Mechanism Design

1. Mechanism Design Ruta Mehta

2. Game design (not video games!) to achieve a desired goal, like fairness, social welfare maximization, etc.

3. Widely Applicable

4. Lets Focus on Voting • Setting • Candidates a, b and c • Voters with preference order over {a, b, c} • Select the majority choice Need for more complex mechanisms -> Condorcet’s Paradox!

5. First Preference Majority • Example: • Winner: a Encourages Strategic Voting! Tie between a and c

6. Borda Count • Give weight to -th candidate in a preference order. • Take sum over all the voters. • The one with highest sum wins. Is this strategy proof? NO!

7. In General: Social Choice • Setting (Recall): • Set of choices. agents (voters). • Set of all complete preference orders over • Social Choice Function: • Function

8. Desired Properties of f • Incentive-compatible(IC) (strategy proof) • Monotone • then and IC iff Monotone • No dictators • No such that is always the first choice as per .

9. Gibbard-Satterthwaite Theorem: If a social choice function is incentive-compatible, where |A|>=3, then it is dictatorship. “Field of Mechanism Design attempts escaping from this impossibility result using various modifications in the model.”

10. Introducing Payment Single Item Auction: • N bidders. • The item is worth to bidder . • Bid of bidder is . • Incentive Compatible: • If then is best. • IC (in Dominant Strategy): • is the best bid regardless of what others do.

11. Auction • Item goes to the bidder with highest bid. • Payment: • No payments Best to bid • The winner pays his bid, rest nothing Suppose and then • The winner pays the second highest bid, rest nothing (Vickery) Not even IC

12. Vickery Auction is IC (in DS) Proof on board

13. Open-Outcry Auctions • English Auction: Start with a low price and keep increasing until only one buyer is interested. • Equivalent to Vickery auction! • Dutch Auction: Start with a very high price where no one is interested. Keep decreasing until someone gets interested. • Equivalent to first price auction!

14. Extensions? • What if identical items to be auctioned? • What if an item has to be procured? In general how to extract private information from agent to make social optimal decision?

15. General Setting • :Set of alternatives to choose from • : Set of valuation functions () for agent • Each agent reports (may not be true) • The mechanism • Social choice function: • Payments

16. Example: Single Item Auction • ={1-wins, 2-wins, …, n-wins} • True valuation: What is the key idea in Vickery Auction? How to extend that to the general setting?

17. Vickery-Clarke-Groves Mechanism • Let be reported valuations. • Social choice func: Social welfare maximizing • Payments: Align payoff with SW maximization • (Independent of )

18. Properties of VCG Proof on board • Incentive Compatible (in DS) • Individually rational? • Non-negative payoff: • No positive transfers? No if No if

19. Clarke Pivot Rule Pay the damage you cause • captures maximum social welfare excluding • - • IR: YES! • No positive transfer:YES! Others welfare without i Others welfare with i

20. Example: Single Item Auction • Recall: • ={1-wins, 2-wins, …, n-wins} • True valuation: • Report: , zero elsewhere • Social choice: • Highest bid • Payment: • Verify: Winner pays second highest bid

21. Example: Multiunit Auction identical items, each agent wants one unit. • Agent values it at and reports • Alternatives? • Set of valuation functions and bid s? • Report: • Winners? • Bidders with highest bids win • Payments? • Each winner pays highest bid

22. Example: Reverse Auction Procure an item from agents • Agent reports cost Break: • Winner? • Payment?

23. Example: Resource Allocation Buying an s-t path in a network • , each edge is owned by an agent • Edge costs to its agent (value ) • Cost of a path is • Social welfare maximizing path: Shortest path • Payment to edge : • If then zero • Else, let be the shortest path in .

24. Example: Multi Item Auction different items on auction. • Agent values set , nothing more nothing less (Single minded bidder) • Value of set is if , else zero. • Agent reports • Computation of social welfare maximizing allocation is NP-hard! • Even approximation On board

25. Approximate Algorithm • Reorder: . Let • For each • If then • Payments: For • ; is smallest s.t. and • Observe • is the lowest possible bid to win.

26. Incentive Compatible • Suppose (S’,w’) gets better payoff than (S,w) • and (S’,w’) should win • Then (S,w’) should be better • betters the chances of winning and may decrease payment • If (S,w) winning bid then for (S,w’) • w’>w does not help; w’<w may loose • If (S,w) loosing bid then for (S,w’) • w’<w does not help; w’>w may fetch –ve payoff

27. approximate social welfare Break:Show that Recall:

28. Pros and Cons of VCG • Best for bidders • Government auctions like road contract, bandwidth allocation • May not be efficiently computable • Multi item auction • Worst for auctioneer • May get zero payment!

29. Sponsored Search Auctions

31. Ad Auctions • Generalized Second Price (GSP) • Google, Yahoo, Bing • Bid on keywords • If the user query contains your keyword, your bid qualifies for the auction

32. GSP Auction Setting • bidders, slots • Agent values a click with and bids • Suppose • is the probability that user clicks slot • Assumptions: • Same click probability for every agent • Same valuation for every slot

33. GSP Auction • Allocation: Highest bids win. Bidder gets slot . • Payment: Pay the bid of i+1 • No negative transfers • Per query payoff: o.w.

34. GSP Properties • , or else may lead to negative payoff • Individually rational • Incentive Compatible? • Exactly one slot => Vickery’ Second price auction

35. Locally Envy Free Equilibrium • A bid profile such that no one wants to switch the slot with the person above. • Existence of locally envy-free equilibrium with VCG payments • Payment from locally envy-free equilibrium is at least as large as the payment from VCG (same bids)