Internet Economics כלכלת האינטרנט

1. Internet Economicsכלכלת האינטרנט Class 6 – more VCG, and ascending price auctions.

2. Today • VCG (cont.) • Example: the roommate problem • Selling multiple items.

3. Example 1: Roommates buy TV • Consider two roommates who would like to buy a TV for their apartment. • TV costs $100 • They should decide: • Do they want to buy a TV together? • If so, how should they share the costs? רק אירוויזיון! I only watch sports

4. Mechanism Design scheme types Bids/reports t1 b1 t2 b2 outcome payments t3 b3 Social planner p1,p2,p3,p4 t4 b4

5. VCG basic idea (cont.) In more details: • You can maximize efficiency by: • Choosing the efficient outcome (given the bids) • Each player pays his “social cost” (how much his existence hurts the others). pi = Optimal welfare (for the other players) if player i was not participating. Welfare of the other playersfrom the chosen outcome

6. VCG in 5-item auctions Optimal welfare (for the other players) if player i was not participating. Welfare of the other playersfrom the chosen outcome • pi= =30+27+25+12+5 The five winners when iis not playing. =30+27+25+12. The other four winners. What is my VCG payment? pays 5 pays ?? $70 $30 $27 $25 $5 $12 $2

7. Formal model • 2 players • w1 = “1 wins”, w2 = “2 wins” • ti=vi (willingness to pay) • v1(v1, w1) = v1v1(v1, w2) = 0 • Goal: choose a winner with the highest vi. • n players • possible outcome w1,w2,…,wm • Each player has private info ti • Each player has a value per each outcome (depends on ti) • vi(ti,w) w is from {w1,…,wm} • Goal of social planner: choose w that maximizes

8. Formal model w*=w5 maximal

9. VCG – formal definition • Bidders are asked to report their private values ti • Terminology: (given the reportedti’s) • w*outcome that maximizes the efficiency. • Let w*-ibe the efficient outcome when i is not playing. • The VCG mechanism: • Outcome w* is chosen. • Each bidder pays: The total value for the other when player i is not participating The total value for the others when i participates

10. Truthfulness Theorem (Vickrey-Clarke-Groves): In the VCG mechanism, truth-telling is a dominant strategy for all players.

11. Now, proof. We will show: no matter what the others are doing, lying about my type will not help me.

12. Truthfulness of VCG - Proof • The VCG mechanism: • Outcome w* is chosen. • Each bidder pays:

13. Truthfulness of VCG - Proof • Buyer’s utility (when w* is chosen): • Assume: bidder i reports a lie t’ outcome x is chosen. • Buyer’s utility (when x is chosen):

14. Truthfulness of VCG - Proof • Buyer’s utility from truth (w* is chosen): • Buyer’s utility from lying (x is chosen): • Lying is good when: > • Impossible since w* maximizes social welfare!

15. Truthfulness of VCG - intuition • The trick is actually quite simple: • By lying, players may be able to change the outcome. • But their utility depends not only on the outcome, but also on their payments. • With VCG payments, the utility of each player is the total efficiency. •  Therefore, players want the efficient outcome to be chosen. Lying my ruin this.

16. VCG in general Conclusion: when our goal is to maximize the social welfare, we can do it in a dominant-strategy equilibrium • Using VCG mechanisms. • We make the agents’ utilities be aligned with the designer’s goal.

17. The VCG family • From the proof, we can see that the VCG mechanism is actually a family of mechanisms. • The VCG mechanism: • Outcome w* is chosen. • Each bidder pays: This could be any function of the other bids.

18. Today • VCG • Example: the roommate problem • Selling multiple items.

19. Example 1: Roommates buy TV • TV cost $100 • Bidders are willing to pay v1and v2 • Private information. • VCG ensures: • Efficient outcome (buy if v1+v2>100) • Truthful revelation. In our model:Welfare when buying: v1+v2Welfare when not buying: 100(saved the construction cost)

20. Example 1: Roommates buy TV • Let’s compute VCG payments. • Consider values v1=70, v2=80. • With player 1: value for the others is 80. • Without player 1: welfare is 100.  p1= 100-80=20 • Similarly: p2 = 100-70 = 30 • Total payment received: 20+30 < 100 • Cost is not covered! In general, p1=100-v2, p2=100-v1 p1+p2= 100-v1+100-v2= 100-(v1+v2-100) < 100 • Whenever we build, cost is not covered.

21. Example 1: Roommates buy TV 100 Payment of agent 1 80 v2 Needed to cover the cost Payment of agent 2 0 0 70 100 v1

22. Example 1: Roommates buy TV Conclusion: in some cases, the VCG mechanism is not budget-balanced. (spends more than it collects from the players.) This is a real problem! There isn’t much we can do:It can be shown that there is no mechanism that is both efficient and budget balanced. • Even in simple settings: one seller and one buyer with private values. • “Myerson-Satterthwaite theorem”

23. Roommates (cont.) Now, assume that the values are v1=110, v2=130. How much each one pays (in VCG)? 0 Reason: agents do not affect the outcome Players that affect the outcome: pivots. Therefore, the VCG mechanism is also known as the pivot mechanism.

24. Context: Public goods • The roommate problem is knows as the “public good” problem. • Consider a government that wants to build a bridge. • When to build? If the total welfare is greater than the cost. • How the cost is shared? • Efficiency vs. Budget Balance (cannot achieve both). • Another: cable infrastructure.

25. Today • VCG • Example: the roommate problem • Selling multiple items.

26. Auctions for non-Identical items • Non identical items: a, b, c, d, e, • Each bidder has a value for each itemvi(a),vi(b),bi(c),.. • Each bidder wants one item only. • How can we sell them? • Separate English auctions? • Next: an ascending-price auction leading to VCG prices. Difference from previous example: items are heterogeneous

27. Simultaneous Ascending Auction • Start with zero prices. • Each bidder reports his favorite item • Price of over-demanded items is raised by $1. • Stop when there are no over-demanded items. • Bidders win their demands at the final prices. Claim: this auction terminates with: (1) Efficient allocation. (2) VCG prices ( ± $1 )

28. Example • What is the optimal social welfare? • Let’s run the ascending-price auction now… • VCG payments? Bidder 1 pays 3 for a Bidder 1 pays 0 for c Bidder 3 pays 3 for b

29. Why does it work? • For bidders that are interested in one item:the goods are substitutes. • If a bidder wants an Apple, and we increase the price of a Banana, the bidder will still demand an Apple. • This procedure actually reaches the efficient outcome for more general preferences. • As long the items are “substitutes”. • Example for complementarities: • TV+DVD • Spectrum: covering contiguous areas. • Acquisition of two firms at the same business

30. Spectrum Auctions • The ascending auction we saw is called:Simultaneous Ascending Auction • Also known in the literature as tatonnement • Was introduced in 1994 for spectrum auctions. • Revolutionized the sale of spectrum. • Has been used since all over the world. • The basis of auctions for complex resource allocation problems. • Eg, transportation.

31. Next week • Online advertising • How search engines sell ads? • How users behave in search engines, and how it affects the ad market?