PREFERENCE NETWORKS IN MATCHING MARKETS

This is a short note to sketch out the ideas for a talk given in CSE 5339: Network Data Analysis. Market interactions between buyers and sellers form an interesting class of problems in network data analysis. To illustrate some of the major results in this field, we will consider three different models in which resources/objects are allocated to users. The three models correspond to three types of preferences: (1) binary (want/do not want), (2) weighted and visible, and (3) weighted and hidden. Here “weighted and visible” means that the users openly state their true valuations for the different objects, and “weighted and hidden” means that the users have a valuation that they internally decide, but what they state openly might be different. The main ideas in this note are presented in [1]. 1. Networks with binary preferences Definition. A graph G pV,Eq is bipartite if there exist X,Y  V such that X \ Y V and all e P E have the form e px, yq, where x P X and y P Y . We will occasionally denote such graphs by G pX,Y,Eq. We adopt the convention that two edges in a graph are disjoint if they do not have a common endpoint. Definition. A matching in a bipartite graph G pX,Y,Eq is a set M  E consisting of disjoint edges. In the case where |X| |Y |, a perfect matching is a matching M such that πXpMq X and πY pMq Y . Here πX : X Y Ñ X and πY : X Y Ñ Y are the natural projections. These minimal definitions allow us to ask the following question: Question 1. For what conditions does a bipartite graph admit a perfect matching? The preceding question admits a result known as Hall’s Matching Theorem. This result is not just interesting from a combinatorial perspective; we now describe an application of this theorem to wireless sensor networks. Such a network consists of sensor nodes that are able to communicate with other nodes within a certain distance, and a sink node that acts as a hub. A sensor node can transmit information directly to the sink if it is close enough, but otherwise, it needs to transmit to another sensor node that is ideally closer to the sink. The objective of a routing protocol is to relay information from a sensor node to the sink node in as few hops as possible. The constraint of having a minimal number of hops is used to minimize the latency in the network. In graph theoretical terms, this problem now becomes equivalent to finding a spanning tree rooted at the sink node. A further constraint would be to minimize the maximum degree of any node in this tree—this consideration is to minimize the energy usage of any node. Thus the problem can now be formulated as follows: Question 2. Given a graph and a sink node, find a spanning tree rooted at the sink node such that: Date: April 5, 2016. 1 2 PREFERENCE NETWORKS IN MATCHING MARKETS (1) The maximum degree of any node in the tree is minimized, and (2) Each edge connects a node to another node that is closer to the sink than itself. Bokal et al solved this problem via a generalization of Hall’s theorem [2]. The idea is that because one does not want to traverse between nodes at the same distance from the sink, these nodes can be viewed as belonging to one partition of a bipartite graph, whereas the other partition consists of nodes that are closer to the sink node. Finding a perfect matching thus allows one to satisfy both conditions in Question 2. Before formally stating Hall’s theorem, we need one more definition: Definition. Let G pX,Y,Eq be a bipartite graph. A set S  X (resp. S  Y ) is constricted if |πY pπ 1 X pSqq| |S|. Alternatively, S is constricted if its neighbor set NpSq satisfies |NpSq| |S|. Here NpSq is defined as: NpSq ty P Y : x P S, px, yq P Eu . Theorem 1 (Hall’s Matching Theorem, 1935). Let G pX,Y,Eq be a bipartite graph such that |X| |Y |. Then there exists a perfect matching M  E if and only if G does not contain any constricted sets. Hall’s theorem is equivalent to another theorem attributed to König and Egerváry. Berge and Tutte also have related theorems. We do not discuss those results here, but point the reader towards notes by Wildstrom on the subject [3]. Finally, we remark that a preference network is any bipartite graph where one partition consists of users, the other to products/services, and the edges correspond to preferences. 2. Networks with visible weighted preferences Suppose we now have a bipartite graph consisting of buyers and sellers, where each seller has an item for which they wish to obtain a particular price. We view this as a weighted graph G pB,S,E, ωq, where |B| |S|, and the weights ω are defined as follows: ωpbi, sjq vij , the valuation of seller j’s item by buyer i, ωpbi, biq 0 for all i, and ωpsi, siq pi, the price of seller i’s item. Occasionally, a buyer may be willing to pay more for a seller’s item than the price set by the seller. The payoff of buyer i for seller j’s item is given by vij pi. The preferred seller of buyer bi is the seller j for whom bi’s payoff is maximized. Note that a buyer may have multiple preferred sellers. We also stipulate that if a buyer is unable to obtain a nonnegative payoff for any seller, then the buyer will simply be better off not transacting. Finally, given a set of valuations and prices, the preferred seller graph is simply the graph of buyers and sellers consisting of edges connecting a buyer to their preferred seller(s). In an ideal situation, each buyer would have a unique preferred seller, and no two buyers would prefer the same seller. Then the buyers would walk away with the items that maximize their payoffs, without having any conflicts with each other. It may also be the case that some buyers have multiple overlapping preferred sellers—in this case, the tie could be broken with some coordination between the buyers, and they would still be able to maximize their payoffs. A set of prices for which either of these two situations holds is called a set of market clearing prices. With the definitions we have already introduced, we can equivalently say that a set of prices is market clearing if the corresponding preferred seller graph contains a perfect matching. We illustrate this situation in Figure 1. One may ask what the conditions on the valuations and prices needs to be for a set of market clearing prices to exist. A result of Egerváry, also known as the Hungarian method, gives an answer to this problem. PREFERENCE NETWORKS IN MATCHING MARKETS 3 Theorem 2 (Egerváry, 1918). Given a graph pB,S,E, ωq and a set of buyer valuations, there always exists a set of market clearing prices. This theorem allows for the following simple way of assigning items to buyers: given a set of buyer valuations, check for a constricted set in the preferred seller graph. If there is none, then a perfect matching can be found, and we are done. If there is a constricted set S  B, then raise the prices of the items in NpSq by one unit, and repeat the preceding steps. Egerváry proved that this process always terminates, i.e. we can obtain a preferred seller graph containing a perfect matching by successively increasing prices. Remark 3. Market-clearing prices are beneficial to buyers by construction, because they allow the buyers to maximize their individual payoffs. But they are also beneficial to the sellers, in the sense that they maximize the amount of money that is changing hands. To see why this is true, let M be a perfect matching in a preferred seller graph corresponding to a set of market clearing prices. Then we have: Total payoff to buyers ̧