Game Theory in Electricity Markets

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Lecture 8: Extensive games with perfect information

Athanasios Papakonstantinou mail: [email protected] web: athpap.net

Centre for Electric Power and Energy Lecture key points

The matching Subgame The extensive Backward pennies: now perfect game Induction and and then Nash equilibrium

DTU Electrical Engineering Centre for Electric Power and Energy Lecture Agenda

Non cooperative game theory

Strategic (static, Extensive (multi- Repeated (many- one-shot) games stage) games shots) games

Stackelberg games (leader-follower) a specific case of extensive games

DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 1)

• Two players choose side of a coin simultaneously

• What type of game and what is the equilibrium?

P1 P2 w1: Heads (q) w2: Tails (1-q)

t1: Heads (p) 1,-1 -1,1

t2: Tails (1-p) -1,1 1,-1

• If pennies match (HH, TT) P1 receives P2’s penny (+1 for P1 and -1 for P2) • If pennies do not match (HT, TH) P1 gives his penny to P2 (-1 for P1 and +1 for P1)

DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 1)

• Two players choose side of a coin simultaneously

• A zero sum game with mixed strategy Nash equilibria (p=0.5)

P1 P2 w1: Heads (q) w2: Tails (1-q)

t1: Heads (p) 1,-1 -1,1

t2: Tails (1-p) -1,1 1,-1

• If pennies match (HH, TT) P1 receives P2’s penny (+1 for P1 and -1 for P2) • If pennies do not match (HT, TH) P1 gives his penny to P2 (-1 for P1 and +1 for P1)

DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 2)

• The two players do not choose simultaneously!!

• Player 1 reveals (truthfully), then player 2 reveals after he has observed player 1: now there is a timeline and a tree:

P1 P2 H T

H 1, -1 -1, 1

T -1, 1 1, -1

Histories for P1 and P2:

H1={H,T}: one action information sets H2={HH,HT,TH,TT}: two action info sets

H2={HH}: P2’s reaction (H) to P1’s H

DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 2)

• The two players do not choose simultaneously!!

• Player 1 reveals (truthfully), then player 2 reveals without having observed player 1: forgetfulness (imperfect recall)

P1 P2 H T

H 1, -1 -1, 1

T -1, 1 1, -1

Histories for P1 and P2:

H1={H,T} H2={H,T}

Same matrix, different (hi)story History: a player’s sequence of actions

DTU Electrical Engineering Centre for Electric Power and Energy Extensive games

• Strategic games suppress the sequential nature of decision- making: agents choose their plans irrespective of what happens

• Extensive games allow agents to change/ adapt as events unfold and can be represented as game trees

• The extensive game: – Players – Actions

– Payoff

DTU Electrical Engineering Centre for Electric Power and Energy Extensive games

• Strategic games suppress the sequential nature of decision- making: agents choose their plans irrespective of what happens

• Extensive games allow agents to change/ adapt as events unfold

• The extensive game: – Players – Actions – Terminal histories: a set of sequences of actions (if-then-else scenarios / scenario trees) – Payoff

DTU Electrical Engineering Centre for Electric Power and Energy Extensive games

• Extensive games with perfect information: tic-tac-toe, chess

* From wikipedia, under creative commons • Games with finite horizon (unofficial chess rules: draw if a position is repeated tree times) • Strictly competitive games: there can be either a clear winner (and a loser) or a draw for both

DTU Electrical Engineering Centre for Electric Power and Energy

Extensive games

• Extensive games with perfect information: tic-tac-toe, chess

• Games with finite horizon (unofficial chess rules: draw if a position is repeated tree times) • Strictly competitive games: there can be either a clear winner (and a loser) or a draw for both

DTU Electrical Engineering Centre for Electric Power and Energy

Extensive games

• For strictly competitive games there can be two outcomes: a) One player has a strategy that guarantees he wins b) Both players have a strategy that guarantees at leas a draw

• What is the case for tic-tac-toe*? a) I play first: I play center, you play corners, I wedge else I play corner, you play center, I play corner (diagonal) b) You play first: my best outcome is to keep blocking you until we reach a draw (or you make a mistake)

• What about chess?

* https://www.quora.com/Is-there-a-way-to-never-lose-at-Tic-Tac-Toe DTU Electrical Engineering Centre for Electric Power and Energy

Extensive games

• Besides tic-tac-toe and chess there are several situations that can be modelled as extensive games

• Incentives: Employer signs workers, workers decide on their behaviour, benefits and promotions are given accordingly

• RnD: Firms decide on which technologies to invest

• Monetary policy: Central banks decide on the monetary supply to create / counter inflation and individuals act accordingly

• Market entry: Firms decide on whether they enter

DTU Electrical Engineering Centre for Electric Power and Energy Market entry

• The race for influence (exercise 1): The “Let it burn” association pushes for a policy change in Windville and Solartown, pioneers in adopting renewables.

• From allocating limited resources to modelling the incumbent- challenger relation: break (or not) a monopoly

• A challenger chooses to enter or not (in, out) in a race and the incumbent chooses to acquiesce (accept reluctantly) or fight if the challenger enters Players: challenger, incumbent Actions: in, out, acquiesce, fight Terminal histories: (in, acquiesce), (in, fight), out

Payoffs: uc(in,acq.)=2, uc(out)=1, uc(in,fight)=0,

ui(out)=2, ui(in,acq.)=1, ui(in,fight)=0

DTU Electrical Engineering Centre for Electric Power and Energy

Market entry

• The diagram illustration highlights the game’s sequential nature

• What is the solution of this game? (for both the challenger and the incumbent?)

DTU Electrical Engineering Centre for Electric Power and Energy The entry game

Representing the extensive form as a strategic game: P1 the Challenger P2 the Incumbent (monopolist)

P1 P2 Acquiesce Fight

In 2,1 0,0

Out 1,2 1,2

Two pure strategy Nash equilibria: (in, acq.), (out, fight)

Note that fight in (out, fight) does not appear if P1 chooses out

(out, fight) is sustained by a non-credible threat of P2

DTU Electrical Engineering Centre for Electric Power and Energy The entry game

Representing the extensive form as a strategic game: P1 the Challenger P2 the Incumbent (monopolist)

Two pure strategy Nash equilibria: (in, acq.), (out, fight)

Note that fight in (out, fight) does not appear if P1 chooses out

(out, fight) is sustained by a non-credible threat of P2

Non-credible: P1 stays out because it expects that P2 will fight

DTU Electrical Engineering Centre for Electric Power and Energy Extensive form equilibria

• The non-credible NE addresses a complication in the equilibrium concept, visible when deriving the payoff matrix

• In the strategic form all actions are observed simultaneously i.e. even if challenger chooses out, it observes the incumbent’s actions: same when using the payoff matrix for the extensive

• The (out, fight) of the strategic form does not exist in extensive

• In the extensive form there is a time-line, hence a challenger cannot observe the incumbent if it chooses out (because the action will never happen)

• There has to be a process that rules out the non-credible equilibria: backwards induction followed by the definition of subgame equilibria

DTU Electrical Engineering Centre for Electric Power and Energy Backward induction

• For finite horizon games (like those we discussed) we start from the bottom of the tree and go backwards node by node

• During this process we eliminate actions that are not sequentially rational e.g. fight in (out, fight)

• The challenger reasons that if he enters, the incumbent will accommodate, because it is better than fighting. Given the incumbent’s behaviour, the challenger chooses to enter

• Backward induction: a player in motion deduces for each of his avail. actions, the actions others (inc. himself) will take (remember tic-tac-toe and chess) and yields the most prefered terminal history (in, acq.) REMEMBER: challenger moves first

DTU Electrical Engineering Centre for Electric Power and Energy The entry game (ver 2)

• Backward induction not always sufficient for determining the solution of a game

• If the challenger enters, the incumbent is indifferent between acquiescing or fighting • If C knows (or assumes) that I will acquiesce he will choose In (else out) or mixed strategy equilibrium (chance moves) • Equilibrium concept similar between strategic and extensive games: model a player’s behaviour in a steady state DTU Electrical Engineering Centre for Electric Power and Energy Subgame perfect equilibrium

• A game tree consists of a set of nodes and a set of edges

• Each edge connects two nodes (these nodes are adjacent) • For any pair of nodes there is a unique path that connects two nodes (??) • The length of a path is given by the number of edges contained • The beginning of the tree is the root and the adjacent nodes the successors