# 10.1 Bipartite Graphs and Perfect Matchings

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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.

3 10.4 Constructing a Set of Market-Clearing 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 preferred-seller graph and check whether there is a perfect matching. 3. If there is, we're done: the current prices are market-clearing. 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 market-clearing prices How Does this Relate to Single-Item 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 market-clearing 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 non-matching 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 W-B-Y-C-Z or paths that are not alternating like W-B-Z- C-Y-D. There's a natural procedure we can use to search for an augmenting path. It works by simply adapting the breadth-first 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 right-hand 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.

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.

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 1-exchange 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 free-form 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 1-exchange 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 four-node path B should have some amount of power over A, but it is weaker kind of power than in the three-nodes path. Experiments showed that B gets roughly between 7/12 and 2/3 of the money, but not more. In five-node 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 Two-Person 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=1-x-y. 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+1-y)/2 to A, and y+s/2 = (y+1-x)/2 to B. Status effects. Benefits about differential status can lead to deviations from theoretical prediction in bargaining.

17 Modeling Two-Person 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 three-node path and the four-node path: (a) unstable and (b) stable outcomes on the three-node path, (c) unstable and (d, e) stable outcomes on the four-node 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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