Internet Economics כלכלת האינטרנט

1. Internet Economicsכלכלת האינטרנט Class 5 – VCG mechanisms

2. Previously… • We studied single-item auctions • Bidders have values vifor an item • A winning bidder gets a utility of ui=vi-pi • A losing bidder pays nothing and get ui=0

3. Previously… • Seller possible goals: • Maximize social welfare (efficiency) • 2nd-price (Vickrey) auction • Maximize revenue • 2nd-price auction with a reserve price (Myerson) • For example, reserve-price=1/2 for the unifom distribution on [0,1] • Reserve price is independent of the number of players. • Optimality assumes a technical assumption on the distributions. • Revenue equivalence

4. Previously … • We saw that in single-item auctions we can maximize efficiency with dominant strategies. • Can this be achieved in other models?

5. Today • This class: math is trivial, but modeling is abstract… • Moving from a specific example (single-item auctions) to a more general mechanism design setting. • Main goal: in the presence of incomplete information, design the right incentive such that the desired outcome will be chosen.

6. Outline • Some examples • VCG idea – intuition • Formal part: • Mechanism design model • The VCG mechanism • Proof: VCG is truthful • Roommates example

7. Auctions scheme values bids v1 b1 v2 b2 winner $$$ payments v3 b3 v4 b4

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

9. 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

10. Example 2: Selling multiple items • Each bidder has a value of vi for an item. • But now we have 5 items! • Each bidder want only one item. • An efficient outcome: sell the items to the 5 bidders with the highest values. $70 $30 $27 $25 $5 $12 $2

11. Vickrey-Clarke-Groves (VCG) mechanisms • Goal: implement the efficient outcome in dominant strategies. • A general method to do this: VCG • 2nd-price auction is a special case • Solution (intuitively):players should pay the “damage” they impose on society.

12. 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

13. Vickrey-Clarke-Groves (VCG) mechanisms • Let’s see how this payment rule works on our examples: Pi = Optimal welfare (for the other players) if player i was not participating. Welfare of the other playersfrom the chosen outcome

14. VCG idea in single item auctions Optimal welfare (for the other players) if player i was not participating. Welfare of the other playersfrom the chosen outcome • Pi= = 2nd-highest value. When i is not playing, the welfare will be the second highest. = 0. When i wins, the total value of the other is 0.  By VCG payments, winners pay the 2nd-highest bid

15. 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

16. VCG in k-item auctions • VCG rules for k-item auctions: • Highest k bids win. • Everyone pay the (k+1)st bid. And truthfulness is a dominant strategy here too. (we will prove it later)

17. Outline • Some examples • VCG idea – intuition  Formal part: • Mechanism design model • The VCG mechanism • Proof: VCG is truthful • VCG: the negative side

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

19. 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

20. Formal model w*=w5 maximal

21. 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

22. Truthfulness Theorem (Vickrey-Clarke-Groves): In the VCG mechanism, truth-telling is a dominant strategy for all players. • Conclusion:welfare maximization can always be achieved in dominant strategies. • No Bayesian distributional assumptions. • No multiple-equilibria as in Nash. • Very simple strategy for the bidders.

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

24. Truthfulness of VCG - Proof • First observation: a payment that does not depend on the player’s bid will not affect her behavior. • Like a sunk cost… • Therefore, we can ignore this term: • This can be a function of the other bids. • It can also be zero – behavior will be the same! • The VCG mechanism: • Outcome w* is chosen. • Each bidder pays:

25. Truthfulness of VCG - Proof • So let’s assume the payment is:In words, each player is paid the sum of the values of the others. • Buyer’s utility: • Bidder i may change the chosen outcome to x by reporting a false type t’. ( but his value is vi(ti,x) not vi(t’,x)) • If lying is beneficial then Impossible! w* maximizes the social welfare.

26. 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.

27. 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: • Choosing ensures individual rationality (when values are positive)(the utility of each player is never negative)and no positive transfers (players are not paid to participate). This could be any function of the other bids.

28. 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 this example:Welfare when buying: v1+v2Welfare when not buying: 100(saved the construction cost)

29. 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.

30. 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 efficient and budget balanced.

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

32. 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 (some hardly use the bridge).

33. More examples • Auctions with externalities • Collusions • False name bids • Revenue monotonicity