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Social Networks 101

Social Networks 101. Prof. Jason Hartline and Prof. Nicole Immorlica. Lecture Twenty-One : Market Equilibria . Dorm-room assignment. Laura. Sunny. Luke. Large. Lana. Quiet. Dorm rooms. Students. Bipartite graphs. Dorm rooms. Students.

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Social Networks 101

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  1. Social Networks 101 Prof. Jason Hartline and Prof. Nicole Immorlica

  2. Lecture Twenty-One: Market Equilibria.

  3. Dorm-room assignment Laura Sunny Luke Large Lana Quiet Dorm rooms Students

  4. Bipartite graphs Dorm rooms Students Definition: A bipartite graph consists of two sets of nodes A and B such that all edges have one endpoint in A and one endpoint in B.

  5. Dorm-room assignment Laura Sunny Luke Large Lana Quiet Dorm rooms Students

  6. Perfect matchings Dorm rooms Students Definition: A perfect matching is an assignment of nodes on the left to nodes on the right such that 1. each node has an edge to its assigned partner, 2. and no two nodes have the same assignment.

  7. Questions Is there always a way to assign students to dorm rooms? (Translation: Does every bipartite graph have a perfect matching?)

  8. Laura Sunny Luke Large Lana Quiet

  9. Questions Can we test when a feasible assignment exists?

  10. Laura Sunny Luke Large Lana Quiet No perfect matching: two people want the same room.

  11. Laura Sunny Luke Large Lana Quiet No perfect matching: three people want the same two rooms.

  12. Constricted sets Laura Sunny Luke Large Lana Quiet Constricted Definition. A set of nodes is constricted if it is strictly larger than its neighbor set.

  13. If constricted, then no perfect matching. (each node in constricted set needs to be matched to a different node in neighbor set, but # constricted nodes > # neighbors)

  14. In fact, constricted sets are the only obstacles to matching! Claim: If no constricted set, then perfect matching. (for proof, take Discrete Math 310)

  15. Incorporating values Idea: Allow students to specify how much they like each room.

  16. Incorporating values 1 Laura Sunny 12, 2, 4 2 Luke Large 8, 7, 6 3 Lana Quiet 7, 5, 2 Quality of assignment: 12 + 6 + 5 = 23

  17. Definition: The assignment that maximizes quality is the optimal assignment.

  18. Opt assignments & perfect matching 1 Laura Sunny 1, 0, 1 2 Luke Large 0, 1, 0 3 Lana Quiet 1, 1, 1 There is a perfect matching if and only if the optimal assignment has value equal to the # of students.

  19. Can we always find an optimal assignment?

  20. Values 1 1 vij = value of buyer j for house of seller i 2 2 (a non-negative whole number) 3 3 House sellers House buyers

  21. Prices 1 vij = value of buyer j for house of seller i 1 pi = price of house i 2 2 3 3 House sellers House buyers

  22. Payoffs 1 vij = value of buyer j for house of seller i 1 pi = price of house i 2 2 vij– pi = payoff of buyer j if she buys house of seller i 3 3 House sellers House buyers

  23. For Sale: $100,000 v12 = $400,000 1 v22 = $500,000 For Sale: $400,000 2 2 For Sale: $250,000 3 v32 = $300,000 Which house will buyer 2 buy?

  24. For Sale: $100,000 v12 = $400,000 1 v22 = $500,000 For Sale: $400,000 2 2 For Sale: $250,000 3 v32 = $300,000 Buyer will buy from a seller who maximizes his payoff (if positive).

  25. Definition. The preferred sellers of a buyer are those sellers whose prices maximize his payoff.

  26. Who are the preferred sellers? Values Prices p1 = 2 1 1 12, 4, 2 p2 = 1 8, 7, 6 2 2 p3 = 0 7, 5, 2 3 3 No perfect matching -> no way to sell all the houses.

  27. Who are the preferred sellers? Values Prices p1 = 3 1 1 12, 4, 2 p2 = 1 8, 7, 6 2 2 p3 = 0 7, 5, 2 3 3 All houses sell (breaking ties appropriately).

  28. Market-clearing Definition. A market clears if all houses sell, that is, if the preferred-seller graph has a perfect matching.

  29. Market-clearing prices Important results: • For any buyer valuations, market-clearing prices exist. • For any set of market-clearing prices, matching in resulting graph is optimal (i.e., has maximum quality).

  30. Market-clearing prices Important results: Market-clearing prices alwaysexist and produce optimal outcomes.

  31. Producing optimal outcomes Let M be matching of market-clearing prices. Total payoff(M) = Total value(M) – sum of all prices • Each buyer maximizes payoff, so M maximizes total payoff. • Sum of prices constant, so M maximizes total value and is therefore optimal.

  32. Constructing prices Question. What should a seller do if his house is over-demanded? Answer. Raise his price!

  33. Constructing prices Values Prices p1 = 0 p1 = 2 p1 = 1 p1 = 3 1 1 12, 4, 2 p2 = 0 8, 7, 6 p2 = 1 2 2 p3 = 0 7, 5, 2 3 3

  34. Constructing prices • Construct preferred seller graph. If there is a perfect matching, we are done. • If not, find constricted set of buyers and raise prices of neighboring houses by one.

  35. The prices settle • Define the energy of the market. • Show that the energy is decreasing, so must run out. • (for proof, read book)

  36. Single-item auction case Values Prices p1 = 1 p1 = 0 p1 = 2 1 1 3, 0, 0 3 p2 = 0 2 2, 0, 0 2 2 p3 = 0 1 1, 0, 0 3 3 Computes the second-price auction!

  37. Next time Advertising auctions.

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