Multi-unit auctions refer to the auctions where multiple homogeneous items are available for sale in a single auction. They are commonly used in many real-world markets to allocate scarce resources such as emissions permits, electricity and spectrum licenses [Hortaçsu and Puller2008, Huang et al.2006]. A common feature of these markets is that a seller sells items to a fixed group of buyers. The seller can hold an auction among a group of buyers, e.g. the classic Vickrey-Clarke-Groves (VCG) mechanism [Vickrey1961, Clarke1971, Groves1973] to get a good outcome. In order to further improve the social welfare and revenue, a common method is to advertise the sale to involve more buyers. However, the seller normally needs to pay for the advertisements and if the advertisements can’t attract enough valuable buyers, the seller’s revenue may decrease. To solve this problem, diffusion mechanism design has been proposed in recent years [Li et al.2017].
Diffusion mechanisms utilize the participants’ connections to attract more buyers. They incentivize every participant who is aware of the mechanism to further diffuse information to all his neighbors. zhao2021mechanism zhao2021mechanism and DBLP:conf/ijcai/GuoH21 DBLP:conf/ijcai/GuoH21 gave comprehensive reviews of diffusion mechanism design. The diffusion incentive can be introduced to many traditional mechanism design settings such as auctions, task allocation, matching and voting. For auctions, all the previous diffusion mechanisms assumed that participants have single-dimensional value functions.
In this paper, we design the very first diffusion mechanism for buyers with multi-dimensional value functions. More specifically, we propose a diffusion mechanism called layer-based diffusion mechanism (LDM) for buyers with diminishing marginal utilities. Diminishing marginal utility refers to the phenomenon that the marginal utility of adding one more unit is non-increasing as the total number of units allocated to the buyer is increasing. Our mechanism satisfies the diffusion incentive compatibility which requires that reporting both valuation and their connections is a dominant strategy for every buyer. The following are the techniques proposed to achieve this.
In single-item diffusion auction, a buyer who can win without inviting others may lose after inviting buyers with higher valuations, but we can reward him the social welfare increase due to his invitation, which is greater than his utility of winning the item (kind of “reselling” the item to his invitees). Offering this reward to all buyers will have the diffusion incentives [Li et al.2017, Zhang et al.2020].
To extend the setting to multi-unit-supply and unit-demand case, the challenge becomes much more difficult. Since there are multiple units for sale, the simple “resale” technique cannot be applied. The reason is that a buyer can control the number of items allocated to his invitees by whether holding one item by himself or not, because the payments from invitees may change according to the number of items they receive. The techniques we have seen to solve this problem is to change the reward scheme such that a buyer’s reward doesn’t depend on his invitees’ payments [Zhao et al.2018] or completely remove the resale option [Kawasaki et al.2020].
Our setting further extends to multi-unit-supply and multi-unit-demand case, this is more challenging and none of the above techniques can be applied. The difficulty is that a buyer now can misreport his valuation to mimic the same demand from both himself and his invitees. For example, a buyer with a single-unit demand of valuation , who has a neighbor with a single-unit demand of valuation , can misreport his valuation as for two units, which is the same as inviting in terms of demand. This is a clear conflict between inviting more buyers and reporting valuation truthfully. Such problem does not exist in unit-demand settings as a buyer can only misreport one value.
Our solution follows the idea of “resale” to design the invitation incentives. To do so, the allocation and payment for a buyer shouldn’t depend on any of his invitees who is a potential competitor. Otherwise, a buyer’s invitees may misreport to affect the buyer’s allocation to gain. Hence, we have to remove the buyers whose potential utility is positive from a buyer’s invitees to calculate the buyer’s allocation and payment. However, this will also incentivize them to misreport to avoid removal which will be prevented by removing more buyers.
2 The Model
We consider a setting where a seller sells homogeneous items via a social network. In addition to the seller, the social network consists of potential buyers denoted by . Each has a private marginal decreasing utility function for the
items which is denoted by a value vectorwhere . Then ’s valuation for receiving units is represented by , and the valuation for receiving nothing is . Each buyer has a set of neighbors denoted by and he does not know the existence of the others except for . Also the seller is only aware of his neighbors initially.
Let be the type of buyer and be the type profile of all buyers. can also be written as where is the type profile of all buyers except for . Let be the type space of buyer and be the type profile space of all buyers. Since the seller initially only connect to a few buyers, we want to design auction mechanisms which ask each buyer to both report his valuation on the items and invite his neighbors to join the mechanism. This is mathematically modelled by reporting his type. Let be buyer ’s type report where because ’s invitation is only among his actual neighbors. Let be the report profile of all buyers.
A general auction mechanism consists of an allocation policy and a payment policy . Given a report profile , is the number of items receives and . is the payment that pays to the mechanism. If , then receives from the mechanism.
Since we assume that each participant is only aware of his neighbors, initially only the seller’s neighbors are invited to join the auction. Other buyers who are not properly invited by early joined buyers cannot join the auction, i.e., their reports cannot be used by the mechanism. Therefore, we have some additional constraints for the mechanism, which will be defined as diffusion auction mechanism.
Given a report profile , an invitation chain from the seller to buyer is a buyer sequence of such that and for all , , and no buyer appears twice in the sequence.
Given a report profile , if there is an invitation chain from the seller to buyer , then we say buyer is valid in the auction. Let be the set of all valid buyers under . Let be the shortest length of all the invitation chains from seller to for each buyer . We denote the set of valid buyers whose shortest length of invitation chains is , i.e., . We also call layer .
A diffusion auction mechanism is an auction mechanism, where for all :
for all invalid buyers , and .
for all valid buyers , and are independent of the reports of all invalid buyers.
Given a buyer of type and a report profile , the utility of under a diffusion auction mechanism is defined as . For simplicity, we will use to represent when the mechanism is clear.
We say a diffusion auction mechanism is individually rational(IR) if for each buyer, his utility is non-negative when he truthfully reports his valuation, no matter how many neighbors he invites and what the others do. It means that buyers’ invitation behaviour will not make his utility negative as long as he reports his valuation truthfully. That is, we cannot force buyers to invite all their neighbors to be IR.
A diffusion auction mechanism is individually rational (IR) if for all , all , and all .
We say a diffusion auction mechanism is incentive compatible (IC) if for each buyer, truthfully reporting his valuation and inviting all his neighbors (i.e. reporting type truthfully) is a dominant strategy.
A diffusion auction mechanism is incentive compatible (IC) if for all , all and all .
3 Diffusion Auction Mechanism on Trees
Different from single-dimensional value functions, we consider the multi-dimensional value functions where buyers will have more strategies to misreport. Let’s see an example in Figure 1. In Figure 1(a), considering the single-unit demand setting, a buyer can gain from inviting . For the same network without , if we go to a multi-unit-demand setting with two units, then can misreport his valuation to mimic the same demand from both himself and to gain. The existing mechanisms [Li et al.2017, Zhao et al.2018, Kawasaki et al.2020] can’t be directly extended to prevent buyers from such misreporting.
In this section, we first design an IR and IC mechanism for the multi-dimensional value functions on tree networks, called layer-based diffusion mechanism on trees (LDM-Tree). Given a report profile , a tree can be constructed where seller is the root and . For each and his neighbor , there is an edge . We use to represent buyer ’s children in for any . We use to represent the number of layers in . Intuitively, LDM-Tree will prioritize the buyers by the layers. Layers closer to the seller have higher priority to be considered in the allocation and we decide the winners layer by layer according to their priority.
When computing allocation and payment of layer , we need to remove the buyers below layer who are potential competitors of layer , because these buyers have positive potential utility and have the motivation to act untruthfully to manipulate the allocation of high-priority buyers. They can be divided into two parts: buyers who are potential winners and buyers who diffuse information to potential winners. The latter can misreport his valuation to take more items from the high-priority layer and get more rewards through resale. To avoid such manipulation, for each buyer , we first remove a set of buyers including all children of who have child. If these buyers are removed, all buyers in will not get information and also be removed. In the remaining children of , we should find out the buyers who are potential winners. An intuitive method is to remove the top ranked buyers in ’s remaining children according to their valuation report for the first item.
However, removing and the top remaining children of does not guarantee IC. If buyers in have high valuations, they can misreport that they have no neighbors to replace some buyers of the top ranked buyers. The replaced buyers will not be removed and they will compete with the high-priority layers to get more items for the misreporting buyers. Therefore, we need to remove a larger set of buyers which includes the top ranked buyers in ’s remaining children where is the upper bound of (i.e. ). We assume is prior information of the seller. Removing ensures that the replaced buyers will still be removed even if buyers in misreport. Let be the total removed set for buyer .
Following the above discussion, when LDM-Tree considers the buyers in layer , it will remove the set of buyers and computes optimal social welfare in the remaining buyers for the allocation. The payment of is calculated according to the social welfare increase due to his participation. In order to compute the constrained social welfare of the others when is absent, LDM-Tree will remove the set of buyers . A formal definition of the mechanism is given in the following algorithm.
Here we give an example to illustrate our mechanism. We first show the removed sets through Figure 2. Suppose and . Since is given in the figure, we will omit it to simplify the description. It is easy to see and . For the buyer with , we have . Thus i.e. . We have marked in Figure 2. Taking the union of these two sets can get . For the buyer with , we have , . and are also marked in Figure 2. Taking the union of these three sets can get . For other buyers, the removed sets are .
Then, let’s compute the allocation and payment in Figure 2. For the first layer, the constrained optimization problem in step 4 of algorithm 1 will be computed among . Since we now focus on the first layer, the constraint means that no items have been allocated. The solution is and . For buyer , and his payment is . For buyer , and his payment is , i.e., buyer receives 4 from the seller. For buyer , and his payment is . Update .
For the second layer, the constrained optimization problem in step 4 of algorithm 1 will be computed among . According to the allocation of the first layer, the constraint means that two items have been allocated to buyer . There is only one item left and the solution is to allocate the last item to buyer , i.e., . Then . For buyer , and his payment is . For other buyers in , their payment are 0. Update and the auction is completed.
The seller’s revenue is under LDM-Tree. If the seller only sells to his neighbors by VCG, it has and . The revenue of VCG is 3 and LDM-Tree achieves a higher revenue than VCG in this example.
In the proofs of LDM-Tree’s properties, we will formally analyze the necessity of the removed sets in detail.
The LDM-Tree is individually rational (IR).
If and , then or . From the definition, the only difference between and is that different sets of buyers are considered when doing optimization. Since . Thus , i.e., the LDM-Tree is individual rational. ∎
Now we prove LDM-Tree is IC. To simplify the following analysis, we extend some definitions to the first layer. For seller , let .
We first show the following observation that buyers with positive potential utility have no incentive to change the removed set by misreporting.
For any buyer and , given any , if and , then .
Case 1. If when , is still in after changing report type to . Then all sets remain the same and will not change.
Case 2. If when . Then after changing type to , can not be in . Given , . Then all sets remain the same and will not change.
Using the above observation, we prove one equilibrium of LDM-Tree in the following lemma.
Given any and fix the invitation of to be , we have for any .
Suppose and his parent is .
Case 1. when reports truthfully.
has no incentive to misreport to be in because in that case after misreporting. If misreports to and still in . From Observation 1, buyers in will not change which indicates that for and , allocation will not change. Thus and are computed under the same constrains and so do and . When computing , buyer is removed and we have shown the constrains are independent of . Thus is a constant given . If reports truthfully, no matter what the is, .
If misreports valuation and , then . If misreports and , then . If , then it contradicts to that is optimal. Thus .
Case 2. when reports truthfully.
Since when reporting truthfully, and . If misreports to and does not get any items, then . If misreports to and , then because there are at least buyers in whose value of the first item is larger than ’s. ∎
The following fact shows that buyers with positive potential utility have no incentive to change the removed set by stopping information diffusion.
For any buyer and , given any , if and , then .
Case 1: If when , may belong to two different sets after changing report type to .
(1) If , then no sets will change and will not change as well.
(2) If , then will decrease by and will increase by . The size increase of will absorb , which makes unchanged.
Case 2: If when , then do not have children, which means . In that case, will not change. ∎
Using the above observation, we prove the other equilibrium of LDM-Tree in the following lemma.
Given any and fix the valuation report of to be , we have for any .
Suppose and his parent is . If , then when diffuses information to all neighbors, . If , is still in after diffusing the information to less neighbors. By Observation 2, no matter how many neighbors diffuses information to, and are computed under the same constrains. When computing , buyers in are all removed. Thus is a constant given . where the first item of is monotonic increasing with while the second item is independent of . So diffusing information to all neighbors will maximize utility of . ∎
Combining Lemma 1 and Lemma 2, we prove LDM-Tree is IC.
The LDM-Tree is incentive compatible (IC).
Although the exact social welfare of LDM-Tree depends on the tree structure, the following proposition gives the tight lower bound of the social welfare of LDM-Tree.
The social welfare of LDM-Tree is no less than the social welfare of the VCG in the first layer.
Now we will show the revenue improvement of LDM-Tree. Let be the allocation and payment of under VCG mechanism among the first layer. Let be the payment of under LDM-Tree mechanism. Let , is the sum of the last values in top ranked value in set . In the following analysis, we use as the abbreviation for .
For , Let . Let . Let . Then
We will first show the sum of for all buyers in is larger than revenue of VCG mechanism among . Then, we will show the sum of for can be compensated by the sum of for for any . Combining these two results, we will show the revenue of LDM-Tree is better than the revenue of VCG mechanism among seller’s neighbors.
We divide buyers in into three groups and according to and .
Let all buyers with be in the group . For these buyers, we have
Let all buyers with be in the group . For these buyers, we have
Let all buyers with be in the group and let . For these buyers, we have , where it has because all values in the right side are not in and are considered when computing the left side. Then