Mechanism Design without Money

1. Mechanism Design without Money Lecture 12

2. Individual rationality and efficiency: an impossibility theorem with a (discouraging) worst-case bound • For every k> 3, there exists a compatibility graph such that no k-maximum allocation which is also individually rational matches more than 1/(k-1) of the number of nodes matched by a k-efficient allocation.

3. a1 a3 Proof (for k=3) b e c a2 d

4. “Cost” of IR is very small - Simulations

5. But the cost of not having IR could be very high if it causes centralized matching to break down

6. But current mechanisms aren’t IR for hospitals • Current mechanisms: Choose (~randomly) an efficient allocation. Proposition: Withholding internal exchanges can (often) be strictly better off (non negligible) for a hospital regardless of the number of hospitals that participate. A-O O-A And hospitals can withhold individual overdemanded pairs

7. What if we have a prior? • Infinite horizon • In each timestep, a hospital samples its patients from some known distribution • Then there exists a truthful mechanism with efficiency 1 – o(1)

8. Matching • Initially the hospital has zero credits • In the beginning of the round, if the hospital has zero credits, each patients enters the match with probability 1 – 1/k1/6 • For each positive credit, the hospital increases this probability by 1/k2/3and the credit is gone • For each negative credit, the hospital decreases this probability by 1/k2/3 and the credit is gone. The probability is always > ½

9. Gaining credit • For each patient over k, the hospital gets 1 credit • For each patient below k, the hospital looses 1 credit • These credits only affect the next rounds

10. Proof idea • Hiding a patient can give an additive advantage, but causes a multiplicative loss • Number of credit doesn’t matter – you always care about the future • Can work for every distribution of patients

11. Voting

12. Voting rule Social choice: mapping of a profile onto a winner(s) Social welfare: mapping of a profile onto a total ordering Agent Sometimes assume odd number of agents to reduce ties Vote Total order over outcomes Profile Vote for each agent Terminology Extensions include indifference, incomparability, incompleteness

13. Voting rules: plurality • Otherwise known as “majority” or “first past the post” • Candidate with most votes wins • With just 2 candidates, this is a very good rule to use • (See May’s theorem)

14. Voting rules: plurality • Some criticisms • Ignores preferences other than favourite • Similar candidates can “split” the vote • Encourages voters to vote tactically • “My candidate cannot win so I’ll vote for my second favourite”

15. Two rounds Eliminate all but the 2 candidates with most votes Then hold a majority election between these 2 candidates Consider 25 votes: A>B>C 24 votes: B>C>A 46 votes: C>A>B 1st round: B knocked out 2nd round: C>A by 70:25 C wins Voting rules: plurality with runoff

16. Voting rules: plurality with runoff • Some criticisms • Requires voters to list all preferences or to vote twice • Moving a candidate up your ballot may not help them (monotonicity) • It can even pay not to vote! (see next slide)

17. Consider again 25 votes: A>B>C 24 votes: B>C>A 46 votes: C>A>B C wins easily Two voters don’t vote 23 votes: A>B>C 24 votes: B>C>A 46 votes: C>A>B Different result 1st round: A knocked out 2nd round: B>C by 47:46 B wins Voting rules: plurality with runoff

18. STV If one candidate has >50% vote then they are elected Otherwise candidate with least votes is eliminated Their votes transferred (2nd placed candidate becomes 1st, etc.) Identical to plurality with runoff for 3 candidates Example: 39 votes: A>B>C>D 20 votes: B>A>C>D 20 votes: B>C>A>D 11 votes: C>B>A>D 10 votes: D>A>B>C Result: B wins! Voting rules: single transferable vote

19. Voting rules: Borda • Given m candidates • ith ranked candidate score m-i • Candidate with greatest sum of scores wins • Example • 42 votes: A>B>C>D • 26 votes: B>C>D>A • 15 votes: C>D>B>A • 17 votes: D>C>B>A • B wins Jean Charles de Borda, 1733-1799

20. Voting rules: positional rules • Given vector of weights, <s1,..,sm> • Candidate scores si for each vote in ith position • Candidate with greatest score wins • Generalizes number of rules • Borda is <m-1,m-2,..,0> • Plurality is <1,0,..,0>

21. Voting rules: approval • Each voters approves between 1 and m-1 candidates • Candidate with most votes of approval wins • Some criticisms • Elects lowest common denominator? • Two similar candidates do not divide vote, but can introduce problems when we are electing multiple winners

22. Voting rules: other • Cup (aka knockout) • Tree of pairwise majority elections • Copeland • Candidate that wins the most pairwise competitions • Bucklin • If one candidate has a majority, they win • Else 1st and 2nd choices are combined, and we repeat

23. Voting rules: other • Coomb’s method • If one candidate has a majority, they win • Else candidate ranked last by most is eliminated, and we repeat • Range voting • Each voter gives a score in given range to each candidate • Candidate with highest sum of scores wins • Approval is range voting where range is {0,1}

24. Voting rules: other • Maximin (Simpson) • Score = Number of voters who prefer candidate in worst pairwise election • Candidate with highest score wins • Veto rule • Each agent can veto up to m-1 candidates • Candidate with fewest vetoes wins • Inverse plurality • Each agent casts one vetor • Candidate with fewest vetoes wins

25. Voting rules: other • Dodgson • Proposed by Lewis Carroll in 1876 • Candidate who with the fewest swaps of adjacent preferences beats all other candidates in pairwise elections • NP-hard to compute winner! • Random • Winner is that of a random ballot • …

26. Voting rules • So many voting rules to choose from .. • Which is best? • Social choice theory looks at the (desirable and undesirable) properties they possess • For instance, is the rule “monotonic”? • Bottom line: with more than 2 candidates, there is no best voting rule

27. Axiomatic approach • Define desired properties • E.g. monotonicity: improving votes for a candidate can only help them win • Prove whether voting rule has this property • In some cases, as we shall see, we’ll be able to prove impossibility results (no voting rule has this combination of desirable properties)

28. May’s theorem • Some desirable properties of voting rule • Anonymous: names of voters irrelevant • Neutral: name of candidates irrelevant

29. May’s theorem • Another desirable property of a voting rule • Monotonic: if a particular candidate wins, and a voter improves their vote in favour of this candidate, then they still win • Non-monotonicity for plurality with runoff • 27 votes: A>B>C • 42 votes: C>A>B • 24 votes: B>C>A • Suppose 4 voters in 1st group move C up to top • 23 votes: A>B>C • 46 votes: C>A>B • 24 votes: B>C>A

30. May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • May, Kenneth. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", Econometrica, Vol. 20, pp. 680–68 • Since these properties are uncontroversial, this about decides what to do with 2 candidates!

31. May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • Proof: Plurality rule is clearly anonymous, neutral and monotonic • Other direction is more interesting

32. May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • Proof: Anonymous and neutral implies only number of votes matters • Two cases: • N(A>B) = N(B>A)+1 and A wins. • By monotonicity, A wins whenever N(A>B) > N(B>A)

33. May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • Proof: Anonymous and neutral implies only number of votes matters • Two cases: • N(A>B) = N(B>A)+1 and A wins. • By monotonicity, A wins whenever N(A>B) > N(B>A) • N(A>B) = N(B>A)+1 and B wins • Swap one vote A>B to B>A. By monotonicity, B still wins. But now N(B>A) = N(A>B)+1. By neutrality, A wins. This is a contradiction.

34. Condorcet’s paradox • Collective preference may be cyclic • Even when individual preferences are not • Consider 3 votes • A>B>C • B>C>A • C>A>B • Majority prefer A to B, and prefer B to C, and prefer C to A! Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet (1743 – 1794)

35. Condorcet principle • Turn this on its head • Condorcet winner • Candidate that beats every other in pairwise elections • In general, Condorcet winner may not exist • When they exist, must be unique • Condorcet consistent • Voting rule that elects Condorcet winner when they exist (e.g. Copeland rule)

36. Condorcet principle • Plurality rule is not Condorcet consistent • 35 votes: A>B>C • 34 votes: C>B>A • 31 votes: B>C>A • B is easily the Condorcet winner, but plurality elects A

37. Condorcet principle • Thm. No positional rule with strict ordering of weights is Condorcet consistent • Proof: Consider • 3 votes: A>B>C • 2 votes: B>C>A • 1 vote: B>A>C • 1 vote: C>A>B • A is Condorcet winner

38. Condorcet principle • Thm. No positional rule with strict ordering of weights is Condorcet consistent • Proof: Consider • 3 votes: A>B>C • 2 votes: B>C>A • 1 vote: B>A>C • 1 vote: C>A>B • Scoring rule with s1 > s2 > s3 • Score(B) = 3.s1+3.s2+1.s3 • Score(A) = 3.s1+2.s2+2.s3 • Score(C) = 1.s1+2.s2+4.s4 • Hence: Score(B)>Score(A)>Score(C)

39. Arrow’s theorem • We have to break Condorcet cycles • How we do this, inevitably leads to trouble • A genius observation • Led to the Nobel prize in economics

40. Arrow’s theorem • Free • Every result is possible • Unanimous • If every votes for one candidate, they win • Independent to irrelevant alternatives • Result between A and B only depends on how agents preferences between A and B • Monotonic

41. Arrow’s theorem • Non-dictatorial • Dictator is voter whose vote is the result • Not generally considered to be desirable!

42. Arrow’s theorem • Thm: If there are at least two voters and three or more candidates, then it is impossible for any voting rule to be: • Free • Unanimous • Independent to irrelevant alternatives • Monotonic • Non-dictatorial

43. Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose not the case and result has A>B>C • By IIA, this would not change if every voter moved C above A: • B>A>C => B>C>A • B>C>A => B>C>A • A>C>B => C>A>B • C>A>B => C>A>B • Each AB and BC vote the same!

44. Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose not the case and result has A>B>C • By IIA, this would not change if every voter moved C above A • By transitivity A>C in result • But by unanimity C>A • B>A>C => B>C>A • B>C>A => B>C>A • A>C>B => C>A>B • C>A>B => C>A>B

45. Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose not the case and result has A>B>C • A>C and C>A in result • This is a contradiction • B can only be top or bottom in result

46. Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose voters in turn move B from bottom to top • Exists pivotal voter from whom result changes from B at bottom to B at top

47. Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose voters in turn move B from bottom to top • Exists pivotal voter from whom result changes from B at bottom to B at top • B all at bottom. By unanimity, B at bottom in result • B all at top. By unanimity, B at top in result • By monotonicity, B moves to top and stays there when some particular voter moves B up

48. Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose voters in turn move B from bottom to top • Exists pivotal voter from whom result changes from B at bottom to B at top • Pivotal voter is dictator (need to show)

49. Proof of Arrow’s theorem • Pivotal voter is dictator between A and C • Consider profile when pivotal voter has just moved B to top (and B has moved to top of result) • For any AC, let pivotal voter have A>B>C • By IIA, A>B in result as AB votes are identical to profile just before pivotal vote moves B (and result has B at bottom) • By IIA, B>C in result as BC votes are unchanged • Hence, A>C by transitivity

50. Proof of Arrow’s theorem • Each two alternatives {A,C} have a voter which dictates which one of them will be higher. • Let i be the dictator for {A,C} • Let j be the dictator for {A,B} • Let k be the dictator for {B,C} • If ij and jk and ik we can create a cycle: • i prefers A to C • k prefers C to B • j prefers B to A • Similar argument for ij=k, i=j  k, ji=k