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
Augustgoods
2014
1 / 21
Second price / uniform price auctions
Suppose we sell
one unit
of
one 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
Augustgoods
2014
2 / 21
Second price / uniform price auctions
Suppose we sell
one unit
of
one 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
Augustgoods
2014
2 / 21
Second price / uniform price auctions
Suppose we sell
one unit
of
one 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
Augustgoods
2014
2 / 21
Second price / uniform price auctions
Suppose we sell many units of
one 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
Augustgoods
2014
2 / 21
Second price / uniform price auctions
Suppose we sell many units of
one good
in a sealed bid auction.
The highest bidders win.
They pay the highest losing bid.
Willingness to Pay
Your bid for unit i + 1 might
affect your price on units 1 to i.
1
2
3
4
5
Units
‘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
Augustgoods
2014
2 / 21
Second price / uniform price auctions
Suppose we sell many units of
one good
in a sealed bid auction.
The highest bidders win.
They pay the highest losing bid.
Willingness to Pay
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.
1
2
3
4
5
Units
‘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
Augustgoods
2014
2 / 21
Second price / uniform price auctions
Suppose we sell many units of
one good
in a sealed bid auction.
The highest bidders win.
They pay the highest losing bid.
Willingness to Pay
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.
1
2
3
4
5
Units
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
Augustgoods
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
Augustgoods
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
Augustgoods
2014
2 / 21
Geometric Analysis of Demand: Model
n indivisible goods.
x1
x2
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
3 / 21
Geometric Analysis of Demand: Model
n indivisible goods. Finite set A ⊂ Zn of bundles of goods available.
x1
x2
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
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.
x1
26
0
35
24
Example
of u(x)
x2
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
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}.
x1
26
0
35
24
Example
of u(x)
x2
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
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{u(x) − p.x}
x∈A
x1
26
0
35
24
Example
of u(x)
p
2
(1,1)
x2
(0,0)
(1,0)
(0,1)
p
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
Augustgoods
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{u(x) − p.x}
x∈A
x1
26
0
35
24
Example
of u(x)
p
2
(0,0)
(1,0)
(1,1)
(0,1)
x2
p
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
Augustgoods
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{u(x) − p.x}
x∈A
x1
13
0
32
12
p
2
(1,0)
(0,0)
Example
of u(x)
(1,1)
(0,1)
x2
p
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
Augustgoods
2014
3 / 21
How does demand change as you cross a facet?
p
2
(2,0) (1,0)
(0,0)
A tropical hypersurface
is composed of facets:
linear pieces in dimension (n − 1).
(1,1)
(0,1)
(0,2)
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
Augustgoods
2014
4 / 21
How does demand change as you cross a facet?
p
2
(2,0) (1,0)
(0,0)
A tropical hypersurface
is composed of facets:
linear pieces in dimension (n − 1).
(1,1)
(0,1)
(0,2)
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
Augustgoods
2014
4 / 21
How does demand change as you cross a facet?
2
(2,0) (1,0)
(
p
(11
(0,0)
(1,1)
(0,1)
(
A tropical hypersurface
is composed of facets:
linear pieces in dimension (n − 1).
(-11
(0,2)
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)
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
Augustgoods
2014
4 / 21
How does demand change as you cross a facet?
p
2
A tropical hypersurface
is composed of facets:
linear pieces in dimension (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
Augustgoods
2014
4 / 21
Economics from Geometry
v3
w
v4 w4 3
w1 w2
v2
Every tropical hypersurface
Pis balanced:
around each (n − 2)-cell, i wi vi = 0.
v1
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
Augustgoods
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
Augustgoods
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
Augustgoods
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.
p
2
(0,0)
p
1
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
6 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
has at most one +ve, one -ve coordinate entry.
(x 1+1,x2-2)
(
1
-2
(
p2
(x 1,x 2)
p1
Increase price i to cross a facet.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
has at most one +ve, one -ve coordinate entry.
(x 1+1,x2-2)
(
1
-2
(
p2
(x 1,x 2)
p1
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 6= i.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
has at most one +ve, one -ve coordinate entry.
p
2
(0,0)
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
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
has at most one +ve, one -ve coordinate entry.
p
2
(0,0)
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
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
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’ (∈
/ Tu ) and increase price i.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
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’ (∈
/ 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, vik < 0 for k = 1, . . . , r.
⇒ xrj ≥ x0j for all j 6= i
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
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’ (∈
/ Tu ) and increase price i.
As price changes, demand in turn x0 , x1 , . . . , xr .
S
At each stage, vk = xk − xk−1 is a facet normal.
IT
T
S
By the strict law of demand, vik < 0 for k = 1, . . . , r.
⇒ xrj ≥ x0j for all j 6= i
E. Baldwin and P. Klemperer
SU
E
UT
B
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
has all positive (or all negative) coordinate entries.
(23
(x 1+2,x2+3)
(
p2
(x 1,x 2)
p1
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, vik < 0 for k = 1, . . . , r.
⇒ xrj ≤ x0j for all j 6= i
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
has all positive (or all negative) coordinate entries.
(23
(x 1+2,x2+3)
(
p2
(x 1,x 2)
p1
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
By the strict law of demand,
⇒ xrj ≤ x0j for all j 6= i
E. Baldwin and P. Klemperer
vik
S
T
N
E
is a facet normal.
EM
L
P
< 0 for k = 1, . . . , r.
M
CO
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
is (1, 4)
(
(3,12)
1
4
(
p2
e.g. (car bodies, car wheels)
(2,8)
(1,4)
(0,0)
p1
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, vik < 0 for k = 1, . . . , r.
Demand fewer (1, 4) bundles
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
is (1, 4)
(
(3,12)
1
4
(
p2
e.g. (car bodies, car wheels)
(0,0)
(1,4)
(2,8)
p1
CT
E
S
F
As price changes, demand in turn x , x , . . . , x .
R
T
E
At each stage, v = x − x
is a facet normal. P
EN
M
By the strict law of demand, v < 0 for k = 1, . . . , r. LE
P
Demand fewer (1, 4) bundles
M
CO
Start in a ‘unique demand region’ (∈
/ Tu ) and increase price i.
0
k
k
1
r
k−1
k
i
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
2014
7 / 21
Economic properties from facets
Suppose every facet normal v to Tu ...
is in set D ⊂ Zn .
(
(3,12)
1
4
(
p2
e.g. (car bodies, car wheels)
(2,8)
(1,4)
(0,0)
p1
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, vik < 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
Augustgoods
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, vik < 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
Augustgoods
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 ConvR A, 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
Augustgoods
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 ConvR A, 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
Augustgoods
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 ConvR A, 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
Augustgoods
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 ConvR A, 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
Augustgoods
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 ConvR A, 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
Augustgoods
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 ConvR A, where u : A → R.
We subdivide it, to join up the sets Du (p).
2
2
0
4
3
2
4
4
2
✲ (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
Augustgoods
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 ConvR A, where u : A → R.
We subdivide it, to join up the sets Du (p).
2
2
0
4
2.5
2
4
4
2
✲ (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
Augustgoods
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 ConvR A, 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
Augustgoods
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 ConvR A, 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
Augustgoods
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
o
nP
P j
j (xj ) | xj ∈ A ,
u
x
=
x
.
U (x) = max
j
j
j
‘Aggregate’ tropical polynomial is tropical product of individual ones.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
Augustgoods
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
o
nP
P j
j (xj ) | xj ∈ A ,
u
x
=
x
.
U (x) = max
j
j
j
‘Aggregate’ tropical polynomial is tropical product of individual ones.
Definition (Standard)
If supply is x, a competitive equilibrium among agents i consists of
P
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
Augustgoods
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
August goods
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
August goods
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 6= j. Intersections.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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 6= j. Intersections.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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 6= j. Intersections.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
10 / 21
Tropical hypersurface of aggregate demand
DU (p) = Du1 (p) + · · · + Dum (p)
Easy to draw TU , just superimpose individual tropical hypersurfaces.
★
Du1 (F) = {(1, 0), (0, 1)}
Du2 (F) = {(0, 0), (1, 1)}
DU (F) =
{(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 6= j. Intersections.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
10 / 21
Tropical hypersurface of aggregate demand
DU (p) = Du1 (p) + · · · + Dum (p)
Easy to draw TU , just superimpose individual tropical hypersurfaces.
★
Du1 (F) = {(1, 0), (0, 1)}
Du2 (F) = {(0, 0), (1, 1)}
DU (F) =
{(1, 0), (0, 1), (1, 2), (2, 1)}.
DU (F) is not integer-convex
U is not concave
Bundle (1, 1) is not aggregate demand at F, 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
August goods
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 lim→0 Tu1 ∩ (Tu2 + 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
August goods
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 lim→0 Tu1 ∩ (Tu2 + 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
August goods
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 lim→0 Tu1 ∩ (Tu2 + 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
August goods
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 lim→0 Tu1 ∩ (Tu2 + 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
August goods
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 lim→0 Tu1 ∩ (Tu2 + w) 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.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
11 / 21
Classic theorems of competitive equilibrium
Theorem (Kelso and Crawford 1982)
Suppose
domain Ai = {0, 1}n
ui
:
Ai
for all agents i.
→ R is a concave substitute
Supply x ∈
valuation for all agents.
{0, 1}n .
Then competitive equilibrium exists.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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
August goods
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
August goods
2014
12 / 21
Characterising when equilibrium always exists.
★
?
Area=2.
The problem is that the bundle is in the middle of the square.
ConvR DU (F) ∩ Zn 6= DU (F)
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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
E. Baldwin and P. Klemperer
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Nσ
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Nσ2
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
E. Baldwin and P. Klemperer
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
[Nσ : Nσ1 + Nσ2 ] = 2
●
●
●
●
●
Nσ2
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
ConvR DU (p) ∩ Zn 6= 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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
ConvR DU (p) ∩ Zn 6= 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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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 ConvR DU (p) ∩ Zn = DU (p).
Sketch proof. Since intersection transversal, can uniquely write
x ∈ ConvR DU (p) ∩ Zn as sum of bundles in ConvR Dui (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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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 ConvR DU (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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
August goods
2014
13 / 21
Characterising when equilibrium always exists.
★
1
2
Let σi (σ) be the individual (aggregate) SNP faces corresp. to price p.
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
August goods
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
August goods
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).
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
13 / 21
When unimodularity fails: 2-D Bézout-Bernstein
Return to substitutes / complements example.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
14 / 21
When unimodularity fails: 2-D Bézout-Bernstein
Return to substitutes / complements example. Modify the valuations.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
14 / 21
When unimodularity fails: 2-D Bézout-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
August goods
2014
14 / 21
When unimodularity fails: 2-D Bézout-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
August goods
2014
14 / 21
When unimodularity fails: 2-D Bézout-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
August goods
2014
14 / 21
When unimodularity fails: 2-D Bézout-Bernstein
★
★
★
Theorem (Tropical Bézout-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
August goods
2014
14 / 21
When unimodularity fails: 2-D Bézout-Bernstein
●
●
●
●
●
●
●
●
●
●
●
●
Theorem (Tropical Bézout-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
August goods
2014
14 / 21
Generalised Bézout-Bernstein (Bertrand and Bihan, 2007)
Given SNP k-face σ ↔ TH (n − k)-cell Cσ
Under Nσ ∼
= Zk ,→ Rk assign volk (σ)
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
15 / 21
Generalised Bézout-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).
● ● ● ● ●
E. Baldwin and P. Klemperer
● ● ● ● ●
vol1()=1
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
15 / 21
Generalised Bézout-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).
● ● ● ● ●
● ● ● ● ●
vol1()=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 ) = M V (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
August goods
2014
15 / 21
Generalised Bézout-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).
● ● ● ● ●
● ● ● ● ●
vol1()=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 ) = M V (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
X
M V (∆1 , ∆2 , (n1 , n2 )).
n1 +n2 =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
August goods
2014
15 / 21
Unimodular examples: Strong / step-wise substitutes
n ⊂ Zn vectors have at most one +1, at most one -1, otherwise 0s.
Dss
Substitutes where trade-offs are locally 1-1.
pB
1 0
1
0 1 −1
pA
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
August goods
2014
16 / 21
Unimodular examples: Strong / step-wise substitutes
n ⊂ Zn vectors have at most one +1, at most one -1, otherwise 0s.
Dss
Substitutes where trade-offs are locally 1-1.
p3


1 0 0
1
1
0
 0 1 0 −1
0
1 
0 0 1
0 −1 −1
p2
(1,1,1)
Unimodular set (classic result).
Equilibrium always exists
p1
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
August goods
2014
16 / 21
Interval package valuations
n via the upper triangular matrix of 1s.
D is the basis change of Dss
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
August goods
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
August goods
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
August goods
2014
18 / 21
The Bank of England’s Product-Mix Auction
Price on "strong"
Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral.
£100m
Price on "weak"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
19 / 21
The Bank of England’s Product-Mix Auction
Price on "strong"
Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral.
£100m
Price on "weak"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
19 / 21
The Bank of England’s Product-Mix Auction
Price on "strong"
Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral.
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
August goods
2014
19 / 21
The Bank of England’s Product-Mix Auction
Price on "strong"
Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral.
£100m
"weak"
Nothing
£100m
£100m
"strong"
Price on "weak"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
19 / 21
The Bank of England’s Product-Mix Auction
Price on "strong"
Bid interest rates to receive liquidity against 2 ‘strengths’ of collateral.
Assume that trade-offs are 1-1: strong substitutes.
£100m
"weak"
Nothing
£100m
£100m
"strong"
Price on "weak"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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 "s"
W
0
S
Price on "w"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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 "s"
WWW
WW W
WS
0
S
SS
WWS
WSS
SSS
Price on "w"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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!
W
0
Price on "s"
WW
S
WS
SS
Price on "w"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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!
W
0
Price on "s"
WW
S
WS
SS
Price on "w"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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!
0
W
Price on "s"
WW
-ve
S
WS
SS
Price on "w"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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!
0
W
Price on "s"
WW
-ve
S
WS
SS
Price on "w"
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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).
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
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
August goods
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.
V. Danilov and V. Grishukhin. Maximal unimodular systems of vectors.
European Journal of Combinatorics, 20(6):507–526, 1999.
V. Danilov and G. Koshevoy. Discrete convexity and unimodularity–I.
Advances in Mathematics, 189(2):301–324, 2004.
V. Danilov, G. Koshevoy, and K. Murota. Discrete convexity and equilibria
in economies with indivisible goods and money. Mathematical Social
Sciences, 41:251–273, 2001.
V. Danilov, G. Koshevoy, and C. Lang. Gross substitution, discrete
convexity, and submodularity. Discrete Applied Mathematics, 131(2):
283–298, 2003.
J. W. Hatfield, S. D. Kominers, A. Nichifor, M. Ostrovsky, and
A. Westkamp. Stability and competitive equilibrium in trading networks.
Journal of Political Economy, 121(5):pp. 966–1005, 2013.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
21 / 21
References
A. S. Kelso and V. P. Crawford. Job matching, coalition formation, and
gross substitutes. Econometrica, 50(6):1483–1504, 1982.
P. Klemperer. A new auction for substitutes: Central bank liquidity
auctions, the U.S. TARP, and variable product-mix auctions. Working
paper, Oxford University, 2008.
P. Klemperer. The product-mix auction: A new auction design for
differentiated goods. Journal of the European Economic Association, 8
(2-3):526–536, 2010.
G. Mikhalkin. Decomposition into pairs-of-pants for complex algebraic
hypersurfaces. Topology, 43(5):1035–1065, 2004.
P. Milgrom and B. Strulovici. Substitute goods, auctions, and equilibrium.
Journal of Economic Theory, 144(1):212–247, 2009.
P. Seymour. Decomposition of regular matroids. Journal of Combinatorial
Theory, Series B, 28(3):305–359, 1980.
B. Sturmfels. Solving systems of polynomial equations. Regional
conference series in mathematics. Published for the Conference Board of
the Mathematical Sciences by the American Mathematical Society,
2002.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
21 / 21
N. Sun and Z. Yang. Equilibria and indivisibilities: Gross substitutes and
complements. Econometrica, 74(5):1385–1402, 2006.
E. Baldwin and P. Klemperer
The geometry of auctions and competitive equilibrium with indivisible
August goods
2014
21 / 21