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.