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The geometry of auctions and competitive equilibrium with indivisible goods

Elizabeth Baldwin Paul Klemperer

London School of Economics Oxford University

August 2014

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 1 / 21

Second price / uniform price auctions

Suppose we sell

• ne unit
• f
• ne good

in a sealed bid auction. The highest bidder wins. They pay the highest losing bid. Your maximum willingness to pay is v. How to bid?

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell

• ne unit
• f
• ne good

in a sealed bid auction. The highest bidder wins. They pay the highest losing bid. Your maximum willingness to pay is v. How to bid? Bid v. Your bid does not affect your price, affects when you win. This way, you win exactly when you want to win.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell

• ne unit
• f
• ne good

in a sealed bid auction. The highest bidder wins. They pay the highest losing bid. Your maximum willingness to pay is v. How to bid? Bid v. Your bid does not affect your price, affects when you win. This way, you win exactly when you want to win. ‘Truthful revelation mechanisms’ are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell many units of

• ne good

in a sealed bid auction. The highest bidders win. They pay the highest losing bid. ‘Truthful revelation mechanisms’ are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell many units of

• ne good

in a sealed bid auction. The highest bidders win. They pay the highest losing bid.

Willingness to Pay Units

1 2 3 4 5

Your bid for unit i + 1 might affect your price on units 1 to i. ‘Truthful revelation mechanisms’ are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell many units of

• ne good

in a sealed bid auction. The highest bidders win. They pay the highest losing bid.

Willingness to Pay Units

1 2 3 4 5

Optimal bidding schedule

Your bid for unit i + 1 might affect your price on units 1 to i. But if you are small relative to market size, then optimal ‘shading’ is small. ‘Truthful revelation mechanisms’ are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell many units of

• ne good

in a sealed bid auction. The highest bidders win. They pay the highest losing bid.

Willingness to Pay Units

1 2 3 4 5

Optimal bidding schedule

Your bid for unit i + 1 might affect your price on units 1 to i. But if you are small relative to market size, then optimal ‘shading’ is small. Nearly truthful revelation mechanisms are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell many units of many goods in a sealed bid auction. Who wins? What do they pay? How can we design a (nearly) truthful revelation mechanism? Nearly truthful revelation mechanisms are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Second price / uniform price auctions

Suppose we sell many units of many goods in a sealed bid auction. Who wins? What do they pay? How can we design a (nearly) truthful revelation mechanism? The uniform price auction for one good: Assumes bidders want the item iff price is below their bid Finds the minimum price such that aggregate demand = supply. To replicate this with more goods, need to understand the geometry of consumer preferences in price space. Nearly truthful revelation mechanisms are useful for auctioneers: Informative Efficient Easy for participants – encourage market entry.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 2 / 21

Geometric Analysis of Demand: Model

n indivisible goods.

x2 x1

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

Geometric Analysis of Demand: Model

n indivisible goods. Finite set A ⊂ Zn of bundles of goods available.

x2 x1

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

Geometric Analysis of Demand: Model

n indivisible goods. Finite set A ⊂ Zn of bundles of goods available. Valuation u : A → R on bundles. Example

• f u(x)

x2 x1

26 35 24

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

Geometric Analysis of Demand: Model

n indivisible goods. Finite set A ⊂ Zn of bundles of goods available. Valuation u : A → R on bundles. If prices are p then ‘indirect utility’ is V (p) := maxx∈A{u(x) − p.x}. Example

• f u(x)

x2 x1

26 35 24

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

Geometric Analysis of Demand: Model

n indivisible goods. Finite set A ⊂ Zn of bundles of goods available. Valuation u : A → R on bundles. If prices are p then ‘indirect utility’ is V (p) := maxx∈A{u(x) − p.x}. What is demanded? Anything in set Du(p) := arg max

x∈A

{u(x) − p.x} Example

• f u(x)

x2 x1

26 35 24

p

1

p

2 (0,1) (1,0) (0,0) (1,1)

To investigate what is demanded where, study where demand changes. Where Du(p) contains more than one bundle

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

Geometric Analysis of Demand: Model

n indivisible goods. Finite set A ⊂ Zn of bundles of goods available. Valuation u : A → R on bundles. If prices are p then ‘indirect utility’ is V (p) := maxx∈A{u(x) − p.x}. A ‘tropical’ polynomial using ‘max-plus’ algebra. What is demanded? Anything in set Du(p) := arg max

x∈A

{u(x) − p.x} Example

• f u(x)

x2 x1

26 35 24

p

1

p

2 (0,1) (1,0) (0,0) (1,1)

Definition: “Tropical Hypersurface (TH)” Tu={ prices p ∈ Rn where #Du(p) > 1}.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

Geometric Analysis of Demand: Model

n indivisible goods. Finite set A ⊂ Zn of bundles of goods available. Valuation u : A → R on bundles. If prices are p then ‘indirect utility’ is V (p) := maxx∈A{u(x) − p.x}. A ‘tropical’ polynomial using ‘max-plus’ algebra. What is demanded? Anything in set Du(p) := arg max

x∈A

{u(x) − p.x} Example

• f u(x)

x2 x1

13 32 12

p

1

p

2 (0,1) (1,0) (0,0) (1,1)

Definition: “Tropical Hypersurface (TH)” Tu={ prices p ∈ Rn where #Du(p) > 1}.

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 3 / 21

How does demand change as you cross a facet?

A tropical hypersurface is composed of facets: linear pieces in dimen- sion (n − 1).

(0,0) (0,1) (0,2) (1,1) (1,0) (2,0) p

2

p

1

If p is in a facet then the agent is indifferent between two bundles: u(x) − p.x = u(y) − p.y

• 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?

A tropical hypersurface is composed of facets: linear pieces in dimen- sion (n − 1).

(0,0) (0,1) (0,2) (1,1) (1,0) (2,0) p

2

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)

• 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?

A tropical hypersurface is composed of facets: linear pieces in dimen- sion (n − 1).

• 1

1

(

(0,0) (0,1) (0,2) (1,1) (1,0) (2,0) p

2

p

1

(

1 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?

A tropical hypersurface is composed of facets: linear pieces in dimen- sion (n − 1).

p

1

p

2

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

1

v

2

v

3

v

4

w

1 w 2

w

3

w

4

Every tropical hypersurface is balanced: around each (n − 2)-cell,

i wivi = 0.

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∗

i

∂pj > 0 means goods are ‘substitutes’ (tea, coffee). ∂x∗

i

∂pj < 0 means goods are ‘complements’ (coffee, milk).

• 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∗

i

∂pj > 0 means goods are ‘substitutes’ (tea, coffee). ∂x∗

i

∂pj < 0 means goods are ‘complements’ (coffee, milk).

With THs, look first at discrete price changes that cross one facet.

(0,0) p

2

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 Tu... has at most one +ve, one -ve coordinate entry.

1

• 2

(

(x ,x )

1 2

(x +1,x -2)

1 2

p1 p2

(

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 Tu... has at most one +ve, one -ve coordinate entry.

1

• 2

(

(x ,x )

1 2

(x +1,x -2)

1 2

p1 p2

(

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, vi < 0. ⇒ vj ≥ 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 Tu... has at most one +ve, one -ve coordinate entry.

(0,0)

p

2

p

1

Start in a ‘unique demand region’ (/ ∈ Tu) 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 Tu... has at most one +ve, one -ve coordinate entry.

(0,0)

p

2

p

1

Start in a ‘unique demand region’ (/ ∈ Tu) 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 Tu... has at most one +ve, one -ve coordinate entry.

(0,0)

p

2

p

1

(2,0) (0,2) (1,0)

Start in a ‘unique demand region’ (/ ∈ Tu) 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 Tu... has at most one +ve, one -ve coordinate entry.

(0,0)

p

2

p

1

(2,0) (0,2) (1,0)

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

i < 0 for k = 1, . . . , r.

⇒ xr

j ≥ x0 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 Tu... has at most one +ve, one -ve coordinate entry.

(0,0)

p

2

p

1

(2,0) (0,2) (1,0)

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

i < 0 for k = 1, . . . , r.

⇒ xr

j ≥ x0 j for all j = i

S U B S T I T U T E 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 Tu... has all positive (or all negative) coordinate entries.

(x ,x )

1 2

(x +2,x +3)

1 2

p1 p2

2 3

( (

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

i < 0 for k = 1, . . . , r.

⇒ xr

j ≤ x0 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 Tu... has all positive (or all negative) coordinate entries.

(x ,x )

1 2

(x +2,x +3)

1 2

p1 p2

2 3

( (

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

i < 0 for k = 1, . . . , r.

⇒ xr

j ≤ x0 j for all j = i

C O M P L E M E N T 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 Tu... is (1, 4)

(0,0) p1 p2

1 4

( (

(1,4) (2,8) (3,12) e.g. (car bodies, car wheels)

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

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 Tu... is (1, 4)

(0,0) p1 p2

1 4

( (

(1,4) (2,8) (3,12) e.g. (car bodies, car wheels)

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal.

P E R F E C T

By the strict law of demand, vk

i < 0 for k = 1, . . . , r.

Demand fewer (1, 4) bundles

C O M P L E M E N T 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 Tu... is in set D ⊂ Zn.

(0,0) p1 p2

1 4

( (

(1,4) (2,8) (3,12) e.g. (car bodies, car wheels)

Start in a ‘unique demand region’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

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 Tu... is in set D ⊂ Zn. Definition: “Demand Type” u is of demand type D if every facet of Tu 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’ (/ ∈ Tu) and increase price i. As price changes, demand in turn x0, x1, . . . , xr. At each stage, vk = xk − xk−1 is a facet normal. By the strict law of demand, vk

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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(p).

3 2 2 2 2 4 4 4

✲(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(p).

2.5 2 2 2 2 4 4 4

✲(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 Tu lives in price space. The dual space is quantity space. The Newton Polytope of Tu is ConvRA, where u : A → R. We subdivide it, to join up the sets Du(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 uj : Aj → R. Definition (Standard) Aggregate demand at p is the Minkowski sum of individual demands: Du1(p) + · · · + Dum(p) and not hard to see aggregate demand is DU(p) where U(x) = max

• j uj(xj) | xj ∈ Aj,

j xj = x

• .

‘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 uj : Aj → R. Definition (Standard) Aggregate demand at p is the Minkowski sum of individual demands: Du1(p) + · · · + Dum(p) and not hard to see aggregate demand is DU(p) where U(x) = max

• j uj(xj) | xj ∈ Aj,

j xj = x

• .

‘Aggregate’ tropical polynomial is tropical product of individual ones. Definition (Standard) If supply is x, a competitive equilibrium among agents i consists of allocations xi such that

i xi = x.

a price p such that xi ∈ Dui(p) for all i. So require some p such that x ∈ DU(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

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU,

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21

Tropical hypersurface of aggregate demand

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU, 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

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU, just superimpose individual tropical hypersurfaces. Then what is DU(p)? If p / ∈ TU, easy: use “facet normal × weight = change in demand”. If p ∈ Tui, only one i, and individual valuations concave, also easy. Interesting case: p ∈ Tui, Tuj 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

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU, just superimpose individual tropical hypersurfaces.

★

Then what is DU(p)? If p / ∈ TU, easy: use “facet normal × weight = change in demand”. If p ∈ Tui, only one i, and individual valuations concave, also easy. Interesting case: p ∈ Tui, Tuj 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

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU, just superimpose individual tropical hypersurfaces.

★

Then what is DU(p)? If p / ∈ TU, easy: use “facet normal × weight = change in demand”. If p ∈ Tui, only one i, and individual valuations concave, also easy. Interesting case: p ∈ Tui, Tuj for i = j. Intersections.

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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21

Tropical hypersurface of aggregate demand

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU, just superimpose individual tropical hypersurfaces.

★

Du1(⋆) = {(1, 0), (0, 1)} Du2(⋆) = {(0, 0), (1, 1)} DU(⋆) = {(1, 0), (0, 1), (1, 2), (2, 1)}. Then what is DU(p)? If p / ∈ TU, easy: use “facet normal × weight = change in demand”. If p ∈ Tui, only one i, and individual valuations concave, also easy. Interesting case: p ∈ Tui, Tuj 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

DU(p) = Du1(p) + · · · + Dum(p) Easy to draw TU, just superimpose individual tropical hypersurfaces.

★

Du1(⋆) = {(1, 0), (0, 1)} Du2(⋆) = {(0, 0), (1, 1)} DU(⋆) = {(1, 0), (0, 1), (1, 2), (2, 1)}. DU(⋆) 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.

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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 10 / 21

Stable intersections

After a small generic translation, Tu1, Tu2 intersect transversally i.e. if C1, C2 are intersecting ‘cells’ of Tu1, Tu2, then dim(C1 + C2) = n. 2 Definition The stable intersection is limǫ→0 Tu1 ∩ (Tu2 + ǫw) for generic w. It is well-defined (with also multiplicities) by the balancing condition.

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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 11 / 21

Stable intersections

After a small generic translation, Tu1, Tu2 intersect transversally i.e. if C1, C2 are intersecting ‘cells’ of Tu1, Tu2, then dim(C1 + C2) = n. 2 Definition The stable intersection is limǫ→0 Tu1 ∩ (Tu2 + ǫw) 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, Tu1, Tu2 intersect transversally i.e. if C1, C2 are intersecting ‘cells’ of Tu1, Tu2, then dim(C1 + C2) = n. 2

★ ★ ★

Definition The stable intersection is limǫ→0 Tu1 ∩ (Tu2 + ǫw) 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, Tu1, Tu2 intersect transversally i.e. if C1, C2 are intersecting ‘cells’ of Tu1, Tu2, then dim(C1 + C2) = n. 2

★ ★ ★

Definition The stable intersection is limǫ→0 Tu1 ∩ (Tu2 + ǫw) 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, Tu1, Tu2 intersect transversally i.e. if C1, C2 are intersecting ‘cells’ of Tu1, Tu2, then dim(C1 + C2) = n. 2

★ ★ ★

Definition The stable intersection is limǫ→0 Tu1 ∩ (Tu2 + ǫw) for generic w. It is well-defined (with also multiplicities) by the balancing condition. Translations ↔ modifications of valuations: T(u(•)+•.w) = Tu + {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.

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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 Ai = {0, 1}n for all agents i. ui : Ai → R is a concave substitute valuation for all agents. Supply x ∈ {0, 1}n. Then competitive equilibrium exists.

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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 Ai = A, a fixed product of intervals, for all agents i. ui : Ai → R is a concave strong substitute valuation for all agents. Supply x ∈ A. Then competitive equilibrium exists.

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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 domain Ai ⊂ {−1, 0, 1}n for all agents i. ui : Ai → 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 domain Ai ⊂ {−1, 0, 1}n for all agents i. ui : Ai → 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. ConvRDU(⋆) ∩ Zn = DU(⋆) 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. det 1 −1 1 1

• = 2
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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 13 / 21

Characterising when equilibrium always exists.

★

• 1

2

Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Consider [Nσ : Nσ1 + · · · + Nσm].

• 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Consider [Nσ : Nσ1 + · · · + Nσm].

• Nσ1
• ●
• ●

Nσ

• Nσ2
• 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Consider [Nσ : Nσ1 + · · · + Nσm].

• Nσ1
• ●
• ●

[Nσ : Nσ1 + Nσ2] = 2

• Nσ2
• 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Proposition If

• Nσ : Nσ1 + · · · + Nσm
• > 1 and dim σi = 1 for all i then

ConvRDU(p) ∩ Zn = DU(p). Sketch proof. DU(p) is the vertices of a parallellepiped parallel to the fundamental parallelepiped of Nσ1 + · · · + Nσm. 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Proposition If

• Nσ : Nσ1 + · · · + Nσm
• > 1 and dim σi = 1 for all i then

ConvRDU(p) ∩ Zn = DU(p). Corollary If

• Nσ : Nσ1 + · · · + Nσm
• > 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Proposition If

• Nσ : Nσ1 + · · · + Nσm
• = 1 then ConvRDU(p) ∩ Zn = DU(p).

Sketch proof. Since intersection transversal, can uniquely write x ∈ ConvRDU(p) ∩ Zn as sum of bundles in ConvRDui(p). If x ∈ Zn 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Proposition If

• Nσ : Nσ1 + · · · + Nσm
• = 1 then ConvRDU(p) ∩ Zn = DU(p).

Corollary If

• Nσ : Nσ1 + · · · + Nσm
• = 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Proposition (Standard) If dim Nσ = n and v1, . . . , vn is the union of bases for Nσ1, . . . , Nσm

• Nσ : Nσ1 + · · · + Nσm
• = | det(v1, . . . , vn)| = volume fund’l p’ped
• 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 σi (σ) be the individual (aggregate) SNP faces corresp. to price p. Transversal intersections means m

i=1 dim σi = dim σ (m agents).

Let Nτ ⊂ Zn be integer vectors parallel to lattice polytope τ. Proposition (Standard) If dim Nσ = n and v1, . . . , vn is the union of bases for Nσ1, . . . , Nσm

• Nσ : Nσ1 + · · · + Nσm
• = | det(v1, . . . , vn)| = volume fund’l p’ped

Say D ⊂ Zn is unimodular if | det(v1, . . . , vn)| = 1 for any linearly independent v1, . . . , vn ∈ D. (Small tweak when D does not span Zn.)

• 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 = {v1, . . . , vr} 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 Tu are in D.) Proposition (Standard) If dim Nσ = n and v1, . . . , vn is the union of bases for Nσ1, . . . , Nσm

• Nσ : Nσ1 + · · · + Nσm
• = | det(v1, . . . , vn)| = volume fund’l p’ped

Say D ⊂ Zn is unimodular if | det(v1, . . . , vn)| = 1 for any linearly independent v1, . . . , vn ∈ D. (Small tweak when D does not span Zn.)

• 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 = {v1, . . . , vr} 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 Tu 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).

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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.

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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.

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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 One intersection.

• Corresp. SNP face has area 2.

After the shift Two intersections.

• 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 One intersection.

• Corresp. SNP face has area 2.

After the shift Two intersections.

• Corresp. SNP faces have area 1.

Call this SNP area the multiplicity of the intersection. See # intersections is constant, up to multiplicity.

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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 Cσ Under Nσ ∼ = Zk ֒ → Rk assign volk(σ)

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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 Cσ Under Nσ ∼ = Zk ֒ → Rk assign volk(σ) Define cell weight w(Cσ) := k!volk(σ).

• vol ()=1

1

• 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 Cσ Under Nσ ∼ = Zk ֒ → Rk assign volk(σ) Define cell weight w(Cσ) := k!volk(σ).

• vol ()=1

1

• If cells Cσ1, Cσ2 of Tui, Tu2 intersect transversally at Cσ, then in Tu1 ∩ Tu2:

mult(Cσ) := w(Cσ1)w(Cσ2)

• Nσ1+σ2 : Nσ1 + Nσ2
• Theorem (Bertrand and Bihan 2007)

mult(Cσ) = MV (σ1, σ2, (n1, n2)) where ni = dim σi (i.e. repeat σi in the mixed volume ni 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 Cσ Under Nσ ∼ = Zk ֒ → Rk assign volk(σ) Define cell weight w(Cσ) := k!volk(σ).

• vol ()=1

1

• If cells Cσ1, Cσ2 of Tui, Tu2 intersect transversally at Cσ, then in Tu1 ∩ Tu2:

mult(Cσ) := w(Cσ1)w(Cσ2)

• Nσ1+σ2 : Nσ1 + Nσ2
• Theorem (Bertrand and Bihan 2007)

mult(Cσ) = MV (σ1, σ2, (n1, n2)) where ni = dim σi (i.e. repeat σi in the mixed volume ni times). Theorem Competitive equilibrium exists if the number of 0-cells of the intersection, weighted only by cell weights, equals

• n1+n2=n

MV (∆1, ∆2, (n1, n2)). 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

Dn

ss ⊂ Zn vectors have at most one +1, at most one -1, otherwise 0s.

Substitutes where trade-offs are locally 1-1. 1 1 1 −1

• p

A

p

B

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.

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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 16 / 21

Unimodular examples: Strong / step-wise substitutes

Dn

ss ⊂ Zn vectors have at most one +1, at most one -1, otherwise 0s.

Substitutes where trade-offs are locally 1-1.   1 1 1 1 −1 1 1 −1 −1  

(1,1,1)

p1 p2 p3 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.

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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 16 / 21

Interval package valuations

D is the basis change of Dn

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.

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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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1        front-line workers

• 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.

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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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1        front-line workers

• 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 "weak" Price on "strong" £100m

• 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 "weak" Price on "strong" £100m

• 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 "weak" Price on "strong" Bid for "weak" OR "strong" whichever has "better" price £100m

• 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 "weak" Price on "strong" £100m Nothing £100m "weak" £100m "strong"

• 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 "weak" Price on "strong" £100m Nothing £100m "weak" £100m "strong"

• 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 "w" Price on "s" W S

• 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.

S Price on "w" Price on "s" W WW WWW SS SSS WS WWS WSS

• 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. Add and subtract simple “either-or” bids = tropical factorisation!

S Price on "w" Price on "s" W WW SS WS

• 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. Add and subtract simple “either-or” bids = tropical factorisation!

S Price on "w" Price on "s" W WW SS WS

• 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. Add and subtract simple “either-or” bids = tropical factorisation!

S Price on "w" Price on "s" W WW SS WS

• ve
• 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. Add and subtract simple “either-or” bids = tropical factorisation!

S Price on "w" Price on "s" W WW SS WS

• ve
• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 19 / 21

Further auction development

Know when equilibrium existence guaranteed ⇒ allow new preferences in new Product-Mix auctions. E.g. indivisible goods with complementarities of use in: adjacent bands of radio spectrum; U.K. Dept of Energy and Climate Change (DECC) “buying” electricity capacity. Geometric methods also help develop other auction details. See Baldwin and Klemperer (soon).

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The geometry of auctions and competitive equilibrium with indivisible goods August 2014 20 / 21

Summary

Geometric analysis helps us understand Individual Demand

See an agent’s demand and their trade-offs. Classify valuations via ‘demand types’. Relate one structure of trade-offs to another.

Aggregate Demand

Always have competitive equilibrium iff ‘demand type’ is unimodular. Count intersections to check for equilibrium in other cases.

Matching with transferable utility

Stability = equilibrium = unimodularity of set of putative coalitions. New models of multiparty stable matching: see Seymour (1980).

Auctions

Product-Mix Auction (Klemperer 2008, 2010, Baldwin and Klemperer, soon).

• E. Baldwin and P. Klemperer

The geometry of auctions and competitive equilibrium with indivisible goods August 2014 21 / 21

References

• L. M. Ausubel and P. Milgrom. Ascending auctions with package bidding.

Frontiers of Theoretical Economics, 1(1):1–42, 2002.

• B. Bertrand and F. Bihan. Euler characteristic of real nondegenerate

tropical complete instersections. Available on Arxiv.org arXiv:0710.1222, 2007.

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