Game Theory in Electricity Markets
Lecture 3: Mixed Strategy Equilibria
Athanasios Papakonstantinou mail: [email protected] web: athpap.net
Centre for Electric Power and Energy Overview of the Lecture
• More games: The goalkeeper and the penalty shoot-out • Mixed Strategy Equilibria • What happens after an arms race: failed deterrence and the Hawk Dove game • Game-theoretic complexities of Bertrand and Cournot Nash Equilibria
2 DTU Electrical Engineering Centre for Electric Power and Energy Lecture key points
Implications The penalty Theory: mixed Theory: mixed of Matching shoot-out: No strategy NE strategy NE pennies & pure Nash Hawk Dove
3 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•No tricks penalty shoot-out: players move simultaneously •The intuition: both players choose a side…. Goalie wants to match sides
Striker wants to mismatch sides
If a player succeeds in his intention he receives +1
If a player fails in his intention he receives -1
4 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•P1: Colums model the actions of the goalkeeper •P2: Rows model the actions of the striker
P1 P2 w1: Left (q) w2: Right (1-q) 1,-1 -1, 1 t1: Left (p) -1, 1 1,-1 t2: Right (1-p) •We calculate best responses for P1 and P2, but we cannot match them – What does this mean?
5 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•P1: Colums model the actions of the goalkeeper •P2: Rows model the actions of the striker
P1 P2 w1: Left (q) w2: Right (1-q) 1,-1 -1, 1 t1: Left (p) -1, 1 1,-1 t2: Right (1-p) •We calculate best responses for P1 and P2, but we cannot match them
No pure strategy Nash Equilibria
6 DTU Electrical Engineering Centre for Electric Power and Energy More Theory for Game Theory
•A mixed strategy αi for agent i is a probability distribution over the pure strategies ai
• pi=αi(ai) is the probability assigned by agent’s i mixed strategy αi to his action ai
•For goalkeeper if p is going left i.e. α1(t1)=p & α1(t2)=1-p
•For p=1, mixed strategy is a pure strategy i.e. agent
chooses ai (and not αi)
• Bi(α-i): the set of agent’s i best mixed strategies when the list of the other agents’ mixed strategies is α-i Did you notice that we replace actions with strategies?
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More Theory for Game Theory
•A mixed strategy NE captures the stochasticity in the game, given that agents’ actions are not deterministic
•Due to the stochasticity utilities u are replaced by expected utilities U: representing preference over the set of all given outcomes
•A mixed strategy profile α* is a mixed strategy Nash equilibrium iff for every agent’s equilibrium mixed strategy is a best response to the other agents’ equilibrium mixed strategy
for every mixed strategy αi of player i
8 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•For the goalkeeper p is going left and 1-p is going right •For the striker q is hitting left and 1-q is hitting right
P1 P2 w1: Left (q) w2: Right (1-q) 1,-1 -1, 1 t1: Left (p) -1, 1 1,-1 t2: Right (1-p) Goalkeeper’s expected payoff:
= goalie’s exp. payoff when L is a pure goalie’s exp. payoff when R is a pure strategy and striker uses a mixed strategy strategy and striker uses a mixed strategy
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=
P1 P2 w1: Left (q) w2: Right (1-q) 1,-1 -1, 1 t1: Left (p) -1, 1 1,-1 t2: Right (1-p) The goalkeeper’s expected payment to the mixed strategy pair is a weighted average of his expected payoff to L & R when the striker uses mixed strategy α2
10 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•Goalie’s exp. payoff when following L as pure strategy and striker uses a mixed one
P1 P2 w1: Left (q) w2: Right (1-q)
t1: Left (p) 1,-1 -1 ,1
t2: Right (1-p) -1, 1 1,-1 •Goalie’s exp. payoff when following R as pure strategy and striker uses a mixed one
11 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
• For q< ½: 2q-1 for L and 1-2q for R show that: Goalie’s exp. payoff to R exceeds his exp. payoff to L, hence also exceeds his exp. payoff to every mixed strategy that assigns a positive probability to L Goalie’s best response to striker’s strategy is to assign 0 to L • For q> ½: 2q-1 for L and 1-2q for R show that: Goalie’s exp. payoff to L exceeds his exp. payoff to R, hence also exceeds his exp. payoff to every mixed strategy that assigns a positive probability to R Goalie’s best response to striker’s strategy is to assign 1 to L • For q= ½: 2q-1 for L and 1-2q for R show that: L = R, hence all mixed strategies yield the same exp. payoff
12 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
• The best response for the goalkeeper:
• Similarly, the best response for the striker:
• Best response lines form a cross centered at p=q=1/2: • For goalie’s choice of ½ striker’s choice cannot be deterministic and vice versa. Hence, mixed strategy NE is for the goalie and the striker to assign probability ½ to their action
13 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•This game is under the class of “matching pennies” games
•It is a strictly competitive game and attempting to predict the probability of the opponent has the opposite effects: 1. If the goalie favours L, the striker hits R with certainty and vice versa 2. If the striker favours L, the goalie dives L with certainty
• There can be no equilibrium if probabilities are not ½
• The “paradox” of randomisation: can one assign decision making to “luck”? Even if dice are biased? (alternatively why bother with mixed strategies?)
14 DTU Electrical Engineering Centre for Electric Power and Energy Penalty shoot-out
•This game is under the class of “matching pennies” games
•It is a strictly competitive game and attempting to predict the probability of the opponent has the opposite effects: 1. If the goalie favours L, the striker hits R with certainty and vice versa 2. If the striker favours L, the goalie dives L with certainty
• There can be no equilibrium if probabilities are not ½
• The “paradox” of randomisation: can one assign decision making to “luck”? Even if dice are biased?
• Yes! It becomes pointless for your opponent to analyse and exploit your systematic behavioural patterns 15 DTU Electrical Engineering Centre for Electric Power and Energy Arms race: when deterrence fails
Two countries are engaged in a combination of stalemate and mutual destruction. Countries can attack or pass.
They perceive victory if one attacks and the other passes, tie if they both pass, and destruction comes if they both attack. Their preferences are: – Each country prerfers winning (1) to tying (0) – Each country prefers tying (0) to losing (-1) – Each country prefers losing to destruction (-A)
The strategic game: Players Actions Payoff
16 DTU Electrical Engineering Centre for Electric Power and Energy Arms race: when deterrence fails
•Group A: Identify pure NE (if they exist) and check Pareto Efficiency (Lecture2 s. 7,8,17,18)
P1 P2 w1: attack (q) w2: pass (1-q)
t1: attack (p) -A,-A 1,-1
t2: pass (1-p) -1, 1 0,0
17 DTU Electrical Engineering Centre for Electric Power and Energy Arms race: when deterrence fails
•Group B: Determine mixed strategy NE (Lecture2 s. 35,36,37)
P1 P2 w1: attack (q) w2: pass (1-q)
t1: attack (p) -A,-A 1,-1
t2: pass (1-p) -1, 1 0,0
18 DTU Electrical Engineering Centre for Electric Power and Energy Arms race: when deterrence fails
•This game falls under the Hawk Dove games, a class of anti-coordination games
•Hawk Dove game: models Darwinian competition i.e. two subtypes of a species can exhibit hawkish or dovish behaviour as they compete for a resource
•A hawk will fight to its death, a dove compromises
•Probabilities can be expressed as function of payoffs (i.e. the impact of an action)
•There are two pure strategy equilibria and a mixed strategy: At a certain probability pure NE are abandoned
19 DTU Electrical Engineering Centre for Electric Power and Energy The reality of zero marginal cost markets
•What if there is so much stoch. renewable production with capacity consistently maxing out demand?
20 DTU Electrical Engineering Centre for Electric Power and Energy The reality of zero marginal cost markets
•What if there is so much stoch. renewable production with capacity consistently maxing out demand?
21 DTU Electrical Engineering Centre for Electric Power and Energy The reality of zero marginal cost markets
•Cournot (quantity) v Bertrand (price) competition: Who will be the sheriff of Windville?
22 DTU Electrical Engineering Centre for Electric Power and Energy The reality of zero marginal cost markets
•Cournot competition: quantity competition
•Originally described a market consisting of 2 water sellers (each owning a spring)
•Zero (or symmetric) marginal costs, elastic demand, sellers are profit maximisers: for homogeneous goods optimise quantities and asking price
•Cournot output and Nash equilibrium the same: hence Cournot-Nash: a quantity and a price as functions of demand parameters
•Extensions: Collusion (illegal), aggregation of supply (legal), dependence of NE given the # of firms
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The reality of zero marginal cost markets
•Bertrand competition: price competition
•Originally aimed as criticism of to Cournot’s work by Bertrand: who argued that in most markets firms set their prices and market clears quantities. What do you say?
•Under the same assumptions Bertrand output and Nash equilibrium are the same: Bertrand-Nash leads to perfect competition
•Bertrand being a “realistic” extension of Cournot, how can Bertrand be made more “realistic” ?
24 DTU Electrical Engineering Centre for Electric Power and Energy The reality of zero marginal cost markets
Extensions for Bertrand competition
•Bertrand paradox (perfect competition for oligopolists): breaks with asymmetric or zero costs )
•To apply Bertrand in a realistic setting we introduce capacity constraints which bring the Edgeworth paradox: no pure strategies as firms cannot satisfy demand on their own
Strategic Bidding in Electricity Markets with Only Renewables. J. A. Taylor and J. L. Mathieu (uploaded in P II in NB)
25 DTU Electrical Engineering Centre for Electric Power and Energy Thank you for your attention!
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