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The basics of Game Theory Understanding strategic behaviour

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The basics of Game Theory As we saw last week, oligopolies are a problem for classical theory The best strategy for a firm depends on what the other firm decides to do Unless some assumption is made, the solution can’t be found... Game theory is the study of the strategic behaviour of agents Not just useful in economics, but also in international relations, games of money, etc.

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The basics of Game Theory The prisoner’s dilemma Nash equilibrium and welfare Mixed strategy equilibria Retaliation

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The prisoner’s dilemma The prisoner’s dilemma is the “historical” game that founded game theory as a specific area of study: This is because the solution to this game is sub- optimal from the point of view of the players. This means that there is a solution that makes both players better off, but the rationality of the agents does not lead to it. The prisoner’s dilemma shows quite elegantly how difficult it is to get agents to cooperate, even when this cooperation is beneficial to all agents.

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The prisoner’s dilemma A typical prisoner’s dilemma: Two suspected criminals are caught by the police, but the police lacks the hard evidence to charge them. They can only sentence them to 1 year for minor misdemeanours. The police needs to get them to confess their crimes in order to be able to charge them both to 20 years. How do the police get the suspects to confess ?

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The prisoner’s dilemma They offer the criminals a “deal”... If one of them “spills the beans” on his colleague, he gets a reduced sentence (6 months), and the other guys gets a extended one (25 years) Payoff Matrix 1 st criminal ConfessDeny 2 nd criminal Confess 20 25 0.5 Deny 0.5 25 1111

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The prisoner’s dilemma The prisoner’s dilemma applied to a duopoly Two firms competing on a market can: Compete (This leads, for example, to the Cournot solution) Collude and share monopoly profits (cartel). Profit in a cartel > profit in a duopoly. If collusion is not illegal, then it is clearly the optimal situation from the point of view of these two firms. But is it the equilibrium the market ends up in ?

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The prisoner’s dilemma 2 players : 2 firms (A and B) producing the same good (Airbus/Boeing fits well!!) 2 strategies : Produce at the duopoly level Produce at the cartel level (which is lower) Given 2 players and 2 strategies, there are 4 possible market configurations These are listed in the payoff matrix

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The prisoner’s dilemma Let’s put some numbers on the different possible profits: For the Cartel case: Each firm earns a share of the monopoly profits: Π c = 10 For the duopoly competition case : Each firm earns duopoly profits, which are lower: Π d = 2 For the “cheating” case: The firm producing at duopoly level captures the market share of the other firm, and makes very high profits : Π t = 15 The other firm is penalised and earns minimum profits : Π m = 0

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The prisoner’s dilemma Payoff Matrix Firm B QdQd QcQc Firm A QdQd 2222 0 15 QcQc 0 10 For firm A: Q d if firm B chooses Q d Q d if firm B chooses Q c Note: the game is symmetric, so the dominant strategy is to produce the duopoly quantity. What is the best strategy for each firm? For firm B: Q d if firm A chooses Q d Q d if firm A chooses Q c

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The basics of Game Theory The prisoner’s dilemma Nash equilibrium and welfare Mixed strategy equilibria Retaliation

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Nash equilibrium and welfare Definition of a Nash equilibrium: A situation where no player can improve his outcome by unilaterally changing his strategy Central properties: The Nash equilibrium is generally stable Every game has at least one Nash equilibrium: Either in pure strategies : Players only play a single strategy in equilibrium Or in mixed strategies : Players play a combination of several strategies with a fixed probability The proof of this result is the main contribution of John Nash (and the reason why it is called a Nash equilibrium)

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Nash equilibrium and welfare Let’s go back to the Duopoly example: Payoff Matrix Firm B QdQd QcQc Firm A QdQd 2222 0 15 QcQc 0 10 Is the “Qd-Qd” equilibrium a Nash equilibrium ? Can firm A or B improve their outcome by shifting alone to the cartel quantity Q c ? “Qd-Qd” is indeed a Nash equilibrium

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Nash equilibrium and welfare Payoff Matrix Firm B QdQd QcQc Firm A QdQd 2222 0 15 QcQc 0 10 So the dominant strategy is to produce “Qd” But the “Qd-Qd” equilibrium is not socially optimal With a small number of agents, individual rationality does not necessary lead to a social optimum

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The basics of Game Theory The prisoner’s dilemma Nash equilibrium and welfare Mixed strategy equilibria Retaliation

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Mixed strategy equilibria A pure-strategy Nash equilibrium does not exist for all games… Example of a penalty shoot-out: 2 players: a goal-keeper and a striker 2 strategies : shoot / dive to the left or the right We assume that the players are talented: The striker never misses and the goalkeeper always intercepts if they choose the correct side. This is not required for the game, but it simplifies things a bit! What is the payoff matrix?

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Mixed strategy equilibria For the striker: R if the keeper goes L L if the keeper goes R Payoff Matrix Goalkeeper LR Striker L 1010 0101 R 0101 1010 For the goalkeeper: L if the striker shoots L R if the striker shoots R No pure-strategy Nash equilibrium ! Whatever the outcome, one of the players can increase his sucess by changing strategy

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Mixed strategy equilibria Payoff Matrix Goalkeeper LR Striker L 1010 0101 R 0101 1010 There is, however, a mixed strategy equilibrium Strategy for both players: Go L and R 50% of the time (1 out of two, randomly) That way : oEach outcome has a probability of 0.25 oThe striker scores one out of two, the other is stopped by the goalkeeper

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Mixed strategy equilibria Let’s check that this is actually a Nash equilibrium: The goalkeeper plays L and R 50% of the time. Can the striker increase his score by changing his strategy? The striker decides to play 60% left and 40% right. His new success rate is: (0.6 ✕ 0.5) + (0.4 ✕ 0.5) = 0.5 (0.3) + (0.2) = 0.5 By choosing 60-40, the striker scores more on the left hand side, but less on the right. His success rate is the same, his situation has not improved. This corresponds to a Nash equilibrium !

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The basics of Game Theory The prisoner’s dilemma Nash equilibrium and welfare Mixed strategy equilibria Retaliation

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Finally, the stability of the equilibrium also depends on whether the game is repeated or not. The very concept of a mixed strategy equilibrium depends on the repetition of the game through time. Even for a pure strategy equilibrium, the ability to replay the game can influence the outcome Players can retaliate, and thus influence the decisions of other players

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Retaliation Back to the duopoly case: The 2 firms agree to form a cartel, and maximise joint profits. There is, however, the temptation to cheat on this agreement Imagine now that the game is played several times If one firm cheats, it captures all the profits for that period What do you think happens in the next period?

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Retaliation Actually, this depends on whether the game is repeated a fixed number of times or indefinitely (open-ended)... Let’s say that our 2 firms decide to play the game 5 times (5 years) What is the best strategy on year 5 ? What about year 4, given what we know about year 5 ? This process shows that the equilibrium cannot be stable

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Retaliation Lets imagine now that our 2 firms have an open-ended agreement. The threat of retaliation can bring the social optimum The optimal retaliation strategy is also the simplest one: “tit for tat” Robert Axelrod: just choose what your opponent did last period: cooperate if he cooperated, cheat if he cheated. But the threat needs to be credible i.e. the opponent needs to believe that it will effectively be carried out.

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