Uniform-price auctions versus
pay-as-bid auctions
Andy Philpott
The University of Auckland
www.esc.auckland.ac.nz/epoc
(joint work with Eddie Anderson, UNSW)
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Summary
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Uniform price auctions
Market distribution functions
Supply-function equilibria for uniform-price case
Pay-as-bid auctions
Optimization in pay-as-bid markets
Supply-function equilibria for pay-as-bid markets
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Uniform price auction (single node)
price
T1(q)
price
T2(q)
p
quantity
quantity
price
combined offer stack
p
demand
quantity
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Residual demand curve for a generator
S(p) = total supply curve from other generators
D(p) = demand function
p
c(q) = cost of generating q
R(q,p) = profit = qp – c(q)
Residual demand curve = D(p) – S(p)
Optimal dispatch point to maximize profit
q
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A distribution of residual demand curves
p
e
D(p) – S(p) + e
(Residual demand shifted by random demand shock e )
Optimal dispatch point to maximize profit
q
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One supply curve optimizes for all demand realizations
p
The offer curve is a “wait-and-see”
solution. It is independent of the
probability distribution of e
q
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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
S(p) = supply curve from
other generators
D(p) = demand function
e
= random demand
shock
F
= cdf of random shock
quantity
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Symmetric SFE with D(p)=0
[Rudkevich et al, 1998, Anderson & P, 2002]
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Example: n generators, e~U[0,1], pmax=2
n=5 n=4
n=3
n=2
p
Assume cq = q, qmax=(1/n)
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Example: 2 generators, e~U[0,1], pmax=2
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T(q) = 1+2q in a uniform-price SFE
Price p is uniformly distributed on [1,2].
Let VOLL = A.
E[Consumer Surplus] = E[ (A-p)2q ]
= E[ (A-p)(p-1) ]
= A/2 – 5/6.
• E[Generator Profit] = 2E[qp-q]
= 2E[ (p-1)(p-1)/2 ]
= 1/3.
• E[Welfare] = (A-1)/2.
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Pay-as-bid pool markets
• We now model an arrangement in
which generators are paid what they
bid –a PAB auction.
• England and Wales switched to NETA
in 2001.
• Is it more/less competitive?
(Wolfram, Kahn, Rassenti,Smith & Reynolds
versus Wang & Zender, Holmberg etc.)
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Pay-as-bid price auction (single node)
price
T1(q)
price
T2(q)
p
quantity
quantity
price
combined offer stack
p
demand
quantity
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Modelling a pay-as-bid auction
Offer curve p(q)
price
(q, p(q ))
qmax
quantity
• Probability that the quantity between q
and q + dq is dispatched is 1  y (q, p(q ))
• Increase in profit if the quantity between q
and q + dq is dispatched is ( p(q )  c '(q ))d q
• Expected profit from offer curve
q
is
0 [ p(q)  c '(q)][1 y (q, p(q))]dq
max
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Calculus of variations
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Necessary optimality conditions (I)
Z(q,p)<0
p
Z(q,p)>0
qA
x
x
qB
q
( the derivative of profit with respect to
offer price p of segment (qA,qB) = 0 )
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Example: S(p)=p, D(p)=0, e~U[0,1]
S(p) = supply curve from
other generator
D(p) = demand function
e
= random demand
shock
q+p=1
Z(q,p)<0
Optimal offer (for c=0)
Z(q,p)>0
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Finding a symmetric equilibrium
[Holmberg, 2006]
• Suppose demand is D(p)+e where e has distribution
function F, and density f.
• There are restrictive conditions on F to get an upward
sloping offer curve S(p) with Z negative above it.
• If
–f(x)2 – (1 - F(x))f’(x) > 0
then there exists a symmetric equilibrium.
• If
–f(x)2 – (1 - F(x))f’(x) < 0 and costs are close to
linear then there is no symmetric equilibrium.
• Density of f must decrease faster than an exponential.
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Prices: PAB vs uniform
1000
Price
800
600
Uniform bid = price
PAB marginal bid
PAB average price
400
200
Demand shock
0
0
2000
4000
6000
8000
10000
Source: Holmberg (2006)
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Example: S(p)=p, D(p)=0, e~U[0,1]
S(p) = supply curve from
other generator
D(p) = demand function
e
= random demand
shock
q+p=1
Z(q,p)<0
Optimal offer (for c=0)
Z(q,p)>0
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Consider fixed-price offers
• If the Euler curve is downward sloping then horizontal
(fixed price) offers are better.
• There can be no pure strategy equilibria with horizontal
offers – due to an undercutting effect…
• .. unless marginal costs are constant when Bertrand
equilibrium results.
• Try a mixed-strategy equilibrium in which both players
offer all their power at a random price.
• Suppose this offer price has a distribution function G(p).
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Example
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Two players A and B each with capacity qmax.
Regulator sets a price cap of pmax.
D(p)=0, e can exceed qmax but not 2qmax.
Suppose player B offers qmax at a fixed price p with
distribution G(p). Market distribution function for A is
B undercuts A
A undercuts B
• Suppose player A offers qmax at price p
• For a mixed strategy the expected profit of A is a constant
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Determining pmax from K
Can now find pmax for any K, by solving G(pmax)=1.
Proposition: [A&P, 2007] Suppose demand is inelastic, random
and less than market capacity. For every K>0 there is a price
cap in a PAB symmetric duopoly that admits a mixed-strategy
equilibrium with expected profit K for each player.
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Example (cont.)
Suppose c(q)=cq
Each generator will offer at a price p no less than pmin>c, where
and (qmax,p) is offered with density
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Example
Suppose c=1, pmax= 2, qmax= 1/2. Then pmin= 4/3, and K = 1/8
g(p) = 0.5(p-1)-2
Average price = 1 + (1/2) ln (3) > 1.5 (the UPA average)
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Expected consumer payment
Suppose c=1, pmax=2.
g(p) = 0.5(p-1)-2
Generator 1 offers 1/2
at p1 with density g(p1).
Generator 2 offers 1/2
at p2 with density g(p2).
Demand e ~ U[0,1].
If e < 1/2, then clearing price = min {p1, p2}.
If e > 1/2, then clearing price = max {p1, p2}.
E[Consumer payment] = (1/2) E[e|e < 1/2] E[min {p1, p2}]
+(1/2) E[e|e > 1/2] E[max {p1, p2}]
= (1/4) + (7/32) ln (3)
( = 0.49 )
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Welfare
Suppose c=1, pmax=2.
g(p) = 0.5(p-1)-2
E[Profit] = 2*(1/8)=1/4.
< E[Profit] = 1/3 for UPA
E[Consumer surplus] = A E[e] – E[Consumer payment]
= (1/2)A – E[Consumer payment]
= (1/2)A – 0.49
E[Welfare] = (1/2)A – 0.24
> (1/2)A – 5/6 for UPA
> (1/2)A – 0.5
for UPA
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Conclusions
• Pay-as-bid markets give different outcomes from uniformprice markets.
• Which gives better outcomes will depend on the setting.
• Mixed strategies give a useful modelling tool for studying
pay-as-bid markets.
• Future work
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N symmetric generators
Asymmetric generators (computational comparison with UPA)
The effect of hedge contracts on equilibria
Demand-side bidding
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The End
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