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Visualizing Combinatorial Auctions Vis Comput (2011) 27: 633–643 DOI 10.1007/s00371-011-0576-9 ORIGINAL ARTICLE Visualizing combinatorial auctions Joe Ping-Lin Hsiao · Christopher G. Healey Published online: 19 April 2011 © Springer-Verlag 2011 Abstract We propose a novel scheme to visualize combina- This ability to bid on bundles makes combinatorial opti- torial auctions; auctions that involve the simultaneous sale of mization and mathematical programming relevant for solv- multiple items. Buyers bid on complementary sets of items, ing the distribution of items that maximizes all buyers’ utili- or bundles, where the utility of securing all the items in the ties. Thus, combinatorial auctions are studied in economics, bundle is more than the sum of the utility of the individ- operations research, and computer science [10, 12, 17]. ual items. Our visualizations use concentric rings divided Combinatorial auctions allow buyers to fully express into arcs to visualize the bundles in an auction. The arcs’ their preferences, particularly when items are complemen- positions and overlaps allow viewers to identify and follow tary. A complement is a set of items S ={i1,...,in } that has bidding strategies. Properties of color, texture, and motion a greater utility than the sum of the utility for each individ- n are used to represent different attributes of the auction, in- ual item, u(S) > j=1 u(ij ). For example, a pair of shoes cluding active bundles, prices bid for each bundle, winning is more valuable than a left shoe or a right shoe alone. The ∈ bids, and bidders’ interests. Keyframe animations are used value of an individual item ij S may differ between bid- to show changes in an auction over time. We demonstrate ders, particularly if a bidder already possesses other items our visualization technique on a standard testbed dataset in S. generated by researchers to evaluate combinatorial auction Combinatorial auctions have been employed in a variety bid strategies, and on recent Federal Communications Com- of industries: transportation routing, airport arrival and de- mission (FCC) auctions designed to allocate wireless spec- parture scheduling, and allocating radio spectrums for com- trum licenses to cell phone service providers. munications services [3, 11]. In each case, the motivation for using a combinatorial auction is the complementary nature of the desired resources. For example, a trucker’s cost for · · Keywords Combinatorial auction Ecommerce handling shipping to a new destination depends on his orig- · Perception Visualization inal routes. If the new destination is near his existing path, the cost for shipping goods there will be relatively lower than shipping to a location not near the path. 1 Introduction To date, combinatorial auction results are normally pre- sented as data tables showing active bundles, bid prices, bid- Combinatorial auctions are auctions where multiple items ders’ demand sets, competitive allocations of items to win- are sold simultaneously. Buyers place bids on bundles of ners, and so on. Table 1 shows two rounds of an auction, and the final bundle prices, involving four bidders and three items rather than bidding separately on individual items. items. The table is effective for showing quantitative results of the auction. However, it can be difficult and time consum- ing to gain an overall understanding of the relationships be- · J.P.-L. Hsiao C.G. Healey ( ) tween different bidders, their interests, and their bid strate- Department of Computer Science, North Carolina State University, Raleigh, NC, USA gies, particularly as these properties change during the life- e-mail: [email protected] time of the auction. Our goal for this project is to provide a 634 J.P.-L. Hsiao, C.G. Healey visualization that complements the data table. Specifically, we want to: 1. Produce a concise spatial layout to increase the amount of information contained in our visualizations. 2. Visualize bundle prices, competitive allocations, and bid- ders’ interests using colors and textures selected using guidelines from visual perception. 3. Visualize changes in an auction over time using motion, again selected based on guidelines from human percep- tion. Fig. 1 The same two steps from Table 1, displayed using our visual- Researchers generally analyze results in two ways: (1) as ization strategy a post-processing operation after an auction completes; or (2) as an auction runs. Our technique is designed to support both needs, allowing viewers to track an auction’s state in generate profit from the bundle, he will pass for the next real-time as it unfolds, or to visualize auction logs to see how round. an auction progressed. Our expectation is that the visualiza- To simplify the auction, certain rules are applied to our tions will be used to quickly explore an auction to identify testbed. An agent may have multiple bundles in his demand interesting bundle prices, bid strategies, allocations of items, set, but he can increase his bid on only one bundle per round. changes in a bidder’s interests, and so on. These events can The bid increment adds a fixed amount to the preferred then be examined in more detail by studying related values bundle’s current price. Given these constraints, during each in the data table. round an agent can: (1) continue biding on the most recently bid-on bundle; (2) start bidding on a different bundle in the demand set; (3) add a new bundle to the demand set and bid 2 Iterative proxy combinatorial auctions on it; or (4) pass for the round. After receiving a round’s bids, the auctioneer assigns Our first testbed dataset involves a type of combinatorial bundles to winning agents to produce a competitive allo- auction known as an iterative proxy combinatorial (IPC) cation that: (1) maximizes the sum of the bid prices— auction [16]. The auction iterates in rounds, with proxy the revenue—that the auctioneer receives; and (2) assigns agents acting in place of the actual buyers. Each buyer sub- each item to at most one agent. If multiple allocations with mits to his agent the amounts that he is willing to pay for the same revenue exist, the auctioneer chooses one at ran- each bundle before the auction begins. Agents bid on be- dom. half of their buyers based on these amounts. An agent’s Set notation is used to represent competitive allocations. profit for each bundle is the difference between the price the Positions in the set correspond to an ordered list of agents. buyer pays and the price the agent pays to win the bundle. Bundles appearing at position i imply that agent i has Since agents will never bid above the buyer’s price, profit won the bundle. For example, assume four agents are bid- for any bundle won is always greater than or equal to zero. ding on three items D ={A,B,C}. In the first round, each All agents act to try to increase their profit. agent bids on the bundle containing all three items (Ta- At the start of each round, agents submit offers for bun- ble 1 and Fig. 1a). The possible competitive allocations are dles where they think they can maximize their profit. The therefore {ABC, –, –, –}, {–,ABC–, –}, {–, –,ABC,–},or auctioneer receives the bids and solves the winner determi- {–, –, –,ABC}. The auctioneer will choose the allocation nation problem (WDP) to assign items to agents. The auc- corresponding to the agent who offers the highest bid price. tioneer announces the winners and winning prices to end the The auction itself is represented as a subset of rounds round. This process continues until the auction ends. The where one of the following inflection points occurs: (1) a new iterative nature of the auction allows agents to offer pro- bundle enters an agent’s demand set; (2) competitive alloca- visional prices, then adjust their allocations based on other tions change; or (3) an agent withdraws as an active bid- agents’ bids. der. Each agent maintains a demand set of bundles that will Figure 1 shows the two timesteps in Table 1, displayed maximize the agent’s profit, based on the bundles’ current using our visualization technique. Bundles of decreasing prices. As the auction progresses and bundle prices change, size are represented as concentric rings, with all the bun- an agent’s demand set will also change. At the beginning of dles of a given size occupying a common ring. Sharp- a round, an agent may increase the bid on a preferred bundle ness represents bidders’ interests, color represents the bid or pass. If an agent wins a bundle, or if the agent cannot price for each bundle, saturation represents winning bids, Visualizing combinatorial auctions 635 Table 1 Two steps and the final prices for four agents {1, 2, 3, 4} and three items {A,B,C} showing current bundle prices, agents’ active bundle sets, demand sets Di , and probabilities of bidding on a bundle Step 2, t = 1 Prices: 0 0 0 0 0 0 0 Di AB ABC ACBCABCpass 3 1 {ABC,–,–,–}1 ABC 4 2 3 1 {–, ABC,–,–}2 ABC 4 2 3 1 {–, –, ABC,–}3 ABC 4 2 3 1 {–,–,–,ABC}4 ABC 4 2 ··· = 5 Step 10, t 36 6 Prices: 8 7.25 16 8.25 16 16.25 24.25 Di AB ABC ACBCABCpass {A, BC,–,–}1 A, AB, AC 00 0 1 1 1 1 {A,–,BC,–}2 B, BC 4 4 2 1 1 1 {–, –, ABC,–}3 C, BC, ABC 4 0 4 2 {–, –, C, AB}4 ∅ {AB,–,C,–}4 = 5 Step 11, t 39 6 Prices: 8 8 16 9 16 17 25 ABABCACBCABCpass and textured links connect items in a common bundle 2. The growth of the cardinality produces complex superset– set. subset relationships between consecutive Di .
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