The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20)
Strategy-Proof and Non-Wasteful Multi-Unit Auction via Social Network1
Takehiro Kawasaki,1 Nathanael¨ Barrot,2 Seiji Takanashi,3 Taiki Todo,1,2 Makoto Yokoo1,2 1Kyushu University, Japan, [email protected], {todo, yokoo}@inf.kyushu-u.ac.jp 2RIKEN AIP, Japan, [email protected] 3Kyoto University, Japan, [email protected]
Abstract buyers to forward the information to as many followers as possible, as well as truthfully reporting their valuation func- Auctions via social network, pioneered by Li et al. (2017), tions. For selling a single unit of an item, Li et al. (2017) have been attracting considerable attention in the litera- ture of mechanism design for auctions. However, no known developed an auction mechanism in which each buyer is in- mechanism has satisfied strategy-proofness, non-deficit, non- centivized to forward the information to her followers. wastefulness, and individual rationality for the multi-unit Zhao et al. (2018) studied a multi-unit unit-demand auc- unit-demand auction, except for some na¨ıve ones. In this pa- tion via social network, where each unit is identical and each per, we first propose a mechanism that satisfies all the above buyer requires a unit. They proposed the generalized infor- properties. We then make a comprehensive comparison with mation diffusion mechanism (GIDM) and argued that it is two na¨ıve mechanisms, showing that the proposed mecha- strategy-proof. However, Takanashi et al. (2019) pointed out nism dominates them in social surplus, seller’s revenue, and an error in their proof and argued that GIDM is not strategy- incentive of buyers for truth-telling. We also analyze the char- proof. They also proposed a strategy-proof mechanism for acteristics of the social surplus and the revenue achieved by the same model, which however violates a revenue condi- the proposed mechanism, including the constant approxima- bility of the worst-case efficiency loss and the complexity of tion called non-deficit, i.e., the seller might suffer a deficit. optimizing revenue from the seller’s perspective. To the best of our knowledge, for the multi-unit unit-demand auction via social network, no mechanism satisfying both strategy-proofness and non-deficit has been developed, ex- 1 Introduction cept for some na¨ıve ones. Auction theory has attracted much attention in artificial in- The main objective of this paper is to propose a mecha- telligence as a foundation of multi-agent resource alloca- nism that satisfies both strategy-proofness and non-deficit, tion. One of the mainstreams in the literature is analyzing as well as some other properties. As Takanashi et al. (2019) auctions from the perspective of mechanism design. In par- pointed out, no mechanism satisfies those properties and ticular, several works studied how to design strategy-proof Pareto efficiency, i.e., maximizing the social surplus, un- auctions, which incentivize each buyer to truthfully report der certain natural assumptions. They thus considered weak- her valuation function, regardless of the reports of the other ening the non-deficit condition. In this paper, on the other buyers. One critical contribution in the literature is the devel- hand, we consider a weaker efficiency property called non- opment of the Vickrey-Clarke-Groves mechanism (VCG), wastefulness, which only requires the allocation of as many which satisfies strategy-proofness and various other proper- units as possible. Non-wastefulness has its own importance ties (Vickrey 1961; Clarke 1971; Groves 1973). in practice. For example, in a spectrum auction, it is impor- Li et al. (2017) proposed a new model of auctions, in tant to allocate as much frequency range as possible to car- which buyers are distributed in a social network and the in- riers in order to guarantee a sufficient number of services. formation on the auction propagates over it. Utilizing a so- We propose a new mechanism, called dinstance-based cial network, the seller can advertise the auction to more po- network auction mechanism for multi-unit, unit-demand tential buyers beyond her followers, as many works studied buyers (DNA-MU), for a multi-unit unit-demand auction in network science (Emek et al. 2011; Borgatti et al. 2009; via social network, which satisfies strategy-proofness, non- Jackson 2008; Kempe, Kleinberg, and Tardos 2003). From deficit, non-wastefulness, and individual rationality, i.e., no the buyers’ perspective, however, forwarding the informa- buyer receives negative utility under truth-telling, and whose tion increases the number of buyers, which reduces the pos- description is much simpler than GIDM. It is inspired by the sibility that they will get the item. Therefore, the main chal- concept of the diffusion critical tree, originally proposed in lenge in the auction via social network is how to incentivize Li et al. (2017), which specifies, for each buyer i, the set of critical buyers for i’s participation. If a buyer j is critical for Copyright c 2020, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. 1A full version is available at http://arxiv.org/abs/1911.08809
2062 another buyer i’s participation, i.e., if i cannot participate in For notation simplicity, we introduce additional technical the auction without j’s forwarding of information, j must terms regarding the auction network. A buyer i is connected receive a higher priority in the competition. if a path s →···→i in G is formed based on the reported We then make a comprehensive comparison with two r. Let Nˆ denote the set of connected buyers. For each i, na¨ıve mechanisms that also satisfy (most of) the above prop- let d(i) denote the distance of the shortest path from s to erties. One is based on VCG, being applied only to the i.Ifi is not connected, we assume d(i)=∞.Givenθ , buyers who are directly connected to the seller. The other a buyer j ∈ Nˆ is a critical parent of i ∈ Nˆ if, without mechanism simply allocates the units in the first-come-first- j’s participation, i is not connected, i.e., j appears in any served manner with no payment. We show that the DNA- ˆ path from s to i in G. Let Pi(θ ) ⊆ N denote the set of all MU dominates both of these na¨ıve ones in terms of social critical parents of i under θ . The buyer j ∈ Pi(θ ) closest surplus and the seller’s revenue. Furthermore, in those mech- to i is called the least critical parent of i. An allocation x is anisms, hiding the information, combined with reporting the ˆ feasible if xi ≤ k, and xi =1implies i ∈ N for each true value, is also a dominant strategy, while this is not the i∈N i ∈ N. Let X be the set of all feasible allocations. case in our mechanism when k ≥ 2. This indicates that each Now we are ready to give a formal description of (di- buyer has a stronger incentive for truth-telling in the DNA- rect revelation) mechanisms2. A mechanism (f,t) consists MU. of two components, an allocation rule f and a profile of We further analyze the characteristics of the social sur- transfer rules (ti)i∈N . An allocation rule f maps a profile plus and the revenue of the DNA-MU. About social surplus, θ of reported types to a feasible allocation f(θ ) ∈ X.We it guarantees that each winner is in the set of top-k buyers sometimes use the notation of f(θi, θ−i) instead, especially except for her followers. It also has a constant worst-case i θ efficiency loss, based on a measure proposed by Nath and when we focus on the report of a specific buyer , where −i indicates the profile of types reported by the others. Given Sandholm (2018), when an optimal reserve price is intro- θ , fi(θ ) ∈{0, 1} denotes the assignment to buyer i. Each duced. About revenue, we show that a revenue monotonicity transfer rule ti maps a profile θ to a real number ti(θ ) ∈ R, condition fails and that maximizing revenue by optimally i sending the information is NP-complete. which indicates the amount that buyer pays to the seller. Here, we define several properties that mechanisms should satisfy. Feasibility requires that for any input, the al- 2 Preliminaries location returned by the mechanism is feasible. We first define the standard notations for multi-unit unit- Definition 1. A mechanism (f,t) is feasible if for any θ , demand auctions. Let s be a seller who is willing to sell the f(θ ) is feasible. K k N n set of identical units. Let be the set of buyers, Strategy-proofness is an incentive property, requiring that, where each buyer i ∈ N has a unit-demand valuation func- K x =(x ) ⊂{0, 1}n for any buyer, reporting its true valuation and forwarding the tion for . Let i i∈N be an allocation, information to all of its followers is a dominant strategy. which specifies who obtains a unit, where xi =1indicates (f,t) i that buyer i obtains a unit under allocation x, and xi =0 Definition 2. Given a mechanism and a buyer with θ =(v ,r ) θ∗ =(v∗,r∗) ∈ R(θ ) otherwise. Let vi ∈ R≥0 indicate the true unit-demand value true type i i i , a report i i i i is a θ θ ∈ R(θ ) of buyer i for a single unit. We assume that each buyer’s util- dominant strategy if for any −i and i i , ity is quasi-linear, i.e., the utility of buyer i under allocation ∗ ∗ vi · f(θi , θ−i) − ti(θi , θ−i) ≥ vi · f(θi, θ−i) − ti(θi, θ−i) x, when she pays pi ∈ R,isgivenasvi · xi − pi. Next, we define additional notations for the auction via holds. A mechanism (f,t) is strategy-proof if reporting θi is social network. For each buyer i ∈ N, let ri ⊆ N \{i} a dominant strategy for any i under (f,t). be the set of buyers to whom buyer i can forward the infor- Individual rationality is a property related to the incen- mation, called i’s followers. Also, let rs ⊆ N be the set of tives of the buyers for participation, which requires that direct buyers, i.e., those to whom the seller s can directly truth-telling guarantees a non-negative utility. send the information. Given (ri)i∈N∪{s}, we define the auc- Definition 3. A mechanism (f,t) is individually rational if tion network as a digraph G =(N ∪{s},E), where for each for any i, θi, and θ−i, vi ·f(θi, θ−i)−ti(θi, θ−i) ≥ 0 holds. i ∈ N ∪{s} and each j ∈ ri, a directed edge, from i to j, is added to the set E. Note that ri is also private informa- Non-deficit is a property about seller’s revenue, which re- tion of buyer i in our model, so the auction network is de- quires that the seller’s revenue cannot be negative. Note that fined according to reported r =(ri)i∈N , where ri indicates it does not consider each individual transfer and thus does the set of i’s followers to whom i forwards the information. not imply the non-negativity of each buyer’s payment. i To summarize, for each , the private information is given Definition 4. A mechanism (f,t) satisfies non-deficit if for θ =(v ,r ) i as i i i , called the true type of , consisting of the any θ , i∈N ti(θ ) ≥ 0 holds. true value vi and the set ri of the followers. Any reportable Non-wastefulness is a property about the efficiency of type θi =(vi,ri) of i with true type θi =(vi,ri) satisfies allocation, which requires that the mechanism allocate as ri ⊆ ri, i.e., a buyer can only forward the information to her followers. Let R(θi) be the set of all reportable types by i 2As discussed in the full version, the revelation principle holds with θi. Also, let θ denote the profile of types reported by under our assumption R on reportable subset ri ⊆ ri. Therefore, all buyers and Θ denote the set of all possible type profiles. focusing on direct revelation mechanisms is w.l.o.g.
2063 i i i i i 1 many units as possible. Note that the traditional definition 1 3 5 7 30 of non-wastefulness ignores the network structure, and thus 30 34 50 40 |N| i3 i5 i7 the second term in RHS is replaced with . s i6 s 34 50 40 i2 i4 66 (f,t) i2 i4 i6 Definition 5. A mechanism is non-wasteful if for any 72 45 ˆ 72 45 66 θ , i∈N fi(θ ) ≥ min{k, |N|} holds. 2.1 Two Na¨ıve Mechanisms for Comparison Figure 1: Example of Buyers Network and Corresponding One might expect that those properties hold in na¨ıve mech- Diffusion Critical Tree. anisms. Indeed, we can easily find the following two candi- dates. The formal definitions are in the full version. We com- 8: fi(θ ) ← 0,ti(θ ) ← 0 pare their performances with that of our new mechanism in 9: end if the following sections. 10: end for The first mechanism applies VCG to only the direct buy- The following example demonstrates how the DNA-MU ers rs. It satisfies strategy-proofness, individual rationality, works. The network of buyers, defined by r, as well as their non-deficit, and non-wastefulness for |rs|≥k. We refer to this mechanism as No-Diffusion-VCG (ND-VCG in short). true values, is shown in Fig. 1. Such a mechanism is also considered in Li et al. (2017), al- Example 1. Consider three units and seven buyers N = though they focused on single-item auctions. {i1,i2,...,i7}. Each vertex in the left figure of Fig. 1 cor- The second mechanism gives the units to buyers for free, responds to a buyer, and the number in each vertex denotes in the first-come-first-served manner, which is referred to as her true valuation. The priority order is given as i1 i2 FCFS-F. It satisfies individual rationality, non-deficit, and ··· i7. Assume that every buyer forwards the informa- non-wastefulness, and it is strategy-proof when earlier ar- tion to all of her followers, i.e., Nˆ = N. The corresponding rivals are not allowed, e.g., based on ascending order of d(·), diffusion critical tree is given on the right in Fig. 1. as usually assumed in online mechanism design (Hajiaghayi, The assignment to buyers is computed one-by-one, in the i p = Kleinberg, and Parkes 2004; Todo et al. 2012). priority order. For buyer 1, the price is given as i1 ∗(Nˆ \ W, k −|W |)=v∗({i ,i ,i ,i ,i ,i }, 3) = v −i1 2 3 4 5 6 7 v =50 p >v 3 Distance-based Network Auction i5 . Since i1 i1 , she does not win a unit. For buyer i p = v∗({i ,i ,i ,i ,i }, 3) = mechanism for Multi-unit, Unit-demand 2, the price is given as i2 1 3 5 6 7 v =40 v i buyers i7 . Note that i4 is ignored because 2 is a criti- cal parent of i4. Since vi >pi , she wins a unit; W is The definition of the new mechanism is given in Defini- 2 2 updated to {i2}, and k is decremented to 2.Fori3, the tion 6. A key concept in describing the mechanism is the p = v∗({i ,i }, 2) = v =30 price is given as i3 1 4 i1 . Since diffusion critical tree T (θ ), originally introduced in Zhao v >p W {i ,i } i3 i3 , she wins a unit; is updated to 2 3 , et al. (2018). Given θ , the diffusion critical tree T (θ ) is and k is decremented to 1.Fori4, the price is given as a rooted tree, where s is the root, the nodes of T (θ ) are p = v∗({i ,i ,i ,i }, 1) = v =66 p >v ˆ ˆ i4 1 5 6 7 i6 . Since i4 i4 , all the connected buyers N, and for each node i ∈ N and she does not win a unit. j ∈ P (θ) (j, i) ∗ its least critical parent , an edge is drawn. For i5, the price is given as pi = v ({i1,i4}, 1) = P (θ)=∅ (s, i) 5 If i , we draw an edge . Furthermore, given vi =45. Since vi >pi , she wins a unit; W is updated to T (θ) i T (θ) 4 5 5 and a node , all of the nodes in the subtree of {i2,i3,i5} and k is decremented to 0. Since no unit remains, i i rooted at are called ’s descendants. Also, given a report the prices for the remaining buyers, i6 and i7, become infin- ˆ ∗ θ , a subset S ⊆ N, and an integer k ≤ k, v (S, k ) de- ity, and thus neither of the buyers wins a unit. To sum up, i2, notes the k -th highest value in S under θ .Fork ≤ 0, let i3, and i5 are winners, who pay 40, 30, and 45 respectively. v∗(S, k)=∞ |S| 2064 which is not successfully manipulable; no buyer can make The only way for i to decrease her price is to turn such a the distance shorter by not forwarding the information to her winner into a loser by over-bidding. Assume j (where j i) followers, which is shown by Lemma 1. is such a winner. If j is i’s ancestor, i cannot affect j’s price. Thus, j and i are in different branches in T (θ ). Since j is ∗ ˆ ˆ ˆ 3.1 Properties of DNA-MU a winner, vj ≥ v (N−j \ Wj,k −|Wj|) holds. Also, We show feasibility, individual rationality, and non-deficit to increase j’s price, vi must be smaller than or equal to ∗ ˆ ˆ ˆ ˆ in Theorem 1, non-wastefulness in Theorem 2, and strategy- v (N−j \ Wj,k−|Wj|). Note that i is included in N−j \ ˆ ˆ ˆ proofness in Theorem 3. The proofs of Theorems 1 and 2 are Wj.Ifi is within the top k −|Wj|−1 buyers in N−j \ in the full version due to space limitations. Let Wˆ denote a ˆ Wj,evenifi over-bids, she cannot change j’s price. Thus, {w ,w ,...} Wˆ {w ∈ Wˆ | set of winners 1 2 and j denote vj ≥ vi holds. However, we assume vj ≤ πi 2065 dummy buyers i1 Example 2. See Fig. 2. Since there are three units, the mech- 40 30 anism first adds three dummy vertices. The price for i1 is p =50 i3 i5 i7 given as i1 , and she is not allocated a unit. The price s i p =45 40 34 50 40 for 2 is given as i2 , and she wins a unit. The price i p =40 i2 i4 i6 for 3 is given as i3 , which comes from the valua- 40 72 45 66 tion of the dummy buyer. Since her value is strictly less than p i i3 , she is not allocated a unit. The price for 4 is given as p =50 i i4 , and she is not allocated a unit. The price for 5 is Figure 2: Implementation of Reserve Price vh(= 40) by given as pi =40, and she wins a unit. At this moment one Adding k(= 3) Dummy Buyers to Diffusion Critical Tree. 5 unit remains. For buyer i6, the price is given as pi6 =45, which is strictly less than her value of 66. Thus, she wins a unit. Now that no unit remains, the price for buyer i7 is set This property is useful to show other characteristics of our to be infinity, To sum up, i2, i5, and i6 win a unit, and each mechanism, e.g., Proposition 2. One can also easily observe pays 40, 40, and 45, respectively. that the two na¨ıve mechanisms violate this property. Nearly identical proofs work for feasibility, non-deficit, 4.2 Social Surplus Domination individual rationality, and strategy-proofness. However, the introduction of a reserve price obviously breaks down non- A mechanism (f,p) is said to dominate another mechanism wastefulness. Actually, for any non-zero vh, there is a case (f ,p) in terms of social surplus if for any N and any θ ,it v 2066 v i1 i4 In particular, if the value of each buyer is less than h, 5 15 becomes 0. Thus, the worst case efficiency loss is k · vh.If i2 ≥ k, the DNA-MU allocates k units from Lemma 3. The s 20 maximum efficiency loss is bounded by k(¯v−vh), which can i3 occur, for instance, when there are 2k buyers, forming a path 6 graph, and those k buyers closer to s have the value of vh, v¯ k while the rest have the value of : the DNA-MU allocates Figure 3: Violation of Revenue Monotonicity. units to the closest k buyers. From the above, the maximum efficiency loss is given as max(k · vh,k(¯v − vh)). This is bounded from the bottom by k · v/¯ 2, which is achieved by ity have been studied, including bidder revenue monotonic- setting vh to v/¯ 2. Thus, the DNA-MU is 1/2-inefficient for ity (Rastegari, Condon, and Leyton-Brown 2011; Todo, vh =¯v/2. Iwasaki, and Yokoo 2010) and item revenue monotonicity, a.k.a. destruction-proofness (Muto and Shirata 2017). Observe that there is a tradeoff between achieving non- wastefulness and guaranteeing a better worst-case perfor- The condition studied in this section is weaker than bid- mance by the mechanism with a reserve price, where the der revenue monotonicity. A mechanism is follower revenue monotonic if the seller’s revenue is monotonically increasing former is achieved by vh =0and the latter by vh =¯v/2. with respect to the number of direct buyers. Let ti(θ | rs) Obtaining the lower bound of α that a strategy-proof mech- 0 be the payment from buyer i when θ is reported and s sends anism achieves remains an open question. However, since - r inefficiency implies Pareto efficiency, the impossibility sug- the information of the auction to a subset s of direct buyers. gested by Takanashi et al. (2019) implies that no strategy- Definition 8. A mechanism (f,t) is follower revenue mono- proof mechanism that also satisfies non-deficit and individ- tonic if for any rs, θ , and rs ⊆ rs, it holds that ual rationality achieves 0-inefficiency. i∈N ti(θ | rs) ≥ i∈N ti(θ | rs). For a multi-unit auction without any network among buy- 5 Revenue Analysis ers, there are several bidder revenue monotonic mechanisms, The seller’s revenue is also an important evaluation criterion and thus follower revenue monotonic ones. However, the ex- for auction mechanisms. In this section, we first show that ample below shows that, for auction via social network, our the seller’s revenue in the DNA-MU is no less than those of mechanism is not even follower revenue monotonic. the two na¨ıve ones. We also show that maximizing the rev- Example 3. Consider two units and four buyers i1, i2, i3, enue by optimally choosing the set of its followers to whom and i4; see Fig. 3. The priority order is i1 i2 i3 i4. it sends the information is NP-complete. Assume all of the buyers behave sincerely. When the seller s i i 5.1 Revenue Domination sends the information to all of its followers, i.e., 1, 2, and i3, all buyers become connected. The price for each buyer We define the domination in terms of the seller’s revenue is given as p1 =6, p2 =6, p3 =15, and p4 =6, where (f,p) analogously. A mechanism dominates another mech- i2 and i4 win a unit and the revenue is 12. When s sends (f ,p) N anism in terms of the seller’s revenue if for any the information only to i1 and i2, i3 becomes unconnected. θ t (θ) ≥ t (θ) and any , it holds that i∈N i i∈N i . The price for each buyer is given as p1 =0, p2 =15, and Proposition 3. The DNA-MU dominates both ND-VCG and p4 = ∞, where i1 and i2 win a unit and the revenue is 15.As FCFS-F in terms of the seller’s revenue, but not vice versa. a result, the revenue is not maximized when the seller sends the information to all the direct buyers. Proof. The DNA-MU obviously dominates ND-VCG when Therefore, a question rises to the seller: to which set of |rs|≤k, since the price for each winner in ND-VCG direct buyers should she send the information to maximize is zero. When |rs| >k, each winner in ND-VCG pays ∗ her revenue? We define a simplified form of this problem, so v (rs \{i},k). On the other hand, the price pi for each win- ∗ ˆ called OPTIMAL DIFFUSION, from the perspective of com- ner i in the DNA-MU satisfies pi ≥ v (N−i,k) by defi- ˆ putational complexity: Assuming that the seller knows the nition. For every winner i, N−i is a superset of rs \{i}. exact network among buyers and that all buyers behave sin- ∗ Therefore, from the monotonicity of v on the first argu- cerely, she tries to find an optimal set of direct buyers to ∗ ˆ ∗ ment, pi ≥ v (N−i,k) ≥ v (rs \{i},k) holds. Also, there whom she should send the information. exists a case where the inequality becomes strict. Thus, the Definition 9 (OPTIMAL DIFFUSION). Given a number k of DNA-MU dominates ND-VCG but not vice versa. Since the units, a profile θ , a set rs of direct buyers, and a threshold revenue of FCFS-F is always zero, while the revenue of the K, is there a subset rs ⊆ rs s.t. i∈N ti(θ | rs) ≥ K DNA-MU is non-negative for any input and can be strictly holds under the DNA-MU? positive, the DNA-MU also dominates FCFS-F but not vice versa. We show that OPTIMAL DIFFUSION is NP-complete by a reduction from PARTITION. Due to space limitations, we 5.2 Revenue Monotonicity present a proof sketch below; a full proof is in the full ver- The seller’s revenue is required to have some specific form sion. of monotonicity. Several forms of such revenue monotonic- Definition 10 (PARTITION). Given a set A where each i ∈ 2067 a1 1 v 1 6 Incentive Analysis a0 1 . Now we show that, compared with those two na¨ıve mech- . v(a1) . 1 buyers anisms, our mechanism also has its own strength on buy- av(a1) . v1 ers incentive; in those mechanisms, hiding the information, . az N A combined with the report of the true value, is also a domi- z v 1 a0 1 nant strategy, while this is not the case in our mechanism for . z any k ≥ 2. This indicates that the incentive for each buyer s . v(a ) buyers z to report her type truthfully in the DNA-MU is stronger than av(az ) v1 that in both of those na¨ıve ones. b1 b2 bm+1 bm+2 Proposition 4. Assume k ≥ 2. For each i, reporting (vi, ∅) B v2 v3 ... v3 v4 N is not a dominant strategy in the DNA-MU. c c1 c2 cm m+1 ... C v5 N Proof. Consider k units and k +2buyers i1,...,ik+2, such r = {i ,i ,i ,i ,...,i } θ =(15, {i }) θ = that s 1 3 5 6 k+2 , i1 2 , i2 (20, ∅), θi =(10, {i4}), θi =(9, ∅), and θi =(30, ∅) Figure 4: Reduction from Partition: Network of Buyers. 3 4 j for all 5 ≤ j ≤ k +2. The priority is given as i5 i6 ··· ik+2 i3 i1 i4 i2. The first k − 2 units are sold to {ij}5≤j≤k+2, regardless of i1’s forwarding strategy. A has a value v(i) ∈Z+, does there exist a subset A ⊆ A Under i1’s sincere forwarding to i2, i1 wins a unit and pays such that v(i)=m, where m = v(i)/2? i∈A i∈A 9.Ifi1 does not forward the information to i2, then i1 would Theorem 5. OPTIMAL DIFFUSION is NP-complete. win a unit and pay 10. So not forwarding the information is dominated by a sincere forwarding in this case. Proof Sketch.First, OPTIMAL DIFFUSION is in NP since we Proposition 5. For each i, reporting (vi, ∅) is a dominant can compute i∈N ti(θ | rs) in polynomial time. Given an instance of PARTITION, we construct an instance of OPTI- strategy in both ND-VCG and FCFS-F. A B C MAL DIFFUSION as follows, with N = N ∪ N ∪ N : Proof. In ND-VCG, only the reports from the direct buyers • i∈A (ai ) N A affect the outcome. Therefore, for each direct buyer i ∈ rs, For all , we create set j 0≤j≤v(i) in such that i any valuation report vi, and any action by other buyers, the θai =(, (aj)1≤j≤v(i)), and θai =(v1, ∅) for 1≤j ≤v(i). 0 j choice of ri ⊆ ri does not change the outcome at all. B i ∈ N i • Set N =(bj)1≤j≤m+2 is such that θb1 =(v2, {b2}), In FCFS-F, for each , any follower of who origi- θb =(v3, {bj+1}) for 2≤j ≤m+1, and θb =(v4, ∅). nally arrives after i under i’s sincere forwarding ri is still ar- j m+2 C riving after i under any manipulation ri ⊂ ri. Thus, whether • Set N =(cj)1≤j≤m+1 is such that θc =(, {cj+1}) j i wins a unit does not depend on the choice of ri. for 1≤j ≤m, and θcm+1 =(v5, ∅). i • The seller’s direct followers are (a0)i∈A ∪{b1,c1}. 7 Conclusions The network is illustrated in Fig. 4. Buyers are labelled with The DNA-MU satisfies strategy-proofness, non- wastefulness, non-deficit, and individual rationality. any ascending order of d(·) satisfying bm+1 cm+1. The The performance is comprehensively analyzed; it dominates prices satisfy 2068 References Zhao, D.; Li, B.; Xu, J.; Hao, D.; and Jennings, N. R. 2018. Borgatti, S. P.; Mehra, A.; Brass, D. J.; and Labianca, G. Selling multiple items via social networks. In Proceedings 2009. Network analysis in the social sciences. Science of the 17th International Conference on Autonomous Agents 323(5916):892–895. and MultiAgent Systems, AAMAS 2018, Stockholm, Sweden, July 10-15, 2018, 68–76. Clarke, E. H. 1971. Multipart pricing of public goods. Pub- lic Choice 2:19–33. 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