Designing Refund Bonus Schemes for Provision Point Mechanism in Civic Crowdfunding

10/27/2018 ∙ by Sankarshan Damle, et al. ∙ IIIT Hyderabad koinearth 0

Civic crowdfunding is a practice with which interested players can raise funds for a civic project. With Blockchains gaining traction, civic crowdfunding can be implemented in a reliable, transparent and secure manner with smart contracts, thus becoming a powerful tool for social planners and governments. One fundamental challenge in civic crowdfunding is free riding - once the civic project is provisioned, all players, irrespective of their contribution can enjoy its benefits; hence, strategic players may free ride. Researchers have addressed this challenge through the game theory lens. The proposal by Zubrickas et. al. of refund bonus to the contributors in the case of the project not getting provisioned has interesting properties. As observed by Chandra et. al. however, this approach faces a challenge of race condition. To address this, their proposal, PPS considers the temporal aspects of a contribution in civic crowdfunding. However, PPS is computationally complex and is difficult to explain to a layperson. In this work, we look for all important properties a refund bonus scheme must satisfy in order to discourage free riding while avoiding the race condition. We identify Contribution Monotonicity and Time Monotonicity as sufficient and necessary conditions for this. We also propose three simple refund bonus schemes which satisfy these two conditions. Further, we introduce three novel mechanisms for civic crowdfunding deploying these schemes - PPRG, PPRP and PPRE. We show that PPRG is the most cost effective mechanism amongst these, as well as PPS, when deployed as a smart contract. We then prove that under certain assumptions on valuations of the players, in PPRG, PPRE and PPRP, the project is funded at equilibrium. We simulate these mechanisms in Reinforcement Learning enviornment to show that they do not trade off cost efficiency for provision accuracy.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Crowdfunding is the practice of funding a project by raising voluntary contributions from a large pool of interested players and is an active area of research [Alaei:2016:DMC:2940716.2940777, strausz2017theory, DBLP:conf/wine/2017, shen2018information]. Players are incentivized to contribute towards crowdfunding for private projects by offering them rewards. Using crowdfunding in order to raise funds for civic (non-excludable) goods, however, introduces the free riding problem - since players cannot be excluded from enjoying the benefits of the public project, strategic players may not contribute. If this challenge can be addressed, civic crowdfunding (CC) can lead to greater democratic participation. It also contributes to citizens’ empowerment as it allows them to collectively increase their well-being by solving societal issues. In this paper, we focus on solving the challenge of free riding in CC implemented using Blockchain based smart contracts.

With the advancement of the blockchain technology, crowdfunding projects are now being deployed using smart contracts. A smart contract is a computer protocol intended to digitally facilitate, verify, or enforce the negotiation or performance of a contract [wiki:SmartContract]. Since a crowdfunding project as a smart contract is on a trusted publicly distributed ledger, it is open and auditable, making the contributions of the players and the execution of the payments transparent as well as anonymous. In addition, as there is no need for any centralized, trusted third party, this reduces the cost incurred in setting up the project. [weifund] and [starbase] are examples of decentralized crowdfunding platforms on public blockchains like Ethereum. In this paper, our focus is to study game-theoretic challenges in CC, especially over blockchain. Our work builds on the literature which studies the lack of proper incentives for contributions towards public goods. Over the years, researchers have addressed such interaction as a game and analyzed equilibrium strategies of the players in it [bagnoli1989provision, chandra2016crowdfunding, chandra2017referral, zubrickas2014provision].

In the baseline approach, the social planner uses the voluntary contribution mechanism with a provision point, provision point mechanism ([bagnoli1989provision]). The social planner sets up a target amount, referred as provision point, to be raised. If the contributions by the players crosses this provision point, the project is executed. Otherwise, the contributions are returned. The mechanism has had a long history of applications. However, it has been shown to consist of several inefficient equilibria [bagnoli1989provision, brubaker1975free, schmidtz1991limits].

Provision Point mechanism with Refund bonus (PPR) by [zubrickas2014provision] introduces an additional refund bonus to be paid to the contributing players, along with their contributions, in case the project is not provisioned. This refund bonus induces a simultaneous move game in PPR in which the project is provisioned at equilibrium. PPR fails in settings such as Internet-based online platforms since in such a setting player can observe the current amount of funds raised. Hence, in online settings, strategic players in PPR would choose to wait and free ride till the end to check if the project is provisioned and would contribute only in the end in anticipation of a refund bonus. This leads to a scenario where every strategic player is trying to compete for a refund bonus at the deadline. We refer to this scenario as a race condition. In online settings, as the players can observe the history of the contributions, it induces a sequential game, and hence we refer to these settings as sequential settings.

Provision Point mechanism with Securities (PPS) by [chandra2016crowdfunding] introduced a class of mechanisms using complex prediction markets [abernethy2013efficient] which incentivizes a player to contribute as soon as it arrives at the crowdfunding platform, thus resolving the race condition. The challenge with the practical implementation of PPS is, as it uses complex prediction markets, it is not only difficult to explain to a layperson but also computationally expensive to implement, primarily as a smart contract.

The introduction of refund bonus is essential in these mechanisms as it incentivizes players to contribute and helps avoid free riding. Hence, in this paper, we focus on provision point mechanisms with refund bonus for CC. We look for refund bonus schemes that can avoid free riding as well as the race condition. The goal is to identify a class of refund bonus schemes satisfying a set of conditions i.e., Contribution Monotonicity and Time Monotonicity, which are sufficient to implement crowdfunding projects in a sequential setting such that the project is provisioned at equilibrium.

We propose three novel refund bonus schemes which satisfy these conditions and are clear to explain to a layperson as well as computationally efficient to implement as a smart contract. With these three schemes, we design novel mechanisms for CC, namely Provision Point mechanism with Refund through Geometric Progression (PPRG); Provision Point mechanism with Refund based on Exponential function (PPRE), and Provision Point mechanism with Refund based on Polynomial function (PPRP). We analyze the cost effectiveness of these mechanisms, as well as PPS when deployed as smart contracts and show that PPRG is the most cost effective. We measure the performance of these mechanisms by provision accuracy, the fraction of the projects that are successfully provisioned using the mechanism. We further simulate PPRG, PPRE, PPRP, and PPS and show that PPRG has similar provision accuracy as PPS. In the next section, we present the required preliminaries.

2 Preliminaries

We focus on CC projects which involve provisioning of projects without coercion where players arrive over time and not simultaneously i.e., CC in a sequential setting. Similar to [bagnoli1989provision, chandra2016crowdfunding, zubrickas2014provision], we also assume that apart from knowing the history of contributions, i.e., the provision point and the total amount remaining towards the project’s provision at any time, players do not have any information regarding the project’s provision. Thus, every player’s belief is symmetric towards the project’s provision. Further, no player has any information about the sequence of player arrivals.

  • Model

    A Project Maker (PM) puts a proposal for crowdfunding of a civic project on web based crowdfunding platform; that is, we are dealing with sequential settings. PM seeks voluntary contributions towards it. The proposal specifies a target amount necessary for the project to be provisioned, referred to as the provision point. It also specifies deadline by which the funds need to be raised. If the target amount is not achieved by the deadline, the project is not provisioned and the contributions are returned.

    A set of players are interested in the crowdfunding of . A Player has value if the project is provisioned. It arrives at time to the project, observes its valuation for it and can contribute at time , such that , towards its provision. Let , be the sum of the contributions and as the amount that remains to be funded at time .

    A project is provisioned if and not provisioned if at the end of deadline . is the budget kept aside by the PM to be distributed as a refund bonus among the contributors, if the project is not provisioned. This setup induces a game among the players.

    Let

    be the vector of strategy profile of every player where Player

    ’s strategy consists of the tuple , such that is its voluntary contribution to the project at time . We use the subscript to represent vectors without Player . The payoff for a Player with valuation for the project, when all the players play the strategy profile is .

    Let

    be an indicator random variable which takes the value 1 if

    is true and 0 otherwise. Then the payoff structure for a provision point mechanism with a refund bonus scheme with budget , for every Player contributing and at time , will be

    (1)

    where is the share of refund bonus for Player as per the refund bonus scheme . Let be the Refund Bonus Scheme for a provision point mechanism. We use to denote a refund bonus scheme and to denote Player ’s share of the refund bonus as per whenever the inputs are obvious.

  • Important Definitions

    Definition 1 (Pure Strategy Nash Equilibrium (PSNE)).

    A strategy profile is said to be a Pure Strategy Nash equilibrium (PSNE) if for every Player , it maximizes the payoff i.e., ,

    The strategy profile for the Nash Equilibrium is useful in a simultaneous move game. However, for sequential settings, where the players can see the actions of the other players, they may not find it best to follow the PSNE strategy. For this, we require a strategy profile which is the best response of every player at any time during the project i.e., the best response for every sub-game induced during it. Such a strategy profile is said to be a Sub-game Perfect Equilibrium.

    Definition 2 (Sub-game Perfect Equilibrium (SPE)).

    A strategy profile is said to be a sub-game perfect equilibrium if for every Player , it maximizes the payoff i.e. ,

    Here, is the history of the game till time , constituting the players’ arrivals and their contributions and indicates that the players who arrive after follow the strategy specified by . Informally it means that, at every stage of the game, irrespective of what has happened, it is Nash Equilibrium to follow the SPE strategy for every player.

3 Related Work

We focus on the class of mechanisms which require the project to aggregate a minimum level (Provision Point) of funding before the PM can claim it. There is an extensive literature on the design for mechanisms with provision points for CC. [morgan2000financing] incentivizes player contribution for civic projects using state lotteries such that a higher contribution leads to a higher likelihood of winning. The game induced attains a unique equilibrium. In [marx2000dynamic], players contribute in a round-robin fashion such that an equilibrium exists where a player contributes iff other players make their equilibrium contributions. Our work is most closely related to the PPM, PPR and PPS mechanisms.

3.1 Provision Point Mechanism (PPM)

PPM [bagnoli1989provision] is the simplest mechanism in this class where players contribute voluntarily. Players gain a positive payoff only when the project gets provisioned and a payoff of zero otherwise i.e., . Then the payoff structure of PPM, for every Player , is,

where, are Player ’s payoff and contribution respectively. PPM has been shown to have multiple equilibria and also does not guarantee strictly positive payoff to the players. It has led the mechanism to report under-provisioning of the projects [healy2006learning].

3.2 Provision Point Mechanism With Refund (PPR)

PPM does not guarantee strictly positive payoff for players. PPR [zubrickas2014provision] improved upon this by offering refund bonuses to the players in case the project doesn’t get provisioned and rewarded payoff like PPM otherwise. The refund bonus scheme is directly proportional to player’s contribution and is given as , where is the total budget. Then the payoff structure of PPR, for every Player , can be expressed as,

In PPR, a player has no knowledge of other players’ contribution. This results in a simultaneous move game. PPR applied in a sequential setting where players can see contributions from everyone, would collapse to a one shot simultaneous game which leads to the race condition, which we define as,

Definition 3 (Race Condition).

A strategy profile is said to have a race condition if , such that , the payoff is maximum where is the PSNE of the induced game i.e., ,

Here, .

For PPR, and , i.e., the strategy constitutes a set of PSNE of PPR in sequential game. This is because the refund bonuses here are independent of time of contribution. Thus, players have no incentive to contribute early. Such strategies lead to the project not getting provisioned in practice and are therefore undesirable.

3.3 Provision Point Mechanism With Securities (PPS)

PPS [chandra2016crowdfunding] addresses the shortcomings of PPR by offering early contributors higher refund than a late contributor for the same amount. The refund bonus of a contributor is determined using securities from a cost based complex prediction market [abernethy2013efficient] and is given as where, are Player ’s time of contribution and the number of securities allocated to it, respectively. depends on the contribution and the total number of securities issued in the market at the time contribution denoted by . Then the payoff structure of PPS, for every Player , can be expressed as,

To set up a complex prediction market in the context of CC, PPS requires a cost function () satisfying [chandra2016crowdfunding, CONDITIONS 1-4,6-7]. PPS awards every contributing player securities for the project not getting provisioned. These securities are dependent on the player contribution i.e., greater the contribution, more the number of securities allocated to the player. Each of these securities pay out an unit amount if the project is not provisioned. However, setting up such a market and computing securities to be alloted is computationally expensive and costly to implement as a smart contract. Hence, we want to look for more desirable refund bonus schemes.

4 Desirable Properties of Refund Bonus Schemes

width=1 Mechanism Refund Scheme Parameters Covergence of Sum Based On PPRG Geometric Progression (GP) PPRE Exponential Function (EF) PPRP Polynomial Function (PF)

Table 1: Various Refund schemes satisfying Condition 1 and Condition 2 for a Player .

A desirable refund bonus scheme should not just restrict the set of strategies in a way that the project is provisioned at equilibrium, but should also incentivize greater as well as early contributions, so as to avoid the race condition, from all interested players. We constitute these desirable properties as the following two conditions for a refund bonus scheme where such that , with budget and which is continuous and differentiable over :

Condition 1 (Contribution Monotonicity).

The refund must always increase with the increase in contribution so as to incentivize greater contribution i.e., must be a monotonically increasing function with respect to contribution or

(2)
Condition 2 (Time Monotonicity).

The refund must always decrease with the increase in the duration of the project so as to incentivize early contribution i.e., must be a monotonically decreasing function with respect to time or

(3)

We now analyze the consequence of such a refund bonus scheme on the characteristics of the game induced by it.

4.1 Sufficiency of the Refund Bonus Scheme

Let, be the game induced by the refund bonus scheme . We require to satisfy the following properties:

Property 1.

In , at equilibrium, the total contribution equals the provision point i.e., .

Property 2.

must avoid the race condition.

Property 3.

is a sequential game and possesses sub-game perfect equilibria (SPE).

With these properties, we present the following theorem.

Theorem 1.

is a game induced by a refund bonus scheme with , in which if satisfies Conditions 1 and 2 then Properties 1, 2 and 3 hold.

Proof: In Steps 1, 2 and 3, we show that satisfying Condition 1 is sufficient to satisfy Property 1 and Condition 2 is sufficient to satisfy Properties 2 and 3.

  • Step 1: As , from Eq. 1, at equilibrium cannot hold, as with , at least, that could obtain a higher refund bonus by marginally increasing its contribution since satisfies Condition 1 and . For , any player with a positive contribution could gain in payoff by marginally decreasing its contribution. Thus, at equilibrium or satisfies Property 1.

  • Step 2: Every Player contributes as soon as it arrives, since satisfies Condition 2 i.e., ,

    In other words, the best response is the strategy . Thus, as per Definition 3, avoids the race condition or satisfies Property 2.

  • Step 3: Since satisfies Property 2, it avoids the race condition. Hence, it can be implemented in a sequential setting or is a sequential game.

    Now, when a Player enters the project and , its best response would be contributing . However, if , then its best response is that contribution in which its provisioned payoff is equal to its not provisioned payoff. With backward induction, it is the best response for every player to follow the same strategy in which their provisioned payoffs are equal to their not provisioned payoffs irrespective of .

    For a Player entering the project such that , its best response will be contributing . This is because for a contribution , its provisioned payoff will be greater than its not provisioned payoff. Player will also contribute the maximum contribution required, , since its not provisioned payoff increases as its contribution increases. Therefore, contributing an amount less than will result in a lesser not provisioned payoff for the player. Thus, these strategies form a set of sub-game perfect equilibria in or satisfies Property 3. ∎

4.2 Necessity of the Refund Bonus Scheme

Theorem 1 shows that Condition 1 is sufficient to satisfy Property 1 and Condition 2 is sufficient to satisfy Properties 2 and 3. We believe that these conditions are not necessary and provide an argument through the following claims. However, a formal proof remains illusive.

Claim.

Condition 1 may not be necessary to satisfy Property 1.

Proof: Observe that, if does not satisfy Condition 1 then, s.t.

However, because of the sequential arrival of the players to the crowdfunding platform, it is trivial to see that players need not arrive/contribute at . Thus, the project may still get funded as

where may still hold. Thus, Condition 1 may not be necessary to satisfy Property 1. ∎

Claim.

Condition 2 may not be necessary to satisfy Property 2.

Proof: Observe that, if does not satisfy Condition 2 then, for a Player for which,

However, this may not imply that for every Player arriving in the interval , the equation

will hold as it depends on the magnitude of the contribution as well. Thus, from Definition 3, there is no guarantee that and hence the race condition may be avoided. Hence, Condition 2 may not be necessary to satisfy Property 2. ∎

Claim.

Condition 2 may not be necessary to satisfy Property 3.

Proof: As shown in the previous claim, need not satisfy Condition 2 to avoid the race condition. Thus, the notion of a sequential game in may still hold.

The argument for sub-game perfect equilibria follows similar to Step 3, Theorem 1. Thus, Condition 2 may not be necessary to satisfy Property 3. ∎

Through this generalized result on refund bonus schemes, we show the following proposition:

Proposition 1.

PPS satisfies Condition 1 and Condition 2.

Proof: Since every cost function used in PPS for crowdfunding must satisfy , [chandra2016crowdfunding, CONDITION-7], PPS satisfies Condition 1.

For Condition 2, observe that , from [chandra2016crowdfunding, Eq. 6]

(4)

In Eq. 4, as , as it is a monotonically non-decreasing function of and thus R.H.S. of Eq. 4 decreases since R.H.S. of Eq. 4 is a monotonically decreasing function of [chandra2016crowdfunding, Theorem 3 (Step 2)]. Thus, PPS also satisfies Condition 2.∎

Corollary 1.

PPS avoids the race condition and thus can be implemented sequentially.

Proof: The authors prove in [chandra2016crowdfunding, Theorem 3] that PPS can be implemented sequentially without using Condition 1 and 2. However, from Proposition 1, and the fact that PPS payoff structure follows Eq. 1, we see from Theorem 1 that PPS can be implemented in a sequential setting. ∎

In the next subsection, we present three novel refund schemes satisfying Conditions 1 and 2 and the novel provision point mechanisms based on them.

4.3 Refund Bonus Schemes

Table 1 presents three novel refund schemes for a Player contributing at time as well as the mechanisms which deploy them. Note that, we require all the refund bonus schemes to converge to a particular sum that can be pre-computed. This convergence allows these schemes to be budget balanced. The parameters are mechanism parameters (for their respective mechanisms) which the PM is required to announce at the start of the project.

The refund schemes presented deploy three mathematical functions: geometrical, exponential and polynomial decay. and refunds the contributing players based on the sequence of their arrivals (similar to PPS), while the refund scheme refunds them on the basis of their time of contribution. This allows us to compare the evolution in the refund share, in comparison to PPR and PPS, with respect to the increase in time, for a Player .

Figure 1 depicts the comparison.The evolution in the refund share of PPRG, PPRE and PPRP, in comparison to PPR and PPS, with respect to the increase in time, for a Player is depicted in Figure 1. To compare the refund shares of different schemes we keep Player ’s contribution , the budget and the provision point same for all, with .

The horizontal axis in Figure 1 represents the time at which Player contributes. For PPRG and PPRP, this is equivalent to the sequence in which the players contribute, i.e., the axis represents

, as defined in Claim 2. For PPRE, the horizontal axis is the epoch of time at which Player

contributes, i.e., . For PPS, the horizontal axis is also the sequence of players contributing, just like in PPRG and PPRP. Each Player () is issued a constant number of securities, i.e., the number of outstanding securities in the market increases by a constant number as the number of players contributing increases.

As evident in Figure 1, the refund scheme of PPRG decreases gradually when compared to refund schemes of PPRE and PPRP. Thus, PPRG can provide significant refund share for a greater number of players for the same bonus budget. Thus, it increases the contribution of players towards the project and the higher chance of it getting provisioned. We now show that PPRG satisfies Conditions 1 and 2.

Figure 1: Evolution of the refund share for a Player for different provision point mechanisms.
Claim 1.

satisfies Condition 1 .

Proof: Observe that ,

Therefore, satisfies Condition 1 . ∎

Claim 2.

satisfies Condition 2.

Proof: For every Player arriving at time , its share of the refund bonus given by will only decrease from that point in time, since its position in the sequence of contributing players can only go down, making it liable for a lesser share of the bonus, for the same contribution. Let be the position of the player arriving at time , when it contributes at time . While will take discrete values corresponding to the position of the players, for the purpose of differentiation, let . Now, we can argue that at every epoch of time , Player will contribute to the project. With this, can be written as,

Further observe that ,

Therefore, satisfies Condition 2. ∎

Claim 3.

satisfies Condition 1 .

Proof: Observe that ,

Therefore, satisfies Condition 1 . ∎

Claim 4.

satisfies Condition 2 .

Proof: Observe that ,

Therefore, satisfies Condition 2 . ∎

Claim 5.

satisfies Condition 1 .

Proof: Observe that ,

Therefore, satisfies Condition 1 . ∎

Claim 6.

satisfies Condition 2.

Proof: For every Player arriving at time , its share of the refund bonus given by will only decrease from that point in time, since its position in the sequence of contributing players can only go down, making it liable for a lesser share of the bonus, for the same contribution. Let be the position of the player arriving at time , when it contributes at time . While will take discrete values corresponding to the position of the players, for the purpose of differentiation, let . Now, we can argue that at every epoch of time , Player will contribute to the project. With this, can be written as,

Further observe that ,

The inequality follows from the fact that as . Therefore, satisfies Condition 2. ∎

4.4 Gas Comparisons

Every smart contract is compiled to a bytecode and is then executed on EVM (Ethereum Virtual Machine). EVM is sandboxed and completely isolated from the rest of the network and thus, every node runs each instruction independently on EVM. For executing every instruction, there is a specified cost, expressed in the number of gas units. Gas is the name for the execution fee that senders of transactions need to pay for every operation made on an Ethereum blockchain. Gas and ether are decoupled deliberately since units of gas align with computation units having a natural cost, while the price of ether fluctuates as a result of market forces. The Ethereum protocol charges a fee per computational step that is executed in a contract or transaction to prevent deliberate attacks and abuse on the Ethereum network [ethereum_docs].

We present a hypothetical cost comparison between PPS, PPRG, PPRE and PPRP based on the Gas usage statistics given in [buterin2014next, wood2014ethereum]. Towards it, the cost in Gas units is as follows for the relevant operations: ADD: 3, SUB: 3, MUL: 5, DIV: 5, EXP(): and LOG(): . Note that, in PPRG, we can replace an exponential operation with multiplication operation which is significantly less expensive, by storing the previous GP terms in a temporary variable.

width=1 Operation PPS PPRG PPRE PPRP Operations Gas Consumed Operations Gas Consumed Operations Gas Consumed Operations Gas Consumed ADD 2 6 2 6 2 6 2 6 SUB 2 6 0 0 0 0 0 0 MUL 2 10 2 10 2 10 3 15 DIV 2 10 1 5 1 5 2 10 EXP() 2 0 0 1 0 0 LOG() 2 0 0 0 0 0 0 Total Gas: 407 (at least) Total Gas: 21 Total Gas: 31 (at least) Total Gas: 31

Table 2: Gas Consumption comparison between PPS, PPRG, PPRE and PPRP for a player. All values are in Gas units.

Table 2 provides a hypothetical cost comparison between PPS, PPRG, PPRE and PPRP based on the Gas usage statistics given in [buterin2014next, wood2014ethereum]. The cost in Gas units is as follows for the relevant operations: ADD: 3, SUB: 3, MUL: 5, DIV: 5, EXP(): and LOG(): .

Note that, we need not require any exponential calculations in PPRG. Towards this, the PM can have a variable (say ) to store the previous GP term. For instance, when the first player contributes it is allocated . Post this, . The second player to contribute is then allocated or after which is updated with this value. Thus, in PPRG, we can replace an exponential operation with multiplication operation which is significantly less expensive.

For every player, PPRG takes gas units, PPRP takes gas units, PPRE takes at least gas units and PPS takes at least gas units. When implemented on smart contract, PPS is an expensive mechanism because of its logarithmic scoring rule for calculating payment rewards. PPRG, PPRP, and PPRE, on the other hand, use simpler operations and therefore have minimal operational cost

5 Provision Point mechanism with Refund through Geometric Progression (PPRG)

PPRG incentivizes an interested player to contribute as soon as it arrives at the crowdfunding platform. In PPRG, for the same contribution of Player and Player i.e., , the one who contributed earlier obtains a higher share of the refund bonus. This difference in shares is allocated using the terms of an infinite GP series with common ratio . From Table 1, the refund bonus scheme in PPRG is,

(5)

, as the total bonus budget allocated for the project by the PM and where . The values and are mechanism parameters which the PM is required to announce at the start of the project, with .

Equilibrium Analysis of PPRG: We now provide the equilibrium analysis of this mechanism as the following theorem,

Theorem 2.

For PPRG, with the refund as described by Eq. 5 , satisfying and with the payoff structure as given by Eq. 1, a set of strategies are sub-game perfect equilibria, such that at equilibrium . In this, is the contribution towards the project, is the arrival time to the project of Player , respectively.

Proof. First we claim in Step 1 that induces a sequential move game, which possesses sub-game perfect equilibria. In Step 2 and 3, we characterize the equilibria strategy of Player . We derive the condition for the existence of Nash Equilibrium in PPRG in Step 4. Finally, we discuss the sub-game perfect equilibria strategies in Step 5.

Step 1: Since satisfies Condition 1 (Claim 1) and Condition 2 (Claim 2) and has a payoff structure as given by Eq. 1, from Theorem 1 we get the result that PPRG induces a sequential move game and thus, can be implemented in a sequential setting. Further, PPRG possesses sub-game perfect equilibria in which the project is provisioned at equilibrium.
Step 2: From Claim 2, the best response for any player is to contribute as soon as he arrives i.e., at time .
Step 3: For any player, it’s equilibrium strategy is that for which its provisioned payoff is no less than its not provisioned payoff, since the players have symmetric belief. Now,

The result follows from the fact that at equilibrium .
Step 4: Summing over , we get,

as . From the above equation, we get

as the condition for existence of Nash Equilibrium for PPRG.

Step 5: Consider a Player arriving at , then its best response is contributing . If , then irrespective of the value of , the set of strategies , as defined in the theorem, form the set of sub-game perfect strategies as shown in Theorem 1. ∎

5.1 Equilibrium Analysis of PPRE and PPRP

Theorem 3.

For PPRE, with the refund as described in Table 1 (in the paper) , , which satisfies and has the payoff structure as given by Eq. 1, the set of strategies are sub-game perfect equilibria. In this, is the contribution towards the project, is the arrival time to the project of Player , respectively.

Proof. The proof for the theorem follows similar to as presented for Theorem 2. The condition for the existence of Nash Equilibrium for PPRE is given as,

Theorem 4.

For PPRP, with the refund as described in Table 1 (in the paper) , , which satisfies and has the payoff structure as given by Eq. 1, the set of strategies are sub-game perfect equilibria. In this, is the contribution towards the project, is the arrival time to the project of Player , respectively.

Proof. The proof for the theorem follows similar to as presented for Theorem 2. The condition for the existence of Nash Equilibrium for PPRP is given as,

(6)

In the next section, we look at implementation aspects of PPRG, PPRE and PPRP in terms of its provision accuracy with respect to PPS. We also look at the effect of on the provision accuracy of all these mechanisms.

6 Simulation Analysis

In this section, we compare PPRG, PPRE, PPRP, and PPS for provision accuracy using a CC proprietary simulator built in partnership with industry (name hidden for review). In this simulator, we create a Reinforcement Learning environment for PPRG, PPRE, PPRP, and PPS where players learn to participate in the mechanisms. Players go through repetitive iterations and learn their best strategy through rewards distributed by the corresponding mechanism. We run the simulation of 25 players for all the mechanisms and obtain comparison results between PPRG, PPRE, PPRP with respect to PPS. The results are shown in Figure 4.

Among PPRG, PPRE, and PPRP, it is clear to see that PPRG shows better provision accuracies. PPRP shows slightly better accuracies for when the total expected valuation () is low (5 times the provision point), but the gain in the accuracy only comes at the expense of a budget very close to the maximum possible budget () which is difficult to get in realistic circumstances.

(a)
(b)
Figure 4: Comparison of provision accuracy of PPRG, PPRE and PPRP with PPS for (a) (top) (a) (bottom) and (b) .

When compared to PPS, PPRG shows significantly good provision accuracies when is high (10 times provision point, for instance). Even when PPS shows slightly higher accuracies, it again comes at the expense of a budget close to the maximum possible budget, . However, for a reasonable budget of approximately or less, both the mechanisms share similar accuracies, therefore, it is safe to claim that PPS and PPRG performs equally well in terms of provision accuracy for a rational budget.

7 Conclusion

Motivated by the theoretical guarantees of PPR [zubrickas2014provision] and PPS [chandra2016crowdfunding], we looked for provision point mechanisms for CC with refund bonus schemes. We introduced two conditions, namely Contribution Monotonicity and Time Monotonicity, for refund bonus schemes in provision point mechanisms. We proved that these two conditions are sufficient to implement provision point mechanisms with refund bonus to possess an equilibrium that avoids free riding and race condition (Theorem 1). With this, we proposed three simple refund bonus schemes based on geometric progression, exponential and polynomial functions. With these schemes, we designed novel mechanisms, namely, PPRG, PPRE and PPRP. We showed that PPRG has much less cost when implemented as a smart contract over Ethereum framework. We identified a set of sub-game perfect equlibria for PPRG in which the project is provisioned at equilibrium (Theorem 2). To measure the performance of these mechanisms, we introduced a notion of provision accuracy. Our simulations showed that, whenever there is a hefty valuation for the project under consideration, with small refund bonus budgets, PPRG achieves the same provision accuracy as PPS. We leave it for future work to explore other refund bonus schemes having simplicity and efficiency as PPRG and much higher provision accuracies when the aggregate of the players’ valuations is just over target value.