1. VCG Computational game theory Fall 2010 by Inna Kalp and Yosef Heskia

2. Mechanisms with money group of all valuation functions for player i. A – group of alternatives. ( is a single alternative) Each player is determined by his valuation function: A model for n players:

3. Mechanisms with money – cont. A mechanism with money consists of: Social choice function: Payments: - payment for player i. … Mechanism … The utility function of player i:

4. Truthful mechanism: Definition: Incentive compatible (IC) (or truthful) mechanism: i.e. – the utility function of player i maximizes when he tells the truth.

5. Our goal: Find a mechanism that: 1)Incentive Compatible. 2)Maximizes the social welfare. Definition: social welfare: Where is the true valuation of player i, i.e. no one is lying..

6. VCG Mechanism Definition: VCG (Vichery Clarke Grove) mechamism: a mechanism with money s.t: Given valuations, returns:

7. Theorem: VCG mechamism is: (1) IC (2) Maximizes social welfare VCG mechanism. Cont.

8. Proof: first, it follows by definition that VCG maximizes social welfare, since: We now have to prove that VCG is Incentive Compatible. VCG maximizes social welfare

9. We shall fix player i and valuations vector for the other players. We’ll define as the real valuation, and as the false valuation. We also define: VCG is IC The player’s utility when he tell the truth : The player’s utility when reports false valuation:

10. VCG is IC- Cont. Since a maximizes social welfare, We conclude that, and thus VCG mechanism is IC.

11. Generalization of VCG Definition: a function f is called Affine maximizer: note: if f is called Maximal in Range

12. Affine Maximizer - Cont. Claim: we can turn an Affine maximizer to be IC. We define:

13. Affine maximizer is IC- Cont. proof: lets calculate the utility functions for vi (true) and vi’ (false):

14. Affine maximizer is IC- Cont. Since maximizes social welfare, We conclude that Affine maximizer is IC.

15. Roberts Theorem Roberts Theorem: if Vi can be any function, then there are no other mechanisms that are Incentive Compatible.

16. Definitions: A mechanism is Individually Rational (IR) if: i.e. every player benefits from the game, no mater what the outcome is. A mechanism has No Positive Transfers (NPT) if:

17. Clarke Pivot Payments Clarke Pivot Payments (CPP) determine the payments in VCG (they determine hi): = “what could have been the social welfare if player i didn’t participate”

18. VCG with CPP has NPT: Using CPP the payments are: All the payments are positive -> the mechanism has no Positive Transfers!

19. VCG with CPP is IR: Claim: and the mechanism is Individually rational. proof: Since f maximizes social welfare

20. Example 1: Single Item Auction We define a single item auction by: In this case, f is reduced into:

21. Example 1: cont. The mechanism will give the item to the highest bidder, and the players will receive vi$ The mechanism is IC (no one will be better off by lying)

22. Example 1: using CPP Second Price Auction

23. Example 2: selling k items with CPP Each player wants only one item. There are k (identical) items for sale: K highest bidders win and pay

24. Example 3: Bilateral Trade Each player wants only one item. There are k (identical) items for sale:

25. Example 3: Bilateral Trade Since Vs<Vb, The mechanism subsidizes the project

26. Example 4: public project In a public project like bridge, the players benefit from building. The problem is how to divide the cost of building. The mechanism subsidizes the project

27. Uniqueness Of Prices Describe each type (valuation function) v i ∊ V i as |A| = m dimensional vector v i =(v i (a 1 ), …, v i (a m )). Note that V i ⊂ R |A| Uniqueness Of Prices: If for every i: V i ⊂ R |A| is con, and mechanism (f, p 1, …, p n ) is IC then: Mechanism (f, p 1 ’, …, p n ’) is IC iff For every i, v i there exists h(v -i ) s.t. p i ’(v i, v -i ) = p i (v i, v -i ) + h(v -i )

28. Uniqueness Of Prices (cont.)  Trivial – given a certain v -i, player i can’t influence the prices in (f, p 1 ’, …, p n ’). That is, since prices are off by a constant h(v -i ) compared to prices of (f, p 1, …, p n ), thus: for every v i (“the truth”) and every other v i ’: v i (f(v i ’, v -i )) – p i ’(v i ’, v -i ) = v i (f(v i ’, v -i )) - p i (v i ’, v -i ) + h(v -i ) (assumption) <= v i (f(v i, v -i )) - p i (v i, v -i ) + h(v -i ) (truthful) = v i (f(v i, v -i )) – p i ’(v i, v -i ) (assumption)  (f, p 1 ’, …, p n ’) is truthful as well.

29. Uniqueness Of Prices (Lemma)  : First we prove the following lemma: A mechanism that maximizes social welfare is IC  (1) For every i, v -i, every v i, v i ’: f(v i, v -i ) = f(v i ’, v -i )  p i (v i, v -i ) = p i (v i ’, v -i ) Proof: Assume otherwise – so given i, v -i, v i, v i ’ s.t. f(v i, v -i ) = f(v i ’, v -i ) and p i (v i, v -i ) < p i (v i ’, v -i ) (w.l.o.g.): -p i (v i, v -i ) > -p i (v i ’, v -i ) (add v i ’(f(v i ’, v -i ))) v i ’(f(v i ’, v -i ))- p i (v i, v -i ) > v i ’(f(v i ’, v -i ))- p i (v i ’, v -i ) but f(v i, v -i ) = f(v i ’, v -i ) - switch in left side:

30. Lemma (cont.) v i ’(f(v i, v -i ))- p i (v i, v -i ) > v i ’(f(v i ’, v -i )) - p i (v i ’, v -i ) thus mechanism is not truthful (“if v i ’ is the truth – better saying v i ”) Note that if combined with (1), we have: (2) f(v i, v -i ) ∊ argmax a ∊ A (v i (a)-P a ) (P a “price for player i if a is the result” is uniform for every v i that gives: f(v i,v -i ) = a – along with condition (1)) It’s (1), (2) iff IC, but it’s not needed for Uniqueness Of Prices.

31. Uniqueness Of Prices (  side) We have V i ⊂ R |A| connected, m=(f, p 1, …, p n ) and m’=(f, p 1 ’, …, p n ’) IC mechanisms. We fix i, v-i. We need to show for every v i, v i ’ ∊ V i that p i ’(v i, v -i ) = p i (v i, v -i ) + h(v -i ) thus p i ’(v i, v -i ) - p i (v i, v -i ) = h(v -i ) (fixed, for every v i )  for every v i, v i ’ : p i ’(v i,v -i ) - p i (v i,v -i ) = p i ’(v i ’,v -i ) - p i (v i ’,v -i )) Define division of V to fragments: V a = {v i ∊ V i |f(v i, v -i ) = a} – thus price for i on every v i ∊ V a is fixed at P a in m and P a ’ in m’ (both truthful, according lemma)

32.  side (cont) So if v i,v i ’ ∊ V a we get: p i ’(v i,v -i ) - p i (v i,v -i ) = P a ’ – P a = p i ’(v i ’,v -i ) - p i (v i ’,v -i )) But if v i ∊ V a, v i ’ ∊ V b for a != b: Definition: V a, V b are “close” if: Now, if a,b are close, for the v i a,v i b that are epsiolon apart: m is truthful, v i a ∊ V a thus: v i a (a) – P a >= v i a (b) – P b Similarily: v i b ∊ V b  v i b (a) – P a <= v i b (b) – P b

33.  side (cont) Combine them to get: v i a (a) - v i a (b) >= P a – P b >= v i b (a) - v i b (b) Now P a – P b is caught in a sandwich, whose size is: v i a (a) - v i a (b) – (v i b (a) - v i b (b)) = (v i a (a) - v i b (a)) + (v i b (b) - v i a (b)) a,b are “close” so each red expression <= epsilon (by definition)  v i a (a) - v i a (b) = P a – P b = v i b (a) - v i b (b) We do the same for m’ and get: P’ a – P’ b = v i b (a) - v i b (b) = P a – P b

34.  side (cont) that’s since v i b (a) - v i b (b) doesn’t change in m’ (valuation functions are the same) P’ a – P’ b = P a – P b  P’ a – P a = P’ b – P b Meaning, for v i ∊ V a, v i ’ ∊ V b where a,b are close we got the needed p i ’(v i,v -i ) - p i (v i,v -i ) = P’ a – P a = P’ b – P b = p i ’(v i ’,v -i ) - p i (v i ’,v -i )). Note that V is connected and continuous, so between every V a, V b ⊂ V i there must be chain a, a 1, …, a m, b close in couples

35. Uniqueness Of Prices conclusion  P’ a – P a = P’ a 1 – P a 1 ’ = … = P’ a m – P a m = P’ b – P b  We get the needed result: for each v i, v i ’ ∊ V i : p i ’(v i,v -i ) - p i (v i,v -i ) = p i ’(v i ’,v -i ) - p i (v i ’,v -i ))  for each v i ∊ V i : p i ’(v i, v -i ) = p i (v i, v -i ) + h(v -i ) We showed “  side” as well, and thus “Uniqueness Of Prices”

36. Weak Monotonicity Definition: function f: (V 1 * …* V n )  A is WMON if: Note that it’s not a complete mechanism, but just its alternative picking function. Also note that if a=b, it trivially holds as well.

37. Saks Yu Theorem (1) For m=(f, p 1, …, p n ) IC mechanism  f is WMON. (2) If every V i ⊂ R |A| is convex and given f WMON  exists p 1, …, p n Payments functions s.t. m=(f, p 1, …, p n ) is IC (alternatively f is “implementable”) Proof for (1): assume otherwise, thus exists f(v i, v -i ) = a != b = f(v i ’, v -i ) s.t. v i (a) - v i (b) < v i ‘(a) - v i ‘(b)

38. Saks Yu Theorem (cont.) but f is IC, so Uniqueness of prices: v i (a)-P a >=v i (b)-P b as well as v i ‘(a)-P a <=v i ‘(b)-P b Add inequations: v i (a)-P a + v i ‘(b)-P b >=v i (b)-P b +v i ‘(a)-P a  v i (a) + v i ‘(b) >=v i (b)+v i ‘(a)  v i (a) - v i (b) >=v i ‘(a) - v i ‘(b) – contradiction.  f is WMON The (2) proof of Saks Yu will be skipped for now.

39. Weak Monotonicity (examples) One Item Auction: Under the assumptions that non zero value for player i exists only if chosen alternative is that it gets the item. Under m players auction V i ⊂ R m but non zero value exists only on the axis of “player i won” – so V i is a segment from (0,…,0) to the direction of axis i. Note that Saks Yu (2) applies as every V i is convex  f is implementable (e.g. VCG)

40. WMON (One Item Auction) For example with 2 players V 1, V 2 ⊂ R 2 looks like that: If we fix v -1 = v 2 and observe v 1 on its possible axis, then according to WMON, if there are 2 possible outcomes (1 wins (a) and 2 wins (b)) then if f(v 1, v 2 ) = a, f(v 1 ’, v 2 ) = b  v 1 (a) – v 1 (b) >= v 1 ‘(a) – v 1 ‘(b) but for every v ∊ V 1 v(b) = 0  v 1 (a) >= v 1 ‘(a) thus if 1 won, its valuation is greater than

41. WMON (One Item Auction cont) any possible valuationwhich led to b (1 didn’t win), so there has to be some threshold in V 1 over which all v i emit f(v i ) = a, and under which f(v i ) = b. Obviousely, the threshold is v 2 (b) – valuation of the item by the other player.

42. WMON (public project example) Another example: Public project. For player i, fix v -i has again only 2 alternatives – project constructed – a, and not constructed – b. Again valuation for b is zero, leaving with similar layout – out of R 2 space V i occupies the positive axis of alternative a. Again, if the 2 alternatives are possible, we’ll get the same pattern with the threshold as the WMON forced on the one item auction previousely.

43. Cyclic Monotonicity Definition: function f: (V 1 * …* V n )  A is Cyclic Monotone (CMON) if: Note that when limited only to k=2, it’s exactly the WMON condition.

44. CMON (Alternative definition) An alternative way to represent CMON is by fixing some v -i, and defining weighted, complete directed graph G=(V i,E), W: E  R. Each vertex v i ∊ V i has also its corresponding f value – f(v i, v -i ). Now define W(v k,v j ) (edge directed from v k to v j ), where f(v k, v -i ) = a, f(v j, v -i ) = b: W(v k,v j ) = v j (b)-v j (a) Intuition: Edge (v k,v j ), if the truth is v j how much damage will be caused if you’ll say you’re v k

45. CMON (Alternative def. cont.) Now, CMON condition in graph representation for each i of f is that the graph doesn’t contain any directed negative cycles in terms of summing the cycle’s edges.

46. Rochet Theorem Given f: (V 1 * …* V n )  A: f is CMON iff f is implementable ( = there exists payment functions p 1, …, p n s.t. m=(f, p 1, …, p n ) yields IC mechanism) Proof:  : We’ll build explicitly the payment functions p 1, …, p n based on CMON f. For p i We’ll pick a some v i ∊ V i, and based on it for every v i ’ ∊ V i : p i (v i ’, v -i ) will be the shortest path on the weighted (f, v -i ) graph from v i to v i ’.

47. Rochet Theorem (  side proof) Shortest paths on the graph is well defined since f is CMON. Shortest paths also has the triangle inequality over the distances as follows (for every v i ’’, v i ’ ∊ V i ): dist(v i, v i ’’) <= dist(v i, v i ’) + w(v i ’, v i ’’) [Recall dist(v i, x) is p i (x, v -i ) and w(v i ’, v i ’’) is v i ’’(f(v i ’’,v -i )) - v i ’’(f(v i ’,v -i ))]  p i (v i ’’, v -i ) <= p i (v i ’, v -i ) + v i ’’(f(v i ’’,v -i )) – v i ’’(f(v i ’,v -i ))(move sides)  v i ’’(f(v i ’,v -i )) - p i (v i ’, v -i ) <= v i ’’(f(v i ’’,v -i )) – p i (v i ’’, v -i )  f is IC

48. Rochet Theorem (  side proof) Now, let f be implementable, assume in contradiction that the graph contains negative cycle: Thus we get: (v j (b)-v j (a)) + (v k (c)-v k (b)) + (v l (d)-v l (c)) + (v i (a)-v i (d)) < 0

49.  side proof (cont.) But f is implementable  exists m=(f, p 1, …, p n ) IC mechanism. Uniqueness of prices – denote p i (v i, v -i ), where f(v i, v -i )=a to be P a IC: v j (b)-P b >=v j (a)-P a  v j (b)-v j (a)>= P b -P a v k (c)-P c >=v k (b)-P b  v k (c)-v k (b)>= P c -P b v l (d)-P d >=v l (c)-P c  v l (d)-v l (c)>= P d -P c v i (a)-P a >=v i (d)-P d  v i (a)-v i (d)>= P a -P d

50.  side proof (cont.) Now we sum the columns: (v j (b)-v j (a)) + (v k (c)-v k (b)) + (v l (d)-v l (c)) + (v i (a)-v i (d)) >= P b - P a + P c - P b + P d - P c + P a – P d = 0 Thus we a, b, c, d is not a negative cycle  contradiction to the existence of negative cycles  f is CMON.