The Product-Mix Auction
Elizabeth Baldwin Paul Goldberg Paul Klemperer
Oxford University
May 2016
Covering material from Klemperer (2008, 2010), and much furtherwork in collaboration
This work was supported by ESRC grant ES/L003058/1.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 1 / 35
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?
are useful for auctioneers:
Informative
Efficient
Easy for participants – encourage market entry.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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.
are useful for auctioneers:
Informative
Efficient
Easy for participants – encourage market entry.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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
Units1 2 3 4 5
Your bid for unit i + 1 mightaffect your price on units 1 to i .
But if you are small relative to marketsize, then optimal ‘shading’ is small.You are unlikely to be ‘marginal’.
‘Truthful revelation mechanisms’ are useful for auctioneers:
Informative
Efficient
Easy for participants – encourage market entry.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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
Units1 2 3 4 5
Optimal biddingschedule
Your bid for unit i + 1 mightaffect your price on units 1 to i .
But if you are small relative to marketsize, then optimal ‘shading’ is small.You are unlikely to be ‘marginal’.
‘Truthful revelation mechanisms’ are useful for auctioneers:
Informative
Efficient
Easy for participants – encourage market entry.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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
Units1 2 3 4 5
Optimal biddingschedule
Your bid for unit i + 1 mightaffect your price on units 1 to i .
But if you are small relative to marketsize, then optimal ‘shading’ is small.You are unlikely to be ‘marginal’.
Nearly truthful revelation mechanisms are useful for auctioneers:
Informative
Efficient
Easy for participants – encourage market entry.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
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 ofconsumer preferences in price space.
Nearly truthful revelation mechanisms are useful for auctioneers:
Informative
Efficient
Easy for participants – encourage market entry.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 2 / 35
Bank of England Problem
After Northern Rock bank run, Bank of England urgently wants to loanfunds to banks, etc., – willing to take weaker-than-usual collateral, butonly in return for higher interest rate.
i.e., wanted to sell related goods to banks (loans against different kinds ofcollateral: “strong” (UK / US sovereign debt), “weak” (mortgage-backedsecurities?!), etc.
Turn to Paul Klemperer for help
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 3 / 35
Bank of England Problem
After Northern Rock bank run, Bank of England urgently wants to loanfunds to banks, etc., – willing to take weaker-than-usual collateral, butonly in return for higher interest rate.
i.e., wanted to sell related goods to banks (loans against different kinds ofcollateral: “strong” (UK / US sovereign debt), “weak” (mortgage-backedsecurities?!), etc.
Turn to Paul Klemperer for help
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 3 / 35
US Treasury problem – September 2008
After Lehman’s collapse, U.S. “TARP” plans to spend up to $700 billionbuying “Toxic” Assets.
i.e., wanted to buy related goods from banks (Alt-A and subprimenon-agency mortgage-backed securities originally rated AAA)
Turn to Paul Klemperer (and others) for help
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 4 / 35
US Treasury problem – September 2008
After Lehman’s collapse, U.S. “TARP” plans to spend up to $700 billionbuying “Toxic” Assets.
i.e., wanted to buy related goods from banks (Alt-A and subprimenon-agency mortgage-backed securities originally rated AAA)
Turn to Paul Klemperer (and others) for help
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 4 / 35
High-Frequency Trading problems
Wasteful investment in speed
“Flash crashes”, etc
have led to calls to replace continuous trading with Batch auctions, sayone per second (see e.g. Budish et al. 2015).
With time between trades, bidders may wish to make trades contingent onthe outcome of other trades,
e.g., “buy X and sell Y iff price difference < z”
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 5 / 35
UK Department of energy and climate change problem
DECC wants to procure commitments to build new electricity-supplycapacity and promote use of renewables. “to replace 25% of existingcapacity by 2020. . . . . .we need around £200bn. . .”
Has preferences about mix of gas, nuclear, wind etc.
Contracts projects with long lead-in times – wants to distinguish by year-ofdelivery.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 6 / 35
General Problem
Supplier wants to sell multiple versions of a product: multiple “goods”.
Seller costs depend on bundle of goods sold. So their preferred bundle tosell depends on prices on all goods.
Bidders’ demand depends on prices on all goods.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 7 / 35
Standard Approaches
Separate auction for each good: fix quantities
Single auction with fixed relative prices
True seller preferences in general may be more like:
p2 − p1
q2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 8 / 35
Standard Approaches
Separate auction for each good: fix quantities
Single auction with fixed relative prices
True seller preferences in general may be more like:
p2 − p1
q2Problems:
1. Bidders have to guess which auction to bid in, may regret after theevent.
2. Market Power: too little competition between bidders.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 8 / 35
Standard Approaches
Separate auction for each good: fix quantities
Single auction with fixed relative prices
True seller preferences in general may be more like:
p2 − p1
q2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 8 / 35
Standard Approaches
Separate auction for each good: fix quantities
Single auction with fixed relative prices
True seller preferences in general may be more like:
p2 − p1
q2Problems
1. End result may not reflect seller preferences.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 8 / 35
Standard Approaches
Separate auction for each good: fix quantities
Single auction with fixed relative prices
True seller preferences in general may be more like:p2 − p1
q2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 8 / 35
Dynamic mechanisms: SMRA and clock auctions
Used in many auctions of mobile-phone licenses
Under some conditions, creates ‘healthy competition’: competitiveequilibrium, all bidders, and auctioneer, get same as they would havechosen at the final prices, and fully efficient.
But
1. Time taken
interactions with other markets, manipulationscostly for bidders, so lower participation
2. Collusion and predation (because bidders can respond to others’signals).
3. (Not at present developed to reflect non-trivial seller preferences)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 9 / 35
Dynamic versus Static Mechanisms
Dynamic (multi-round) auction Static (single-round) ‘proxy’auction
Single-unit ascending Sealed-bid 2nd-price
Multi-unit ascending Uniform-price
Multi-unit multi-variety (simulta-neous multiple-round auction orclock variant)
Product-mix auction (proxySMRA)
SlowFast
Information revealed; communica-tion possible
No information until process ends
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 10 / 35
Dynamic versus Static Mechanisms
Dynamic (multi-round) auction Static (single-round) ‘proxy’auction
Single-unit ascending Sealed-bid 2nd-price
Multi-unit ascending Uniform-price
Multi-unit multi-variety (simulta-neous multiple-round auction orclock variant)
Product-mix auction (proxySMRA)
SlowFast
Information revealed; communica-tion possible
No information until process ends
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 10 / 35
Product-mix auction versus SMRA
Product-Mix Auction restricts bidders’ strategies: their “preferences” can’tdepend on others’ bids (so they can’t condition their behaviour on others)But
expands the preferences that auctioneer can express about howquantities bought / sold depend on bidding
is quicker
is often easier to understand
is less vulnerable to collusion and predation.
Both are designed assuming all bidders have strong substitute / M\
preferences.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 11 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
Price (interest rate) on "weak"Pri
ce (
inte
rest
rate
) on "
str
ong"
100m
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
Price (interest rate) on "weak"Pri
ce (
inte
rest
rate
) on "
str
ong"
100m
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
Price (interest rate) on "weak"Pri
ce (
inte
rest
rate
) on "
str
ong"
100m
Bid for "weak" OR "strong"
whichever has "better" price
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
Price (interest rate) on "weak"Pri
ce (
inte
rest
rate
) on "
str
ong"
100m
Nothing100m
"weak"
100m
"strong"
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
Price (interest rate) on "weak"Pri
ce (
inte
rest
rate
) on "
str
ong"
100m
Nothing100m
"weak"
100m
"strong"
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
0W
S
Price on "w"
Pri
ce o
n "
s"
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
S
0WWWWWW
SS
SSS
WS
WWSWSS
Pri
ce o
n "
s"
Price on "w"
‘Just’ need to find prices so that supply equals demand!
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
The Product-Mix Auction
Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.
S
0WWWWWW
SS
SSS
WS
WWSWSS
Pri
ce o
n "
s"
Price on "w"
‘Just’ need to find prices so that supply equals demand!E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35
Seller preferences: Method 1
Seller has preferences e.g. q1 + q2 is constant; q2 as a function ofp2 − p1. This is the ‘supply curve’.
For a set of relevant values of (q1, q2), find (minimum) prices (p1, p2)such that (q1, q2) is demanded.
This allows us to derive a ‘demand curve’.
Intersect supply and demand to find the equilibrium.
p2 − p1
q2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 13 / 35
Seller preferences: Method 1
Seller has preferences e.g. q1 + q2 is constant; q2 as a function ofp2 − p1. This is the ‘supply curve’.
For a set of relevant values of (q1, q2), find (minimum) prices (p1, p2)such that (q1, q2) is demanded.
This allows us to derive a ‘demand curve’.
Intersect supply and demand to find the equilibrium.
p2 − p1
q2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 13 / 35
Seller preferences: Method 1
Seller has preferences e.g. q1 + q2 is constant; q2 as a function ofp2 − p1. This is the ‘supply curve’.
For a set of relevant values of (q1, q2), find (minimum) prices (p1, p2)such that (q1, q2) is demanded.
This allows us to derive a ‘demand curve’.
Intersect supply and demand to find the equilibrium.
Advantages:
People in business and central bankers understand.
Can use for a wide range of seller preferences (not necessarily strongsubstitute).
Disadvantage:
Could be ad-hoc and computationally inefficient.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 13 / 35
Seller preferences: Method 2
Suppose the seller has strong substitute preferences also.That is, seller has a valuation uS : AS → R, where typically AS ( Zn
−.This valuation is concave and of the strong substitute demand type.
Definition
There exists competitive equilibrium between this seller and buyers withaggregate valuation U if there exists p such that 0 ∈ Dus (p) + DU(p).
But we can let the ‘maximum supply’ be
y := sup−AS , i.e. yi = max{−xi : x ∈ AS}.
Define a shifted seller valuation with domain AS ′ = {y}+ AS ( Zn.
uS′
: AS ′ → R via uS′(y + x) = uS(x).
Now let U ′ be the aggregate valuation of U and uS′.
Competitive equilibrium exists iff y ∈ DU′(p).
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 14 / 35
Seller preferences: Method 2
Suppose the seller has strong substitute preferences also.That is, seller has a valuation uS : AS → R, where typically AS ( Zn
−.This valuation is concave and of the strong substitute demand type.
Definition
There exists competitive equilibrium between this seller and buyers withaggregate valuation U if there exists p such that 0 ∈ Dus (p) + DU(p).
But we can let the ‘maximum supply’ be
y := sup−AS , i.e. yi = max{−xi : x ∈ AS}.
Define a shifted seller valuation with domain AS ′ = {y}+ AS ( Zn.
uS′
: AS ′ → R via uS′(y + x) = uS(x).
Now let U ′ be the aggregate valuation of U and uS′.
Competitive equilibrium exists iff y ∈ DU′(p).
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 14 / 35
Seller preferences: Method 2
p2 − p1
q2
Auctioneer’s Supply Curve
p2
p1
Corresponding Tropical Hypersurfaceshowing Auctioneer’s “demand”
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 15 / 35
Bank of England implementation
Originally
2 goods, fixed quantity
‘Method 1’
Since February 2014
3 goods, endogenous total quantity
Combination of ‘Method 1’ and ‘Method 2’.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 16 / 35
Bank of England implementation
Originally
2 goods, fixed quantity
‘Method 1’
Since February 2014
3 goods, endogenous total quantity
Combination of ‘Method 1’ and ‘Method 2’.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 16 / 35
More general strong substitute preferences
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
0WWWWWW
SS
SSS
WS
WWSWSS
Pri
ce o
n "
s"
Price on "w"
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 17 / 35
More general strong substitute preferences
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"
Pri
ce o
n "
s"
0W
WW
SS
WS
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 17 / 35
More general strong substitute preferences
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"
Pri
ce o
n "
s"
0W
WW
SS
WS
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 17 / 35
More general strong substitute preferences
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
0W
WW
SS
WS
-ve
Price on "w"
Pri
ce o
n "
s"
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 17 / 35
More general strong substitute preferences
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
0W
WW
SS
WS
-vePri
ce o
n "
s"
Price on "w"
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 17 / 35
Interpreting “Dot Bids”:
A single (positive) dot bid at r represents valuation
u(0) = 0, u(ei ) = riThis has tropical hypersurface Tr, consisting of n + n(n − 1)/2 facets:
Hods: Agent indifferent between buying nothing, and buying ei .
Hri = {p ∈ Rn : pi = ri , pk ≥ rk for k 6= i}.
Flanges: Agent indifferent between buying ei and ej , i 6= j
F rij = {p ∈ Rn : pi − pj = ri − rj , pi ≤ ri , pk ≥ rk for k 6= i , j}.
p1
p2
Associate rational polyhedral complex Πr, weight 1 on each facet.E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 18 / 35
Interpreting “Dot Bids”:
A single (positive) dot bid at r represents valuation
u(0) = 0, u(ei ) = riThis has tropical hypersurface Tr, consisting of n + n(n − 1)/2 facets:
Hods: Agent indifferent between buying nothing, and buying ei .
Hri = {p ∈ Rn : pi = ri , pk ≥ rk for k 6= i}.
Flanges: Agent indifferent between buying ei and ej , i 6= j
F rij = {p ∈ Rn : pi − pj = ri − rj , pi ≤ ri , pk ≥ rk for k 6= i , j}.
(1,1,1)
p1
p2
p3
Associate rational polyhedral complex Πr, weight 1 on each facet.E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 18 / 35
Interpreting “Dot Bids”: many bids
A collection of positive dot bids R = (r1, . . . , ra)
⇔ Aggregate valuation of corresponding u1, . . . , ua
⇔ Tropical hypersurface TR = Tr1 ∪ · · · ∪ Tra .
⇔ Balanced weighted rational polyhedral complex (ΠR,w) in whichthe weights are the number of dot bids associated with each facet.
p1
p2
Write (TR,w) = (Tr1 , 1) + · · ·+ (Tra , 1)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 19 / 35
Interpreting “Dot Bids”: many bids
A collection of positive dot bids R = (r1, . . . , ra)
⇔ Aggregate valuation of corresponding u1, . . . , ua
⇔ Tropical hypersurface TR = Tr1 ∪ · · · ∪ Tra .
⇔ Balanced weighted rational polyhedral complex (ΠR,w) in whichthe weights are the number of dot bids associated with each facet.
p1
p2(0,0)
(0,1)
(0,3)
2
Write (TR,w) = (Tr1 , 1) + · · ·+ (Tra , 1)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 19 / 35
Interpreting “Dot Bids”: many bids
A collection of positive dot bids R = (r1, . . . , ra)
⇔ Aggregate valuation of corresponding u1, . . . , ua
⇔ Tropical hypersurface TR = Tr1 ∪ · · · ∪ Tra .
⇔ Balanced weighted rational polyhedral complex (ΠR,w) in whichthe weights are the number of dot bids associated with each facet.
p1
p2(0,0)
(0,1)
(0,3)
2
Write (TR,w) = (Tr1 , 1) + · · ·+ (Tra , 1)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 19 / 35
Interpreting “Dot Bids”: many bids
A collection of positive dot bids R = (r1, . . . , ra)
⇔ Aggregate valuation of corresponding u1, . . . , ua
⇔ Tropical hypersurface TR = Tr1 ∪ · · · ∪ Tra .
⇔ Balanced weighted rational polyhedral complex (ΠR,w) in whichthe weights are the number of dot bids associated with each facet.
p1
p2(0,0)
(0,1)
(0,3)
2
Write (TR,w) = (Tr1 , 1) + · · ·+ (Tra , 1)E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 19 / 35
‘Arithmetic’ of weighted tropical hypersurfaces
Given weighted (Tu1 ,w1), (Tu2 ,w2), aggregate (TU ,wU) satisfies:
TU = Tu1 ∪ Tu2If F is a facet of TU then wU(F ) =
∑w1(F 1) +
∑w2(F 2) where F j
are facets of T uj containing F .
Now write (TU ,wU) = (Tu1 ,w1) + (Tu2 ,w2) in this case.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 20 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
Tu1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
Tu1Tu2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
1
1
1
-1
-1
0
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
1
1
1
-1
-1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
1
1
1
-1
-1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
1
1
1
-1
-1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2 Tu1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
Tu2
Tu1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
0
0
0
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
p1
p2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
‘Arithmetic’ of weighted tropical hypersurfaces II
Identify Tu1 ∪ Tu2 and associated complex Π̃.
If F is a facet of Π̃ then wU(F ) =∑
w1(F 1)−∑
w2(F 2) where F j
are facets of T uj containing F .
Let Π contain all facets F of Π̃ with w(F ) 6= 0, and all cells in theboundaries of these facets. Let T be the support of Π.
(T ,w) := (Tu1 ,w1)− (Tu2 ,w2). ‘Z-weighted’ support of poly’l complex.
Π is balanced (difference between balanced complexes).
So if w ≥ 0, then T is a tropical hypersurface of some valuation.
Lemma
The arithmetic of supports of balanced weighted rational polyhedralcomplexes defined in this way is commutative and associative.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 21 / 35
Interpreting “Dot Bids”: positive and negative bids
Given positive dot bids R = (r1, . . . , ra) and negative dot bidsS = (s1, . . . , sb).
(TR,wR) = (Tr1 , 1) + · · ·+ (Tra , 1) and(TS ,wS) = (Ts1 , 1) + · · ·+ (Tsb , 1) as before.Define (TR,S ,w
R,S) = (TR,wR)− (TS ,wS).
Definition
Bids are valid if (TR,S ,wR,S) is a (Z+-weighted) tropical hypersurface.
In this case, the associated valuation is a strong substitute valuation, byconstruction.
p1
p2 r1
r2
r3 s1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 22 / 35
Interpreting “Dot Bids”: positive and negative bids
Given positive dot bids R = (r1, . . . , ra) and negative dot bidsS = (s1, . . . , sb).
(TR,wR) = (Tr1 , 1) + · · ·+ (Tra , 1) and(TS ,wS) = (Ts1 , 1) + · · ·+ (Tsb , 1) as before.Define (TR,S ,w
R,S) = (TR,wR)− (TS ,wS).
Definition
Bids are valid if (TR,S ,wR,S) is a (Z+-weighted) tropical hypersurface.
In this case, the associated valuation is a strong substitute valuation, byconstruction.
p1
p2 r1
r2
r3 s1
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 22 / 35
Interpreting “Dot Bids”: positive and negative bids, II
Translating R,S to valuation uR−S is convoluted.Translating R,S to DuR−S (p) is (generically) easy!
Suppose pi 6= ri , pi − pj 6= ri − rj for all r ∈ R ∪ S.
(DuR−S (p))i =
|{r ∈ R : ri − pi = max{0, rk − pk : k = 1, . . . , n}|−|{s ∈ S : si − pi = max{0, sk − pk : k = 1, . . . , n}|
p1
p2 r1
r2
r3 s1p Green lines show
thresholds: hereri − pi = 0 orr1 − p1 = r2 − p2.
DuR−S (p) = (1, 0)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 23 / 35
Interpreting “Dot Bids”: positive and negative bids, II
Translating R,S to valuation uR−S is convoluted.Translating R,S to DuR−S (p) is (generically) easy!
Suppose pi 6= ri , pi − pj 6= ri − rj for all r ∈ R ∪ S.
(DuR−S (p))i =
|{r ∈ R : ri − pi = max{0, rk − pk : k = 1, . . . , n}|−|{s ∈ S : si − pi = max{0, sk − pk : k = 1, . . . , n}|
p1
p2 r1
r2
r3 s1p Green lines show
thresholds: hereri − pi = 0 orr1 − p1 = r2 − p2.
DuR−S (p) = (0, 1)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 23 / 35
Interpreting “Dot Bids”: positive and negative bids, II
Translating R,S to valuation uR−S is convoluted.Translating R,S to DuR−S (p) is (generically) easy!
Suppose pi 6= ri , pi − pj 6= ri − rj for all r ∈ R ∪ S.
(DuR−S (p))i =
|{r ∈ R : ri − pi = max{0, rk − pk : k = 1, . . . , n}|−|{s ∈ S : si − pi = max{0, sk − pk : k = 1, . . . , n}|
p1
p2 r1
r2
r3 s1
p
Green lines showthresholds: hereri − pi = 0 orr1 − p1 = r2 − p2.
DuR−S (p) = (1, 1)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 23 / 35
Interpreting “Dot Bids”: positive and negative bids, II
Translating R,S to valuation uR−S is convoluted.Translating R,S to DuR−S (p) is (generically) easy!
Suppose pi 6= ri , pi − pj 6= ri − rj for all r ∈ R ∪ S.
(DuR−S (p))i =
|{r ∈ R : ri − pi = max{0, rk − pk : k = 1, . . . , n}|−|{s ∈ S : si − pi = max{0, sk − pk : k = 1, . . . , n}|
p1
p2 r1
r2
r3 s1
p
Green lines showthresholds: hereri − pi = 0 orr1 − p1 = r2 − p2.
DuR−S (p) = (1, 1)
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 23 / 35
Interpreting “Dot Bids”: positive and negative bids, II
Translating R,S to valuation uR−S is convoluted.Translating R,S to DuR−S (p) is (generically) easy!
Suppose pi 6= ri , pi − pj 6= ri − rj for all r ∈ R ∪ S.
(DuR−S (p))i =
|{r ∈ R : ri − pi = max{0, rk − pk : k = 1, . . . , n}|−|{s ∈ S : si − pi = max{0, sk − pk : k = 1, . . . , n}|
For non-generic p, find DuR−S (p + t) for (n + 1)! sufficiently small t,covering all ways of strictly ordering 0, t1, . . . , tn.
These are the bundles demanded in all unique demand regions adjacent top, and thus DuR−S (p + t) is the convex hull of these bundles.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 23 / 35
Representation of Strong Substitute Valuations
Say A is triangular if A = {x ∈ Zn+ :
∑i xi ≤ d} for some d .
Theorem (Characterisation of Strong Substitutes)
A valuation u : A→ R is a strong substitute (M\-concave) valuation withtriangular domain iff it can be presented using finite collections of positiveand negative dot bids.
If the domain is not triangular, we can extend to the minimal triangulardomain containing it, with arbitrarily low negative values.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 24 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
p1
p2
p3
r
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
p1
p2
p3
r
H1
H2
H3
F 23
F 12
F 13
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
p1
p2
p3
rH2
H3
F 23
F 12
F 13
H2 ∩ H3 ∩ F 23
H1 ∩ H3 ∩ F 13H1 ∩ H2 ∩ F 12
F 12 ∩ F 23 ∩ F 13
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
Definition
A set C̃ ⊂ Tu is a maximal facet continuation if it is the maximal union offacets sharing an affine span and connected along (n − 2)-cells.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
Definition
A set C̃ ⊂ Tu is a maximal facet continuation if it is the maximal union offacets sharing an affine span and connected along (n − 2)-cells.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
Definition
A set C̃ ⊂ Tu is a maximal facet continuation if it is the maximal union offacets sharing an affine span and connected along (n − 2)-cells.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
Definition
A set C̃ ⊂ Tu is a maximal facet continuation if it is the maximal union offacets sharing an affine span and connected along (n − 2)-cells.
Again, these are H i -style or F ij -style
All boundaries of maximal facet continuations must be (n − 2)-cells whichare either H i ∩ H j ∩ F ij -style or F ij ∩ F jk ∩ F ki -style.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Strong Substitute (n − 2)-cells, and extended facets
If u is strong substitutes, all facets are either
normal vector ei : “H i -style” “hod-style”
normal vector ei − ej : “F i ,j -style” “flange-style”
Possible pairs of normals: every (n − 2)-cell is exactly one of
H i ∩ H j ∩ F ij -style
F ij ∩ F jk ∩ F ki -style
locally the intersection of two or more hyperplanes.
Definition
A set C̃ ⊂ Tu is a maximal facet continuation if it is the maximal union offacets sharing an affine span and connected along (n − 2)-cells.
Again, these are H i -style or F ij -styleAll boundaries of maximal facet continuations must be (n − 2)-cells whichare either H i ∩ H j ∩ F ij -style or F ij ∩ F jk ∩ F ki -style.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 25 / 35
Local Minimal Points
A hod-style maximal facet C continuation is bounded below in everycoordinate because A is triangular.
H i -style, then (n − 2)-cells bounding it are H i ∩ H j ∩ F ij -style.
So if r ∈ C is locally minimal in all coordinates, is intersection ofH j -style facets for j = 1, . . . , n, and in boundary of the facetcontinuations.
Must be balanced around H i ∩ H j ∩ F ij -style (n − 2)-cells. So r alocal minima for each H j -style facet continuation in the intersection.
Similarly consider all possible boundaries to F ij :
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 26 / 35
Local Minimal Points
A hod-style maximal facet C continuation is bounded below in everycoordinate because A is triangular.
H i -style, then (n − 2)-cells bounding it are H i ∩ H j ∩ F ij -style.
So if r ∈ C is locally minimal in all coordinates, is intersection ofH j -style facets for j = 1, . . . , n, and in boundary of the facetcontinuations.
Must be balanced around H i ∩ H j ∩ F ij -style (n − 2)-cells. So r alocal minima for each H j -style facet continuation in the intersection.
Similarly consider all possible boundaries to F ij :
????
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 26 / 35
Local Minimal Points
A hod-style maximal facet C continuation is bounded below in everycoordinate because A is triangular.
H i -style, then (n − 2)-cells bounding it are H i ∩ H j ∩ F ij -style.
So if r ∈ C is locally minimal in all coordinates, is intersection ofH j -style facets for j = 1, . . . , n, and in boundary of the facetcontinuations.
Must be balanced around H i ∩ H j ∩ F ij -style (n − 2)-cells. So r alocal minima for each H j -style facet continuation in the intersection.
Similarly consider all possible boundaries to F ij :
Lemma
In a sufficiently small neighbourhood of r, have Tu equal to Tr
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 26 / 35
Minimal Points for Facets II
For every hod-style maximal facet continuation assoc with Tu, put‘dot bids’ at every local minimal point r, commensurate with weightson facets.
These generate Tr consisting of ‘hods’ and ‘flanges’.
Every facet is now contained in Tr for one of these r
Lemma (Covering Lemma)
If u is a strong substitute valuation, there exists a minimal finite setR ⊂ Rn such that
Tu ⊆ TRwu(F ) ≤ wR(F ′), where F ,F ′ facets of Tu, TR resp., s.t. F ⊆ F ′
For an open neighbourhood D of every r ∈ R, haveD ∩ (Tu,wu) = D ∩ (TR,wR).
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 27 / 35
Algorithm to generate TuStart with strong substitute u and Tu.
Find R1 as in covering lemma: R1 minimal such thatTu ⊆ TR1
wu(F ) ≤ wR1
(F ′), where F ,F ′ facets of Tu, TR1 resp., s.t. F ⊆ F ′
(Tu,wu) and (TR1 ,wR1
) equal locally around every r ∈ R1.
Write (T 2,w2) := (TR1 ,wR1)− (Tu,wu).
This is a (Z+-weighted) tropical hypersurface for strong substitutes.
Find R2 as in covering lemma, for (T 2,w2).
(T 3,w3) := (TR2 ,wR2)− (T 2,w2). Find R3 covering (T 3,w3)
. . .
Suppose this terminates: at some stage T l+1 = ∅. Then
(Tu,wu) = (TR1 ,wR1)− (TR2 ,wR
2) + · · ·+ (−1)l−1(TRl ,wR
l)
= (TR,wR)− (TS ,wS)
where R = R1 ∪R3 ∪ · · · and S = R2 ∪R4 ∪ · · · .
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 28 / 35
Algorithm to generate TuStart with strong substitute u and Tu.
Find R1 as in covering lemma: R1 minimal such thatTu ⊆ TR1
wu(F ) ≤ wR1
(F ′), where F ,F ′ facets of Tu, TR1 resp., s.t. F ⊆ F ′
(Tu,wu) and (TR1 ,wR1
) equal locally around every r ∈ R1.
Write (T 2,w2) := (TR1 ,wR1)− (Tu,wu).
This is a (Z+-weighted) tropical hypersurface for strong substitutes.
Find R2 as in covering lemma, for (T 2,w2).
(T 3,w3) := (TR2 ,wR2)− (T 2,w2). Find R3 covering (T 3,w3)
. . .
Suppose this terminates: at some stage T l+1 = ∅. Then
(Tu,wu) = (TR1 ,wR1)− (TR2 ,wR
2) + · · ·+ (−1)l−1(TRl ,wR
l)
= (TR,wR)− (TS ,wS)
where R = R1 ∪R3 ∪ · · · and S = R2 ∪R4 ∪ · · · .
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 28 / 35
Algorithm to generate TuStart with strong substitute u and Tu.
Find R1 as in covering lemma: R1 minimal such thatTu ⊆ TR1
wu(F ) ≤ wR1
(F ′), where F ,F ′ facets of Tu, TR1 resp., s.t. F ⊆ F ′
(Tu,wu) and (TR1 ,wR1
) equal locally around every r ∈ R1.
Write (T 2,w2) := (TR1 ,wR1)− (Tu,wu).
This is a (Z+-weighted) tropical hypersurface for strong substitutes.
Find R2 as in covering lemma, for (T 2,w2).
(T 3,w3) := (TR2 ,wR2)− (T 2,w2). Find R3 covering (T 3,w3)
. . .
Suppose this terminates: at some stage T l+1 = ∅. Then
(Tu,wu) = (TR1 ,wR1)− (TR2 ,wR
2) + · · ·+ (−1)l−1(TRl ,wR
l)
= (TR,wR)− (TS ,wS)
where R = R1 ∪R3 ∪ · · · and S = R2 ∪R4 ∪ · · · .
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 28 / 35
Algorithm to generate TuStart with strong substitute u and Tu.
Find R1 as in covering lemma: R1 minimal such thatTu ⊆ TR1
wu(F ) ≤ wR1
(F ′), where F ,F ′ facets of Tu, TR1 resp., s.t. F ⊆ F ′
(Tu,wu) and (TR1 ,wR1
) equal locally around every r ∈ R1.
Write (T 2,w2) := (TR1 ,wR1)− (Tu,wu).
This is a (Z+-weighted) tropical hypersurface for strong substitutes.
Find R2 as in covering lemma, for (T 2,w2).
(T 3,w3) := (TR2 ,wR2)− (T 2,w2). Find R3 covering (T 3,w3)
. . .
Suppose this terminates: at some stage T l+1 = ∅. Then
(Tu,wu) = (TR1 ,wR1)− (TR2 ,wR
2) + · · ·+ (−1)l−1(TRl ,wR
l)
= (TR,wR)− (TS ,wS)
where R = R1 ∪R3 ∪ · · · and S = R2 ∪R4 ∪ · · · .
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 28 / 35
Algorithm to generate TuStart with strong substitute u and Tu.
Find R1 as in covering lemma: R1 minimal such thatTu ⊆ TR1
wu(F ) ≤ wR1
(F ′), where F ,F ′ facets of Tu, TR1 resp., s.t. F ⊆ F ′
(Tu,wu) and (TR1 ,wR1
) equal locally around every r ∈ R1.
Write (T 2,w2) := (TR1 ,wR1)− (Tu,wu).
This is a (Z+-weighted) tropical hypersurface for strong substitutes.
Find R2 as in covering lemma, for (T 2,w2).
(T 3,w3) := (TR2 ,wR2)− (T 2,w2). Find R3 covering (T 3,w3)
. . .
Suppose this terminates: at some stage T l+1 = ∅. Then
(Tu,wu) = (TR1 ,wR1)− (TR2 ,wR
2) + · · ·+ (−1)l−1(TRl ,wR
l)
= (TR,wR)− (TS ,wS)
where R = R1 ∪R3 ∪ · · · and S = R2 ∪R4 ∪ · · · .E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 28 / 35
Illustration of the algorithm
p1
p2
Tu
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
Tu
r11
r12
r13
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
TuTR1
r11
r13
r12
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
T 2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
T 2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
T 2
r21
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
T 2
r21TR2
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Illustration of the algorithm
p1
p2
T 3
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 29 / 35
Termination of the algorithm
If we write T 1 := Tu then
Find Rk as in covering lemma: Rk minimal such that
T k ⊆ TRk
wu(F ) ≤ wRk
(F ′), where F ,F ′ facets of Tu, TRk resp., s.t. F ⊆ F ′
(T k ,wk) and (TRk ,wRk
) equal locally around every r ∈ Rk .
(T k+1,wk+1) := (TRk ,wRk)− (T k ,wk).
Every type H j facet of TRk contained in affine span of facets of T k
Every type H j facet of T k+1 contained in affine span of facets of T k
Every type H j facet of T k contained in affine span of facets of Tu.
Every r ∈ Rk is contained in n-way intersections of these affine spans.But this is a finite set of points.
The minimum r in any of these affine spans strictly increases at eachstage.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 30 / 35
Termination of the algorithm
If we write T 1 := Tu then
Find Rk as in covering lemma: Rk minimal such that
T k ⊆ TRk
wu(F ) ≤ wRk
(F ′), where F ,F ′ facets of Tu, TRk resp., s.t. F ⊆ F ′
(T k ,wk) and (TRk ,wRk
) equal locally around every r ∈ Rk .
(T k+1,wk+1) := (TRk ,wRk)− (T k ,wk).
Every type H j facet of TRk contained in affine span of facets of T k
Every type H j facet of T k+1 contained in affine span of facets of T k
Every type H j facet of T k contained in affine span of facets of Tu.
Every r ∈ Rk is contained in n-way intersections of these affine spans.But this is a finite set of points.
The minimum r in any of these affine spans strictly increases at eachstage.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 30 / 35
Termination of the algorithm
If we write T 1 := Tu then
Find Rk as in covering lemma: Rk minimal such that
T k ⊆ TRk
wu(F ) ≤ wRk
(F ′), where F ,F ′ facets of Tu, TRk resp., s.t. F ⊆ F ′
(T k ,wk) and (TRk ,wRk
) equal locally around every r ∈ Rk .
(T k+1,wk+1) := (TRk ,wRk)− (T k ,wk).
Every type H j facet of TRk contained in affine span of facets of T k
Every type H j facet of T k+1 contained in affine span of facets of T k
Every type H j facet of T k contained in affine span of facets of Tu.
Every r ∈ Rk is contained in n-way intersections of these affine spans.But this is a finite set of points.
The minimum r in any of these affine spans strictly increases at eachstage.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 30 / 35
Termination of the algorithm
If we write T 1 := Tu then
Find Rk as in covering lemma: Rk minimal such that
T k ⊆ TRk
wu(F ) ≤ wRk
(F ′), where F ,F ′ facets of Tu, TRk resp., s.t. F ⊆ F ′
(T k ,wk) and (TRk ,wRk
) equal locally around every r ∈ Rk .
(T k+1,wk+1) := (TRk ,wRk)− (T k ,wk).
Every type H j facet of TRk contained in affine span of facets of T k
Every type H j facet of T k+1 contained in affine span of facets of T k
Every type H j facet of T k contained in affine span of facets of Tu.
Every r ∈ Rk is contained in n-way intersections of these affine spans.But this is a finite set of points.
The minimum r in any of these affine spans strictly increases at eachstage.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 30 / 35
Termination of the algorithm
If we write T 1 := Tu then
Find Rk as in covering lemma: Rk minimal such that
T k ⊆ TRk
wu(F ) ≤ wRk
(F ′), where F ,F ′ facets of Tu, TRk resp., s.t. F ⊆ F ′
(T k ,wk) and (TRk ,wRk
) equal locally around every r ∈ Rk .
(T k+1,wk+1) := (TRk ,wRk)− (T k ,wk).
Every type H j facet of TRk contained in affine span of facets of T k
Every type H j facet of T k+1 contained in affine span of facets of T k
Every type H j facet of T k contained in affine span of facets of Tu.
Every r ∈ Rk is contained in n-way intersections of these affine spans.But this is a finite set of points.
The minimum r in any of these affine spans strictly increases at eachstage.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 30 / 35
Termination of the algorithm
If we write T 1 := Tu then
Find Rk as in covering lemma: Rk minimal such that
T k ⊆ TRk
wu(F ) ≤ wRk
(F ′), where F ,F ′ facets of Tu, TRk resp., s.t. F ⊆ F ′
(T k ,wk) and (TRk ,wRk
) equal locally around every r ∈ Rk .
(T k+1,wk+1) := (TRk ,wRk)− (T k ,wk).
Every type H j facet of TRk contained in affine span of facets of T k
Every type H j facet of T k+1 contained in affine span of facets of T k
Every type H j facet of T k contained in affine span of facets of Tu.
Every r ∈ Rk is contained in n-way intersections of these affine spans.But this is a finite set of points.
The minimum r in any of these affine spans strictly increases at eachstage.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 30 / 35
Next Step: Finding Equilibrium Prices
Given relevant supply y, write
gy(p) = u(y)− y.p−maxx∈A{u(x)− x.p}.
This is concave, piecewise-linear, maximised when y ∈ Du(p).
It is easy to find a subgradient of gy(p) at any price: compute an elementof Du(p) from dot bids.
Future work (with Paul Klemperer and Paul Goldberg) will fill in detailshere.
Related work: Paes Leme and Wong (2015).
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 31 / 35
Validity of Bids
Recall positive and negative bids R,S is ‘valid’ if TR,S is Z≥0-weighted.
Recall that every dot bid generates n semi-infinite hod sections, andn(n − 1)/2 semi-infinite flange sections.
p1
p2
p3
r
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 32 / 35
Validity of Bids
Recall positive and negative bids R,S is ‘valid’ if TR,S is Z≥0-weighted.
Recall that every dot bid generates n semi-infinite hod sections, andn(n − 1)/2 semi-infinite flange sections.
Bids valid ⇔ ∀p ∈ Rn, and ∀i 6= j = 1, . . . , n
# jth hod sections of +ve bids that contain x≥ # jth hod sections of -ve bids that contain x.
# (i , j)-flanges of +ve bids that contain x≥ # (i , j)-flanges of -ve bids that contain x.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 32 / 35
Validity of Bids
Recall positive and negative bids R,S is ‘valid’ if TR,S is Z≥0-weighted.
Recall that every dot bid generates n semi-infinite hod sections, andn(n − 1)/2 semi-infinite flange sections.
Bids valid ⇔ ∀p ∈ Rn, and ∀i 6= j = 1, . . . , n
# jth hod sections of +ve bids that contain x≥ # jth hod sections of -ve bids that contain x.
# (i , j)-flanges of +ve bids that contain x≥ # (i , j)-flanges of -ve bids that contain x.
Unfortunately:
Claim
The problem of determining whether a given set of positive and negativebids are valid is co-NP-complete.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 32 / 35
Next Steps
Easily-identifiable clusters of dot bids
Fixed combinatorial structure: validity not in question.Identify ‘meaning’ to make use easier for bidders.
Implementation of algorithm, and simplified variants.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 33 / 35
Open Questions about Product-Mix Auctions
Market Power
Have assumed bidders behave competitively.
How well does PMA work with “small” numbers of bidders?e.g., for DECC, small number of large electricity providers, all knownto each other.
How much better is Product-Mix Auction than standard approaches?
Discriminatory versus Uniform Pricing
How well does PMA work with ‘pay your bid’ pricing?
Game-theoretic analysis?
Experiments?
Further extensions?
Design auctions for other unimodular demand types!
Handle different ‘sizes’ of indivisible goods
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 34 / 35
Summary
We need sealed-bid auctions covering multiple goods
Product-mix auction ‘dot bids’
Represent any strong substitute preferencesProvide a computationally-efficient way to query demand at genericprices
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 35 / 35
References
E. Baldwin and P. Klemperer. Tropical geometry to analyse demand.Mimeo. Available from www.paulklemperer.org andelizabeth-baldwin.me.uk, May 2014.
E. Baldwin and P. Klemperer. Understanding preferences: “demandtypes”, and the existence of equilibrium with indivisibilities. Mimeo.,February 2015.
E. Baldwin, P. Goldberg, and P. Klemperer. The multi-dimensionalproduct-mix auction. In preparation.
E. Budish, P. Cramton, and J. Shim. The high-frequency trading armsrace: Frequent batch auctions as a market design response. TheQuarterly Journal of Economics, 130(4):1547–1621, 2015.
P. Klemperer. A new auction for substitutes: Central bank liquidityauctions, the U.S. TARP, and variable product-mix auctions. Workingpaper, Oxford University, 2008.
P. Klemperer. The product-mix auction: A new auction design fordifferentiated goods. Journal of the European Economic Association, 8(2-3):526–536, 2010.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 35 / 35
R. Paes Leme and S. C.-w. Wong. Computing walrasian equilibria: Fastalgorithms and economic insights. preprint 1511.04032, ArXiv, 2015.
E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 35 / 35