Andy Philpott

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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)

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Uniform price auction
price
T2(q)
p
quantity
price
combined offer stack
p
quantity
3
Discriminatory price (pay-as-bid) auction
price
T2(q)
p
quantity
price
combined offer stack
p
quantity
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Motivation which is better?
Source Regulatory Impact Statement, Cabinet
Paper on Ministerial Review of Electricity Market
(2010)
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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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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 dont exist.
• Characterize equilibria with mixed strategies

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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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Residual demand curve with shock
D(p)
D(p) e
D(p) e - S(p)
price
p
q
q
q
quantity
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The market distribution functionAnderson P,
2002

• S(p) supply curve from
• other generators
• D(p) demand curve
• random demand
• F cdf of demand
• f density of demand

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The offer distribution functionAnderson,
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 competitors
mixture passes below (q,p) Pr S(p) gt q
Note G(q,p) 1 for qlt0
price
G(q,p)1
p
G(q,p)0
q
quantity
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F G equals y
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Example 1 no mixing, D(p)0, demande
( probability that demand lt qq(p) )
price
G(q,p)1
q(p)
p
G(q,p)0
q(p)
quantity
q
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Evaluating a pay-as-bid offer
price
Offer curve p(q)
quantity
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A calculus of variations lesson
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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
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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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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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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 pgt2 has the same
profit (1/2), and so all horizontal curves have
this profit.
G0.8
G0.6
G0.0
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Example 2 Mixed strategy equilibrium
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Example 2 Mixed strategy equilibrium
G0.6
G0.4
G0.2
G0.0
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When do these mixtures exist?

• These only exist when demand is inelastic
  (D(p)0) and each players 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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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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Slope-constrained optimality conditions
Z(q,p)lt0
p
( the derivative of profit with respect to offer
price p of segment (qA,qB) 0 )
Z(q,p)gt0
q
x
x
qB
qA
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Examples 3 and 4 D(p)0
Two identical players each with capacity qm lt
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 competitors 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, qm0.5778
p18.205
G(p1)1
p02
G(p0)0
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Recall optimality conditions
Z(q,p)lt0
p
( the derivative of profit with respect to offer
price p of segment (qA,qB) 0 )
Z(q,p)gt0
q
x
qB
qA
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Example 3 K 0.719, qm0.5778
Plot of Z(q,p) for K 0.719, qm0.5778
p3
p2.5
p2
p2
p2.5
p3
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Example 4 K 0.349, qm0.5778
p14.06
p01.1
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Example 4 K 0.349, qm0.5778
Plot of Z(q,p) for K 0.34933, qm0.5778
p1.1
p1.2
p1.3
p1.5
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Example 4 Hockey-stick
4.06
1.5
1.1
0.5778
0.5156
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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 VgtU 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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What we know for increasing marginal costs
Inelastic Demand Elastic demand
Pivotal suppliers Equilibrium with horizontal mixtures and price cap. Equilibrium with hockey-stick mixtures and price cap. Equilibrium with horizontal mixtures and price cap. In special cases, equilibrium with horizontal mixtures and no price cap.
Non pivotal suppliers Equilibrium with sloping mixtures. No known mixed equilibrium
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Listener, July 4-10, 2009
Professor Andy Philpotts letter (June 20)
supporting marginal-cost electricity pricing
contains statements that are worse than wrong, to
use Wolfgang Paulis 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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The End