Gross Substitutes Tutorial

Transcription

1 Gross Substitutes Tutorial Part I: Combinatorial structure and algorithms (Renato Paes Leme, Google) Part II: Economics and the boundaries of substitutability (Inbal Talgam-Cohen, Hebrew University)

2 Three seemingly-independent problems

3 Three seemingly-independent problems [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes

4 Three seemingly-independent problems [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes [Dress-Wenzel 91] generalize Grassmann-Plucker relations valuated matroids matroidal maps

5 Three seemingly-independent problems [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes [Dress-Wenzel 91] generalize Grassmann-Plucker relations valuated matroids matroidal maps [Murota-Shioura 99] generalize convexity to discrete domains M-discrete concave

6 Three seemingly-independent problems [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes Discrete Convex Analysis [Dress-Wenzel 91] generalize Grassmann-Plucker relations valuated matroids matroidal maps [Murota-Shioura 99] generalize convexity to discrete domains M-discrete concave

7 Some notation to start Discrete sets of goods: [n] ={1,...,n} Valuation function v :2 [n]! R Given prices p 2 R n define v p (S) =v(s) p(s) Demand correspondence D(v; p) = argmax S v p (S) Demand oracle O D (v, p) 2 D(v; p) Value oracle O V (v, S) =v(s) Marginals v(s T )=v(s [ T ) v(t )

8 Walrasian equilibrium n goods m buyers

9 Walrasian equilibrium n goods m buyers v 1 v 2 v 3 v 4 Valuations v i :2 N! R

10 Walrasian equilibrium p 1 p 2 p 3 p 4 p 5 p 6 n goods m buyers v 1 v 2 v 3 v 4 Valuations v i :2 N! R

11 Walrasian equilibrium p 1 p 2 p 3 p 4 p 5 p 6 n goods m buyers v 1 v 2 v 3 v 4 Valuations Demands v i :2 N! R D(v i,p) = argmax S N [v i (S) P i2s p i]

12 Walrasian equilibrium p 1 p 2 p 3 p 4 p 5 p 6 n goods S 1 2 D(v 1,p) m buyers v 1 v 2 v 3 v 4 Valuations Demands v i :2 N! R D(v i,p) = argmax S N [v i (S) P i2s p i]

13 Walrasian equilibrium p 1 p 2 p 3 p 4 p 5 p 6 n goods S 1 2 D(v 1,p) S 2 2 D(v 2,p) m buyers v 1 v 2 v 3 v 4 Valuations Demands v i :2 N! R D(v i,p) = argmax S N [v i (S) P i2s p i]

14 Walrasian equilibrium p 1 p 2 p 3 p 4 p 5 p 6 n goods S 1 2 D(v 1,p) S 2 2 D(v 2,p) ;2D(v 3,p) m buyers v 1 v 2 v 3 v 4 Valuations Demands v i :2 N! R D(v i,p) = argmax S N [v i (S) P i2s p i]

15 Walrasian equilibrium p 1 p 2 p 3 p 4 p 5 p 6 n goods S 1 2 D(v 1,p) S 2 2 D(v 2,p) S 4 2 D(v 4,p) ;2D(v 3,p) m buyers v 1 v 2 v 3 v 4 Valuations Demands v i :2 N! R D(v i,p) = argmax S N [v i (S) P i2s p i]

16 Walrasian equilibrium Market equilibrium: prices p 2 R n s.t. S i 2 D(v i,p) i.e. each good is demanded by exactly one buyer. First Welfare Theorem: in equilibrium the welfare P i v i(s i ) is maximized. (proof: LP duality) When do equilibria exist? How do markets converge to equilibrium prices? How to compute a Walrasian equilibrium?

17 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

18 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

19 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

20 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

21 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

22 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

23 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

24 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

25 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

26 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

27 Walrasian tatonnement n goods p i j = p j if j 2 S i p i j = p j + if j/2 S i m buyers v 1 v 2 v 3 v 4 Initialize S 1 =[n], S i = ; and prices p j =0 While there is S i /2 D(v i,p i ) assign X i 2 D(v i ; p i i) to i and increase the prices in X i \ S i by.

28 Walrasian tatonnement This process always ends, otherwise prices go to infinity. When it ends S i 2 D(v i ; p i )

29 Walrasian tatonnement This process always ends, otherwise prices go to infinity. When it ends S i 2 D(v i ; p) in the limit! 0

30 Walrasian tatonnement This process always ends, otherwise prices go to infinity. When it ends S i 2 D(v i ; p) in the limit! 0 What else?

31 Walrasian tatonnement This process always ends, otherwise prices go to infinity. When it ends S i 2 D(v i ; p) in the limit! 0 What else? The only condition left is that [ i S i =[n] For that we need: S i X i 2 D(v i ; p i )

32 Walrasian tatonnement This process always ends, otherwise prices go to infinity. When it ends S i 2 D(v i ; p) in the limit! 0 What else? The only condition left is that [ i S i =[n] For that we need: S i X i 2 D(v i ; p i ) Definition: A valuation satisfied gross substitutes if for all prices p apple p 0 and S 2 D(v; p) there is X 2 D(v; p 0 ) s.t. S \ {i; p i = p 0 i} X

33 Walrasian tatonnement This process always ends, otherwise prices go to infinity. When it ends S i 2 D(v i ; p) in the limit! 0 What else? The only condition left is that [ i S i =[n] For that we need: S i X i 2 D(v i ; p i ) Definition: A valuation satisfied gross substitutes if for all prices p apple p 0 and S 2 D(v; p) there is X 2 D(v; p 0 ) s.t. S \ {i; p i = p 0 i} X With the new definition, the algorithm always keeps a partition.

34 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists.

35 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S

36 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S

37 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S

38 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S

39 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S

40 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S matroid-matching

41 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S matroid-matching

42 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Some examples of GS: additive functions v(s) = P i2s v(i) unit-demand v(s) = max i2s v(i) matching valuations v(s) = max matching from S matroid-matching Open: GS?= matroid-matching

43 Walrasian equilibrium Theorem [Kelso-Crawford]: If all agents have GS valuations, then Walrasian equilibrium always exists. Theorem [Gul-Stachetti]: If a class C of valuations contains all unit-demand valuations and Walrasian equilibrium always exists then C GS

44 Valuated Matroids Given vectors v 1,...,v m 2 Q n define p(v 1,...,v n )=n if det(v 1,...,v n )=p n a/b for p prime a, b, p 2 Z. Question in algebra: min v i 2V p(v 1,...,v n )s.t. det(v 1,...,v n ) 6= 0 Solution is a greedy algorithm: start with any nondegenerate set and go over each items and replace it by the one that minimizes p(v 1,...,v n ). [DW]: Grassmann-Plucker relations look like matroid cond

45 Valuated Matroids Definition: a function v :! R is a valuated [n] k matroid if the Greedy is optimal.

46 Matroidal maps Definition: a function v :2 [n]! R is a matroidal map if for every p 2 R n a set in D(v; p) can be obtained by the greedy algorithm : S 0 = ; and S t = S t 1 [ {i t } for i t 2 argmax i v p (i S t )

47 Matroidal maps Definition: a function v :2 [n]! R is a matroidal map if for every p 2 R n a set in D(v; p) can be obtained by the greedy algorithm : S 0 = ; and S t = S t 1 [ {i t } for i t 2 argmax i v p (i S t ) Definition: a subset system M 2 [n] is a matroid if for every p 2 R n the problem max p(s) can be solved S2M by the greedy algorithm.

48 Discrete Concavity A function f : R n! R is convex if for all p 2 R n, a local minimum of f p (x) =f(x) hp, xi is a global minimum. Also, gradient descent converges for convex functions. We want to extend this notion to function in the hypercube: v :2 [n]! R (or lattice v : Z [n]! R or other discrete sets such as the basis of a matroid)

49 Discrete Concavity A function f : R n! R is convex if for all p 2 R n, a local minimum of f p (x) =f(x) hp, xi is a global minimum. Also, gradient descent converges for convex functions. We want to extend this notion to function in the hypercube: v :2 [n]! R (or lattice v : Z [n]! R or other discrete sets such as the basis of a matroid)

50 Discrete Concavity A function f : R n! R is convex if for all p 2 R n, a local minimum of f p (x) =f(x) hp, xi is a global minimum. Also, gradient descent converges for convex functions. We want to extend this notion to function in the hypercube: v :2 [n]! R (or lattice v : Z [n]! R or other discrete sets such as the basis of a matroid)

51 Discrete Concavity A function v :2 [n]! R is discrete concave if for all p 2 R n all local minima of v p are global minima. I.e. v p (S) v p (S [ i), 8i /2 S v p (S) v p (S \ j), 8j 2 S v p (S) v p (S [ i \ j), 8i /2 S, j 2 S then v p (S) v p (T ), 8T [n]. In particular local search always converges. [Murota 96] M-concave (generalize valuated matroids) [Murota-Shioura 99] M \ -concave functions

52 Equivalence [Fujishige-Yang] A function v :2 [n]! R is gross substitutes iff it is a matroidal map iff it is discrete concave. [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes [Murota-Shioura 99] generalize convexity to discrete domains M-discrete concave [Dress-Wenzel 91] generalize Grassmann-Plucker relations valuated matroids matroidal maps

53 Equivalence [Fujishige-Yang] A function v :2 [n]! R is gross substitutes iff it is a matroidal map iff it is discrete concave. [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes [Murota-Shioura 99] generalize convexity to discrete domains M-discrete concave [Dress-Wenzel 91] generalize Grassmann-Plucker relations valuated matroids matroidal maps In particular S 2 D(v; p) in poly-time.

54 Equivalence [Fujishige-Yang] A function v :2 [n]! R is gross substitutes iff it is a matroidal map iff it is discrete concave. [Kelso-Crawford 82] necessary / sufficient condition for price adjustment to converge gross substitutes [Murota-Shioura 99] generalize convexity to discrete domains M-discrete concave [Dress-Wenzel 91] generalize Grassmann-Plucker relations valuated matroids matroidal maps In particular S 2 D(v; p) in poly-time. Proof through discrete differential equations

55 Discrete Differential Equations Given a function v :2 [n]! R we define the discrete derivative with respect to i 2 [n] as the i v :2 [n]\i! R which is given i v(s) =v(s [ i) v(s) (another name for the marginal)

56 Discrete Differential Equations Given a function v :2 [n]! R we define the discrete derivative with respect to i 2 [n] as the i v :2 [n]\i! R which is given i v(s) =v(s [ i) v(s) (another name for the marginal) If we apply it twice we ij v(s) :=@ i v(s) =v(s [ ij) v(s [ i) v(s [ j)+v(s) ij v(s) apple 0

57 Discrete Differential Equations [Reijnierse, Gellekom, Potters] A function v :2 [n]! R is in gross substitutes iff it ij v(s) apple max(@ ik kj v(s)) apple 0 condition on the discrete Hessian. Idea: A function is in GS iff there is not price such that: D(v; p) ={S, S [ ij} or D(v; p) ={S [ k, S [ ij} If v is not submodular, we can construct a price of the first type. ij v(s) > max(@ ik kj v(s)) then we can find a certificate of the second type.

58 Algorithmic Problems Welfare problem: given m agents with v 1,...,v m :2 [n]! R P find a partition S 1,...,S m of [n] maximizing i v i(s i ) Verification problem: given a partition S 1,...,S m find whether it is optimal. Walrasian prices: given the optimal partition (S 1,...,S m) find a price such that S i 2 argmax S v i (S) p(s)

59 Algorithmic Problems Techniques: Tatonnement Linear Programming Gradient Descent Cutting Plane Methods Combinatorial Algorithms

60 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 {0, [0, 1] 1}

61 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1]

62 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual

63 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal For GS, the IP is integral: min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual W IP apple W LP = W D-LP Consider a Walrasian equilibrium and p the Walrasian prices and u the agent utilities. Then it is a solution to the dual, so: W D-LP apple W eq = W IP

64 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual

65 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual In general, Walrasian equilibrium exists iff LP is integral.

66 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual In general, Walrasian equilibrium exists iff LP is integral. Separation oracle for the dual: is the demand oracle problem. u i max S v i(s) p(s)

67 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual

68 Linear Programming [Nisan-Segal] Formulate this problem as an LP: max P i v i(s)x is P S x is =1, 8i 2 [m] P P i S3j x is =1, 8j 2 [n] x is 2 [0, 1] primal min P i u i + P j p j P u i v i (S) j2s p j8i, S p j 0,u i 0 dual Walrasian equilibrium exists + demand oracle in poly-time = Welfare problem in poly-time [Roughgarden, Talgam-Cohen] Use complexity theory to show non-existence of equilibrium, e.g. budget additive.

69 Gradient Descent We can Lagrangify the dual constraints and obtain the following convex potential function: (p) = P i max S[v i (S) p(s)] + P j p j Theorem: the set of Walrasian prices (when they exist) are the set of minimizers of. j (p) =1 i 1[j 2 S i]; S i 2 D(v i ; p) Gradient descent: increase price of over-demanded items and decrease price of over-demanded items. Tatonnement: p j p j j (p)

70 Comparing Methods method oracle running-time

71 How to access the input

72 How to access the input Value oracle: given i and S: query v i (S).

73 How to access the input Value oracle: given i and S: query v i (S). Demand oracle: given i and p: query S 2 D(v i,p).

74 How to access the input Value oracle: given i and S: query v i (S). Demand oracle: given i and p: query S 2 D(v i,p). Aggregate Demand: given p, query. P i S i; S i 2 D(v i,p)

75 Comparing Methods method oracle running-time tatonnement/gd aggreg demand pseudo-poly

76 Comparing Methods method oracle running-time tatonnement/gd aggreg demand pseudo-poly linear program demand/value weakly-poly

77 Comparing Methods method oracle running-time tatonnement/gd aggreg demand pseudo-poly linear program demand/value weakly-poly cutting plane aggreg demand weakly-poly [PL-Wong]: We can compute an exact equilibrium with Õ(n) calls to an aggregate demand oracle.

78 Comparing Methods method oracle running-time tatonnement/gd aggreg demand pseudo-poly linear program demand/value weakly-poly cutting plane aggreg demand weakly-poly

79 Comparing Methods method oracle running-time tatonnement/gd aggreg demand pseudo-poly linear program demand/value weakly-poly cutting plane aggreg demand weakly-poly combinatorial value strongly-poly

80 Comparing Methods method oracle running-time tatonnement/gd aggreg demand pseudo-poly linear program demand/value weakly-poly cutting plane aggreg demand weakly-poly combinatorial value strongly-poly [Murota]: We can compute an exact equilibrium for gross susbtitutes in Õ((mn + n 3 )T V ) time.

81 Algorithmic Problems Welfare problem: given m agents with v 1,...,v m :2 [n]! R P find a partition S 1,...,S m of [n] maximizing i v i(s i ) Verification problem: given a partition S 1,...,S m find whether it is optimal. Walrasian prices: given the optimal partition (S 1,...,S m) find a price such that S i 2 argmax S v i (S) p(s)

82 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items.

83 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items.

84 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items. w jk = v i (S i ) v i (S i [ k \ j)

85 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items. w i k = v i (S i ) v i (S i [ k)

86 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items. w j i 0 = v i (S i ) v i (S i \ j)

87 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items. w i i 0 =0

88 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items.

89 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items.

90 Computing Walrasian prices Given a partition S 1,...,S m we want to find prices such that S i 2 argmax S v i (S) p(s) For GS, we only need to check that no buyer want to add, remove or swap items.

91 Computing Walrasian prices Theorem: the allocation is optimal if the exchange graph has no negative cycle. Proof: if no negative cycles the distance is well defined. So let p j = dist(,j) then: dist(,k) apple dist(,j)+w jk v i (S i ) v i (S i [ k \ j) p k + p j And since S i is locally-opt then it is globally opt. Conversely: Walrasian prices are a dual certificate showing that no negative cycles exist.

92 Computing Walrasian prices Theorem: the allocation is optimal if the exchange graph has no negative cycle. Proof: if no negative cycles the distance is well defined. So let p j = dist(,j) then: dist(,k) apple dist(,j)+w jk v i (S i ) v i (S i [ k \ j) p k + p j And since S i is locally-opt then it is globally opt. Conversely: Walrasian prices are a dual certificate showing that no negative cycles exist. Nice consequence: Walrasian prices form a lattice.

93 Algorithmic Problems Welfare problem: given m agents with v 1,...,v m :2 [n]! R P find a partition S 1,...,S m of [n] maximizing i v i(s i ) Verification problem: given a partition S 1,...,S m find whether it is optimal. Walrasian prices: given the optimal partition (S 1,...,S m) find a price such that S i 2 argmax S v i (S) p(s)

94 Algorithmic Problems Welfare problem: given m agents with v 1,...,v m :2 [n]! R P find a partition S 1,...,S m of [n] maximizing i v i(s i ) Verification problem: given a partition S 1,...,S m find whether it is optimal. Walrasian prices: given the optimal partition (S 1,...,S m) find a price such that S i 2 argmax S v i (S) p(s)

95 Algorithmic Problems Welfare problem: given m agents with v 1,...,v m :2 [n]! R P find a partition S 1,...,S m of [n] maximizing i v i(s i ) Verification problem: given a partition S 1,...,S m find whether it is optimal. Walrasian prices: given the optimal partition (S 1,...,S m) find a price such that S i 2 argmax S v i (S) p(s)

96 Incremental Algorithm For each t =1..n we will solve problem to find the optimal allocation of items [t] ={1..t} to m buyers. Problem is easy. W 1 W t W t Assume now we solved getting allocation S 1,...,S m and a certificate p = maximal Walrasian prices. w jk = v i (S i ) v i (S i [ k \ j)+p k p j w j i 0 = v i (S i ) v i (S i \ j) p j w i k = v i (S i ) v i (S i [ k)+p k

97 Incremental Algorithm For each t =1..n we will solve problem to find the optimal allocation of items [t] ={1..t} to m buyers. Problem is easy. W 1 W t W t Assume now we solved getting allocation S 1,...,S m and a certificate p = maximal Walrasian prices.

98 Incremental Algorithm For each t =1..n we will solve problem to find the optimal allocation of items [t] ={1..t} to m buyers. Problem is easy. W 1 W t W t Assume now we solved getting allocation S 1,...,S m and a certificate p = maximal Walrasian prices.

99 Incremental Algorithm Algorithm: compute shortest path from to t +1 Update allocation by implementing path swaps

100 Incremental Algorithm Algorithm: compute shortest path from to t +1 Update allocation by implementing path swaps

101 Incremental Algorithm Algorithm: compute shortest path from to t +1 Update allocation by implementing path swaps

102 Incremental Algorithm Algorithm: compute shortest path from to t +1 Update allocation by implementing path swaps Graph has O(t 2 + mt) non-negative edges After n iterations of Dijkstra we get Õ(n 3 + n 2 m)

103 Incremental Algorithm Proof that new allocation S 1... S m is optimal Define the new prices p j = dist(,j) (1) New prices are also a certificate for S 1...S m (2) v i (S i ) p(s i )=v i ( S i ) p( S i ) Hence, S 1... S m and p are Walrasian prices.

104 Closure properties If v 1,v 2 2 GS we might not have v 1 + v 2 2 GS

105 Closure properties If v 1,v 2 2 GS we might not have v 1 + v 2 2 GS Some preserving operations: affine transformation ṽ(s) =v(s)+p 0 Pi2S p i endowment convolution ṽ(s) =v(s X) v 1 v 2 (S) = max T S v 1 (T )+v 2 (S \ T ) strong-quotient-sum tree-concordant-sum

106 Closure properties If v 1,v 2 2 GS we might not have v 1 + v 2 2 GS Some preserving operations: affine transformation ṽ(s) =v(s)+p 0 Pi2S p i endowment convolution ṽ(s) =v(s X) v 1 v 2 (S) = max T S v 1 (T )+v 2 (S \ T ) strong-quotient-sum tree-concordant-sum Open question: can we construct all gross substitutes from matroid rank functions and those operations? Some progress: See talk by Eric Balkanski on Thu

107 End of Part I