1 A Look at Simple, Learnable Pricing Policies in Electricity Markets Steve Kimbrough Fred Murphy INFORMS, November 2005 File: sok-murphy-informs-2005.ppt

2 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

3 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)

4 Figure 1

5 Figure 2

6 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

7 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

8 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

9 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

10 Monopolist Agent Searches stochastically Parameters: –initialPrice –Delta –Epsilon –History

11 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)

12 Example Results: Deterministic Demand

13 Random Demand

14 Monopoly: Random Walk Demand

15 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

16 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.

17 Typical Results: Random Walk

18 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?

19 Firm 2 Has Non-Zero Costs

20 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).

21 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.

22 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.