1 Introduction
Social networks play a vital role in our daily activities, especially in electronic commerce. How to utilize social networks for promotions is widely studied by both researchers and practitioners [5]
. Mechanism design on social networks, aiming to establish specific policies satisfying some desirable properties, has received much attention from the fields of economics, computer science and artificial intelligence
[2].The fixedprice mechanism has been the mostly applied trading rules for a seller to sell products in our daily life. The seller sets a fixed price before the buyers buy the product and a buyer would buy it if the price is not over her valuation for the product. Due to its simplicity, there does not exist much research on it [7].
However, determining an optimal price under a fixedprice mechanism is still challenging. If the seller aims to maximize her revenue, she could find the optimal price given the number of buyers in the market and their valuation distributions. We will show that the number of buyers can significantly influence the seller’s revenue. Specifically, even under fixedprice mechanisms, the more buyers participating in the mechanism, the higher revenue for the seller. Therefore, the seller is incentivized to promote the sale to attract more buyers to increase her revenue, even if there is only one item to sell.
This paper aims to design a novel fixedprice mechanism to help the seller to attract more buyers via a social network. We assume that the seller is located on the social network and she can sell the item with a fixedprice mechanism to her neighbours without promotion. We want the seller’s neighbours to help her attract more buyers from their neighbours. However, her neighbours would not do so without giving them a proper incentive (why should they bring competitors for the item). The goal of our mechanism is to design such an incentive for the neighbours to invite more buyers.
A similar problem has been studied recently in [6, 15] where they proposed an auction with a dedicated payment scheme to attract more buyers via a social network. The payment requires the whole structure of the network and also the valuations of all buyers, while our proposed mechanism is fixedpriced and computationally easy with a guaranteed revenue improvement for the seller.
Our mechanism is also inspired by the mechanism with rewards used by the MIT Red Balloon Challenge Team to win the 2009 DARPA Networks Competition sponsored by a research organization of the United States Department of Defense [11]. The challenge is to locate 10 balloons located at undisclosed locations across the U.S.. With the dedicated rewarding scheme, the MIT team only began with four initial participants but eventually attracted over 5,000 participates and won the challenge. We extended this rewarding scheme into our mechanism to incentivize buyers to invite their neighbours.
As closely related work, fixedprice mechanisms have also been investigated in other mechanism design settings [14, 12]. Goldberg et al. used an optimal fixed pricing as a benchmark for evaluating the expected revenue of singleround and sealedbid auctions [4]. Emek et al. analyzed the properties of the wellknown geometric reward mechanism that incentivizes the participation for multilevel marketing [3]. McAfee showed under the condition that the buyer’s distribution dominates the seller’s distribution for a bilateral trade, a fixedprice mechanism gets at least half of the maximum social welfare gain from the trade [8]. Badanidiyuru et al. presented a constantcompetitive fixedprice mechanism for online procurement markets with submodular utility [1].
However, to the best of our knowledge, no work has considered introducing information diffusion into fixedprice mechanisms. Thus, in this work, we propose a novel mechanism by integrating the merits of both fixedprice mechanism and information diffusion. Our fixedprice diffusion mechanism distinguishes itself with the following.

It helps the seller to attract more buyers via social networks without paying them in advance.

It does not require buyers to reveal their valuations. The payments for buyers are relatively fixed and computationally easy.

The seller’s revenue improvement is guaranteed and it is bounded above half of the optimal.
The rest of the paper is structured as follows. Section 2 describes the model of our study. Section 3 investigates the traditional fixedprice mechanism. Section 4 defines the proposed Fixedprice Diffusion Mechanism. Section 5 studies the properties of the proposed mechanism and we conclude in Section 6.
2 The Model
We consider a market where a seller sells one item via a social network. We assume that the social network is a rooted tree, and the root is and there are other nodes in the tree denoted by . The other nodes are the potential buyers of the item sold by . Each agent has a set of neighbours denoted by , and she can only exchange information with her neighbours. Each buyer has a valuation
for the item. We assume, as suggested by Myerson, buyers’ valuations are independently derived from the uniform distribution on the interval
[9].Without any advertising, the seller can only sell the item among her neighbours since she can not communicate with the rest buyers from the network directly. Fixedprice mechanisms have been widely applied for this kind of market due to their simplicity. However, determining an optimal fixedprice is challenging. Moreover, the more buyers in the market, the higher chance to sell it with a higher fixedprice. In order to attract more people to buy, the seller may pay thirdparty platforms to promote the sale. This is often very costly with no guarantee to improve the revenue. Therefore, in this paper, we consider another class of fixedprice mechanisms which can improve the seller’s revenue by attracting more people to buy. It achieves this by incentivizing buyers to propagate the sale information to all their neighbours, instead of by the seller’s advertising through other platforms. At the same time, it does not require buyers to report their valuations as standard fixedprice mechanisms would do.
Since our mechanism does not require buyers to report their valuations, the action of a buyer is just diffusing the sale information to their neighbours. Let be the type of buyer , be the type profile of all buyers and be the type profile of all buyers except . Let or be the action of , where represents the neighbours informed by and represents that has not been informed or she does not want to participate in the sale. Furthermore, let be the action profile of all buyers and be the action profile of all buyers except .
Definition 1
Given the type profile of all buyers, an action profile is feasible if for all ,

if and only if can be informed the sale information from the seller following the action profile of .

if , then .
Let be the set of all feasible action profiles of all buyers. The mechanism is defined only on feasible action profiles.
Definition 2
A mechanism in the social network consists of an allocation policy and a payment policy , where and are the allocation and payment functions for buyer .
In the above definition, given the action profile , indicates that receives the item whereas means that does not get the item. indicates pays to the mechanism whereas means receives from the mechanism.
Next, we introduce the desirable properties we want to design for the mechanism.
Definition 3
An allocation is feasible if for all ,

for all , if , then .

.
A feasible allocation cannot allocate the item to a buyer who does not join the sale and it should not allocate the item to multiple buyers. In the rest of the paper, only feasible allocations are considered.
Given a feasible action profile and a mechanism , the utility of is defined as
A mechanism is individually rational if, for each buyer, her utility is always nonnegative no matter how many neighbours she informs.
Definition 4
A mechanism is individually rational (IR) if for all , for all .
In a standard fixedprice mechanism, a mechanism is incentive compatible if and only if for each buyer, reporting her truthful action is a dominant strategy [10]. In our model, the action of each buyer is diffusing the sale information. Hence, incentive compatibility means that for each buyer, diffusing the sale information to all her neighbours is a dominant strategy.
Definition 5
A mechanism is incentive compatible (IC) if for all , for all such that .
In this paper, we design a fixedprice mechanism that is IC and IR, which encourages the buyers to diffuse the sale information without being rewarded in advance. More importantly, it gives the seller a nearly optimal revenue.
3 Fixedprice Mechanism without Diffusion
As we mentioned in Section 1, fixedprice mechanisms are widely used due to their simplicity. However, like many other traditional mechanisms such as the wellknown VickreyClarkeGroves (VCG) mechanism, the seller only sells the item among the people she can communicate with [13].
Given the model defined in Section 2, we can give a brief description of the fixedprice mechanism for selling one item as follows.
Fixedprice Mechanism without Diffusion

The seller sets a fixed price and informs it to all buyers.

The buyers whose valuations are greater than the fixed price claim to buy the item.

The seller selects one of them who claimed to buy (with random tiebreaking), gives the item to her, and receives the payment from the buyer.

If no one claims to buy the item, the seller keeps it.
In order to get higher revenue, the seller can optimize the price of in the above mechanism. Theorem 1 shows that if the seller knows how many buyers participating in the sale, there is an optimal fixed price for her to maximize her revenue.
Theorem 3.1
Under the fixedprice mechanism without diffusion, given the number of buyers in the market, the optimal fixedprice to maximize the seller’s expected revenue is , which is an increasing function of .
Proof
Since the valuations of all buyers on the item have the independent and identical uniform distribution
, the probability that a buyer’s valuation
equals , and the probabilty that all of the buyers’ valuations are smaller than equals , which is also the probability that the item is not sold. So, the probability that the item can be sold is with the seller’s revenue being . Thus we get the seller’s expected revenueand its derivative with respect to is
By letting , we get the optimal fixedprice,
(1) 
Hence, the seller’s maximum expected revenue is
(2) 
Let , then . Therefore . By , which is easy to prove, we have . Having a positive derivative, is an increasing function of , and so is . ∎
Furthermore, since is an increasing function of , we conclude that is an increasing function of . That is more buyers participating in the fixedprice mechanism, more expected revenue for the seller.
Now, we consider an extreme case of the network, where all buyers are the seller’s neighbours. In this case, the maximum of the seller’s expected revenue under the fixedprice mechanism is , where is the total number of buyer in the network.
If is the number of neighbours of the seller , then and define the lower bound and the upper bound of the seller’s expected revenue under our diffusion mechanism that we propose next.
4 Fixedprice Diffusion Mechanism
In the fixedprice mechanism discussed in Section 3, the seller sells the item only among her neighbours . However, the seller will obtain higher expected revenue if the number of participants increases. Therefore, in this section, we design a mechanism, called fixedprice diffusion mechanism, to incentivize the seller’s neighbours to diffuse the sale information to their neighbours for the seller to increase the number of participants.
Our mechanism will iron the conflict between the seller aiming to improve her revenue and the buyers unwilling to diffuse the information to bring more rivals.
Before introducing our mechanism, we need some extra notations. Given the treelike social network, let be the depth of buyer and be the set of all buyers except on the path from the seller to . We assume that the seller has neighbours and let . In addition, we call a subtree rooted at each a branch and denote the set of the buyers in branch with , the cardinality of with , and the number of the rest buyers in the network except branch with . Without loss of generality, we assume . As a final notation, is a factor, which is used to adapt the amount of the rewards in the proposed mechanism.
Our mechanism is defined as follows.
Fixedprice Diffusion Mechanism (FPDM)
Given the buyers’ action profile and , compute the number of buyers in each branch, for each branch , by substituting in Equation (1) with , fixed price is defined for all the buyers in branch if they receive the item.

Allocation:

Starting from , while , repeat the following:
The seller informs all buyers in branch of the price .

Those buyers in branch whose valuations are greater than or equal to claim to buy the item.

If there are multiple claimers, the seller picks the one with the smallest depth. If there are multiple such claimers, select the one with the largest number of neighbours to allocate the item (with random tiebreaking). Allocation ends.

Otherwise, set .


If the item is not allocated in Step 1, the seller keeps it.


Payment: If the item is allocated to a buyer in branch , then for each buyer , her payment is
where is the optimal fixed price to sell among .
Obviously, is irrelevant to all the buyers in . That is to say, is independent of buyers’ action, which will be used in the proof of the IC of our mechanism.
Intuitively, if no buyers want to buy the item, then the seller keeps the item; Otherwise, the seller first selects the branch with the largest number of buyers to sell the item. Later in this branch, among those whose valuations are higher than the fixedprice, the seller allocates the item to the buyer who is closest to the seller and of the most neighbours (with random tiebreaking). The recipient of the item pays her payment to the seller.
Unlike in the fixedprice mechanism defined in Section 3, here the payment varies with different branches.
Figure 1 illustrates the proposed mechanism with an example. The seller sells an item. There has three branches denoted by branch 1, 2 and 3. . Then . The valuations of all buyers from number 1 to number 10 are assumably given to be respectively. The seller first selects branch 1 since it has the largest number of buyers. Then in branch 1, according to , we know only buyer 5 and 6 can buy the item. Since , and since buyer 5 has one neighbour and buyer 6 has none, the seller allocates the item to buyer 5. We can work out that . At the same time, we get the seller’s revenue generated by our mechanism is and . Following the fixedprice mechanism defined in Section 3, we know that buyer 2 receives the item. Therefore . We let . The payments of buyer 1 and 5 are and respectively. So the utility of buyer 1 equals , the utility of buyer 5 equals , and the other utilities equal to zero.
5 Properties of FPDM
In this section, we first show that the seller’s expected revenue generated by our mechanism is greater than that of the fixedprice mechanism without diffusion and approaches the expected revenue of the extreme optimal case defined in Section 3. Second, we prove that our fixedprice diffusion mechanism is individually rational. At last, we prove that our mechanism is incentive compatible. Therefore, the seller is willing to apply our mechanism.
Theorem 5.1
The seller’s expected revenue generated by our mechanism is greater than that of the fixedprice mechanism without diffusion; when the number of buyers in the social network is infinity and , the former is at least of the extreme optimal case defined in Section 3. This bound is independent of the structure of networks and is only related to the number of buyers in the network.
Proof
We first prove the first part of the theorem. According to the payment policy of our mechanism, given the buyers’ action profile , the seller’s expected revenue generated by our mechanism equals
By Equation (2), the seller’s expected revenue generated by the fixedprice mechanism without diffusion equals
It is a tedious manipulation to compare and . Without loss of generality, we assume . Then
, 
. 
It is obvious that , so we have .
To prove the second part of the theorem, for simplicity, we further assume that only one of the seller’s neighbours has descendants.Then, the seller’s expected revenue under our mechanism is
Again, following the Section 3, we know that .
To compare with , the curve of is depicted in Figure 2, from which we see that achieves at least of . ∎
For a clearer understanding of the theorem, we consider two special cases.

The first case is one where the number of the seller’s neighbours is a constant and where the number of buyers in the social network is infinity. For convenience, we let and fix .
From the Figure 3, we see that when , the seller’s expected revenue generated by our mechanism is greater than that of the fixedprice mechanism without diffusion, i.e., .

The second case is one where and , we compute the following limitation
We show the curve of in this case in Figure 4, from which we see that is more than of .
From the above two cases, it can be induced that the seller’s expected revenue generated by our mechanism is not only greater than that of the fixedprice mechanism without diffusion but also can achieve at least of the extreme optimal case defined in Section 3.
Next, we investigate the individual rationality of the proposed mechanism.
Theorem 5.2
The fixedprice diffusion mechanism is individually rational.
Proof
Given the buyers’ action profile , we only need to show that for all , it holds that
for all .
From the definition of FPDM, for any buyer , without loss of generality, we assume that . Then we have
According to the allocation policy of the mechanism, we have , so
By the Theorem 2, we know that . Then following the definition of and , we get
Therefore, we have . ∎
Now we discuss the incentive compatibility of the fixedprice diffusion mechanism. Based on the model defined in Section 2, we only need to analyze the diffusion action of each buyer. We obtain the following theorem by proving that for all buyers, diffusing the information to all their neighbours (i.e., ) maximizes their utilities, which means the mechanism incentivizes them to do so.
Theorem 5.3
The fixedprice diffusion mechanism is incentive compatible.
Proof
Given the buyers’ action profile , to prove that FPDM is incentive compatible, we need to show that for each buyer such that , diffusing the sale information to all ’s neighbours maximises
’s utility. We first classify all the buyers into three categories, then for buyers in each category prove the theorem is true.

The buyer who receives the item, denoted by .

All buyers on the path from the seller to .

All buyers who are not included in the above two categories.
Category (1): Without loss of generality, we assume . Then when reporting type truthfully, ’s utility is , which is independent of her diffusion action. Now when misreports her type, then her utility is also . Therefore, the utility is not better than reporting type truthfully. Therefore, reporting type truthfully maximizes ’s utility.
Category (2): According to the notation introduced in Section 4, for each buyer in this category, we have , where is the buyer who recieves the item. Without loss of generality, we assume . When reporting type truthfully, the buyer ’s utility is . Since that , is independent of ’s action, and and are irrevalent to ’s action, we know that ’s utility is independent of her action. When misreporting her type, then her utility does not change. Therefore it is not worth misreporting type.
Category (3): For each buyer in this category, her utility is 0. If belongs to the branch which the winner belongs to, diffusing the sale information to all her neighbours makes or her descendants have the chance to become the winner, which increases ’s utility. Otherwise, ’s diffusing the sale information to all her neighbours increases the number of buyers in the branch which belongs to, which creates the chance that her branch wins the item, and then will have the chance to become the winner or to stand in the path from the seller to the winner, which increases ’s utility. Therefore, diffusing the sale information to all her neighbours maximizes ’s utility.
Putting the above together, we know that the fixedprice diffusion mechanism is incentive compatible. ∎
6 Conclusions
This paper designed a mechanism called fixedprice diffusion mechanism (FPDM) for a seller to sell an item via a treestructure social network. The mechanism helps the seller to attract more buyers by incentivizing buyers to invite each other on the network. Under this mechanism, the seller’s revenue improvement is guaranteed, which is at least 1/2 of the optimal revenue the seller can get with the fixedprice mechanism. More importantly, the mechanism advances other existing solutions in the sense that it is fixpriced and does not require buyers’ valuations to make the decision.
We have only considered treestructure social networks in this paper. It would be very interesting to look at the problem in a general network structure. We have already had an idea to tackle this problem. Roughly speaking, we transform the graph structure social network into one of tree structure by randomly deleting some edges, which we will prove to maintain the properties of our mechanism. Another interesting topic would be to consider hiding the social network structure from the seller as well. Lastly, there might exist other fixedprice mechanisms to further improve the seller’s revenue.
References
 [1] Badanidiyuru, A., Kleinberg, R., Singer, Y.: Learning on a budget: posted price mechanisms for online procurement. In: Proceedings of the 13th ACM Conference on Electronic Commerce. pp. 128–145. ACM (2012)
 [2] Domingos, P., Richardson, M.: Mining the network value of customers. In: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. pp. 57–66. KDD ’01, ACM, New York, NY, USA (2001). https://doi.org/10.1145/502512.502525, http://doi.acm.org/10.1145/502512.502525
 [3] Emek, Y., Karidi, R., Tennenholtz, M., Zohar, A.: Mechanisms for multilevel marketing. In: Proceedings of the 12th ACM conference on Electronic commerce. pp. 209–218. ACM (2011)
 [4] Goldberg, A.V., Hartline, J.D., Karlin, A.R., Saks, M., Wright, A.: Competitive auctions. Games and Economic Behavior 55(2), 242–269 (2006)
 [5] Jackson, M.O.: The economics of social networks (2005)
 [6] Li, B., Hao, D., Zhao, D., Zhou, T.: Mechanism design in social networks. Proceedings of The ThirtyFirst AAAI Conference on Artificial Intelligence pp. 586–592 (06 2017)
 [7] Mathews, T.: The impact of discounting on an auction with a buyout option: A theoretical analysis motivated by ebay’s buyitnow feature. Journal of Economics 81(1), 25–52 (2004)
 [8] McAfee, R.P.: The gains from trade under fixed price mechanisms. Applied Economics Research Bulletin 1(1), 1–10 (2008)
 [9] Myerson, R.B.: Optimal auction design. Mathematics of operations research 6(1), 58–73 (1981)

[10]
Nisan, E.B.N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic game theory. Communications of the Acm
53(7), 78–86 (2009)  [11] Pickard, G., Pan, W., Rahwan, I., Cebrian, M., Crane, R., Madan, A., Pentland, A.: Timecritical social mobilization. Science 334(6055), 509–512 (2011)
 [12] Tran, T., Desiraju, R.: Groupbuying and channel coordination under asymmetric information. European Journal of Operational Research 256(1), 68–75 (2017)
 [13] Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. The Journal of finance 16(1), 8–37 (1961)
 [14] Wang, X., Montgomery, A., Srinivasan, K.: When auction meets fixed price: A theoretical and empirical examination of buyitnow auctions. Quantitative Marketing and Economics 6(4), 339 (2008)
 [15] Zhao, D., Li, B., Xu, J., Hao, D., Jennings, N.R.: Selling multiple items via social networks. In: Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems. pp. 68–76. International Foundation for Autonomous Agents and Multiagent Systems (2018)
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