Nash Bargaining Game

1. Nash Bargaining Game • Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, they must each make a demand for some portion of the 100 dollars simultaneously. If the sum of their demands is less than or equal to 100, they each receive their demanded sum; if it is greater than 100, the money is lost to both. • What would you do? • Does their exist a pure strategy Nash equilibrium?

2. Nash Bargaining Game • Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, they must each make a demand for some portion of the 100 dollars simultaneously. If the sum of their demands is less than or equal to 100, they each receive their demanded sum; if it is greater than 100, the money is lost to both. • What would you do? • Does their exist a pure strategy Nash equilibrium? • Yes, any outcome where the demands sum to exactly 100

3. Nash Bargaining Game • Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, they must each make a demand for some portion of the 100 dollars simultaneously. If the sum of their demands is less than or equal to 100, they each receive their demanded sum; if it is greater than 100, the money is lost to both. • What are the Pareto optimal outcomes?

4. Nash Bargaining Game • Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, they must each make a demand for some portion of the 100 dollars simultaneously. If the sum of their demands is less than or equal to 100, they each receive their demanded sum; if it is greater than 100, the money is lost to both. • What are the Pareto optimal outcomes? • Also any outcome where the demands sum to exactly 100

5. Ultimatum Game • Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, player A proposes a division of the money. Player B then has the option to accept or reject that proposal. In the event that Player B rejects, both players receive nothing. • Subject of many psychological studies • Find that many have adverse reactions to stingy offers • Both parties know this

6. Nash Bargaining With Outside Options • Two players, A and B, are given 100 dollars to divide between them. According to the rules of the game, they must each make a demand for some portion of the 100 dollars simultaneously. If the sum of their demands is less than or equal to 100, they each receive their demanded sum; if it is greater than 100, the money is lost to both. • In addition, both A and B have “outside options”, a and b such that • a and b are between 0 and 100 • If no agreement is made, then players receive their outside options • Neither A nor B know the other’s outside option

7. Dynamic, Cooperative Nash Bargaining • Recall: • Dynamic games: not simultaneous. Players observe state of the game and can alternate moves • Cooperative game: mutual agreement is required before payoffs are received • Nash Bargaining as a dynamic, cooperative game: • Two players, A and B, are given 100 dollars to divide between them. Each of A and B may have outside options. Players exchange offers until an agreement is reached: A: How about I get $70, and you get $30? B: No, I want $40. A: How about $34? B: I’ll take $36. A: Agreed

8. Power in Social Networks • Notion of how powerful a node is in a network • Considers dependence, betweenness, etc. • See section 12.1 in text • E.g.: Powerful

9. Many-to-Many Bargaining • Equivalence of outside options and power in networks • Imagine allocating a hypothetical $100 to each node • Before it can be earned, an agreement must be reached on how to divide it • Nodes can use dynamic cooperative Nash bargaining with connected nodes • Nodes with more “power” will have more/better outside options • Should retain a larger portion of its $100 Position of power

10. Bipartite Many-to-Many Bargaining • Imagine sellers (top) and buyers (bottom) • Some sellers and more power than other sellers • Some buyers have more power than other buyers • Some sellers have more power than buyers and vice versa

11. Auctions • a form of negotiation between buyers and sellers • common in Internet trading • can have • one seller, many buyers • one buyer, many sellers (reverse auctions) • many buyers, many sellers (double auctions) • one item, many items • we focus on one seller, many buyers, one item • can be open-cry or sealed bid

12. Open Ascending Price Auctions • also known as English auctions • most common type of auction • auctioneer starts by calling out a low purchase amount • purchase amount is raised in small increments • stops when there is only one interested bidder • the last bidder wins the auction, pays price equal to last bid level

13. Open Descending Price Auctions • also known as Dutch auctions • auctioneer starts by calling out a high purchase amount • purchase amount is lowered in small increments • stops when a bidder indicates interest • this bidder wins the auction, pays price equal to last bid level

14. Sealed-Bid Auctions • First-Price Sealed-Bid Auctions • all buyers submit a secret bid to auctioneer • highest bidder wins the auction, pays price equal to bid amount • Second-Price Sealed-Bid Auctions • all buyers submit a secret bid to auctioneer • highest bidder wins the auction, pays price equal to second-highest bid amount • also known as a Vickrey auction

15. Bidding Strategies • Assume each bidder has • a private monetary value for the item being auctioned, known as a reserve price • a utility function over the set of possible prices, given the item being auctioned • How does a bidder determine the optimal bidding strategy? • i.e. how does a buyer decide • whether to bid (in open auctions) • how much to bid (in sealed-bid auctions) with the intention of maximizing utility

16. SNA Class Auctions • Four items to auction off, one-by-one • Each participant has a reserve price r for each item • Your objective is to maximize utility: • Price p: • u(p) = p – r if you win the auction • u(p) = 0 otherwise • E.g. • Say your reserve is 10 • If you win an auction with a bid of 6, your utility is u(6) = 10 – 6 = 4 • If you win an auction with a bid of 12, your utility is u(6) = 10 – 12 = -2 • If you lose the auction, your utility is 0

17. Lot A: Gemstone Ring This is a first-price, sealed-bid auction

18. Lot B: 1897 Stamp This is a second-price, sealed-bid (Vickrey) auction

19. Lot C: NCAA Basketball This is an open, descending price (Dutch) auction

20. Lot D: Tom Petty Tickets This is an open, ascending price (English) auction

21. Optimal Strategies for First-Price Auctions • computing optimal strategies for 1st price sealed and Dutch auctions is difficult, as beliefs on other bidders’ valuations must be considered • these two auctions are said to be strategically equivalent • this means that, for each bidding strategy in one type of auction, there is a strategy in the other type of auction resulting in the same outcomes • In each type of auction, the bidder’s strategy maps his private information to a bid. Although the Dutch auction is open, there is no activity to observe before the bid is submitted • Thus the two auction types are considered equivalent, and are referred to as first-price auctions

22. Optimal Strategies for Second-Price Auctions • optimal strategies for English and Vickrey auctions are more easily computed, as they rely only on one’s own valuation • English: bid until reserve price is reached • Vickrey: bid reserve price • Since the optimal strategy is to bid (up to) reserve value, with the winner paying the reserve price of the second highest bidder (or perhaps a small increment higher in the case of English), these auctions are referred to as second-price auctions.

23. Proof that the optimal strategy in Vickrey auctions is to bid reserve price • Let b and r be the bidder’s bid and reserve price, and let b’ be the highest bid of all other bidders. Two cases: • Case (1): b’ >= r • if b > r, then either • u ≤ 0if b>b’ (i.e. if we win the auction), since u = r – b’ • u = 0otherwise • if b = r, then u=0 • we either lose the auction or win with utility 0 • if b < r, then u=0 • we lose the auction • so the optimal strategy is to bid b <= r

24. Proof that the optimal strategy in Vickrey auctions is to bid reserve price • Let b and r be the bidder’s bid and reserve price, and let b’ be the highest bid of all other bidders. Two cases: • Case (2): b’ < r • if b > r, then u=r-b’ > 0 • if b = r, then u=r-b’ > 0 • if b < r, then either • u=r-b’ if b>b’ (if we win the auction) • u=0 otherwise (if we lose) • so the optimal strategy is to bid b >= r • Case 1 optimal strategy: bid b <= r • Case 2 optimal strategy: bid b >= r • Thus it is always optimal to bid b = r. So this is a dominant strategy

25. Combinatorial Auctions • Multi-item, sealed bid auctions • Participants can place bids on combinations of items, or “bundles” • Four items available: a,b,c,d, no duplicates • Bidder 1 bids $10 for {a,b} • Bidder 2 bids $10 for {a,c}, $8 for {b,d} • Bidder 3 bids $4 for {b}, $4 for {c}, $12 for {b,c}, $5 for {d} • Auctioneer considers bids and computes valid allocations of items, i.e. where no item is awarded to more than one bid • Must solve constraint satisfaction to determine the optimal allocation, referred to as the Winner-Determination Problem (WDP) • WDP is shown to be NP-complete. Many research efforts focus on fast algorithms for practical problem instances