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Kleinberg - Chap10. Matching markets

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This is just a brief content for the chapter 10 of the Kleinberg 's book. Link: http://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch10.pdf

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NETWORKS, CROWDS AND MARKETS CHAPTER 10: MATCHING MARKETS Chapter 10. Matching markets. 1
Abstract • Many practical problems can be seen under network-structured view. • For instance. Issue of traffic, etc. • Market is a prime example of network- structured interaction between many agents. • Matching markets. Chapter 10. Matching markets. 2
Matching markets - Basic principles. 1. People may have different preferences for different kinds of goods. 2. Prices can decentralize the allocation of goods to people. 3. Prices can lead to allocations that are socially optimal. Chapter 10. Matching markets. 3
The 1st scenario. Room Assigning - Assigning rooms to student. - Room is designed for a single student. - Students may have different preferences over rooms. Chapter 10. Matching markets. 4
The 1st scenario. Room Assigning Chapter 10. Matching markets. 5
Definitions - Bipartite graph. - Perfect matchings. - Constricted sets. - The Matching theorem. Chapter 10. Matching markets. 6
Bipartite graph - Nodes are divided into two categories. - Edges connect a node in one category to a node in the other category. Chapter 10. Matching markets. 7
Perfect matchings • A choice of edges in the bipartite graph so that each node is the endpoint of exactly one of the chosen edges. Chapter 10. Matching markets. 8
Constricted sets • A set of nodes that their edges to the other side of the bipartite graph “constrict” the formation of a perfect matching. Chapter 10. Matching markets. 9
The Matching theorem Matching theorem. If a bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching, then it must contain a constricted set. Proof here Chapter 10. Matching markets. 10
The 1bis scenario. Valuations and Optimal Assignment More than binary choice “accept-or-not”. Chapter 10. Matching markets. 11
Definitions - Valuations. - Optimal assignments. Chapter 10. Matching markets. 12
Valuations • A collection of individuals evaluating a collection of objects. Quality of an Sum of each assignment of objects to individuals = individual’s valuations. Chapter 10. Matching markets. 13
Optimal assignment • An assignment that maximizes the total happiness of everyone for what they get. Chapter 10. Matching markets. 14
The 2nd scenario. Market-clearing prices • More standard picture of a market. • Decisions based on prices and own valuations. • Buyers and sellers. Chapter 10. Matching markets. 15
Definitions - Prices and payoffs. - Preferred sellers. - Preferred-seller graph. - Market-clearing prices. Chapter 10. Matching markets. 16
Prices and payoffs • Suppose that each seller i put his house up for a price pi >= 0. • The buyer’s payoff is her valuation for this house minus the amount of money she had to pay: vij – pi. Chapter 10. Matching markets. 17
Prices and payoffs Payoffs of each buyer on each house: a b c X 7 2 2 Y 3 5 6 Z 2 3 2 Chapter 10. Matching markets. 18
Preferred sellers • Seller or sellers that maximize the payoff for buyer j the preferred sellers. a b c X 7 2 2 Y 3 5 6 Z 2 3 2 Chapter 10. Matching markets. 19
Preferred-seller graph • A graph containing edges between buyers and their preferred sellers. Chapter 10. Matching markets. 20
Market-clearing prices • A set of prices is call market-clearing if they cause each house to get bought by a different buyer. Or • A set of prices is market-clearing if the resulting preferred-seller graph has a perfect matching. Chapter 10. Matching markets. 21
Market-clearing prices • Existence of market-clearing prices. For any set of buyer valuations, there exists a set of market-clearing prices. Chapter 10. Matching markets. 22
Market-clearing prices • Optimality of market-clearing prices. For any set of market-clearing prices, a perfect matching in the resulting preferred-seller graph has the maximum total valuation of any assignment of sellers to buyers. Total Payoff of M = Total Valuation of M − Sum of all prices. Chapter 10. Matching markets. 23
Constructing a set of market-clearing prices • A general round of the auction looks like what we’ve just described. 1. At the start of each round, there is a current set of prices, with the smallest one equal to 0. 2. We construct the preferred-seller graph and check whether there is a perfect matching. Chapter 10. Matching markets. 24
Constructing a set of market-clearing prices 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 neighbours N(S). 5. Each seller in N(S) (simultaneously) raises his price by one unit. Chapter 10. Matching markets. 25
Constructing a set of market-clearing prices 6. If necessary, we reduce the prices — the same amount is subtracted from each price so that the smallest price becomes zero. 7. We now begin the next round of the auction, using these new prices. Chapter 10. Matching markets. 26
Constructing a set of market-clearing prices Chapter 10. Matching markets. 27
Constructing a set of market-clearing prices 1. At the start of each round, there is a current set of prices, with the smallest one equal 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 neighbours N(S). 5. Each seller in N(S) (simultaneously) raises his price by one unit. 6. If necessary, we reduce the prices — the same amount is subtracted from each price so that the smallest price becomes zero. 7. We now begin the next round of the auction, using these new prices. LOOP FOREVER ? Chapter 10. Matching markets. 28
A proof of the Matching Theorem The Matching Theorem. The problem. • How can we identify a constricted set in a bipartite graph, knowing only that it contains no perfect matching. Chapter 10. Matching markets. 29
A proof of the Matching Theorem The idea. 1. Give a bipartite graph. 2. Consider a maximum matching. 3. Try to enlarge -> FAIL. 4. Identify the constricted set. Chapter 10. Matching markets. 30
Definitions 1. Matching edges. Edges are used in given matching. 2. Non-matching edges. The other edges. 3. Alternating path. A simple path that alternates between non-matching and matching. Chapter 10. Matching markets. 31
Definitions 4. Augmenting path. An alternating path whose endpoints are unmatched nodes, then the matching can be enlarged. Chapter 10. Matching markets. 32
Searching for an augmenting path • Alternating BFS. • Start any unmatched node on the right. • Explore the rest of the graph layer by layer, add new nodes to the next layer if have connections. • Use non-matching edges to discover new nodes. • If contains an unmatched node from the left- hand size of the graph -> an augmenting path -> enlargeable. Chapter 10. Matching markets. 33
Searching for an augmenting path Alternating BFS Chapter 10. Matching markets. 34
Augmenting paths and constricted sets Chapter 10. Matching markets. 35
Augmenting paths and constricted sets • Claim. Consider any bipartite graph with a matching, and let W be any un-matched node on the right-hand size. Then either there is an augmenting path beginning at W, or there is a constricted set containing W. Chapter 10. Matching markets. 36
Computing a perfect matching 1. Start with an empty matching. 2. Look for an unmatched node W. 3. Use alternating BFS to search for an augmenting path beginning at W. 4. If found, use this path to enlarge the matching. Else, indicate the constricted set. Chapter 10. Matching markets. 37
Computing a maximum matching • Looking for a maximum matching can matter where we start our search for an augmenting path. • If there is no augmenting path beginning at any node on the right-hand size, then in fact the current matching has maximum size. • Revise the alternating BFS by putting all unmatched nodes on the right to layer 0. Chapter 10. Matching markets. 38
Computing a maximum matching - If starting from W, we may fail to find augmenting path. - If starting from Y, we can produce the path Y – B – Z – D. Chapter 10. Matching markets. 39
Q&A Chapter 10. Matching markets. 40

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Kleinberg - Chap10. Matching markets

  • 1. NETWORKS, CROWDS AND MARKETS CHAPTER 10: MATCHING MARKETS Chapter 10. Matching markets. 1
  • 2. Abstract • Many practical problems can be seen under network-structured view. • For instance. Issue of traffic, etc. • Market is a prime example of network- structured interaction between many agents. • Matching markets. Chapter 10. Matching markets. 2
  • 3. Matching markets - Basic principles. 1. People may have different preferences for different kinds of goods. 2. Prices can decentralize the allocation of goods to people. 3. Prices can lead to allocations that are socially optimal. Chapter 10. Matching markets. 3
  • 4. The 1st scenario. Room Assigning - Assigning rooms to student. - Room is designed for a single student. - Students may have different preferences over rooms. Chapter 10. Matching markets. 4
  • 5. The 1st scenario. Room Assigning Chapter 10. Matching markets. 5
  • 6. Definitions - Bipartite graph. - Perfect matchings. - Constricted sets. - The Matching theorem. Chapter 10. Matching markets. 6
  • 7. Bipartite graph - Nodes are divided into two categories. - Edges connect a node in one category to a node in the other category. Chapter 10. Matching markets. 7
  • 8. Perfect matchings • A choice of edges in the bipartite graph so that each node is the endpoint of exactly one of the chosen edges. Chapter 10. Matching markets. 8
  • 9. Constricted sets • A set of nodes that their edges to the other side of the bipartite graph “constrict” the formation of a perfect matching. Chapter 10. Matching markets. 9
  • 10. The Matching theorem Matching theorem. If a bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching, then it must contain a constricted set. Proof here Chapter 10. Matching markets. 10
  • 11. The 1bis scenario. Valuations and Optimal Assignment More than binary choice “accept-or-not”. Chapter 10. Matching markets. 11
  • 12. Definitions - Valuations. - Optimal assignments. Chapter 10. Matching markets. 12
  • 13. Valuations • A collection of individuals evaluating a collection of objects. Quality of an Sum of each assignment of objects to individuals = individual’s valuations. Chapter 10. Matching markets. 13
  • 14. Optimal assignment • An assignment that maximizes the total happiness of everyone for what they get. Chapter 10. Matching markets. 14
  • 15. The 2nd scenario. Market-clearing prices • More standard picture of a market. • Decisions based on prices and own valuations. • Buyers and sellers. Chapter 10. Matching markets. 15
  • 16. Definitions - Prices and payoffs. - Preferred sellers. - Preferred-seller graph. - Market-clearing prices. Chapter 10. Matching markets. 16
  • 17. Prices and payoffs • Suppose that each seller i put his house up for a price pi >= 0. • The buyer’s payoff is her valuation for this house minus the amount of money she had to pay: vij – pi. Chapter 10. Matching markets. 17
  • 18. Prices and payoffs Payoffs of each buyer on each house: a b c X 7 2 2 Y 3 5 6 Z 2 3 2 Chapter 10. Matching markets. 18
  • 19. Preferred sellers • Seller or sellers that maximize the payoff for buyer j the preferred sellers. a b c X 7 2 2 Y 3 5 6 Z 2 3 2 Chapter 10. Matching markets. 19
  • 20. Preferred-seller graph • A graph containing edges between buyers and their preferred sellers. Chapter 10. Matching markets. 20
  • 21. Market-clearing prices • A set of prices is call market-clearing if they cause each house to get bought by a different buyer. Or • A set of prices is market-clearing if the resulting preferred-seller graph has a perfect matching. Chapter 10. Matching markets. 21
  • 22. Market-clearing prices • Existence of market-clearing prices. For any set of buyer valuations, there exists a set of market-clearing prices. Chapter 10. Matching markets. 22
  • 23. Market-clearing prices • Optimality of market-clearing prices. For any set of market-clearing prices, a perfect matching in the resulting preferred-seller graph has the maximum total valuation of any assignment of sellers to buyers. Total Payoff of M = Total Valuation of M − Sum of all prices. Chapter 10. Matching markets. 23
  • 24. Constructing a set of market-clearing prices • A general round of the auction looks like what we’ve just described. 1. At the start of each round, there is a current set of prices, with the smallest one equal to 0. 2. We construct the preferred-seller graph and check whether there is a perfect matching. Chapter 10. Matching markets. 24
  • 25. Constructing a set of market-clearing prices 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 neighbours N(S). 5. Each seller in N(S) (simultaneously) raises his price by one unit. Chapter 10. Matching markets. 25
  • 26. Constructing a set of market-clearing prices 6. If necessary, we reduce the prices — the same amount is subtracted from each price so that the smallest price becomes zero. 7. We now begin the next round of the auction, using these new prices. Chapter 10. Matching markets. 26
  • 27. Constructing a set of market-clearing prices Chapter 10. Matching markets. 27
  • 28. Constructing a set of market-clearing prices 1. At the start of each round, there is a current set of prices, with the smallest one equal 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 neighbours N(S). 5. Each seller in N(S) (simultaneously) raises his price by one unit. 6. If necessary, we reduce the prices — the same amount is subtracted from each price so that the smallest price becomes zero. 7. We now begin the next round of the auction, using these new prices. LOOP FOREVER ? Chapter 10. Matching markets. 28
  • 29. A proof of the Matching Theorem The Matching Theorem. The problem. • How can we identify a constricted set in a bipartite graph, knowing only that it contains no perfect matching. Chapter 10. Matching markets. 29
  • 30. A proof of the Matching Theorem The idea. 1. Give a bipartite graph. 2. Consider a maximum matching. 3. Try to enlarge -> FAIL. 4. Identify the constricted set. Chapter 10. Matching markets. 30
  • 31. Definitions 1. Matching edges. Edges are used in given matching. 2. Non-matching edges. The other edges. 3. Alternating path. A simple path that alternates between non-matching and matching. Chapter 10. Matching markets. 31
  • 32. Definitions 4. Augmenting path. An alternating path whose endpoints are unmatched nodes, then the matching can be enlarged. Chapter 10. Matching markets. 32
  • 33. Searching for an augmenting path • Alternating BFS. • Start any unmatched node on the right. • Explore the rest of the graph layer by layer, add new nodes to the next layer if have connections. • Use non-matching edges to discover new nodes. • If contains an unmatched node from the left- hand size of the graph -> an augmenting path -> enlargeable. Chapter 10. Matching markets. 33
  • 34. Searching for an augmenting path Alternating BFS Chapter 10. Matching markets. 34
  • 35. Augmenting paths and constricted sets Chapter 10. Matching markets. 35
  • 36. Augmenting paths and constricted sets • Claim. Consider any bipartite graph with a matching, and let W be any un-matched node on the right-hand size. Then either there is an augmenting path beginning at W, or there is a constricted set containing W. Chapter 10. Matching markets. 36
  • 37. Computing a perfect matching 1. Start with an empty matching. 2. Look for an unmatched node W. 3. Use alternating BFS to search for an augmenting path beginning at W. 4. If found, use this path to enlarge the matching. Else, indicate the constricted set. Chapter 10. Matching markets. 37
  • 38. Computing a maximum matching • Looking for a maximum matching can matter where we start our search for an augmenting path. • If there is no augmenting path beginning at any node on the right-hand size, then in fact the current matching has maximum size. • Revise the alternating BFS by putting all unmatched nodes on the right to layer 0. Chapter 10. Matching markets. 38
  • 39. Computing a maximum matching - If starting from W, we may fail to find augmenting path. - If starting from Y, we can produce the path Y – B – Z – D. Chapter 10. Matching markets. 39
  • 40. Q&A Chapter 10. Matching markets. 40

Editor's Notes

  1. Agents and behaviors.Implicit network between buyers and sellers.Number of ways of using networks to model interaction among market participant.Extend to the broad notion of social exchange
  2. First class of model as the focus of the current chapter.-They embody, in a very clean and stylized way, a number of basicprinciples:
  3. Rather than expressing preferences simply as binary choice, each individual to express how much they’d like each object, in numerical form.
  4. Of course, while the optimal assignment maximizes total happiness, it does not necessarily give everyone their favourite item; for example, in Figure 10.3(b), all the students think Room 1 is the best, but it can only go to one of them.
  5. Individuals making decisions based on prices and their own valuations.
  6. If this quantity is maximized in a tie between several sellers, then the buyer can maximize her payoff by choosing any one of them.If her payoff vij − pi is negative for every choice of seller i, then the buyer would prefer not to buy any house: we assume she can obtain a payoff of 0 by simply not transacting.
  7. A harder challenge: understanding why market-clearing prices must always exists.+ Take an arbitrary set of buyer valuation.+ Describe a procedure that arrives at market-clearing prices.
  8. - Initially all sellers set their prices to 0. Buyers react by choosing their preferred seller(s) and we look at the resulting preferred-seller graph. If this graph has a perfect matching we’re done. Else -> constricted set.
  9. Potential of a buyer: maximum payoff she can currently get from any seller.Potential of a seller: current price he’s charging.Potential energy of the auction: sum of the potential.
  10. Claim. In a bipartite graph with a matching, if there is an alternating path whose endpoints are unmatched nodes, then the matching can be enlarged.
  11. Even-numbered layers consists of nodes from the right-hand size. Odd-numbered layers consists of nodes from the left-hand sizse.Odd layers contains the same number of nodes as the subsequent even layer.Not counting node W, we have the same number of nodes in odd and even layers.Each nodes in even layers all of its neighbors in the graph present in some layer.
  12. If the alternating breadth-first search fails from any node on the right-hand side, this is enough to expose a constricted set and hence prove there is no perfect matching. However, it is still possible that an alternating breadth-first search could still succeed from some other node. (In this case, the search from W would fail, but the search from Y would succeed.)