Pricing Ad Slots with Consecutive Multi-unit Demand

We consider the optimal pricing problem for a model of the rich media advertisement market, as well as other related applications. In this market, there are multiple buyers (advertisers), and items (slots) that are arranged in a line such as a banner on a website. Each buyer desires a particular number of {\em consecutive} slots and has a per-unit-quality value $v_i$ (dependent on the ad only) while each slot $j$ has a quality $q_j$ (dependent on the position only such as click-through rate in position auctions). Hence, the valuation of the buyer $i$ for item $j$ is $v_iq_j$. We want to decide the allocations and the prices in order to maximize the total revenue of the market maker. A key difference from the traditional position auction is the advertiser's requirement of a fixed number of consecutive slots. Consecutive slots may be needed for a large size rich media ad. We study three major pricing mechanisms, the Bayesian pricing model, the maximum revenue market equilibrium model and an envy-free solution model. Under the Bayesian model, we design a polynomial time computable truthful mechanism which is optimum in revenue. For the market equilibrium paradigm, we find a polynomial time algorithm to obtain the maximum revenue market equilibrium solution. In envy-free settings, an optimal solution is presented when the buyers have the same demand for the number of consecutive slots. We conduct a simulation that compares the revenues from the above schemes and gives convincing results.Comment: 27page

On Revenue Maximization with Sharp Multi-Unit Demands

We consider markets consisting of a set of indivisible items, and buyers that have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a specific number $d_i$ of items; a bundle of size less than $d_i$ has no value, while a bundle of size greater than $d_i$ is worth no more than the most valued $d_i$ items (valuations being additive). We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of "correlated values" where each buyer $i$ has a valuation v_i\qual_j for item $j$, where $v_i$ and \qual_j are positive quantities associated with buyer $i$ and item $j$ respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.Comment: page2

Revenue maximizing envy-free multi-unit auctions with budgets

We study envy-free (EF) mechanisms for multi-unit auctions with budgeted agents that approximately maximize revenue. In an EF auction, prices are set so that every bidder receives a bundle that maximizes her utility amongst all bundles; We show that the problem of revenue-maximizing EF auctions is NP-hard, even for the case of identical items and additive valuations (up to the budget). The main result of our paper is a novel EF auction that runs in polynomial time and provides a approximation of 1/2 with respect to the revenue-maximizing EF auction. A slight variant of our mechanism will produce an allocation and pricing that is more restrictive (so called item pricing) and gives a 1/2 approximation to the optimal revenue within this more restrictive class. © 2012 ACM

On the theory of truthful and fair pricing for banner advertisements

We consider revenue maximization problem in banner advertisements under two fundamental concepts: Envy-freeness and truthfulness. Envy-freeness captures fairness requirement among buyers while truthfulness gives buyers the incentive to announce truthful private bids. A extension of envy-freeness named competitive equilibrium, which requires both envy-freeness and market clearance conditions, is also investigated. For truthfulness also called incentive compatible, we adapt Bayesian settings, where each buyer's private value is drawn independently from publicly known distributions. Therefore, the truthfulness we adopt is Bayesian incentive compatible mechanisms. Most of our results are positive. We study various settings of revenue maximizing problem e.g. competitive equilibrium and envy-free solution in relaxed demand, sharp demand and consecutive demand case; Bayesian incentive compatible mechanism in relaxed demand, sharp demand, budget constraints and consecutive demand cases. Our approach allows us to argue that these simple mechanisms give optimal or approximate-optimal revenue guarantee in a very robust manner

非羨望性を考慮した被覆制約付き複数財市場の価格決定

　近年，PC やスマートフォンの普及により，インターネットを通じた広告配信市場が成長している.この内インターネット広告市場の 40%を占めるリスティング広告には「Real Time Bidding」と呼ばれるシステムが採用され，利用者が web サイトに訪れる度に，どの広告を掲載するかをオークションによって決定する.　広告配信システムに向けて，非羨望性(Envy-Freeness)を考慮したオークションのモデルが数多く研究されている.非羨望性とは，全ての広告主が，自己の利得を最大化する結果を受け取ることを意味する.　本研究では，売り手が「相場は近いが，相異なる財」を複数持つと仮定し，これを満たすモデルとして，「被覆制約付き複数財市場」を提案した. これは，広告配信システムにおいて，広告枠の相場は一定だが，掲載する時間帯や広告閲覧者の属性等が，多様であることに対応する.　提案モデルにおける性質として，(1) 非羨望性を考慮し，財に均一価格を決定するとき，価格に対して割当て可能な財の数が単調非増加であること，(2) ある価格における最適な財の割当は，特殊な重み付けを行ったグラフ上の最小費用流問題を解くことで，計算可能であることを示した.これらにより，均一価格における，非羨望性を考慮した売り手の利得最大化問題に対する多項式 時間アルゴリズムを提案した.　また，既存の DGS オークションを提案モデルに拡張し，配分効率性に関する議論を可能にした.買い手の財の評価額の分布を変化させたデータや，財と買い手の比率を変化させたデータを，乱数を用いて生成し，計算機実験によって，アルゴリズムの性能評価と，提案モデルにおける，非羨望性の考慮と，財の価格を均一に決定することの妥当性の検証を行った.計算機実験の結果より，モデルの仮定を満たすデータにおいては，これらの妥当性を示すことできた.また，買い手の評価額が区間付き一様乱数の場合において，買い手は必ずしも評価額が高いほうが有利ではない傾向が，実験結果から観察できた.電気通信大学201