1 Introduction
Crowdfunding is a process of raising funds from a large pool of interested agents and is an active area of research [Alaei:2016:DMC:2940716.2940777, Itai2017, REPPS2017, strausz2017theory, Shen2018]. 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 ([Bagnoli1989]). 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 [Bagnoli1989, Brubaker1975, Schmidtz1991].
Provision Point mechanism with Refund bonus (PPR) by [PPR2014] 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. [PPS2016] 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. To the best of our knowledge, there are no other refund bonus mechanisms for civic crowdfunding.
Note that in all these mechanisms only those agents with a positive valuation towards the project contribute to its provision. However, several agents may prefer the project to not be provisioned, i.e., their valuation 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 valuation. If addressed, civic crowdfunding with agents with negative valuation 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) valuation or positive (negative) preference 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 preference 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 valuation, 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 preference. 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 preference. 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 freeride, 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.
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 , the Project Maker (PM), sets up a market for its provision. The PM announces a provision point as the target to be reached until a deadline . Let be the set of all agents wherein each Agent has valuation if gets provisioned. The agents contribute to the crowdfunding mechanism. Let ; the total valuation of all the agents and ; the sum of the contributions. A project is provisioned if at the end of deadline and not provisioned otherwise. Such mechanisms are referred to as provision point mechanisms. Note that s are private to the agents and provision point mechanisms are indirect mechanisms to aggregate these. This setup induces a game amongst the agents.
Let
be the vector of strategy profile of every agent. Agent
’s strategy consists of its contribution towards the project’s provision along with other mechanism dependent parameters. We use the subscript to represent vectors without Agent . The utility for Agent with valuation for the project, when all the agents play the strategy profile is .We now define some important gametheoretic definitions necessary for the analysis of the mechanisms presented in this paper.
Definition 1 (Nash Equilibrium (NE)).
A strategy profile is said to be a Nash equilibrium (NE) if for every Agent , it maximizes the utility i.e., ,
For crowdfunding in sequential setting, i.e., when the contributing agents arrive over time, the strategy profile of every Agent also constitutes the time at which it contributes to the mechanism. Let, be the time at which Agent arrives to the mechanism. Further, let denote the amount remaining for the project to be provisioned at time . With this, we define,
Definition 2 (Subgame Perfect Equilibrium (SPE)).
A strategy profile is said to be a subgame perfect equilibrium if for every Agent , it maximizes the utility i.e., ,
Here, is the history of the game till time , constituting the agents’ arrivals and their contributions and indicates that the players who arrive after follow the strategy specified by .
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 [Brubaker1975, Groves1977, Bagnoli1989, Schmidtz1991, Tabarrok1998, Chen2008, REPPS2017]; but our work focuses on PPR [PPR2014] and PPS [PPS2016] mechanisms.
2.2.1 Provision Point Mechanism with Refund (PPR)
To counter the problem of freeriding, 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
be an indicator random variable which takes the value
if is true and otherwise. Then the utility structure of PPR, for each agent is given as,where 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 freeride till the end. Such strategies lead to the project not getting provisioned in practice and are therefore undesirable.
2.2.2 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 is given as,
where, are Agent ’s time of contribution and the number of securities allocated to it, respectively. depends on its contribution at , as well as the total number of securities issued in the market at time , denoted by , i.e. [PPS2016, Eq. 6],
Here, is the cost function governing the underlying prediction market in PPS obtained from the general cost function , by fixing the number of positive outcome securities. A cost function must satisfy [PPS2016, CONDITIONS 14,6] to be used in PPS. The properties of the cost function, , relevant to this paper include:

is an increasing function of , i.e., , [PPS2016, CONDITION7].

is a decreasing function of [PPS2016, Step2 (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 valuation, we categorize agents as follows: positive (negative) agent i.e., Agent with () or with positive (negative) preference towards ’s provision. Let () denote the set of all positive (negative) agents, such that . Further, let as the total valuation for getting provisioned and as the total valuation for not getting provisioned, i.e., . For preference aggregation, the PM’s goal is to determine whether the majority prefers to be provisioned or not, i.e., whether or not .
We also consider agents with asymmetric beliefs towards
’s provision i.e., agents may believe that the project may be provisioned with probability (belief)
or may not be provisioned with probability for some . Let and for some such that , . Let () be the set of agents which believe that will (will not) be provisioned i.e., every agent () has belief () that will be provisioned, such that .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 (ID) i.e., each Agent must have nonnegative utility. Towards this, we make use of peer prediction mechanisms.
2.3 Peer Prediction Mechanisms
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 [Miller2005, Jurca2007, Lambert2008, Dasgupta2013, Radanovic2013].
For illustrative purposes, in this paper, we focus on Robust Bayesian Truth Serum (RBTS) mechanism ([RBTS2014]).
2.3.1 Robust Bayesian Truth Serum
While RBTS mechanism properties hold for an arbitrary number of signals, we present the binary version of the mechanism, since 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 [RBTS2014]:

Information Report: Let be Agent ’s reported signal.

Prediction Report: Let be Agent ’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. The mechanism, for each Agent , selects a reference agent and a peer agent and calculates,
where . Then, the RBTS score for Agent is given by,
(1) 
is the binary quadratic scoring rule [selten1998axiomatic] normalized to give scores between 0 and 1 and is a strictly proper scoring rule.
RBTS mechanism is IC and expost ID ([RBTS2014, Theorem 10]) for the elicitation of binary information for all , without relying on knowledge of the common prior. Note that many other peer prediction mechanisms are IC only when and hence we chose RBTS.
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 preference (valuation) 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 preference.
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 () be the target for provision (rejection) of with () as the total funding received towards (against) its provision. Further, let () denote the amount remaining for the project to be provisioned (not provisioned) at time .
We now present Provision Point Mechanism with Refund for Negative Preference (PPRN) leveraging PPR.
3.1 Provision Point Mechanism with Refunds with Negative Preference (PPRN)
We now propose a mechanism for civic crowdfunding for agents with negative valuation by leveraging PPR, namely PPRN.
3.1.1 Protocol
In PPRN, Agent not only contributes its contribution but also specifies its preference i.e., whether it wants to contribute towards the project getting provisioned or not getting provisioned. Let be a private preference variable for Agent , such that and . Further, let Agent ’s reported preference be . PM adds Agent ’s contribution towards the project’s provision or rejection based on its reported preference . The project is provisioned or not based on whichever target is first reached.
3.1.2 Agent Utility
The utility for Agent with in PPRN is as follows,
(PPRNU1) 
Similarly, the utility for Agent with in PPRN is as follows,
(PPRNU2) 
3.1.3 Equilibrium Analysis
We now provide the equilibrium analysis for PPRN as the following theorem,
Theorem 1.
Proof. In Step 1, we show that the equilibrium contributions are such that at equilibrium either or holds. Step 2 shows that each agent delays its contribution till the deadline . We prove that every agent contributes based on its true preference in Step 3. Step 4 calculates the equilibrium contribution of every agent. Finally, in Step 5 we give the conditions for existence of Nash Equilibrium.
Step 1: As and , at equilibrium and cannot hold, as with and , at least, that could obtain a higher refund bonus by marginally increasing its contribution because of . Likewise, any agent with a positive contribution could gain in utility by marginally decreasing its contribution if or . Thus, at equilibrium, either or holds.
Step 2: As the refund in PPRN is independent of time, no agent has any incentive to contribute early. Thus, each agent delays its contribution till the deadline, .
Step 3: Since every Agent is symmetric in its belief towards the project’s provision, its expected utility is given by . Thus, every Agent has no incentive to deviate from its preference. Thus, .
Step 4:

For Positive agents: At equilibrium, the best response for an Agent is that contribution such that its provisioned utility is not less than not provisioned utility since it prefers the project to be provisioned and is symmetric in its belief, i.e., ,
Last inequality follows from the fact that .

For Negative agents: The best response for an Agent is that equilibrium contribution such that its not provisioned utility is not less than provisioned utility since the agent prefers the project to not be provisioned and is symmetric in its belief, i.e., ,
Step 5: Summing up gives the condition for existence of Nash Equilibrium as,
Similarly, summing up ,
gives the condition for existence of Nash Equilibrium. ∎
In the next subsection, we present Provision Point Mechanism for Securities with Negative Preference (PPSN) by leveraging PPS.
3.2 Provision Point Mechanism for Securities with Negative Preference (PPSN)
We now propose a mechanism for civic crowdfunding for agents with negative valuation by leveraging PPS, namely PPSN.
3.2.1 Protocol
In PPSN, we consider a mechanism with two independent PPS prediction markets  PPS1 and PPS2. In PPS1, agents contribute for the project to be provisioned (and buy negative securities) while in PPS2, 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 PPS1 is reached when the total contribution in it reaches , and rejection point for PPS2 when the total contribution in it reaches . Let, be the total contribution received by the project in PPS1 and be the total contribution received by the project in PPS2. The project is provisioned or not based on whichever target is first reached.
Let , be a private preference variable for Agent , such that and .
3.2.2 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 contributes in any market based on a refund that depends on the minimum of the issued securities present in both the markets i.e., . Based on this, Agent is issued securities () for a contribution given by
from [PPS2016, Eq. 6]. Thus, Agent ’s refund in this scheme is .
However, the number of issued securities only changes for the market in which the agent contributes to, i.e.,
will be the change in the total number of issued securities in the market PPS().
Proposition 1.
The securities allotted to an agent with total issued securities as is always greater than or equal to those it would have received with securities or for the same contribution and the same cost function .
Proof. The statement follows directly from the fact that the number of securities allotted, for the same contribution, is a decreasing function of the total issued securities [PPS2016, Step2 (Theorem 3)]. ∎
Proposition 2.
The refund given by for Agent , is a decreasing function with respect to time .
Proof. The securities allotted to Agent , , decreases as increases (Proposition 1). Further, since and are nondecreasing with respect to time ; is a nondecreasing function of time. Thus, for Agent , is a decreasing function with respect to time . ∎
Let us call PPS1 as , and PPS2 as . Thus, and . Further, let the market in which Agent contributes be .
3.2.3 Agent Utility
The utility for Agent with , in PPSN is as follows,
(PPSNU1) 
The utility for Agent with , in PPSN is as follows,
(PPSNU2) 
3.2.4 Equilibrium Analysis
We now provide the equilibrium analysis of this mechanism as the following theorem,
Theorem 2.
For PPSN, with the utility as given by Eq. PPSNU1 and Eq. PPSNU2 , as the cost function, as the cost function obtained from by fixing the number positive outcome securities satisfying [PPS2016, CONDITION 7] and used in the market satisfying , and as the cost function obtained from by fixing the number of negative outcome securities satisfying^{1}^{1}1It is trivial to see that both the cost functions shall be the same. Hence, from hereon we will refer to both of them without the superscript i.e., as . [PPS2016, CONDITION 7] and used in the market satisfying , with and a set of strategies in the set ,
; are subgame perfect equilibria , such that at equilibrium either holds.
Proof. In Step 1, we show that the equilibrium contributions are such that at equilibrium either or holds. We prove that every agent contributes based on its true preference in Step 2. Step 3 calculates the equilibrium contribution of every agent. In Step 4 we give the conditions for existence of Nash Equilibrium. We show that these set of strategies are subgame perfect in Step 5.
Step 1: As and , at equilibrium and cannot hold, as with and , at least, that could obtain a higher refund bonus by marginally increasing its contribution since [PPS2016, CONDITION 7]. Likewise, any agent with a positive contribution could gain in utility by marginally decreasing its contribution if or . Thus, at equilibrium, either or holds.
Step 2: Since every Agent is symmetric in its belief towards the project’s provision, its expected utility is given by for both the markets. Thus, every Agent has no incentive to deviate from its preference. Therefore, .
Step 3: As the refund scheme is decreasing with respect to time (Proposition 2), Agent contributes as soon as it arrives i.e., at time .
Step 4: Let be the number of total issued securities at market at time , and be the number of total issued securities at market at time , with for Agent . Now,

For Positive agents: At equilibrium, the best response for an Agent is that contribution in market at time such that its provisioned utility is not less than not provisioned utility since it prefers the project to be provisioned and is symmetric in its belief i.e.,
The result follows from [PPS2016, Eq. 6]. Based on this , the number of issued securities changes by in , since a positive agent always contributes in .

For Negative agents: At equilibrium, the best response for an Agent is that contribution at time in market such that its not provisioned utility is not less than provisioned utility since it prefers the project to not be provisioned and is symmetric in its belief i.e.,
The result follows from [PPS2016, Eq. 6]. Based on this , the number of issued securities changes by in , since a negative agent always contributes in .
Step 4: From Proposition 1, can be written as ; or . Summing up , we get the condition for existence of Nash Equilibrium here as, from [PPS2016, Eq. 7],
Similarly for , we have,
Step 5: For Agent entering last, if or , then its best response is contributing . If and , irrespective of the total contribution, its provisioned and not provisioned utility is the same at , defined in the theorem, and it is best response for Agent to follow the equilibrium strategy. With backward induction, by similar reasoning, it is best response for every agent to follow the equilibrium strategy irrespective of the history of the contributions.
For Agent entering the market such that , its best response will be contributing . This is because for the contribution , its provisioned utility will be greater than its not provisioned utility. Agent will also contribute the maximum contribution, , since its not provisioned utility increases as its contribution increases. Therefore, contributing an amount less than will result in a lesser not provisioned utility for the agent. Thus, these strategies form a set of subgame perfect equilibria. ∎
Discussion: For PPSN, it can be seen that . The project is always provisioned if and or is never provisioned if and . Here it must be noted that it can happen that and are simultaneously satisfied. In that case, if , the project attains provision point faster than rejection point and viceversa.
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 . Thus, this methodology allows for truthful aggregation of private preferences 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 submits its belief for the provision of the project for which it is allocated some share (denoted by ) of the reward calculated through Belief Based Reward (BBR) scheme described in the next subsection. In CP, each Agent submits its contribution () 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.
The mechanism requires two separate bonuses for both the phases, which the PM announces at the start of the project. Let be the bonus allocated for the BP and the CP, respectively. Further, let be the time at which Agent arrives to the mechanism for the BP (CP) with as the time at which it reports its belief (contribution). Let the deadline for the BP (CP) be announced at the start of the project.
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 freeride 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 dependent on its belief be . Further, let be the set consisting of all the agents that have reported their belief including the Agent , who reports its belief at time . For as the deadline, consists of all the agents that have reported their belief. Let be the agent scores calculated after the deadline. For,
Agent ’s reward in the scheme is,
(2) 
We refer to the reward scheme given by Eq. 2 as Belief Based Reward (BBR). With this, we show the following proposition:
Proposition 3.
BBR is a decreasing function of time.
Proof. From Eq. 2, BBR is inversely proportional to the order in which agents report their beliefs. Since the arrival of agents is nondecreasing 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. Note that, at the end of the mechanism, only one set of agents, either or , are rewarded.
RBTS Reward Scheme: In this paper, we use RBTS Mechanism to calculate the mechanism score for each Agent . For this, every agent submits its prediction and information report as described earlier.
In this reward scheme, let denote that Agent has belief that the project will be provisioned and denote that Agent 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 or the set .
We now present Provision Point Mechanism with Refunds for Agents with Asymmetric Beliefs (PPRx) by plugging PPR refund bonus scheme for the CP.
4.2 Provision Point Mechanism with Refunds for Agents with Asymmetric Beliefs (PPRx)
In this mechanism, we plugin PPR refund bonus scheme for the Contribution Phase.
4.2.1 Agent Utility
The utility for Agent , in PPRx is as follows,
(PPRxU1) 
Similarly, the utility for Agent , in PPRx are as follows,
(PPRxU2) 
4.2.2 Equilibrium Analysis
We present the equilibrium analysis of PPRx as the following theorem,
Theorem 3.
Proof.
In Step 1, we show that the equilibrium contributions are such that at equilibrium holds. We prove that every agent reports its belief as soon as it arrives to the Belief Phase in Step 2. Step 3 calculates the equilibrium contributions of every agent. Finally, in Step 4 we give the conditions for existence of Nash Equilibrium.
Step 1: At equilibrium, , since the PM stops the protocol as soon as the provision point is reached. Further if , then an agent can increase its utility by contributing more to the project, and receiving a higher utility since . Therefore, the contributions are such that the market is provisioned at equilibrium.
Step 2: Since the reward scheme for the Belief Phase is a decreasing function time (Proposition 3), a rational Agent would report its belief as soon as it arrives to the phase i.e., at time .
Step 3: The equilibrium strategy for each agent is such that its provisioned utility is not less than its not provisioned utility. Now,

For agent :
Since at equilibrium .

For agent :
Since at equilibrium .
Step 4:Note that from Eq. PPRxU1, and from Eq. PPRxU2, . Thus,
At equilibrium we have . Therefore,
is the condition for the existence of Nash Equilibrium. ∎
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.
4.3 Provision Point Mechanism with Securities for Agents with Asymmetric Beliefs (PPSx)
In this mechanism, we plugin PPS refund bonus scheme for the Contribution Phase.
4.3.1 Agent Utility
The utility for Agent , in PPSx is as follows,
(PPSxU1) 
Similarly, the utility for Agent , in PPSx is as follows,
(PPSxU2) 
4.3.2 Equilibrium Analysis
We present the equilibrium analysis of PPSx as the following theorem,
Theorem 4.
For PPSx, with the utility as given by Eq. PPSxU1 and Eq. PPSxU2, , as the cost function, with as the cost function obtained from by fixing the number positive outcome securities satisfying [PPS2016, CONDITION 7], with and Belief Phase reward calculated as per Eq. 2 , a set of strategies in the set,
and a set of strategies in the set,
; are subgame perfect equilibria such that at equilibrium holds.
Proof. In Step 1, we show that the equilibrium contributions are such that at equilibrium holds. We prove that every agent reports its belief as soon as it arrives to the Belief Phase as well as contributes as soon as it arrives to the Contribution Phase in Step 2. Step 3 calculates the equilibrium contributions of every agent. In Step 4, we give the conditions for existence of Nash Equilibrium. We show that these set of strategies are subgame perfect in Step 5.
Step 1: As , at equilibrium cannot hold, as with or with , at least, that could obtain a higher refund bonus by marginally increasing its contribution, since
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