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The Product-Mix Auction Elizabeth Baldwin Paul Goldberg Paul Klemperer Oxford University May 2016 Covering material from Klemperer (2008, 2010), and much further work 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

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Page 1: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 2: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 3: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 4: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 5: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 6: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 7: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 8: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 9: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 10: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 11: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 12: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 13: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 14: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 15: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 16: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 17: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 18: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 19: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 20: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 21: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 22: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 23: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 24: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 25: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 26: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 27: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

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str

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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

Page 28: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 29: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 30: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 31: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 32: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

The Product-Mix Auction

Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.

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‘Just’ need to find prices so that supply equals demand!

E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35

Page 33: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

The Product-Mix Auction

Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.

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E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35

Page 34: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

The Product-Mix Auction

Bids for liquidity against “strong” or “weak” collateral in the Bank ofEngland’s Product-Mix Auction.

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‘Just’ need to find prices so that supply equals demand!E. Baldwin, P. Goldberg, P. Klemperer The Product-Mix Auction May 2016 12 / 35

Page 35: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 36: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 37: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 38: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 39: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 40: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 41: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

Page 42: The Product-Mix Auction - Peoplepeople.math.gatech.edu/~jyu67/HCM/Baldwin2.pdf · The Product-Mix Auction Bids for liquidity against \strong" or \weak" collateral in the Bank of England’s

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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‘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

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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

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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

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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

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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

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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

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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

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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.

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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.

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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

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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

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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

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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.

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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.

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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.

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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.

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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.

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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 :

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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 :

????

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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

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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).

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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 ∪ · · · .

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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 ∪ · · · .

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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 ∪ · · · .

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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 ∪ · · · .

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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

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Illustration of the algorithm

p1

p2

Tu

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Illustration of the algorithm

p1

p2

Tu

r11

r12

r13

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Illustration of the algorithm

p1

p2

TuTR1

r11

r13

r12

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Illustration of the algorithm

p1

p2

T 2

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Illustration of the algorithm

p1

p2

T 2

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Illustration of the algorithm

p1

p2

T 2

r21

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Illustration of the algorithm

p1

p2

T 2

r21TR2

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Illustration of the algorithm

p1

p2

T 3

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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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

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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.

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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.

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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.

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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

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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

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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.

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R. Paes Leme and S. C.-w. Wong. Computing walrasian equilibria: Fastalgorithms and economic insights. preprint 1511.04032, ArXiv, 2015.

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