanism, only one set of agents, either A+ or A-, get the BBR reward.

Equilibrium Analysis

Theorem 4.1 presents the equilibrium analysis of PPSx. We formally prove the theorem in [Damle et al., 2019].

Theorem 4.1. For PPSx, with the utility as given by Eq. 4 and Eq. 5, ∀i ∈ A+, C : R2 → R as the cost function, with C0 : R → R as the cost function obtained from C by fixing the number positive outcome securities satisfying Property 1, C -1 0 (h0 +C0(0)) < ϑ with BB ,BC > 0 and Belief Phase reward calculated as per Eq. 3 ∀i ∈ A, a set of strategies σ∗ i = (x∗i , t1∗ i , t2∗ i ) in the set,

∀i ∈ A+ and a set of strategies σ∗ i = (x∗i , t1∗ i , t2∗ i ) in the set,

∀i ∈ A-; are sub-game perfect equilibria such that at equilibrium X = h0 holds.

Discussion. Trivially, for PPSx, the upper bound for the equilibrium contributions ∀i ∈ A+ (with k1 i as the belief towards project's provision) is greater than the upper bound for the equilibrium contributions ∀i ∈ A- (with k2 i as the belief towards project's provision; k1 i + k2 i = 1), for the same θi and bi. This implies that agents with greater belief towards the project's provision contribute more than agents with lesser belief towards it. Thus, BBR and the utility structure as given by Eqs. 4 and 5 for PPSx, provides a natural way for civic crowdfunding with asymmetric agents such that the project is provisioned at equilibrium. We now discuss the setting of up the markets for these mechanisms.

5 Discussion

For PPSN (PPRN), the PM is required to set up two independent PPS (PPR) markets. The provision point for these projects are determined based on the economics of their construction. The rejection point can be similarly determined. For instance, the rejection point for our garbage dump yard example could be the cost of constructing the dump yard at a different locality. Another method for determining the rejection point could be the cost incurred by the government as a result of the public project not getting provisioned. An instance of this could be the construction of dams. The cost of not setting up the dam, i.e., the rejection point for the project could be the cost incurred by the government in providing electricity or water etc. to the nearby areas which they could have achieved through the dam's construction. Note that, the amount collected if the project is rejected is at the discretion of the government.

The PM should allocate reasonable budget for all these mechanisms. Allocating huge budgets may not guarantee the provision/rejection of the projects. In such a case, the agents may prefer to contribute just enough to get substantial refunds. Likewise, allocating insignificant budgets may prove to not be incentivising enough for agents to contribute to the market. Further, in PPSN, the cost function, C0, used to allocate the securities must also be same for both markets. Additional details for setting up the prediction markets as well as the budget can be found in [Chandra et al., 2016; Chandra et al., 2017].

Mechanisms for Asymmetric Agents with Negative Valuations. Civic crowdfunding mechanisms for agents with information structure consisting of both - their preferences and their belief towards the provision of the project is still illusive as in such a setting, agents have an extra dimension to manipulate the mechanism. For instance, combining PPSN and PPSx (PPRN and PPRx) will not suffice. As formally shown in [Damle et al., 2019], an Agent i ∈ A+ with θi ≥ 0, will always choose to contribute towards the project not getting provisioned, as it believes that the project will be provisioned anyways, making it eligible for the additional refund bonus. Likewise, an Agent i ∈ A- with θi < 0 will always contribute towards the provision of the project. However, an Agent i ∈ A+ with θi < 0 and an Agent i ∈ A- with θi ≥ 0 will always contribute as per its true preference.

Thus, the general method and the general mechanism proposed in this paper for civic crowdfunding for agents with negative valuations and agents with asymmetric beliefs respectively, are not sufficient to incentivize every asymmetric agent to contribute as per their true preferences.

6 Conclusion

In this paper, we explored the limitations of existing literature on civic crowdfunding. We showed that it poses restrictions on the information structure of agents, as it only allows for positive as well as symmetric agents. We broke this barrier on the information structure of an agent by proposing (i) a general methodology for addressing symmetric agents with negative preferences based on which we proposed two mechanisms, PPRN and PPSN (Theorem 3.1); and (ii) a general mechanism for positive agents with asymmetric beliefs based on which we proposed two mechanisms, PPRx and PPSx (Theorem 4.1). We leave it for future work to explore the feasibility of combining negative preferences and asymmetric belief into one framework for civic crowdfunding.