Mixed-strategy equilibria in
discriminatory divisible-good
auctions
Andy Philpott
The University of Auckland
www.epoc.org.nz
(joint work with Eddie Anderson, Par Holmberg)
Marsden Grant UOA0719
EPOC SCOPE seminar, March 19, 2010
Uniform price auction
price
T1(q)
price
T2(q)
p
quantity
quantity
price
combined offer stack
p
demand
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quantity
EPOC SCOPE seminar, March 19, 2010
Discriminatory price (pay-as-bid) auction
price
T1(q)
price
T2(q)
p
quantity
quantity
price
combined offer stack
p
demand
Marsden Grant UOA0719
quantity
EPOC SCOPE seminar, March 19, 2010
Motivation: which is better?
Source: Regulatory Impact Statement,
Cabinet Paper on Ministerial Review of Electricity Market (2010)
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EPOC SCOPE seminar, March 19, 2010
How to model this?
• Construct a model of the auction where
generators offer non-decreasing supply
functions.
• Find a Nash equilibrium in supply functions
(SFE) under the uniform pricing rule (theory
developed by Klemperer and Meyer, 1989).
• Find a Nash equilibrium in supply functions
under the pay-as-bid rule (but difficult to find,
see Holmberg, 2006, Genc, 2008).
• Compare the prices in each setting.
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EPOC SCOPE seminar, March 19, 2010
Recent previous work
• Crampton and Ausubel (1996) show results are
ambiguous in certain demand case.
• Wolfram, Kahn, Rassenti, Smith & Reynolds
claim pay-as-bid is no better in terms of prices
• Wang & Zender, Holmberg claim lower prices in
pay-as-bid.
• Our contribution:
– Describe a methodology for constructing Nash
equilibrium for pay-as-bid case.
– Pure strategies generally don’t exist.
– Characterize equilibria with mixed strategies
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EPOC SCOPE seminar, March 19, 2010
Antoine Augustin Cournot (1838)
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Joseph Louis François Bertrand (1883)
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Francis Edgeworth (1897)
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Symmetric duopoly setting
•
•
•
•
•
•
Two identical players each with capacity qm
Marginal costs are increasing (C’(q) ≥ 0)
Demand D(p) + e
e with distribution F(e) support [e, e].
Each player offers a supply curve S(p)
Essential mathematical tools
– residual demand curve
– market distribution function
– offer distribution function
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EPOC SCOPE seminar, March 19, 2010
Residual demand curve with shock
D(p) + e - S(p)
e
price
S(p)
p
q
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q
q
quantity
EPOC SCOPE seminar, March 19, 2010
The market distribution function
[Anderson & P, 2002]
Define: y(q,p) = Pr [ D(p)+e – S(p) < q]
= F(q + S(p)-D(p))
= Pr [an offer of (q,p) is not fully dispatched]
= Pr [residual demand curve passes below (q,p)]
price
p
( q, p )
q
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S(p) = supply curve from
other generators
D(p) = demand curve
e
= random demand
F
= cdf of demand
f
= density of demand
quantity
EPOC SCOPE seminar, March 19, 2010
The offer distribution function
[Anderson, Holmberg & P, 2010]
Other player mixing over offer curves S(p) results in a random
residual demand.
G(q,p) = Pr [an offer of at least q is made at price below p]
= Pr [offer curve in competitor’s mixture passes below (q,p)]
= Pr [S(p) > q]
price
Note:
G(q,p)=1
p
( q, p )
G(q,p) = 1 for q<0
G(q,p)=0
q
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quantity
EPOC SCOPE seminar, March 19, 2010
F * G equals y
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Example 1: no mixing, D(p)=0, demand=e
(= probability that
demand < q+q(p) )
price
G(q,p)=1
q(p)
p
G(q,p)=0
q
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q(p)
quantity
EPOC SCOPE seminar, March 19, 2010
Evaluating a pay-as-bid offer
price
Offer curve p(q)
(q, p(q ))
qm
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quantity
EPOC SCOPE seminar, March 19, 2010
A calculus of variations lesson
Euler-Lagrange equation
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Replace x by q, y by p
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Summary so far
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Example 1: Optimal response to competitor
Decreasing function!!
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Nash equilibrium is hard to find
Linear cost equilibrium needs
Rapidly decreasing density !!
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Mixed strategies of two types
A mixture G(q,p) so that Z(q,p)≡0 over some region
(“slope unconstrained”)
or
A mixture G(q,p) over curves all of which have Z(q,p)=0
on sloping sections and Z(q,p) with the right sign on
horizontal sections (‘slope constrained”).
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EPOC SCOPE seminar, March 19, 2010
Example 1: Competitor offers a mixture
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Example 1: What is Z for this mixture?
Note that if Z(q,p)≡0
over some region then
every offer curve in this
region has the same
expected profit.
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EPOC SCOPE seminar, March 19, 2010
Example 1: Check the profit of any curve p(q)
Every offer curve p(q) has the same expected profit.
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Example 1: Mixed strategy equilibrium
Each generator offers a horizontal curve at a random price
P sampled according to
Any curve offered in the region p>2 has the same profit
(1/2), and so all horizontal curves have this profit.
G=0.8
G=0.6
G=0.0
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Example 2: Mixed strategy equilibrium
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Example 2: Mixed strategy equilibrium
G=0.6
G=0.4
G=0.2
G=0.0
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EPOC SCOPE seminar, March 19, 2010
When do these mixtures exist?
These only exist when demand is inelastic
(D(p)=0) and each player’s capacity is
more than the maximum demand (neither
is a pivotal producer).
(The cost C(q) and distribution F of e must
also satisfy some technical conditions to
preclude pure strategy equilibria.)
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EPOC SCOPE seminar, March 19, 2010
Mixed strategies of two types
A “slope unconstrained” mixture G(q,p) so that Z(q,p)≡0
over some region: for non-pivotal players.
or
A “slope constrained” mixture G(q,p) over curves all of
which have Z(q,p)=0 on sloping sections and Z(q,p)
with the correct sign on horizontal sections: for pivotal
players.
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EPOC SCOPE seminar, March 19, 2010
Slope-constrained optimality conditions
p
Z(q,p)<0
( the derivative of profit with respect to
offer price p of segment (qA,qB) = 0 )
Z(q,p)>0
qA
x
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x
qB
q
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Examples 3 and 4: D(p)=0
Two identical players each with capacity qm < maximum demand.
(each player is pivotal).
Suppose C(q)=(1/2)q2.
Let strategy be to offer qm at price p with distribution G(p).
G(q,p)= Pr [offer curve in competitor’s mixture passes below (q,p)]
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Examples 3 and 4
Suppose e is uniformly distributed on [0,1].
The expected payoff K is the same for every offer to the mixture.
Choosing K determines G(p). In this example
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Example 3: K = 0.719, qm=0.5778
p1=8.205
G(p1)=1
8
Equilibrium
requires a
price cap p1
6
4
p0=2
0
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G(p0)=0
2
0.1
0.2
0.3
q
0.4
0.5
0.6
EPOC SCOPE seminar, March 19, 2010
Recall optimality conditions
p
Z(q,p)<0
( the derivative of profit with respect to
offer price p of segment (qA,qB) = 0 )
Z(q,p)>0
qA
x
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qB
q
EPOC SCOPE seminar, March 19, 2010
Example 3: K = 0.719, qm=0.5778
Plot of Z(q,p) for K = 0.719, qm=0.5778
3.5
p=3
p=2.5
p=2
3
2.5
2
1.5
p=2
p=2.5
p=3
1
0.5
0
0.1
0.2
0.3
q
0.4
0.5
0.6
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Example 4: K = 0.349, qm=0.5778
5
p1=4.06
4
3
2
p0=1.1
1
0
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0.1
0.2
0.3
q
0.4
0.5
0.6
EPOC SCOPE seminar, March 19, 2010
Example 4: K = 0.349, qm=0.5778
Plot of Z(q,p) for K = 0.34933, qm=0.5778
1.6
1.4
p=1.1
p=1.2
p=1.3
1.2
1
0.8
0.6
p=1.5
0.4
0.2
0
0.1
0.2
0.3
q
0.4
0.5
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0.6
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Example 4: “Hockey-stick”
4.06
1.5
1.1
0.5156
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0.5778
EPOC SCOPE seminar, March 19, 2010
When do we get hockey-stick mixtures?
Assume C is convex, D(p)=0, and generators are pivotal.
• There exists U such that for all price caps p1 greater than or
equal to U there is a unique mixed strategy equilibrium.
• There exists V>U such that for all price caps p1 greater than
or equal to V there is a unique mixed strategy equilibrium
consisting entirely of horizontal offers.
• For price caps p1 greater than or equal to U and less than V
there is a unique mixture of hockey stick bids and
horizontal offers.
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EPOC SCOPE seminar, March 19, 2010
What we know for increasing marginal costs
Pivotal
suppliers
Non pivotal
suppliers
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Inelastic
Demand
Elastic
demand
Equilibrium with
horizontal mixtures
and price cap.
Equilibrium with
hockey-stick
mixtures and price
cap.
Equilibrium with
horizontal mixtures
and price cap.
Equilibrium with
sloping
mixtures.
No known mixed
equilibrium
In special cases,
equilibrium with
horizontal mixtures
and no price cap.
EPOC SCOPE seminar, March 19, 2010
Listener, July 4-10, 2009
Professor Andy Philpott’s letter (June 20) supporting marginal-cost
electricity pricing contains statements that are worse than wrong, to use
Wolfgang Pauli’s words.
He would be well advised to visit any local fish, fruit or vegetable market
where every morning bids are made for what are commodity products, then
the winning bidder selects the quantity he or she wishes to take at that
price, and the process is repeated at a lower price until all the products
have been sold.
The market is cleared at a range of prices – the direct opposite of the
electricity market where all bidders receive the same clearing price,
irrespective of the bids they made. Therefore, there is no competition for
generators to come up with a price – just bid zero and get the clearing price.
John Blundell
(St Heliers Bay, Auckland)
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EPOC SCOPE seminar, March 19, 2010
The End
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