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Stackelberg -leader/follower game 2 firms choose quantities sequentially (1) chooses its output; then (2) chooses it output; then the market clears This changes the game relative to Cournot competition - (2) can respond to (1)’s output, so that there is no need for a conjecture - (1) can anticipate (2)’s reaction when it chooses its output Subgame perfect equilibrium is typically used in this setting - start at the end of the game and work backwards - Since firm (1) moves first, take q 1 as given, find (2)s best response q 2 - then back-up and consider (1)s choice q 1 Firm (2):

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Stackelberg – first subgame Firm (1) solves a different problem, since it knows how (2) will react to its choice.

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Nash Equilibria This is a non-credible threat but it satisfies NE conditions: - the strategies are complete contingent plans - players are best responding on the equilibrium path

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First-mover advantage Note: that if we compute the SPNE, the leader is always better off than in the simultaneous move case (Cournot competition) Since (2) best responds, (1) can get at least the simultaneous move payoff by choosing the Cournot q. There is a first-mover advantage in any game with strategic substitutes. Defn: Strategies are strategic substitutes if best response functions are downward sloping, and strategies are strategic complements if best response functions are upward sloping. In games of strategic complements, first movers have a disadvantage

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Policy game Consider a policy game, where we show how politicians can get voters to spend too much. A politician proposes a policy change and then a voter decides whether to accept or reject the proposal. We should think of the voter as the median voter; the marginal person. Suppose that the policy proposal involves setting some number: - possibly a level of expenditure on education, for example - normalize so that the voter’s preferred point is 0 - suppose the current level of expenditure is below the voter’s preferred point 0 status quo Suppose that the politician has to play a pure strategy, but that the voter can randomize.

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Payoffs We are assuming that the politician’s payoff is increasing in the expenditure level - we do not need to be any more precise than that. - denote the expenditure level by x The voter will - reject any policy proposal that gives a lower utility (is further away from 0 than sq) - accept any policy that has a higher utility - be indifferent between accepting and rejecting any proposal with the same utility as sq, and therefore might mix - at sq and -sq any probability of accepting is a best response

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The SPNE Now back up to the politician’s move. Since the politician wants expenditure to be as high as possible, the politician will propose the highest level of expenditure that the voter is willing to accept. 0

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Infinite NE

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Nash equilibrium Nash equilibrium is defined in terms of strategies, not payoffs Every player is best responding simultaneously (everyone optimizes) This.

Nash equilibrium Nash equilibrium is defined in terms of strategies, not payoffs Every player is best responding simultaneously (everyone optimizes) This.

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