10.1 Bipartite Graphs and Perfect Matchings


 Phillip Turner
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1 10.1 Bipartite Graphs and Perfect Matchings In bipartite graph the nodes are divided into two categories and each edge connects a node in one category to a node in the other category. For example: Administrators of a college dormitory are assigning rooms to students, each room for single student. Students and rooms are categories. When there are an equal number of nodes on each side of a bipartite graph, a perfect matching is an assignment of nodes on the left to nodes on the right, in such a way that I) each node is connected by an edge to the node it is assigned to, and II) no two nodes on the left are assigned to the same node on the right A set of nodes are called constricted set when their edges constrict the formation of a perfect matching. In figure 10.2b Vikram, Wendy and Xin form a constricted set. Matching Theorem: If a bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching, the it must contain a constricted set. Proof is presented in more detail in chapter 10.6.
2 10.2 Valuations and Optimal Assignments To extend this mode we will add weights to edges. In the last example we would tell how much a student likes each choice. We refer to these numbers as students' valuations. This way we can evaluate the quality of an assignment. For example in figure 10.3b we can calculate the quality of the assignment. The maximum possible quality is the optimal assignment Prices and the MarketClearing Property Now lets enter the world of real estate where we have houses, sellers and buyers, where two different buyers may have very different valuations for the same house. Buyers payoff is her valuation for the house, minus the amount of money she had to pay. She will of course try to maximize her payoff by choosing between several sellers. We call the seller or sellers that maximize the payoff for buyer j, the preferred sellers for buyer j IF the payoff is positive. In figure 10.5b (preferredseller graph) all buyers value the house of seller a the highest but for some reason each buyer ends up with different house, why? The answer is price (and payoff). We call such a set of prices market clearing. A set of prices is marketclearing if the resulting preferredseller graph has a perfect matching. x's payoff for a: 125=7 y's payoff for a: 85=3 (but 6 for c) z's payoff for a: 75=2 (but 3 for b) Optimality of Market Clearing Prices: A set of marketclearing prices, and a perfect matching in the resulting preferredseller graph, provides the maximum possible sum of payoffs to all sellers and buyers. Check details from chapters Properties of marketclearing prices, pages
3 10.4 Constructing a Set of MarketClearing Prices Lets look at auctions next. 1. At the start of each round, there is a current set of prices, with the smallest one equal to 0 (initially all sellers set their prices to 0). 2. We construct the preferredseller graph and check whether there is a perfect matching. 3. If there is, we're done: the current prices are marketclearing. 4. If not, we find a constricted set of buyers S and their neighbors N(S). 5. Each seller in N(S) raises his price by one unit. 6. If necessary, we reduce the prices the same amount is substracted from each price so that the smallest price becomes zero. 7. We now begin the next round. We say that auction must come to an end. Only way auction can come to end is if it reaches a set of marketclearing prices. To ease our way there we define potential energy of the auction to be the sum of potential of all participants: potential of a buyer is the maximum payoff potential of a seller is the current prices his charging Auction starts with all sellers with potential of 0 and buyers equal to their maximum payoffs. When sellers in N(S) all raise their prices by one unit in phase 5, their potentials go up by one unit and the potential of each buyer in S goes down by one unit. Since S has strictly more nodes than N(S) does, this means that the potential energy of the auction goes down by at least one unit more than it goes up. This means that we start auction with potential P 0 and cannot drop below 0 so the auction must come to and end within P 0 steps. And when it does, we have marketclearing prices How Does this Relate to SingleItem Auctions? We need equal number of buyers and sellers so we fake additional sellers and give buyers a valuation of 0 for the item offered by each of these fake sellers. Which buyer ends up paired with the real seller in perfect matching is the winner. And from a set of marketclearing prices, we will see what the real item sells for Advanced material: A Proof of the Matching Theorem
4 You might want to read pages by yourself, since I will give a very short view of the material. Matching Theorem: If a bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching, the it must contain a constricted set. Take a bipartite graph with perfect matching, and consider a matching that includes as many nodes as possible we will call this a maximum matching. We will then try to enlarge it. If the search for a larger matching fails, it produces a constructed set. Lets take a look at a simple bipartite graph with matching and nonmatching edges (Fig 10.10). We start the search for a larger matching from node W and we continue the path alternating between matched and nonmatched edges as long as we can while never repeating any nodes. We call this a simple alternating path. If we can find an alternating path that begins and ends at an unmatched node, then we can swap the roles of all edges on this path. We call this an augmenting path, since it gives us a way to augment the matching. We're not interested in alternating paths that don't reach D like WBYCZ or paths that are not alternating like WBZ CYD. There's a natural procedure we can use to search for an augmenting path. It works by simply adapting the breadthfirst search (BFS) procedure to include the requirement of alternation. We call this procedure as alternating BFS. If alternating BFS fails to find and augmenting path, we can in fact extract from this failed search a constricted set that proves there is no perfect matching.
5 The set of nodes on all even layers, at the end of a failed alternating BFS, forms a constricted set. Claim: Consider any bipartite graph with a matching, and let W be any unmatched node on the righthand ride. Then either there is an augmenting path beginning at W, or there is a constricted set containing W. This is indeed enough to prove there is no perfect matching. However, it does not mean that the current matching has maximum size. If there is no augmenting path beginning at any node on the right hand side, then in fact the current matching has maximum size. We can search for maximum size by making all the unmatched nodes on the right constitute layer 0 in the alternating BFS, and otherwise running it as before. The if an unmatched node on the left is ever reached in some layer, we can follow the path from the appropriate node in layer 0 down to it, producing an augmenting path.
6 Trade with Intermediaries (traders). In a wide range of markets, individual buyers and sellers do not interact directly with each other, but instead trade through intermediaries(traders) brokers, market makers, or middlemen who set the prices. To get a sense for how markets with intermediaries typically work, this chapter will uses examples of how buyers and sellers interact on the stock market. New terms: Bid  the highest outstanding offer to buy the stock/goods. (Trader offers bid to seller) Ask  lowest outstanding offer to sell the stock/goods. (Trader offers ask to buyer) Interesting/Further read: Dark pools  p 279 (short) order book/limit orders/market order ( ) A Model of Trade on Networks Our network model will be based on three fundamental principles that exist on the stock market: individual buyers and sellers often trade through traders not all buyers and sellers have access to the same traders and not all buyers and sellers trade at the same price The prices that each buyer and seller commands are determined in part by the range of alternatives that their respective network positions provide. Network structure used in this chapter For the simplest form of the model, we don t try to address the issue of multiple goods for sale, or multiple possible quantities; instead, we assume there is a single type of good that comes in indivisible units. Each seller i initially holds one unit of the good, which he values at vi ; he is willing to sell it at any price that is at least vi. Each buyer j values one copy of the good at vj and will try to obtain a copy of the good if she can do it by paying no more than vj. No individual wants more than one copy of the good, so additional copies are valued at 0. All buyers, sellers, and traders are assumed to know these valuations. Since we assume that the traders act as intermediaries for the possible seller buyer transactions, we require that each edge connects a buyer or seller to a trader.
7 Assumptions: We assume that buyers have the same valuation for all copies of a good. In contrast to chapter 10 the network here is fixed and externally imposed by constraints such as geography (in agricultural markets) or eligibility to participate (in different financial markets) 1 Sample trading network model In all of our figures depicting trading networks we will use the following conventions. Sellers are represented by circles on the left, buyers are represented by circles on the right, and traders are represented by squares in the middle. The value that each seller and buyer places on a copy of the good is written next to the respective node that represents them.
8 Prices and the Flow of Goods. The flow of goods from sellers to buyers is determined by a game in which traders first set prices, and then sellers and buyers react to these prices. Figure 1 Sample trade network Once traders announce prices, each seller and buyer chooses at most one trader to deal with each seller sells his copy of the good to the trader he selects (or keeps his copy of the good if he chooses not to sell it) each buyer purchases a copy of the good from the trader she selects (or receives no copy of the good if she does not select a trader) trader can only sell as many goods to buyers as he receives from sellers Because each seller has only one copy of the good, and each buyer only wants one copy, at most one copy of the good moves along any edge in the network.
9 Payoffs A trader s payoff is the profit he makes from all his transactions: it is the sum of the ask prices of his accepted offers to buyers minus the sum of the bid prices of his accepted offers to sellers For a seller i, the payoff from selecting trader t is b ti, while the payoff from selecting no trader is vi. In the former case, the seller receives b ti units of money, while in the latter he keeps his copy of the good, which he values at v i. For each buyer j, the payoff from selecting trader t is vj atj, while the payoff from selecting no trader is 0. In the former case, the buyer receives the good but gives up atj units of money. Figure 2 Flow of goods For example, with prices and the flow of goods as in Figure 2(b), the payoff to the first trader is ( ) = 0.6 while the payoff to the second trader is ( ) = 1.4. The payoffs to the three sellers are 0.2, 0.3, and 0, respectively, while the payoffs to the three buyers are = 0.2, = 0.3, and 1 1 = 0, respectively. Best Responses and Equilibrium. Because the game in this chapter is in 2 stages (traders set prices; buyers/sellers accept/reject them) this equilibrium is called a subgame perfect Nash equilibrium; in this chapter, we will simply refer to it as an equilibrium. detailed explanation on page 286 Monopoly and perfect competition Monopoly. Buyers and sellers are subject to monopoly in our model when they have access to only a single trader. Perfect Competition. Perfect competition  game condition when both seller and buyer have access to 2 same traders.
10 Whichever trader is performing the trade at equilibrium must have a payoff of 0: he must be offering the same value x as his bid and ask. Because if he offers greater ask or bid (makes profit) other trader can undercut him, by providing better deal to the buyer/seller. Using the previous example it is not hard to work out the equilibrium in it: Sellers S1 and S3, buyers B1and B3 are monopolized and S2&B2 getting benefit from perfect competition. Implicit Perfect Competition. it turns out that traders can make zero profit for reasons based on the global structure of the network, rather than on direct competition with any one trader. In any equilibrium, all bid and ask prices take on some common value x between 0 and 1, and the goods flow from the sellers to the buyers. So all traders again make zero profit.
11 Ripple Effects from Changes to a Network. Here we will explore how small changes to a network can produce effects that ripple to more distant parts of the network (a) (b) Picture 2 all sellers and buyers are monopolized except for B2 we use indifference to assume that B3 will not buy the good but B1 and B4 will Note that there cannot be an equilibrium in which buyer B2 buys from trader T2, since B2 can pay only 2 while trader T2 can sell the unit of the good he is able to buy at a price of 4. Once S2 and T2 form a link, creating the network in example (b), a number of things change. First, and most noticeably, buyer B3 now gets a copy of the good while B1 doesn t. There are other changes as well. Seller S2 is now in a much more powerful position and will command a significantly higher price (since y is at least 1 in any equilibrium). Moreover, the range of possible equilibrium asks to B2 has been reduced from the interval [0, 2] to the interval [1, 2]. So in particular, if we were previously in an equilibrium where the ask to B2 was a value x< 1, then this equilibrium gets disrupted and replaced by one in which the ask is a higher number, y 1. This indicates a subtle way in which B2 was implicitly benefitting from the weak position of the sellers, which has now been strengthened by the creation of the edge between S2 and T2. Social Welfare in Trading Networks Socially optimal solution  solution that maximizes social welfare (sum of payoffs of all players). The social welfare is simply the sum of v j v i over all goods that move from a seller i to a buyer j The maximum value of this quantity over all possible flows of goods the socially optimal value depends not just on the valuations of the sellers and buyers but also on the network structure. Networks that are more richly
12 connected can potentially allow a flow of goods achieving a higher social welfare than networks that are more sparsely connected, with bottlenecks that prevent a desirable flow of goods. If we look at previous case (Picture 2) in each case, the equilibrium yields a flow of goods that achieves the social optimum. In (a), the best possible value of the social welfare is = 7, since there is no way to use the network to get copies of the goods to both B3 and B4. However, when the single edge from S2 to T2 is added, it suddenly becomes possible for both of these buyers to receive copies of the good, and so the value of the social welfare increases to = 9. This provides a simple illustration of how a more richly connected network structure can enable greater social welfare from trade. More to read: Equilibria and Social Welfare (p294) tl;dr; Every equilibrium produces a flow of goods that achieves the social optimum. Trader Profits The examples we ve studied so far suggest the informal principle that, as the network becomes more richly connected, individual traders have less and less power, and their payoffs go down. This example is a little bit counter intuitive. Here, traders T1 and T2 both have monopoly power over their respective sellers, and yet their profits are zero in every equilibrium. We can verify this fact as follows. First, we notice that any equilibrium must look like one of the solutions in (b) or (c). The sellers are monopolized and will get bids of 0. For each buyer, the two asks must be the same; otherwise, the trader making the sale could slightly raise his ask. Now, finally, notice that if the common ask to either buyer were positive, then the trader left out of the trade on the higher one has a profitable deviation by slightly undercutting this ask. Therefore, in this example, all bids and asks equal 0 in any equilibrium, and so neither trader profits. Trader profit: for a given trader T in a network, there exists an equilibrium in which T receives a positive payoff. It turns out that there exists such an equilibrium precisely when T has an edge e to a seller or buyer such that deleting e would change the value of the social optimum. In such a situation, we say that e is an essential edge from T to the other node.
13 Bargaining and Power in Networks In chapter 12.1 the notion of power is introduced. Power is not a property of an individual as it is a property of a relation between two individuals. Figures 1. Social network of five peoples, with node B occupying an intuitively powerful position.
14 Experimental Studies of Power and Exchange 1. A small graph (such as the one in Figure 1) is chosen, and a distinct volunteer test subject is chosen to represent each node. Each person, representing a node, sits at a computer and can exchange instant messages with the people representing the neighboring nodes. 2. The value in each social relation is made concrete by placing a resource pool on each edge let s imagine this as a fixed sum of money, say $1, which can be divided between the two endpoints of the edge. We will refer to a division of this money between the endpoints as an exchange. Whether this division ends up equal or unequal will be taken as a sign of the asymmetric amounts of power in the relationship that the edge represents. 3. Each node is given a limit on the number of neighbors with whom she can perform an exchange. The most common variant is to impose the extreme restriction that each node can be involved in a successful exchange with only one of her neighbors; this is called the 1exchange rule. Given this restriction, the set of exchanges that take place in a given round of the experiment can be viewed as a matching in the graph: a set of edges that have no endpoints in common. However, it will not necessarily be a perfect matching, since some nodes may not take part in any exchange. For example, in the graph in Figure 1, the exchanges will definitely not form a perfect matching, since there are an odd number of nodes. 4. Here is how the money on each edge is divided. A given node takes part in simultaneous sessions of instant messaging separately with each of her neighbors in the network. In each, she engages in relatively freeform negotiation, proposing splits of the money on the edge, and potentially reaching an agreement on a proposed split. These negotiations must be concluded by a fixed time limit; and to enforce the 1exchange rule defined above, as soon as a node reaches an agreement with one neighbor, her negotiations with all other neighbors are immediately terminated. 5. Finally, the experiment is run for multiple rounds. Results of Network Exchange Experiment Figures 2. Path of length (a) 2, (b) 3, (c) 4 and (d) 5 form intstructive examples of different phenomena in exchange networks. In two nodes path the equal split is more reasonable.
15 In three nodes path B receives the overwhelming majority of the money in her exchange (roughly 5/6). If the one exchange rule will be modified to allow B to take part on two exchanges in each round. B will get roughly equal footing with A and C. In fournode path B should have some amount of power over A, but it is weaker kind of power than in the threenodes path. Experiments showed that B gets roughly between 7/12 and 2/3 of the money, but not more. In fivenode path node C which occupies central position, is in fact weak if one exchange rule is used. Experiments showed that C does slightly better than A and E. For example, if we allowed A,C and E to take part in one exchange each, but allowed B and D to take part in two exchange each, C become powerful node.
16 Modeling TwoPerson Interaction: The Nash Bargaining Solution. Two people, A and B, are negotiating over how to split $1 between them. But A also has an outside option of x, and B has an outside option of y. Figures 3. Two nodes bargaining with outside options. Notice that if x+y>1, then no agreement between A and B is possible, since they cannot divide dollar so that one gets at least x and the other gets at least y. Consequently, we will assume that x+y 1. A requires at least x from the negotiation, and B requires at least y. Consequently, the negotiation is really over how to split the surplus s=1xy. The natural prediction is that A and B will split surplus equally, so A gets x+s/2 and B gets y+s/2. Nash Bargaining Solution is x+s/2 = (x+1y)/2 to A, and y+s/2 = (y+1x)/2 to B. Status effects. Benefits about differential status can lead to deviations from theoretical prediction in bargaining.
17 Modeling TwoPerson Interaction: Ultimatum Game. (i) Person A is given a dollar and told to propose a division of it to person B. That is, A should propose how much he keeps for himself, and how much he gives to B. (ii) Person B is then given the option of approving or rejecting the proposed division. (iii) If B approves, each person keeps the proposed amount. If B rejects, then each person gets nothing. Prediction: B s choice is between getting any positive amount of money and getting nothing. Hence, B should accept any positive amount of money. Experimental results: when a player B evaluates an outcome in which she walks away with only 10% of the total, there is a significant negative emotional payoff to being treated unfairly, and hence when we consider B s complete evaluation of the options, B finds a a greater overall benefit to rejecting the low offer and feeling good about it than accepting the low offer and feeling cheated. Moreover, since people playing the role of A understand that this is the likely evaluation that their partner B will bring to the situation, they tend to offer relatively balanced divisions to avoid rejection, because rejection results in A getting nothing as well. Modeling Network Exchange: Stable Outcomes Figure 4. Some examples of stable and unstable outcomes of network exchange on the threenode path and the fournode path: (a) unstable and (b) stable outcomes on the threenode path, (c) unstable and (d, e) stable outcomes on the fournode path. The darkened edges constitute matchings showing who exchanges with whom, and the numbers above the nodes represent the values. Instability: Given an outcome consisting of a matching and values for the nodes, an instability in this outcome is an edge not in the matching, joining two nodes X and Y, such that the sum of X s value and Y s value is less than 1. Stability: An outcome of network exchange is stable if and only if it contains no instabilities.
18 Modeling Network Exchange: Balanced Outcomes Figure 5. The difference between balanced and unbalanced outcomes: (a) not balanced, (b) balanced, and (c) not balanced. Balanced Outcome: An outcome (consisting of a matching and node values) is balanced if, for each edge in the matching, the split of the money represents the Nash bargaining outcome for the two nodes involved, given the best outside options for each node provided by the values in the rest of the network. To read: 12.2 experimental studies of power and exchange 12.3 results of network exchange experiments Advanced material. Exercises:
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