Game theory and market power
Josh Taylor
Section 6.1.3, 6.3 in Convex Optimization of Power Systems.
1
Market weaknesses
Recall
• Optimal power flow:
minimize
X
p,θ
fi (pi )
i
subject to
λi :
pi =
X
bij (θi − θj )
j
χij ≥ 0 :
bij (θi − θj ) ≤ sij
pi ≤ pi ≤ pi
• Prices: λi : the price at node i. Agent i solves:
fi (pi ) − λi pi
minimize
pi
When does nodal pricing / microeconomics fail?
• Nonconvexity - the power flow equations, unit commitment.
• Bounded rationality - agent i has limited time, information, computing
power - can’t find optimal pi .
• Price-taker assumption: agent i oblivious to their influence on λi .
2
2.1
Real-world examples
Enron Scandal, late 1990’s to 2001
Overview:
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Game theory and market power
JAT
• Energy trading, building power plants, natural gas
• Posterchild for electricity/energy markets
• Very shady accounting practices - see Wikipedia
• Gov. Gray Davis’ ruined political career
California Electricity Crisis:
• Making power seem to be from out of state (where does your power come
from?)
• Blocking transmission lines (over scheduling) to raise nodal prices
• Overall bad planning/market design
• Rolling blackouts in 2000, 2001, prices increase by factor of 20.
2.2
JPMorgan, 2010-2012
• Manipulative bidding strategies ...
• JPMorgan pays $410 million in FERC settlement, 2013
2.3
Why?
• Lot’s of markets have problems (healthcare, computer OS, diamonds)
• Failures of price-taker assumption in power
• Should we have power markets? Probably, but cautiously ...
• Were Enron, JPMorgan too smart?
Strategy:
• Markets need physically rigorous design
• Game theory helps identify vulnerabilities mathematically.
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Game theory and market power
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Game theory
Regular optimization:
min f (x).
x∈X
Game theory:
min f (x, y),
x
min g(x, y)
y
Two players, know all about each other.
3.1
Example: prisoner’s dilemma
Setup
• Two players, caught criminals
• Two actions: silence, or betray partner
• Made simultaneously (like rock paper scissors)
Payoffs:
• Both silent: both serve 1 year
• Both betray: both serve 3 years
• 1 silent, 1 betrays: silent 4 years, betrayer 0 years.
Anticipatory decisions:
• Both silent ... either improves by betraying
• 1 silent, 1 betrays ... silent improves by betraying
• Both betray ... no improvement for either
Nash equilibrium:
• Both players betray
• Stable under unilateral actions
• Worse than both silent
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Strategic form games
• Players, i = 1, ..., n
• Pure player strategies, Si .
• Player utility function ui (s), s ∈ S = ×i Si
• Ordinary optimization with just one player
s is a Pure Nash Eq. (PNE) if
ui (s) ≤ ui (t, s−i ) for all t ∈ Si .
PNE exists if
• ui (s) convex in si , continuous in s−i
• Si convex and compact
Mixed strategy:
• Mi set of PDFs over Si
• Mixed strategy m ∈ ×i Mi
• Expected payoff over m:
Z
ui (m) =
ui (s)m(s)ds.
S
• MNE if
ui (m) ≤ ui (t, m−i ) for all t ∈ Mi
(Si works)
Discussion
• PNE often don’t exist.
• Uniqueness uncommon when it does exist.
• MNE describe real situations like sales.
• MNE almost always exist.
• Game theory PPAD complete - easier than NP-complete, still bad.
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3.2
Game theory and market power
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Bertrand competition
• Demand: d
• Prices: λi
• All demand goes to lowest price.
Equilibrium:
• If λ1 = λ2 > 0, λ1 − is profitable for λ1 .
• If λ1 > λ2 , λ1 = λ2 − is profitable for λ1 .
• Nash Eq: λ1 = λ2 = 0 (silly)
4
Load shifting with storage
• Time-varying, inelastic load δ(t), t = 1, ..., T
• Generation cost f (p) = a2 p2 + bp
• Market clearing price:
df (p)
dp
= ap + b
λ =
• N storages inject/extract si (t) - arbitrage
P
• Net load: δ(t) − N
i=1 si (t)
Centralized problem,
minimize
s
subject to
T
X
t=1
T
X
f
δ(t) −
N
X
!
si (t)
i=1
si (t) = 0
t=1
Optimal solutions:
sci (t) = γi δ(t) − δ
N
X
γi = 1
i=1
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where the average demand is
T
1X
δ=
δ(t).
T t=1
• Net load curve: δ
• γ which storage allocation
Remove price-taker assumption. Market price:
λ(t) = a δ(t) −
N
X
!
si (t)
+ b.
i=1
Storage payoffs:
maximize
si
subject to
T
X
t=1
T
X
a δ(t) −
N
X
!
si (t)
!
+ b si (t)
i=1
si (t) = 0
t=1
• Coupling ... N -player game
• Quantity competition - Cournot
• PNE:
sgi (t) =
1
δ(t) − δ
N +1
• Flatter, but less so than centralized.
• Efficiency loss:
PT
Φ = P
T
t=1 f
t=1 f
= PT
a
t=1 2
δ(t) −
PN
δ(t) −
PN
c
i=1 si (t)
g
i=1 si (t)
a 2
2 δ̄ + bδ
2
N
δ
+b
N +1
T
1
N +1 δ(t)
+
1
N +1 δ(t)
+
N
N +1 δ
.
Letting N → ∞, the efficiency loss vanishes, i.e. Φ → 1.
• “Price of anarchy”
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δ (t )
N et l oad , op ti mal
N et l oad , game
t
Figure 1: The nominal load without storage, δ, and the net load in the centralized and game outcomes with
N = 3.
• Worst case - duopoly (only monopoly worse)
• Game shows variations allowed to persist to preserve arbitrage
• More participants flattens net load, approaches true optimum.
5
Bertrand-Edgeworth competition
• Firms submit prices, λi , i = 1, 2 (n player exists)
• Demand d
• Firm capacities: c1 , c2
• Reservation utility: λr
Payoffs:
(
ui (λ) =
λi min{ci , d},
λi < λ−i
λi min{ci , (d − c−i )+ }, λi > λ−i
Expected payoff:
Z
ρi (m) = ui (m) =
Z
ui (λ)m(λ)dλ =
0≤λ≤λr
m1 (λ1 )m2 (λ2 )ui (λ)
0≤λ≤λr
• Case 1: d ≤ c1 , c2 : Always left out player, PNE is λi = 0 and ρ∗i (c) = 0.
Standard Bertrand.
• Case 2: d ≥ c1 + c2 : Both fill up regardless, PNE is λi = λr , ρ∗i (c) = λr ci .
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ρ ∗ (c)
• Case 3: cj < d < c1 + c2 for some j. An MNE exists in which player i’s
profit is
λr min {ci , d}
ρ∗i (c) =
(d + cmax − c1 − c2 ) ,
min {cmax , d}
where cmax = maxi ci ... holding sales.
d
Figure 2: The expected profits of each player, ρ∗i (c), in a capacitated price duopoly for λr = 1, c1 = 1/5,
c2 = 3/5, and d ∈ [0, 1].
5.1
Application to transmission
• 2 nodes, same demand d
• Max price λr
• Generation p1 , p2
• Line capacity s
• Profits: λi pi
d, p1
|p12 | ≤ s
d, p2
Figure 3: Market power in a congested two-node transmission network.
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Payoff function:


 d + min {d, s} if λi < λ−i
λi max {d − s, 0} if λi > λ−i .


d
if λi = λ−i
Equivalent to BE comp. with
• Demand 2d
• capacities d + s̄
Case-wise Analysis
• d ≤ s: Undercutting, PNE λ1 = λ2 = 0.
• s = 0: Two monopolies, PNE λi = λr .
• 0 < s < d: No PNE. MNE:
λr (d − s)
µ(λi ) =
,
2sλ2i
λr (d − s)
λi ∈
, λr ,
d+s
Expected profits: λr (d − s).
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Conclusion
Other popular formats:
• Supply function competition - hard to analyze
• Complementarity models - tractable but crude
Takeaways
• Game theory limited tractability
• More market participants means better competition.
• Transmission congestion → local monopolies (e.g., Enron)!
• Physics matter! Initial market designs ignored transmission.
References
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