Transcript A Look at Simple, Learnable Pricing Policies in Electricity Markets

A Look at Simple, Learnable
Pricing Policies in Electricity
Markets
Steve Kimbrough
Fred Murphy
INFORMS, November 2005
File: sok-murphy-informs-2005.ppt
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Points to Be Raised (Outline)
• Relevance of supply curve equilibria
• Brief review of the literature
• Worries, widely shared, about neoclassical
economic models
• Alternatives? Agent-based (simple) models
• Lessons from a monopoly agent
• Extension to duopoly
• Discussion
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But first…Our Context
• This is work in progress (and not
completed!)
• Context is markets for electric power
– Generally, oligopolies
– Always, heavily iterated (markets for 1 hour
slots of electricity, 24 per day, every day…)
– Simple case: duopoly (see 2 figures following)
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Figure 1
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Figure 2
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Relevance of Supply Curve
Equilibria
• Generators bid step-function supply curves
• This means classic Cournot and Bertrand models
do not apply
• A problem with supply function equilibria is that
they can be characterized analytically only with
heroic assumptions such as symmetry.
• Another problem is that the theory requires
continuous functions, which is not true in
electricity markets
• Still another, multiple equilibria
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Brief Review of the Literature
•
•
•
•
Developed by Klemperer and Meyer in 1989
Applied to electricity by Green and Newbery, 1993
Von de Fehr and Harbord 1993 look at step functions
Baldick and Hogan (2002) and Holmberg (2004) show that
with a price cap the supply function equilibrium is unique
• Baldick and Hogan 2004 affine function only stable
solution
• Anderson and Philpott (2002) show that if the opponent’s
supply function is known and log concave, then there
exists a monotonically increasing supply function for the
player
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Widely Shared Worries about
Neoclassical Models
• Among the main ones:
– Heroic assumptions (e.g., common knowledge,
unlimited computational capacity)
– Too many equilibria (Folk Theorem), and
neglect of repetition/iteration
– Poor track record in predicting behavior
• E.g., behavioral game theory
– OK for perfect competition and monopoly,
weak for oligopoly
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Alternative: Agent-Based Models
• Using simple, plausible rules of behavior
– Characterized by bottom-up and distributed
• Can replace heroic assumptions with
realistic assumptions
• Agents discover solutions, equilibria or not
• Prediction qualities not generally known,
but testable
• Natural for oligopolies
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Monopolist Agent
• Searches stochastically
• Parameters:
–
–
–
–
initialPrice
Delta
Epsilon
History
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Monopolist’s Search Procedure
1) Set currentPrice to initialPrice; samples to 0
2) While (samples < history)
a) Bid price in currentPrice  delta
b) Record return obtained from price
c) Increment samples
3) If returns above currentPrice are greater than
returns below currentPrice, set currentPrice to
currentPrice + epsilon; else…
4) Set samples to 0 and go to 2)
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Example Results: Deterministic
Demand
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Random Demand
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Monopoly: Random Walk
Demand
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Comments
• Not surprising that a simple search
algorithm can approximate the monopoly
price for a linear, deterministic demand
function.
• Heartening that the same algorithm does
reasonably well when demand is stochastic.
• These results set a lower bound on what
agents can achieve in such circumstances
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Duopoly
• Here we model only pure Bertrand
competition
• Viz., figures 1 & 2, assume that all the
action is in the step (quantity) 2 pricing
• Model: each player uses the monopolist’s
search algorithm, with a slight modification
– Players bid independently, but record as data
only the accepted price and associated demand.
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Typical Results: Random Walk
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Discussion & Comments
• This is all quite preliminary, but…
• This demonstrates that the two duopolists have
available workable computational strategies
(learning policies) that will lead them to obtain
prices close to the monopoly prices and to split the
proceeds.
• Will they avail themselves of it or will they yield
to temptation?
• What happens if the two players have different
cost structures?
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Firm 2 Has Non-Zero Costs
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Note (Both Cases)
• From outside the industry, from the
consumer’s perspective, the producers are
obtaining monopoly rents.
• From inside the industry, it may look
brutally competitive, as firms seek to best
each other on the cost dimension (but not on
price in these models).
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Discussion & Comments
• Recalling figures 1 and 2, ample scope
remains for competition: the players can
compete on their stage 1 costs and
quantities, where their interests are opposed
• Further, the stepped supply curve serves to
reduce the temptation to defect from the
mutually-beneficial monopoly search
strategy.
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Discussion & Comments
• Modeling this more complex game
explicitly is on our agenda
• The live possibility is that agent-based
models can show us realistic ways in which
oligopolists may tacitly collude to obtain
monopoly rents.
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