How does demand change as you cross a facet? p ( ( 1 2 (2,0) (1,0) 1 (0,0) A tropical hypersurface (0,1) is composed of facets : (1,1) linear pieces in dimen- ( ( -1 (0,2) sion ( n − 1) . 1 p 1 If p is in a facet then the agent is indifferent between two bundles: u ( x ) − p . x = u ( y ) − p . y ⇐ ⇒ p . ( y − x ) = u ( y ) − u ( x ) The change in bundle is in the direction normal to the facet. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 4 / 21
How does demand change as you cross a facet? p 2 A tropical hypersurface is composed of facets : linear pieces in dimen- sion ( n − 1) . p 1 If p is in a facet then the agent is indifferent between two bundles: u ( x ) − p . x = u ( y ) − p . y ⇐ ⇒ p . ( y − x ) = u ( y ) − u ( x ) Change in bundle is minus ‘weight w ’ times minimal facet normal. Endow all facets with weights: weighted rational polyhedral complex . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 4 / 21
Economics from Geometry v 3 w Every tropical hypersurface is balanced : w 3 v 4 around each ( n − 2) -cell, � 4 i w i v i = 0 . v 1 w 2 w 2 v 1 Theorem (Mikhalkin 2004) A weighted rational polyhedral complex of pure dimension ( n − 1) , connected in codimension 1 , is the tropical hypersurface of a valuation iff it is balanced. A TH corresponds to an essentially unique concave valuation. We need not write down valuations of discrete bundles. We can simply draw tropical hypersurfaces. Project Aim understand economics via geometry. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 5 / 21
Classifying valuations Economists classify valuations by how agents see trade-offs between goods. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 6 / 21
Classifying valuations Economists classify valuations by how agents see trade-offs between goods. For divisible goods, ask how changes in each price affect each demand. Let x ∗ ( p ) be optimal demands of each good at a given price. ∂x ∗ ∂p j > 0 means goods are ‘substitutes’ (tea, coffee). i ∂x ∗ ∂p j < 0 means goods are ‘complements’ (coffee, milk). i E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 6 / 21
Classifying valuations Economists classify valuations by how agents see trade-offs between goods. For divisible goods, ask how changes in each price affect each demand. Let x ∗ ( p ) be optimal demands of each good at a given price. ∂x ∗ ∂p j > 0 means goods are ‘substitutes’ (tea, coffee). i ∂x ∗ ∂p j < 0 means goods are ‘complements’ (coffee, milk). i With THs, look first at discrete price changes that cross one facet. p 2 (0,0) p 1 E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 6 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p 2 (x +1,x -2) 1 2 (x ,x ) ( ( 1 1 2 p 1 -2 Increase price i to cross a facet. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p 2 (x +1,x -2) 1 2 (x ,x ) ( ( 1 1 2 p 1 -2 Increase price i to cross a facet. Demand changes from x to x + v v a facet normal, follows description above. By the strict law of demand, v i < 0 . ⇒ v j ≥ 0 for all j � = i . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p (0,0) 2 p 1 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p (0,0) 2 p 1 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p (0,0) 2 (2,0) (1,0) (0,2) p 1 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p (0,0) 2 (2,0) (1,0) (0,2) p 1 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . As price changes, demand in turn x 0 , x 1 , . . . , x r . At each stage, v k = x k − x k − 1 is a facet normal. By the strict law of demand, v k i < 0 for k = 1 , . . . , r . ⇒ x r j ≥ x 0 j for all j � = i E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has at most one +ve, one -ve coordinate entry. p (0,0) 2 (2,0) (1,0) (0,2) p 1 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . S As price changes, demand in turn x 0 , x 1 , . . . , x r . E T At each stage, v k = x k − x k − 1 is a facet normal. U T I T By the strict law of demand, v k i < 0 for k = 1 , . . . , r . S B U ⇒ x r j ≥ x 0 j for all j � = i S E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has all positive (or all negative) coordinate entries. p 2 ( ( 2 (x ,x ) 3 1 2 (x +2,x +3) p 1 1 2 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . As price changes, demand in turn x 0 , x 1 , . . . , x r . At each stage, v k = x k − x k − 1 is a facet normal. By the strict law of demand, v k i < 0 for k = 1 , . . . , r . ⇒ x r j ≤ x 0 j for all j � = i E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... has all positive (or all negative) coordinate entries. p 2 ( ( 2 (x ,x ) 3 1 2 (x +2,x +3) p 1 1 2 Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . S T As price changes, demand in turn x 0 , x 1 , . . . , x r . N E At each stage, v k = x k − x k − 1 is a facet normal. M E L By the strict law of demand, v k i < 0 for k = 1 , . . . , r . P M O ⇒ x r j ≤ x 0 j for all j � = i C E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... is (1 , 4) p 2 ( ( 1 e.g. (car bodies, car wheels) 4 (0,0) (1,4) (2,8) p 1 (3,12) Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . As price changes, demand in turn x 0 , x 1 , . . . , x r . At each stage, v k = x k − x k − 1 is a facet normal. By the strict law of demand, v k i < 0 for k = 1 , . . . , r . Demand fewer (1 , 4) bundles E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... is (1 , 4) p 2 ( ( 1 e.g. (car bodies, car wheels) 4 (0,0) (1,4) (2,8) p 1 (3,12) T C E Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . F S R T As price changes, demand in turn x 0 , x 1 , . . . , x r . E N P E At each stage, v k = x k − x k − 1 is a facet normal. M E L By the strict law of demand, v k i < 0 for k = 1 , . . . , r . P M O Demand fewer (1 , 4) bundles C E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... is in set D ⊂ Z n . p 2 ( ( 1 e.g. (car bodies, car wheels) 4 (0,0) (1,4) (2,8) p 1 (3,12) Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . As price changes, demand in turn x 0 , x 1 , . . . , x r . At each stage, v k = x k − x k − 1 is a facet normal. By the strict law of demand, v k i < 0 for k = 1 , . . . , r . These facts define structure of trade-offs. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Economic properties from facets Suppose every facet normal v to T u ... is in set D ⊂ Z n . Definition: “Demand Type” u is of demand type D if every facet of T u has normal in D . u is of concave demand type D if it is additionally concave. Set of all such u is “the demand type” or “the concave demand type”. Start in a ‘unique demand region’ ( / ∈ T u ) and increase price i . As price changes, demand in turn x 0 , x 1 , . . . , x r . At each stage, v k = x k − x k − 1 is a facet normal. By the strict law of demand, v k i < 0 for k = 1 , . . . , r . We can ‘break down the demand change in improving D -steps’. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 7 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . SNP ‘faces’ are dual to the cells of the TH. k -dimensional pieces ↔ ( n − k ) -dimensional pieces. Linear spaces parallel to SNP face and corresp. TH cell are dual. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . SNP ‘faces’ are dual to the cells of the TH. k -dimensional pieces ↔ ( n − k ) -dimensional pieces. Linear spaces parallel to SNP face and corresp. TH cell are dual. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . ?? SNP ‘faces’ are dual to the cells of the TH. k -dimensional pieces ↔ ( n − k ) -dimensional pieces. Linear spaces parallel to SNP face and corresp. TH cell are dual. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . ✲ ✲ ✲ ?? ✲ ✲ SNP ‘faces’ are dual to the cells of the TH. k -dimensional pieces ↔ ( n − k ) -dimensional pieces. Linear spaces parallel to SNP face and corresp. TH cell are dual. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . ✲ ✲ ✲ ?? ✲ ✲ SNP ‘faces’ are dual to the cells of the TH. Lemma Bundles which are not SNP vertices are either never demanded or only demanded at prices corresp. to the SNP face(s) they’re in. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . 2 0 2 3 2 4 ✲ (1,1) 2 4 4 SNP ‘faces’ are dual to the cells of the TH. Lemma Bundles which are not SNP vertices are either never demanded or only demanded at prices corresp. to the SNP face(s) they’re in. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . 2 0 2 2.5 2 4 ✲ (1,1) 2 4 4 SNP ‘faces’ are dual to the cells of the TH. Lemma Bundles which are not SNP vertices are either never demanded or only demanded at prices corresp. to the SNP face(s) they’re in. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . ✲ (1,1) SNP ‘faces’ are dual to the cells of the TH. Lemma Bundles which are not SNP vertices are either never demanded or only demanded at prices corresp. to the SNP face(s) they’re in. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Duality: The ‘Subdivided Newton Polytope’ (SNP) Recall that T u lives in price space. The dual space is quantity space . The Newton Polytope of T u is Conv R A , where u : A → R . We subdivide it, to join up the sets D u ( p ) . ✲ (1,1) SNP ‘faces’ are dual to the cells of the TH. Lemma Bundles which are not SNP vertices are either never demanded or only demanded at prices corresp. to the SNP face(s) they’re in. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 8 / 21
Aggregate Demand Many agents: j = 1 , . . . , m , valuations u j : A j → R . Definition (Standard) Aggregate demand at p is the Minkowski sum of individual demands: D u 1 ( p ) + · · · + D u m ( p ) and not hard to see aggregate demand is D U ( p ) where �� � j u j ( x j ) | x j ∈ A j , � j x j = x U ( x ) = max . ‘Aggregate’ tropical polynomial is tropical product of individual ones. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 9 / 21
Aggregate Demand Many agents: j = 1 , . . . , m , valuations u j : A j → R . Definition (Standard) Aggregate demand at p is the Minkowski sum of individual demands: D u 1 ( p ) + · · · + D u m ( p ) and not hard to see aggregate demand is D U ( p ) where �� � j u j ( x j ) | x j ∈ A j , � j x j = x U ( x ) = max . ‘Aggregate’ tropical polynomial is tropical product of individual ones. Definition (Standard) If supply is x , a competitive equilibrium among agents i consists of allocations x i such that � i x i = x . a price p such that x i ∈ D u i ( p ) for all i . So require some p such that x ∈ D U ( p ) . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 9 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , just superimpose individual tropical hypersurfaces. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , just superimpose individual tropical hypersurfaces. Then what is D U ( p ) ? If p / ∈ T U , easy: use “facet normal × weight = change in demand”. If p ∈ T u i , only one i , and individual valuations concave, also easy. Interesting case: p ∈ T u i , T u j for i � = j . Intersections . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , just superimpose individual tropical hypersurfaces. ★ Then what is D U ( p ) ? If p / ∈ T U , easy: use “facet normal × weight = change in demand”. If p ∈ T u i , only one i , and individual valuations concave, also easy. Interesting case: p ∈ T u i , T u j for i � = j . Intersections . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , just superimpose individual tropical hypersurfaces. ★ Then what is D U ( p ) ? If p / ∈ T U , easy: use “facet normal × weight = change in demand”. If p ∈ T u i , only one i , and individual valuations concave, also easy. Interesting case: p ∈ T u i , T u j for i � = j . Intersections . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , just superimpose individual tropical hypersurfaces. D u 1 ( ⋆ ) = { (1 , 0) , (0 , 1) } D u 2 ( ⋆ ) = { (0 , 0) , (1 , 1) } D U ( ⋆ ) = ★ { (1 , 0) , (0 , 1) , (1 , 2) , (2 , 1) } . Then what is D U ( p ) ? If p / ∈ T U , easy: use “facet normal × weight = change in demand”. If p ∈ T u i , only one i , and individual valuations concave, also easy. Interesting case: p ∈ T u i , T u j for i � = j . Intersections . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Tropical hypersurface of aggregate demand D U ( p ) = D u 1 ( p ) + · · · + D u m ( p ) Easy to draw T U , just superimpose individual tropical hypersurfaces. D u 1 ( ⋆ ) = { (1 , 0) , (0 , 1) } D u 2 ( ⋆ ) = { (0 , 0) , (1 , 1) } D U ( ⋆ ) = ★ { (1 , 0) , (0 , 1) , (1 , 2) , (2 , 1) } . D U ( ⋆ ) is not integer-convex U is not concave Bundle (1 , 1) is not aggregate demand at ⋆ , and so not at any price. If supply is (1 , 1) then competitive equilibrium does not exist. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21
Stable intersections After a small generic translation, T u 1 , T u 2 intersect transversally i.e. if C 1 , C 2 are intersecting ‘cells’ of 2 T u 1 , T u 2 , then dim( C 1 + C 2 ) = n . Definition The stable intersection is lim ǫ → 0 T u 1 ∩ ( T u 2 + ǫ w ) for generic w . It is well-defined (with also multiplicities) by the balancing condition. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 11 / 21
Stable intersections After a small generic translation, T u 1 , T u 2 intersect transversally i.e. if C 1 , C 2 are intersecting ‘cells’ of 2 T u 1 , T u 2 , then dim( C 1 + C 2 ) = n . Definition The stable intersection is lim ǫ → 0 T u 1 ∩ ( T u 2 + ǫ w ) for generic w . It is well-defined (with also multiplicities) by the balancing condition. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 11 / 21
Stable intersections ★ After a small generic translation, T u 1 , T u 2 ★ intersect transversally i.e. if C 1 , C 2 are intersecting ‘cells’ of 2 T u 1 , T u 2 , then dim( C 1 + C 2 ) = n . ★ Definition The stable intersection is lim ǫ → 0 T u 1 ∩ ( T u 2 + ǫ w ) for generic w . It is well-defined (with also multiplicities) by the balancing condition. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 11 / 21
Stable intersections ★ After a small generic translation, T u 1 , T u 2 ★ intersect transversally i.e. if C 1 , C 2 are intersecting ‘cells’ of 2 ★ T u 1 , T u 2 , then dim( C 1 + C 2 ) = n . Definition The stable intersection is lim ǫ → 0 T u 1 ∩ ( T u 2 + ǫ w ) for generic w . It is well-defined (with also multiplicities) by the balancing condition. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 11 / 21
Stable intersections ★ After a small generic translation, T u 1 , T u 2 ★ intersect transversally i.e. if C 1 , C 2 are intersecting ‘cells’ of 2 ★ T u 1 , T u 2 , then dim( C 1 + C 2 ) = n . Definition The stable intersection is lim ǫ → 0 T u 1 ∩ ( T u 2 + ǫ w ) for generic w . It is well-defined (with also multiplicities) by the balancing condition. Translations ↔ modifications of valuations: T ( u ( • )+ • . w ) = T u + { w } . So: Lemma If competitive equilibrium fails, it fails at the stable intersection, and it still fails after a sufficiently small translation. Thus we need only study the case of transversal intersections . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 11 / 21
Classic theorems of competitive equilibrium Theorem (Kelso and Crawford 1982) Suppose domain A i = { 0 , 1 } n for all agents i . u i : A i → R is a concave substitute valuation for all agents. Supply x ∈ { 0 , 1 } n . Then competitive equilibrium exists. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 12 / 21
Classic theorems of competitive equilibrium Theorem (Milgrom and Strulovici 2009) Suppose domain A i = A , a fixed product of intervals, for all agents i . u i : A i → R is a concave strong substitute valuation for all agents. Supply x ∈ A . Then competitive equilibrium exists. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 12 / 21
Classic theorems of competitive equilibrium Theorem (Hatfield et al. 2013) Suppose A i ⊂ {− 1 , 0 , 1 } n for all agents i . domain u i : A i → R is a concave substitute valuation for all agents. Supply x = 0 . Then competitive equilibrium exists. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 12 / 21
Classic theorems of competitive equilibrium Theorem (Hatfield et al. 2013) Suppose A i ⊂ {− 1 , 0 , 1 } n for all agents i . domain u i : A i → R is a concave substitute valuation for all agents. Supply x = 0 . Then competitive equilibrium exists. Definition A class of valuations always has a competitive equilibrium if, for every set of agents with valuations in this class, and for every bundle in the Minkowski sum of their domains of valuation, there exist prices such that this bundle is the aggregate demand. Obvious candidate for ‘classes of valuations’: concave demand types. From now on, assume all individual valuations are concave. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 12 / 21
Characterising when equilibrium always exists. ? Area=2. ★ The problem is that the bundle is in the middle of the square. Conv R D U ( ⋆ ) ∩ Z n � = D U ( ⋆ ) i.e. There exists a ‘relevant’ bundle which is never aggregate supply. There exists a bundle there because the area of the square is > 1 . The area is abs. value of the determinant of vectors along its edges. � 1 � − 1 det = 2 1 1 E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Consider [ N σ : N σ 1 + · · · + N σ m ] . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Consider [ N σ : N σ 1 + · · · + N σ m ] . ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● N σ 1 N σ N σ 2 E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Consider [ N σ : N σ 1 + · · · + N σ m ] . ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● N σ 1 [ N σ : N σ 1 + N σ 2 ] = 2 N σ 2 E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Proposition � � If N σ : N σ 1 + · · · + N σ m > 1 and dim σ i = 1 for all i then Conv R D U ( p ) ∩ Z n � = D U ( p ) . Sketch proof. D U ( p ) is the vertices of a parallellepiped parallel to the fundamental parallelepiped of N σ 1 + · · · + N σ m . The subgroup index is greater than 1, so this parallelepiped contains a non-vertex point. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Proposition � � If N σ : N σ 1 + · · · + N σ m > 1 and dim σ i = 1 for all i then Conv R D U ( p ) ∩ Z n � = D U ( p ) . Corollary � � If N σ : N σ 1 + · · · + N σ m > 1 for some 1-dim’l SNP faces of individual valuations, then competitive equilibrium fails for some relevant supply. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Proposition = 1 then Conv R D U ( p ) ∩ Z n = D U ( p ) . � � If N σ : N σ 1 + · · · + N σ m Sketch proof. Since intersection transversal, can uniquely write x ∈ Conv R D U ( p ) ∩ Z n as sum of bundles in Conv R D u i ( p ) . If x ∈ Z n also then each component is integer since subgroup index is 1. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Proposition = 1 then Conv R D U ( p ) ∩ Z n = D U ( p ) . � � If N σ : N σ 1 + · · · + N σ m Corollary � � If N σ : N σ 1 + · · · + N σ m = 1 for all SNP faces of the aggregate valuation, then competitive equilibrium exists for any relevant supply. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Proposition (Standard) If dim N σ = n and v 1 , . . . , v n is the union of bases for N σ 1 , . . . , N σ m = | det( v 1 , . . . , v n ) | = volume fund’l p’ped � � N σ : N σ 1 + · · · + N σ m E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. � � 1 ★ � 2 Let σ i ( σ ) be the individual (aggregate) SNP faces corresp. to price p . Transversal intersections means � m i =1 dim σ i = dim σ ( m agents). Let N τ ⊂ Z n be integer vectors parallel to lattice polytope τ . Proposition (Standard) If dim N σ = n and v 1 , . . . , v n is the union of bases for N σ 1 , . . . , N σ m = | det( v 1 , . . . , v n ) | = volume fund’l p’ped � � N σ : N σ 1 + · · · + N σ m Say D ⊂ Z n is unimodular if | det( v 1 , . . . , v n ) | = 1 for any linearly independent v 1 , . . . , v n ∈ D . (Small tweak when D does not span Z n .) E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. Theorem (cf. Danilov et al. 2001, Danilov and Koshevoy 2004 for ‘if’.) The concave demand type D = { v 1 , . . . , v r } always has a competitive equilibrium iff D is unimodular . (The ‘concave demand type D ’ consists all concave valuations u such that the normals to facets of T u are in D .) Proposition (Standard) If dim N σ = n and v 1 , . . . , v n is the union of bases for N σ 1 , . . . , N σ m = | det( v 1 , . . . , v n ) | = volume fund’l p’ped � � N σ : N σ 1 + · · · + N σ m Say D ⊂ Z n is unimodular if | det( v 1 , . . . , v n ) | = 1 for any linearly independent v 1 , . . . , v n ∈ D . (Small tweak when D does not span Z n .) E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
Characterising when equilibrium always exists. Theorem (cf. Danilov et al. 2001, Danilov and Koshevoy 2004 for ‘if’.) The concave demand type D = { v 1 , . . . , v r } always has a competitive equilibrium iff D is unimodular . (The ‘concave demand type D ’ consists all concave valuations u such that the normals to facets of T u are in D .) From this, follows existence of equilibrium with indivisibilities in: Gross substitutes (Kelso and Crawford, 1982, Ecta). Step-wise / Strong substitutes (Danilov et al., 2003, Discrete Applied Math., Milgrom and Strulovici, 2009, JET). Gross substitutes and complements (Sun and Yang, 2006, Ecta). Full substitutability on a trading network (Hatfield et al. 2013, JPE). E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein Return to substitutes / complements example. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein Return to substitutes / complements example. Modify the valuations. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein Return to substitutes / complements example. Modify the valuations. Now: Bundle (1 , 1) is demanded for some prices. Every bundle is demanded for some prices. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein ★ ★ ★ Before the shift After the shift One intersection. Two intersections. Corresp. SNP face has area 2. Corresp. SNP faces have area 1. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein ★ ★ ★ Before the shift After the shift One intersection. Two intersections. Corresp. SNP face has area 2. Corresp. SNP faces have area 1. Call this SNP area the multiplicity of the intersection. See # intersections is constant, up to multiplicity . E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein ★ ★ ★ Theorem (Tropical B´ ezout-Bernstein Theorem, see Sturmfels 2002) # intersections, with multiplicities, is mixed volume of Newton Polytopes. Theorem When 2-D tropical hypersurfaces intersect transversally, then equilibrium exists for all supply bundles iff # intersections, weighted only by facet weights, equals mixed volume of Newton Polytopes. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
When unimodularity fails: 2-D B´ ezout-Bernstein ● ● ● ● ● ● ● ● ● ● ● ● Theorem (Tropical B´ ezout-Bernstein Theorem, see Sturmfels 2002) # intersections, with multiplicities, is mixed volume of Newton Polytopes. Theorem When 2-D tropical hypersurfaces intersect transversally, then equilibrium exists for all supply bundles iff # intersections, weighted only by facet weights, equals mixed volume of Newton Polytopes. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 14 / 21
Generalised B´ ezout-Bernstein (Bertrand and Bihan, 2007) ● ● ● ● ● Given SNP k -face σ ↔ TH ( n − k ) -cell C σ = Z k ֒ → R k assign vol k ( σ ) Under N σ ∼ ● ● ● ● ● ● ● ● ● ● E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 15 / 21
Generalised B´ ezout-Bernstein (Bertrand and Bihan, 2007) ● ● ● ● ● Given SNP k -face σ ↔ TH ( n − k ) -cell C σ � vol ( � )=1 = Z k ֒ → R k assign vol k ( σ ) Under N σ ∼ 1 ● ● ● ● ● Define cell weight w ( C σ ) := k !vol k ( σ ) . ● ● ● ● ● E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 15 / 21
Generalised B´ ezout-Bernstein (Bertrand and Bihan, 2007) ● ● ● ● ● Given SNP k -face σ ↔ TH ( n − k ) -cell C σ � vol ( � )=1 = Z k ֒ → R k assign vol k ( σ ) Under N σ ∼ 1 ● ● ● ● ● Define cell weight w ( C σ ) := k !vol k ( σ ) . ● ● ● ● ● If cells C σ 1 , C σ 2 of T u i , T u 2 intersect transversally at C σ , then in T u 1 ∩ T u 2 : � � mult( C σ ) := w ( C σ 1 ) w ( C σ 2 ) N σ 1 + σ 2 : N σ 1 + N σ 2 Theorem (Bertrand and Bihan 2007) mult( C σ ) = MV ( σ 1 , σ 2 , ( n 1 , n 2 )) where n i = dim σ i (i.e. repeat σ i in the mixed volume n i times). E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 15 / 21
Generalised B´ ezout-Bernstein (Bertrand and Bihan, 2007) ● ● ● ● ● Given SNP k -face σ ↔ TH ( n − k ) -cell C σ � vol ( � )=1 = Z k ֒ → R k assign vol k ( σ ) Under N σ ∼ 1 ● ● ● ● ● Define cell weight w ( C σ ) := k !vol k ( σ ) . ● ● ● ● ● If cells C σ 1 , C σ 2 of T u i , T u 2 intersect transversally at C σ , then in T u 1 ∩ T u 2 : � � mult( C σ ) := w ( C σ 1 ) w ( C σ 2 ) N σ 1 + σ 2 : N σ 1 + N σ 2 Theorem (Bertrand and Bihan 2007) mult( C σ ) = MV ( σ 1 , σ 2 , ( n 1 , n 2 )) where n i = dim σ i (i.e. repeat σ i in the mixed volume n i times). Theorem Competitive equilibrium exists if the number of 0-cells of the intersection, weighted only by cell weights, equals � MV (∆ 1 , ∆ 2 , ( n 1 , n 2 )) . n 1 + n 2 = n where ∆ 1 , ∆ 2 are the Newton polytopes of the individual valuations E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 15 / 21
Unimodular examples: Strong / step-wise substitutes ss ⊂ Z n vectors have at most one +1, at most one -1, otherwise 0 s. D n Substitutes where trade-offs are locally 1-1. p B � 1 � 0 1 0 1 − 1 p A Unimodular set (classic result). Equilibrium always exists Model of Kelso and Crawford (1982), Danilov et al. (2003), Milgrom and Strulovici (2009), Hatfield et al. (2013). The model of Sun and Yang (2006) is a basis change. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 16 / 21
Unimodular examples: Strong / step-wise substitutes ss ⊂ Z n vectors have at most one +1, at most one -1, otherwise 0 s. D n Substitutes where trade-offs are locally 1-1. p 3 p 2 (1,1,1) 1 0 0 1 1 0 0 1 0 − 1 0 1 0 0 1 0 − 1 − 1 Unimodular set (classic result). p 1 Equilibrium always exists Model of Kelso and Crawford (1982), Danilov et al. (2003), Milgrom and Strulovici (2009), Hatfield et al. (2013). The model of Sun and Yang (2006) is a basis change. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 16 / 21
Interval package valuations D is the basis change of D n ss via the upper triangular matrix of 1s. Consists of vectors with one block of consecutive 1s: People are ordered from 1 to n . Subsets of consecutive people can form coalitions. Where there are ‘gaps’, no complementarity. This can represent: small shops along a street considering a merger; seabed rights for oil / offshore wind. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 17 / 21
Beyond strong substitutes Of course there are many non-isomorphic unimodular demand types (Seymour, 1980, Danilov and Grishukhin, 1999). Smallest example: let D be the columns of: 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 front-line workers 0 0 1 0 0 1 0 1 1 � 0 0 0 1 1 1 1 1 1 manager Interpretation: The first three goods (rows) represent front-line workers. The final good (row) is a manager. ‘Bundles’, i.e. teams, worth bidding for, are: a worker on their own ( not a manager on their own); a worker and a manager; two workers and a manager. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 18 / 21
Beyond strong substitutes Of course there are many non-isomorphic unimodular demand types (Seymour, 1980, Danilov and Grishukhin, 1999). Smallest example: let D be the columns of: 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 front-line workers 0 0 1 0 0 1 0 1 1 � 0 0 0 1 1 1 1 1 1 manager Interpretation: The first three goods (rows) represent front-line workers. The final good (row) is a manager. ‘Bundles’, i.e. teams, worth bidding for, are: a worker on their own ( not a manager on their own); a worker and a manager; two workers and a manager. Interpret as coalitions: model matching with transferable utility. E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 18 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Price on "strong" £100m Price on "weak" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Price on "strong" £100m Price on "weak" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Price on "strong" Bid for "weak" OR "strong" whichever has "better" price £100m Price on "weak" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Price on "strong" £100m Nothing "weak" £100m £100m "strong" Price on "weak" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Assume that trade-offs are 1-1: strong substitutes. Price on "strong" £100m Nothing "weak" £100m £100m "strong" Price on "weak" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Assume that trade-offs are 1-1: strong substitutes. W 0 Price on "s" S Price on "w" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
The Bank of England’s Product-Mix Auction Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral. Assume that trade-offs are 1-1: strong substitutes. 0 WWW WW W S Price on "s" WS SS WWS SSS WSS Price on "w" E. Baldwin and P. Klemperer The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21
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