marsden grant uoa0719 epoc scope seminar, march 19, 2010 andy philpott the university of auckland ...

EPOC SCOPE seminar, March 19, 2010Marsden Grant UOA0719

Andy PhilpottThe University of Auckland

www.epoc.org.nz

(joint work with Eddie Anderson, Par Holmberg)

Mixed-strategy equilibria in discriminatory divisible-good

auctions


Uniform price auction

price

quantity

price

quantity

combined offer stack

demand

p

price

quantity

T1(q) T2(q)

p


Discriminatory price (pay-as-bid) auction

price

quantity

price

quantity

combined offer stack

demand

p

price

quantity

T1(q) T2(q)

p


Source: Regulatory Impact Statement, Cabinet Paper on Ministerial Review of Electricity Market (2010)

Motivation: which is better?


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.


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


Antoine Augustin Cournot (1838)


Joseph Louis François Bertrand (1883)


Francis Edgeworth (1897)


Symmetric duopoly setting

• Two identical players each with capacity qm

• Marginal costs are increasing (C’(q) ≥ 0) • Demand D(p) + • with distribution F() support [• Each player offers a supply curve S(p)• Essential mathematical tools

– residual demand curve– market distribution function– offer distribution function


Residual demand curve with shock

p

q quantity

price

D(p)

S(p)

D(p) + D(p) + - S(p)

q q


The market distribution function[Anderson & P, 2002]

p

q quantity

price

)p,q(

Define: (q,p) = Pr [ D(p)+ – 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)]

S(p) = supply curve from other generatorsD(p) = demand curve = random demand F = cdf of demandf = density of demand


The offer distribution function[Anderson, Holmberg & P, 2010]

p

q quantity

price

)p,q(

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]

Other player mixing over offer curves S(p) results in a random residual demand.

G(q,p)=0

G(q,p)=1 Note:

G(q,p) = 1 for q<0


F * G equals


Example 1: no mixing, D(p)=0, demand=

p

q quantity

price

G(q,p)=0

G(q,p)=1 q(p)

q(p)

(= probability that demand < q+q(p) )


Evaluating a pay-as-bid offer

quantity

price Offer curve p(q)

( , ( ))q p q

mq


A calculus of variations lesson

Euler-Lagrange equation


Replace x by q, y by p


Summary so far


Example 1: Optimal response to competitor

Decreasing function!!


Nash equilibrium is hard to find

Rapidly decreasing density !!

Linear cost equilibrium needs


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”).


Example 1: Competitor offers a mixture


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.


Example 1: Check the profit of any curve p(q)

Every offer curve p(q) has the same expected profit.


Example 1: Mixed strategy equilibriumEach 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=0.0

G=0.8

G=0.6


Example 2: Mixed strategy equilibrium


Example 2: Mixed strategy equilibrium

G=0.0

G=0.4

G=0.6

G=0.2


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 must also satisfy some technical conditions to preclude pure strategy equilibria.)


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.


Slope-constrained optimality conditions

Z(q,p)<0

q

p

Z(q,p)>0

qBqAxx

( the derivative of profit with respect to offer price p of segment (qA,qB) = 0 )


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)]


Examples 3 and 4

The expected payoff K is the same for every offer to the mixture.

Suppose is uniformly distributed on [0,1].

Choosing K determines G(p). In this example


p0=2

p1=8.205

Example 3: K = 0.719, qm=0.5778

0

2

4

6

8

0.1 0.2 0.3 0.4 0.5 0.6q

G(p0)=0

G(p1)=1

Equilibrium requires a price cap p1


Recall optimality conditions

Z(q,p)<0

q

p

Z(q,p)>0

qBqAx

( the derivative of profit with respect to offer price p of segment (qA,qB) = 0 )


0

0.5

1

1.5

2

2.5

3

3.5

0.1 0.2 0.3 0.4 0.5 0.6q

Example 3: K = 0.719, qm=0.5778

p=2p=2.5p=3

Plot of Z(q,p) for K = 0.719, qm=0.5778

p=2p=2.5p=3


p0=1.1

p1=4.06

Example 4: K = 0.349, qm=0.5778

0

1

2

3

4

5

0.1 0.2 0.3 0.4 0.5 0.6q


Example 4: K = 0.349, qm=0.5778

p=1.1

p=1.3

p=1.5

Plot of Z(q,p) for K = 0.34933, qm=0.5778

p=1.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.1 0.2 0.3 0.4 0.5 0.6q


Example 4: “Hockey-stick”

0.57780.5156

1.1

1.5

4.06


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.


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


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)

Listener, July 4-10, 2009


The End

marsden grant uoa0719 epoc scope seminar, march 19, 2010 andy philpott the university of auckland ...

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