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Sybil-Proof Diffusion Auction in Social Networks

11/03/2022
by   Hongyin Chen, et al.
ShanghaiTech University
Peking University
0

A diffusion auction is a market to sell commodities over a social network, where the challenge is to incentivize existing buyers to invite their neighbors in the network to join the market. Existing mechanisms have been designed to solve the challenge in various settings, aiming at desirable properties such as non-deficiency, incentive compatibility and social welfare maximization. Since the mechanisms are employed in dynamic networks with ever-changing structures, buyers could easily generate fake nodes in the network to manipulate the mechanisms for their own benefits, which is commonly known as the Sybil attack. We observe that strategic agents may gain an unfair advantage in existing mechanisms through such attacks. To resist this potential attack, we propose two diffusion auction mechanisms, the Sybil tax mechanism (STM) and the Sybil cluster mechanism (SCM), to achieve both Sybil-proofness and incentive compatibility in the single-item setting. Our proposal provides the first mechanisms to protect the interests of buyers against Sybil attacks with a mild sacrifice of social welfare and revenue.

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1 Introduction

Auction is an important method of selling commodities where the seller collects bids from buyers and allocates commodities according to these bids. Previous works [bulow1996auctions] have shown that more buyers would significantly lead to higher social welfare and revenue in auctions. However, buyers have no incentive to invite others to their auctions because it would cause tougher competition and hurt their own interests. Recently, there has been an emergence of studies on the diffusion auction over social networks [guo:DM-survey], which studies mechanisms that incentivize buyers to invite new agents to an auction via a social network. In these works, while the seller only knows her neighbors, any buyer who is informed of the auction may bid to buy, as well as diffuse the information about the sale to her neighbors to improve her utility.

The first work of diffusion auctions [li:IDM] proposed the information diffusion mechanism (IDM) for selling one item in a social network, focusing on the incentive compatibility of its information propagation action. Under IDM, it’s a dominant strategy for each bidder to truthfully bid her private valuation and to diffuse the auction information to all her neighbors. zhao2019selling and kawasaki2020strategy further designed diffusion mechanisms in selling multiple homogeneous items in a social network. Those works, however, did not consider a common threat known as the Sybil attack.

The first study on Sybil attacks [douceur2002sybil] considered a situation in peer-to-peer systems where malicious agents may gain an unfair advantage by creating fake identities. One such example is presented in Figure 1 for a social network where the agent creates six false-name identities. The Sybil attack is a significant threat to auctions (commonly called false-name bids) and has been extensively investigated in traditional auction settings [yokoo2004effect]. One example proven to be vulnerable to Sybil attacks is the well-known Vickrey-Clarke-Groves (VCG) auction for combinatorial auctions of at least two items.

Our work studies this fundamental issue in diffusion auctions on social networks, where fake nodes can be easily created. In existing mechanisms such as IDM and FPDM [zhang2021fixed], intermediate buyers are rewarded for inviting more buyers. This makes Sybil attack highly profitable and harmful.

We consider a strong adversarial model that allows every buyer to create fake identities, which can link with each other internally or link to the buyer’s neighbors. But they don’t have incoming arcs from other agents, since they are only visible to the creator. Our goal is to incentivize diffusion without encouraging Sybil attacks in diffusion auctions.

1.1 Our Contribution

We propose two Sybil-proof diffusion mechanisms, the Sybil tax mechanism (STM) and the Sybil cluster mechanism (SCM). STM achieves Sybil-proofness by identifying trustworthy agents. In STM, diffusing to “suspicious” agents is not beneficial to buyers. However, agents in STM do not have a strong incentive to diffuse information. In proposing SCM, we provide a stronger incentive for diffusion, where the reachability of each non-Sybil vertex is credited to some selected agents. By a mild sacrifice on the seller’s revenue, SCM creates a strong incentive to invite new buyers.

Our work resolves several difficulties. We are the first to identify and model the Sybil attack in diffusion auctions. Our adversarial model is the most general form of Sybil attack possible without collusion. Existing diffusion mechanisms cannot resist such Sybil attacks, even after removing “suspicious” agents.

Additionally, we discuss the social welfare and revenue of Sybil-proof diffusion auctions. We prove that there is no optimal SP diffusion auction mechanism for social welfare, and all SP mechanisms perform poorly in the worst case. To further exhibit the performance of STM and SCM, we conduct experiments under different settings. Experimental results indicate that STM and SCM do not sacrifice social welfare and revenue much compared with non-SP mechanisms.

Figure 1: (a) the true type of ; (b) a Sybil attack of involving six Sybil identities (in the red rectangle).

1.2 Related Literature

The Sybil attack has become a fundamental issue in traditional social networks [viswanath2010analysis], where nodes are usually divided into two types: honest ones and Sybil ones. In such settings, various protocols have been proposed to identify Sybil nodes and maintain honest nodes through the graph structures [yu2008sybillimit, liu2014defense].

The Sybil attack is also destructive in auctions. The pioneering work to study Sybil attacks on combinatorial auctions [yokoo2004effect] proved that Sybil-proofness and Pareto optimality can’t be achieved simultaneously. Many other works have followed. For example, iwasaki2010worst have shown that a Sybil-proof combinatorial auction mechanism may result in extremely low social welfare in some cases. In dynamic spectrum access auctions, Sybil-proofness has been only achieved when severe restrictions are imposed on Sybil agents. For example, PRAM [dong2021pram] requires that if an agent performs the Sybil attack, the sum of bids given by herself and her Sybil identities is equal to her private valuation.

2 Preliminaries

In a social network, a seller is selling one item to a buyer among the set of potential buyers . The set is unknown to the seller; instead, she only knows some buyers . Likewise, each buyer has her private social connections, represented as a set of neighbors . Each buyer also has a private valuation of the item, which is denoted as . Collectively, each buyer owns a private type .

In a diffusion auction, can only advertise the sale to her neighbors initially. Then, each buyer with the information of the sale may diffuse it to some of her neighbors in . Recursively, many buyers can be informed. Each buyer is asked to give a bid on the item besides diffusing the sale. The mechanism consequently sells the item to an informed buyer and rewards some buyers for their contribution of inviting others.

We model the bid and diffusion of buyer as the report type , where is her bid and is the set of buyers she diffuses to. A buyer can only diffuse to her neighbors (i.e. ). The input of the mechanism is therefore a report profile , and we suppose the set of seller’s neighbors is provided in advance and fixed.

The set of all possible types and reports of buyer is denoted as , and we denote the set of all possible profiles as .

Definition 1 (Diffusion auction mechanism).

A diffusion auction mechanism is defined as a pair of allocation and payment schemes for arbitrary agent set :

  • allocation scheme ;

  • payment scheme .

Given the reported type profile , whose length is not known in advance, means that agent wins the item, and otherwise. She then pays to the seller.

We assume the following feasibility conditions for diffusion auctions throughout the paper:

  1. Allocation feasibility: ,

  2. Anonymity: except for ties, the mechanism output is invariant to any permutation on , and

  3. Ignorance of unreachable vertices: if is unreachable from on the social network represented by , then , and the mechanism output must be invariant with respect to .

We can use graph theory to formalize the third condition above. The social network represented by the true type profile can be denoted as a graph with vertex set and directed edge set . Likewise, a graph can be defined for the report profile . The subgraph of with vertices that are reachable from is denoted as . All vertices unreachable from are excluded from it. When can be inferred from context, we omit it and write as . The ignorance condition means that the mechanism can only use the structural information about and the bids of as inputs. This is a key difference between a diffusion mechanism and the traditional auction mechanism.

The agents have a quasi-linear utility model. Given a buyer’s true type and the report profile of all agents , her utility under mechanism is .

2.1 Non-deficiency, Individually Rationality and Incentive Compatibility

In this section, we define the objectives of diffusion mechanisms.

We say a mechanism is individually rational if any buyer can achieve a non-negative utility by reporting truthfully, no matter what the other agents do. This means that any agent is at least willing to participate.

Definition 2 (Ir).

A diffusion mechanism is ex-post individually rational (IR) if for all , for all with , it is guaranteed that .111The definition of IR in previous literature in diffusion auctions does not require the buyer to truthfully diffuse, which differs from the traditional definition in AGT. In the setting of this paper, the two definitions are equivalent, and the traditional definition is presented.

A desired mechanism encourages agents to behave truthfully, i.e., to bid their private values and to diffuse the information to all their neighbors. In the diffusion auction setting, an agent may be strategic by overbidding, underbidding, or under-diffusion, if such strategies can bring her an advantage. Dominant-strategy incentive compatibility requires that reporting the true type is a dominant strategy for every buyer, ruling out these strategic reports.

Definition 3 (Dsic).

A diffusion mechanism is dominant-strategy incentive compatible (DSIC, or IC for short) if, for any buyer with type , any report profile of other agents and any satisfying , we have , where and .

Some IC diffusion auction mechanisms, like VCG, may give a negative revenue to the seller [li:IDM]. We define the following non-deficiency condition to rule out these mechanisms. For a mechanism and a type profile , the revenue to the seller is defined as .

Definition 4.

A diffusion mechanism is non-deficit, or weakly budget balanced, if its revenue for the seller is always non-negative, or formally, for all .

2.2 The Sybil Attack and Sybil-Proofness

In our setting, we further consider the desiderata of disincentivizing Sybil attacks. When a buyer performs a Sybil attack, she creates an arbitrary number of Sybil identities (or false-name identities) , with a report profile of . The set of all identities of is denoted as . For every , , it must be guaranteed that because does not know any agent besides herself, her neighbors, and her Sybil identities. Consequently, such Sybil identities of buyer cannot have incoming edges from other buyers because the neighbor sets of other buyers cannot be changed by . Refer to Figure 1 for an example of Sybil attacks.

We define Sybil-proofness as a criterion for ruling out such attacks. A mechanism is Sybil-proof if, for every buyer, any form of Sybil attack cannot bring a higher utility.

Definition 5 (Sp).

A diffusion mechanism is Sybil-proof (SP) if, for any type profile , any buyer , and for all satisfying and , we have

where the Sybil-attack report profile is .

A Sybil attacker can bring an arbitrary number of Sybil identities, and each of the identities (including the agent herself) can report arbitrarily. This formulation is the most general form of Sybil attacks without collusion. As a degenerate case, a Sybil attack with is equivalent to a single-agent strategic play in the previous diffusion action setting. Therefore, Sybil-proofness implies incentive compatibility.

2.3 Vulnerability of Existing Mechanisms

To the best of our knowledge, none of the existing diffusion auctions is Sybil-proof except for the trivial Neighbor Second-Price Auction (NSP), where only the seller’s neighbors are considered with a second price auction (see Appendix B for details).222There are confusingly two mechanisms named VCG in the literature of diffusion auctions: 1) the single-item VCG auction among the seller’s neighbors, and 2) the generic VCG mechanism applied to diffusion auction [li:IDM]. To disambiguate, the former is called Neighbor Second-Price (NSP) in this paper. As assumed, the seller’s neighbors are known to the seller, so there is no chance for them to create fake identities to join NSP.

Other existing mechanisms for diffusion auctions are all vulnerable to the Sybil attack. Here we use the two typical mechanisms proposed in [li:IDM], VCG and IDM, to demonstrate the possibility of Sybil attacks. Definitions of these mechanisms are given in Appendix B.

Figure 2: Sybil attack counterexamples of VCG and IDM.
Observation 1.

VCG and IDM are not Sybil-proof.

The classic VCG mechanism can be easily extended as a diffusion auction. Under VCG, the item is sold to the highest bidder, and other agents are paid the social welfare increase due to their participation. In the example shown in Figure 2(a), if the intermediate node does not participate, and will be unable to join, and the social welfare will be . With ’s participation, the social welfare is , so VCG will pay to . Now, if creates a fake identity , then both and will be paid (a successful Sybil attack).

Since VCG paid a lot to the agents connecting the highest bidder to the seller, it cannot be non-deficit. Thus, IDM was proposed to guarantee that the seller’s revenue is non-negative. IDM does not directly sell the item to the highest bidder; it uses a resale process to find the winner. It first allocates the item to the first cut point to reach the highest bidder, and the buyer pays the highest bid without her participation. In the example shown in Figure 2(a), the item is first allocated to and pays . Then can choose to resell it to and has to pay the highest without to , which is . Now, if creates a fake neighbor with bid , then will need to pay to (another successful Sybil attack).

We also proved that the other existing diffusion auction mechanisms [10.5555/3398761.3398947, zhang2021fixed] are not Sybil-proof in Appendix C.

3 Analysis of the Sybil Attack

In this section, we study the features of Sybil attacks. In our model, a Sybil identity created by a real agent can only be connected by her other Sybil identities or by herself. This implies that every path from to contains . In graph theory [lengauer:dominators], this is noted as dominates , or , and is called a dominator of . If a vertex has no dominator except , one can be sure that is not a Sybil identity. Conversely, when vertex has a dominator , there is a chance that is a Sybil identity of .

In previous diffusion auction mechanisms like VCG and IDM, being a dominator can bring the agent profit (i.e., the mechanisms reward her for inviting new agents), leaving room for one to profit from Sybil attacks. This explains why Sybil-proofness is hard to achieve in diffusion auctions.

An immediate dominator of , denoted as , is defined as the unique vertex who dominates and is dominated by every other dominator of .

Theorem 1.

Every vertex on the graph except has an immediate dominator, and the edges form a directed tree with being its root, called the dominator tree of rooted at .

This is exactly Theorem 1 of [lengauer:dominators]. The definition of dominators is identical to diffusion critical nodes in [li:IDM], and the path from to on the dominator tree is the diffusion critical sequence of .

3.1 Graphical Non-Sybil Agents

In this subsection, we use graph theory to characterize the set of vertices that cannot be Sybil identities. Firstly, the seller and her neighbors are not Sybil identities. In a real-world scenario, there are sometimes trustworthy entities like public figures and centralized institutions. Thus, we introduce an optional set of vertices , which is provided externally and guaranteed not to contain any Sybil identities. If no such vertices are provided, . Allowing such external information makes our mechanisms more flexible.

For the convenience of expression, we first give the definition of meeting points.

Definition 6 (Meeting points).

For a pair of vertices , a vertex is defined to be a meeting point of and if there are two vertex-disjoint paths to , from and respectively.

If a vertex is a meeting point of two other non-Sybil vertices , it must not be a Sybil identity. This is because all paths from non-Sybil vertices to a Sybil identity must contain its owner which contradicts the definition of meeting points. Therefore, we have the following definition of graphical non-Sybil agents which iteratively collects meeting points of existing members.

Definition 7 (Graphical non-Sybil agents).

The set is defined as follows:

  1. Initialize the set as .

  2. For each pair of vertices , if is a meeting point of them in graph , then add to the set, i.e. .

  3. Repeat step 2 until there are no more vertices to add.

It can be shown that is precisely the maximal set of vertices that cannot be Sybil identities. This will be proven in Lemma 3 after the introduction of Sybil clusters.

3.2 Overly Sensitive Mechanism

Given the graphical non-Sybil agents, a straightforward idea to achieve Sybil-proofness is to apply the existing diffusion mechanisms on non-Sybil agents. This idea of detection and removal is a common solution to Sybil attacks in social networks [liu2014defense, wang2020structurebased, xu2010resisting]. However, we find that such an approach doesn’t work because an agent can misreport her neighbor set and turn non-Sybil agents into suspicious ones.

We propose the overly sensitive mechanism (OSM) to show why such an idea does not work. In OSM, we ignore all potential Sybil identities (i.e. all ) and focus on the reachable part of the induced subgraph inducing from , denoted as . The subgraph contains only vertices in , and for each vertex in it, there is a path from to that only passing non-Sybil agents. We adopt IDM on .

OSM seems Sybil-proof because Sybil identities are all ruled out. However, we find that OSM is not even incentive compatible. In OSM, the detection-and-removal process can be exploited by malicious agents. In Figure 3a, every vertex will be in . Under IDM, will buy the item with the second-highest price . However, if chooses not to diffuse the information to as in Figure 3b, would be excluded from , and would get the item with a lower payment of .

Since SP implies IC, OSM is not Sybil-proof either. Therefore, we need a new approach to resist Sybil attacks in diffusion auctions.

Figure 3: A counterexample for the overly sensitive mechanism. The set is denoted by the dashed border rectangle.

4 Sybil Tax Mechanism

In this section, we present the first main contribution of this paper, our first Sybil-proof diffusion mechanism, called Sybil Tax Mechanism (STM).

Before describing it, we introduce some notations. We use to denote the highest bid in a set , that is, . We also denote the vertices she dominates as for every vertex . The vertex is critical for these , because without her diffusion, these vertices are not reachable from . It is also known as diffusion critical children in the terminology of previous literature on diffusion auctions.

Sybil Tax Mechanism (STM)   Given the reported type profile , we first calculate the reachable reported graph and the graphical non-Sybil agent set . Let’s write them as and for short. Find the reachable buyer with the highest bid, denoted by , where . Compute the dominator sequence . Specifically, we have , for all . We define , the buying price of , as the highest bid without the participation of . The selling price of , denoted as , is defined as the highest bid among all vertices that are identified as not the Sybil identity of . Formally,
where
Pick a with the lowest index that satisfies . If such does not exist among index , we set . This agent wins the item with . The payment function is calculated as
The payment and allocation of all other buyers are zero.

In this mechanism, the item is sold along the dominator sequence from to as a series of successive transactions between neighboring agents. Agent ’s buying price is set as other agents’ optimal social welfare (i.e. the highest bid of them) when she does not participate in the auction. This ensures that her report cannot lower her buying price. The agent can sell the item further down the critical sequence to reach more potential buyers with a selling price of . To achieve Sybil-proofness, we need to Sybil-attacking to be not profitable, i.e. not able to increase . When the item is passed from to , since the latter may be a Sybil identity of the former, the selling price of must be irreverent to the report of . Indeed, is defined as the highest bid among those who are guaranteed not the Sybil identity of her. The set is defined in a way that it is monotonically increasing with the report of to incentivize diffusion, and that it contains no Sybil identity.

Conceptually, a buyer who gets the item can choose to keep it or to resell. She will pass the item only when her selling price is higher than her private value. STM simulates this choice based on buyers’ bid through the choice of the winner .

This series of transactions are summed up by STM. In a single transaction, buyer will receive units of money and pays for it. The price difference can be considered as a “tax” paid by the intermediate buyers (which we call brokers) to prove their innocence.

Since , we have , so the monetary gain of brokers and the tax are all non-negative. This leads to individual rationality and non-deficiency.

Figure 4: Examples of the Sybil tax mechanism and the Sybil cluster mechanism.

Figure 4(a) illustrates STM with an example. We assume that the externally provided set is empty. When all buyers report their type truthfully, the mechanism runs as follows.

The set of graphical non-Sybil agents is calculated as . The mechanism identifies the highest-bidder and calculates the dominator sequence . The item is sold to because is the only buyer on the dominator sequence that satisfies . Then we calculate the payments. For brokers , , so they get paid . The winner pays . The seller gets a revenue of . In short, the buyer will pay to buy the item. Other buyers get zero utility.

Theorem 2 (Main).

STM is IR, non-deficit and Sybil-proof.

The proof of Theorem 2 will be elaborated in Appendix E. We provide a proof scketch here.

Intuitively, STM is individually rational because and . A non-Sybil-attacking buyer would want to maximize to maximize her utility, which can be achieved by maximal diffusion. By the graph-theoretic properties of Sybil attacks, if a Sybil attack happens on the dominator sequence, the identities of the same buyer must be contiguous on the sequence, and the tax paid by such brokers would disincentivize this attack.

The above theorem shows that STM is incentive compatible because SP implies IC. In a previous work [li:CDM], li:CDM identified one class of diffusion mechanisms called critical diffusion mechanism (CDM) on social graphs, which covers a large class of incentive compatible mechanisms. The successive reselling in STM resembles CDM, but STM is not a member of that class. By introducing non-Sybil agents externally (i.e. ), STM can contribute the occurrence of some “isolated” non-Sybil agents to buyers in the dominator sequence.

Recall the example in Figure 4, and we can see that every buyer other than the item’s winner has zero utility. The following lemma shows that this is not a fluke. In essence, all possible profits of the brokers are taxed by the seller. The proof can also be found in Appendix E.

Lemma 1.

In STM when , every buyer, except the item winner, has a payment of zero, and thus zero utility.

5 Sybil Cluster Mechanism

In STM, we reward the brokers for their contribution to introducing agents in . However, when is empty, no one other than the seller can bring a graphical non-Sybil agent on her own. This leads to zero profit for brokers, as shown in Lemma 1. As a result, their interests would be neither increased nor decreased through diffusions. Therefore, although STM is incentive-compatible, buyers’ incentive to invite other agents is weak.

In this section, we create a positive incentive for inviting without losing Sybil-proofness. By removing some edges from the reachable reported graph while keeping unchanged, we attribute the introduction of non-Sybil agents to some brokers and reward them. We will introduce a clustering process to accomplish this.

5.1 Clustering

Definition 8 (Sybil clusters).

For every , we define its Sybil cluster as below:

The cluster contains vertex if and only if there is a path from to on that does not contain any vertex in other than itself.

The Sybil cluster includes and all vertices that are suspected of being the Sybil identities of . Call the vertex the root of , who is the only member of that is also in the non-Sybil set . The clusters get the name because they form a partition of .

Lemma 2.

Sybil clusters are disjoint, and every vertex in belongs to some Sybil cluster .

The proof can be found in Appendix D.

Using Sybil clusters, one can prove that is the maximal set of guaranteed non-Sybil vertices.

Lemma 3.

Any vertex in may be a Sybil identity of some other vertex in .

Proof.

Given a report profile , we can compute and the Sybil clusters by definition. For any , there exists such that from Lemma 2. Let , and . We can see that, under the true type profile , the agent may create Sybil identities and make the report profile identical to . This shows that may be a Sybil identity of . ∎

5.2 SCM Mechanism

Sybil Cluster Mechanism (SCM)   Given the reported type profile as input, we reconstruct a social network graph with vertices in . Formally, , where
Sample a random shortest-path tree333For every vertex , we denote the shortest-path length from to it on graph as . A spanning tree of is a subgraph of with , which is also a directed tree. A spanning tree is said to be a shortest-path tree if, for every vertex ,

. A uniformly distributed random shortest-path tree can be generated by independently selecting a parent

for each , where is selected from with equal probability.
of

with equal probability and denote it as

.
We construct a subgraph of using . Formally, where is defined as
Specifically, edge on graph is deleted if and . All the remaining edges form a new graph .
Perform STM with , rather than , on the agents’ reports.

In SCM, we remove some edges in according to the randomly selected shortest-path tree and keep as graphical non-Sybil agents. The appearance of some vertices in can be attributed to some brokers, thus increasing their profit.

SCM is also Sybil-proof and individually rational. We will provide a proof sketch here; rigorous proof can be found in Appendix F.

Theorem 3.

Sybil cluster mechanism is IR, non-deficit, and Sybil-proof.

Individual rationality and non-deficiency follow trivially from the fact that STM is IR and non-deficit. By the selection of the shortest-path tree, the diffusion choice of a buyer can affect vertices on the tree whose distance from is higher than her distance from . Maximally diffusing for a buyer would bring her a more favorable tree structure and give her a better income. Moreover, we find that Sybil attacks are completely ineffective in the clustering process. Combined with the Sybil-proofness of STM, we can show that SCM is Sybil-proof.

Figure 5: A visualization of SCM.

The example in Figure 4b shows an example of the Sybil cluster mechanism. The clustering process and a possible edge-removing process are shown in Figure 5. Assuming that all buyers report their true type, SCM runs as follows:

The mechanism divides into five Sybil clusters , where and . The mechanism randomly picks a shortest-path tree and constructs a subgraph . We only show the case when the mechanism picks tree as Figure 5(b), where the mechanism deletes edge and edge in graph . In this case, edges , and are removed from . With , we perform STM on .

STM identifies the buyer with the highest bidder to be and calculates the dominator sequence . Since , and , we select as the winner of the item.

For the payments, broker pays , gets units of money, and gets . The winner pays . The seller gets a revenue of .

6 Discussion

In this paper, we propose two Sybil-proof mechanisms, STM and SCM. In this section, we evaluate their performance on social welfare and revenue. Comparing our mechanism with the non-diffusion mechanism (i.e., NSP), other potential SP mechanisms and existing diffusion mechanisms (e.g., IDM, VCG) which are not SP, we raise three key questions.

  1. Do our diffusion mechanisms have better performance than non-diffusion ones?

  2. Does STM or SCM achieve optimal social welfare and revenue among all SP mechanisms?

  3. Compared with existing diffusion mechanisms, how much do our mechanisms sacrifice to achieve Sybil-proofness?

We conduct theoretical and experimental analysis to answer these questions. For the first question, we prove that our mechanisms always achieve higher (or equal) social welfare and revenue than NSP. Our experimental results indicate that advantages of STM and SCM are significant. For the second question, we conduct worst-case analysis and show that every SP mechanism has extremely lower social welfare and revenue than another in some cases. Therefore, there is no optimal SP mechanism in terms of worst-case performance. For the last question, our experimental results show that STM and SCM do not sacrifice social welfare and revenue much compared with non-SP mechanisms. To eliminate the external effect, we assume that in this section.

6.1 Comparison

We use and to denote the social welfare and revenue of the mechanism under respectively. We have

Recall that we have defined in Section 2.1.

The following theorem shows that both of our mechanisms outperforms the non-diffusion NSP mechanism. Under our mechanisms, agents’ invitations indeed benefit the seller and the society.

Theorem 4.

For all possible type profile , we have

We are curious whether STM achieves higher social welfare and revenue than all SP mechanisms. However, we’ll show in Section 6.2 that none of SP mechanisms always has optimal social welfare and revenue.

The following theorems qualitatively examine the cost of Sybil-proofness. In Theorem 5, we find that STM achieve better revenue than the most cited diffusion auction, IDM [li:IDM]. However, social welfare is sacrificed to achieve Sybil-proofness. Theorem 6 reflects that there is no clear-cut comparison of the seller’s revenue between SCM and IDM, or between SCM and VCG.

Theorem 5.

For any possible type profile , we have

Theorem 6.

There exist two report profiles , such that

The proofs of Theorem 4, 5, and 6 can be found in Appendix G.

6.2 Worst-Case Efficiency Analysis and (No) Optimality

In this subsection, we conduct worst-case analysis on SP mechanisms to explore the optimality of social welfare and revenue. We consider the concept of worst-case efficiency ratio, which is adopted from previous work [iwasaki2010worst] to measure the social welfare of Sybil-proof combinatorial auctions in the worst case. The worst-case efficiency ratio of indicates the ratio of ’s social welfare and the optimal social welfare in the worst-case input.

Definition 9.

Given a type profile , the optimal social welfare is defined to be the highest private value . The worst-case efficiency ratio of a mechanism is defined as follows:

Theorem 7.

The worst-case efficiency ratio of any non-deficit, IR, and Sybil-proof diffusion auction mechanism is zero.

The above theorem shows that the social welfare of every Sybil-proof mechanism is far below the social optimum in some cases. Its proof is included in Appendix H.

Because every SP mechanism is sufficiently bad compared to social optimum, it is natural to compare their social welfare relative to other SP mechanisms. However, this further impossibility result indicates that every SP mechanism would perform extremely worse than another SP mechanism in some cases. Therefore, we cannot find any optimal diffusion auction, even when the optimality is relative to each other. The proof can also be found in Appendix H.

Theorem 8.

For any non-deficit, SP, and IR diffusion auction mechanism , and for any , there exists another non-deficit, SP, and IR diffusion auction mechanism such that

We can derive a similar result in terms of the seller’s revenue.

Theorem 9.

For any non-deficit, SP, and IR diffusion auction mechanism , and for any , there exists another non-deficit, SP, and IR diffusion auction mechanism such that

The theorems above indicate that all SP mechanisms have extremely low social welfare and revenue compared to some other SP mechanisms. These impossibility results are surprising and show the drastic difference between diffusion mechanisms and traditional auctions.

6.3 Experiments

Despite the qualitative comparison results in Section 6.1, we still wonder how much our mechanisms are better than NSP, and how much social welfare and revenue is sacrificed for Sybil-proofness with comparison to other diffusion mechanisms. Therefore, we conduct simulations to analyze the performance of mechanisms in the average case. Such experiments have never been performed on diffusion auctions in previous literature, so we have to be innovative in the settings.

To test the diffusion auction mechanisms, we must specify the private value vector of buyers and the social network structure. Each buyer’s bid is a one-dimensional continuous variable and can be captured with a distribution function. For simplicity, we assume the private values are drawn i.i.d. from a uniform distribution on

. However, the graph structure in diffusion auctions is highly complex. Since diffusion auctions are held on social networks, we take inspirations from network science to create distributions for our input. Price’s model [Price:model] is a simple and classical model for directed networks, used to describe various scale-free networks in the real world [toivonen2009comparative]. It generates a graph of vertices, each with a degree of . Despite IDM and VCG being not Sybil-proof, we assume that all agents act truthfully in the experiment.

The mechanisms are tested with graphs with vertices, and the density can be controlled by changing the parameter . For each , 1,000 inputs are generated as specified above. We test five mechanisms: NSP, STM, SCM, IDM and VCG. We calculate and analyze their social welfare and revenue. The results are visualized with box plots in Figure 6.

We have the following observations. Firstly, our mechanisms achieve significantly higher social welfare and revenue than the non-diffusion NSP mechanism. Secondly, the average-case social welfare distribution of either STM or SCM is very close to the social optimum (VCG), especially when the graph is denser. Thirdly, STM has the highest revenue, which is consistent with theoretical analysis. Finally, seller’s revenue of SCM is slightly lower than IDM, and higher than VCG.

Experimental results indicate that our diffusion mechanisms have significantly better performance than NSP, and we do not sacrifice seller’s revenue and social welfare much to achieve Sybil-proofness.

(a) Social welfare of mechanisms when .
(b) Social welfare of mechanisms when .
(c) Seller’s revenue of mechanisms when .
(d) Seller’s revenue of mechanisms when .
Figure 6:

The welfare and revenue distribution of five mechanisms on graphs of different densities. The orange line is the median, the green triangle is the mean, the box denotes the range between the first and the third quartile, and the whisker represents the range between the 5th and 95th percentile.

7 Conclusions

In this paper, we study an important issue in diffusion auctions, the Sybil attack. We find that previous diffusion mechanisms are vulnerable to Sybil attacks. We have proposed two novel solutions, STM and SCM, and proved that they are incentive compatible and Sybil-proof. We further discuss the social welfare and revenue of these two mechanisms. Theoretical analysis and experiments indicate that STM and SCM achieve Sybil-proofness with little sacrifice in the social welfare and revenue.

We also conduct worst-case analysis on all Sybil-proof diffusion mechanisms. We prove negative conclusions that the social welfare and revenue of every SP mechanism is far below some other SP mechanism in some cases.

Our work raises many open problems in the domain of Sybil-proof diffusion auctions. Firstly, how to develop Sybil-proof diffusion mechanisms for selling multiple items? Secondly, is there any other effective way to achieve Sybil-proofness? Thirdly, since we can’t pick out the optimal Sybil-proof diffusion mechanism in the worst case, can we develop other methods to compare SP mechanisms? Or can we only compare a subset of all SP mechanisms to avoid such negative conclusions? Furthermore, how to reward the intermediate buyers fairly is also worth consideration.

References

Appendix

Appendix A Table of Notations

The notations in this paper are organized in Table 1.

Model:
Seller
Number of buyers
Set of buyers
Neighbors of , for or
Private value of , for
Private type of , for
Set of neighbors that diffuses to, for
Bid of , for
Reported type of , for
Type profile
Report profile
Type space for buyer
Space of all possible profile
Mechanism:
Allocation function of
Payment function of
A diffusion auction mechanism
Utility function of under
Seller’s revenue under mechanism
Social welfare under mechanism
Optimal social welfare
Graphs and graph theoretical constructions:
Social network graph of
Social network graph of
Subgraph of containing only vertices reachable from
Vertex dominates vertex
Immediate dominator of
Set of vertices dominated by
Externally provided trustworthy vertices
Guaranteed non-Sybil vertices on graph
Sybil cluster of for
Used by STM:
Highest bidder
Highest bid in a set
Dominator sequence of vertex
The -th element of dominator sequence
Set of vertices that is guaranteed not to be a Sybil identity of
buying price of when she get the item from
selling price gets when pass the item to
Used by SCM:
Reconstructed social network with vertices in
A random shortest-path tree of
Subgraph of that is constructed according to
Used in the proofs:
The relative complement of with respect to vertex set
Table 1: Notations in this paper

Appendix B NSP, VCG and IDM

Various diffusion auction mechanisms have been proposed [li:IDM] to achieve the incentive compatibility goal. In this section, we will provide three examples.

The most trivial way to achieve IC in diffusion auctions is to ignore diffusion completely, treat it as a single-item auction, and perform a second-price auction on the seller’s neighbors . We use to denote the highest bid in a set , or .

Neighbor Second-Price Auction (NSP)   Input the reported type profile . The item is sold to the highest bidder among the seller’s neighbor . The winner pays to the seller the second-highest bid in , i.e. .

However, this trivial mechanism defeats the purpose of diffusion auctions, i.e. to utilize the social network structure to advertise the auction to a broader audience. To fulfill this goal while still respecting incentive compatibility, we can apply the famous VCG mechanism [AGTbook2007] to our scenario. Recall that in VCG, each agent pays an amount equal to the social cost of their participation. In our scenario, an agent’s diffusion may introduce new agents to the auction. Therefore, these agents’ participation may increase the social welfare, instead of incurring a social cost, and should be rewarded by the mechanism.

The set is defined to be the set of vertices that cannot be reached from if is not present. is the set of vertices that can reach even if is not present.

VCG Mechanism   Input the reported type profile . We can find the highest bidder among the visible agents . This agent wins the item. Formally444We omit the argument in and when presenting mechanisms for better readability., . She also pays the second-highest price among all visible agents . The payment function of a buyer is . Note that this value is equal to the decrease in social welfare due to the participation of , which is non-positive (i.e., a reward instead of a payment).

We can observe that, in VCG, if an agent receives a participation reward, she must be critical for the diffusion to : Since , cannot reach without the participation of . This is the exact definition of . Therefore, is the exact set of agents that can receive rewards.

In [li:IDM], Li et al. observed that VCG mechanism is IC and has optimal social welfare, but may have a budget deficit because of the payments made to each intermediate buyer. They proposed Information Diffusion Mechanism (IDM) to tackle with this problem. IDM rewards the same set of agents as VCG, but with different amounts to remain within budget.

Information Diffusion Mechanism (IDM)   Given the reported type profile , we can find the visible buyer with the highest bid, denoted by , where . Compute the dominator sequence . Specifically, we have , for all . We define as the optimal social welfare when is not present, and as the optimal social welfare when all agents further in the sequence are not present.
Pick a satisfying ; when there are multiple candidates, pick the one with the lowest index . Since , such must exist. This agent wins the item with . The payment of is calculated as follows.
The payment and allocation function of all other buyers are zero. The revenue of the seller is .

It is proven in [li:IDM] that IDM is IC, IR, and non-deficit.

Appendix C Vulnerability of NRM and FPDM

In this section, we use detailed examples to illustrate our observations that NRM [10.5555/3398761.3398947] and FPDM [zhang2021fixed] are not SP.

Figure 7: Sybil attack counterexamples of NRM and FPDM.
Observation 2.

NRM is not Sybil-proof.

We show this by the counterexample in Figure 7(a). The seller’s neighbors are and , and buyer knows the existence of , , and . Their private values are , , , , and respectively. If everyone reports truthfully, will get the item. The utilities of and are zero. Buyers and get and units of money respectively. The winner pays for the item and receives units of money in the redistribution stage, totaling a utility of .

If creates a Sybil identity to raise the percentage of nodes inside its dominant tree, as shown in Figure 2(b), it will receive in the redistribution stage, leading to a higher utility for .

Observation 3.

FPDM is not Sybil-proof.

We still use the pair of examples shown in Figure 7. Assuming that everyone reports truthfully and the fixed price for branch is , buyers , , have equal opportunities to get the item. If again uses a false-name identity , then she will get the item because she has the largest number of neighbors among the claimers.

By creating Sybil identities, the claimer can get an advantage in competing for the item.

Appendix D Proofs on Sybil Clustering

Lemma 2.

is a (disjoint) partition of the set .

Proof.

First, we prove that every belongs to some . Consider any simple path from to ; such a path must exist because every vertex is reachable from . In the sequence , we can find the highest index such that ; such must exist because . Then is a -free path from to , or equivalently, .

Then, we prove that Sybil clusters are disjoint. Assume otherwise, we have where . Furthermore, we assume that among all vertices in , has the shortest shortest-path from . Let be a path from to such that none of is in , and similarly, . There must be a for some because is a meeting point closure and . Such must be in because and are two -free paths. However, has a shorter shortest-path from , contradicting the choice of . ∎

We can find out that any inter-cluster edge must point to the root of the latter cluster. The proofs of these lemmas are omitted here and can be found in Appendix D.

Lemma 4.

If there is a directed edge between and , where , then .

Proof.

We assume otherwise that . We can find two paths and because and . The path and the path form two vertex-disjoint paths from and resp. to . By the definition of , we have , so must be the root of its component, which contradicts the assumption. ∎

Appendix E Proofs on STM

e.1 The Proof of Theorem 2

To ease our expression, we categorize all buyers into four roles by their outcome: the winner , the brokers , the shadowed