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Nash equilibrium

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  1. Nash equilibrium Nash equilibrium is defined in terms of strategies, not payoffs Every player is best responding simultaneously (everyone optimizes) This is a natural generalization of single-person optimization solutions; we keep the idea that everyone is optimizing, but allow for strategic interdependence

  2. Consider conjectures Two players;each player makes some conjecture about what the other will do and optimizes In equilibrium - every player is best responding to the other - conjectures about other player’s moves are correct

  3. Competitive Strategy Consider the problem of standardization -when new products are introduced, different firms’ products might be incompatible VCRs: Beta, VHS, take different cassettes Suppose both types of firms are in the market • 2 equilibria – both adopt VHS • – both adopt Beta • VHS firms prefer the equilibrium where VHS is adopted • Beta firms prefer the equilibrium where Beta is adopted • In general, there is not a unique Nash equilibrium • This is both a strength and a weakness of equilibrium analysis • a standard will emerge, but we do not know which one • which outcome occurs depends on strategies outside this game

  4. Chicken • Two drivers drive toward each other • - if no one turns they crash and burn - if one turns then the other wins - if both turn then no one wins • Again there are two equilibria • - one turns and the other doesn’t We can characterize the equilibrium even when it is not unique: We can say one player will turn, but we do not know which player Which gets played might depend on pre-play signaling, communication, reputation, commitment devices In order to do comparative statics it is nice to have a unique equilibrium

  5. Cournot Competition • Two firms, identical products, each firm decides how much to produce • Comparing the Cournot and Bertrand games shows that behavior matters in industries • In otherwise identical markets: • - if firms compete by choosing quantities then P>MC • - if firms compete by choosing price then P=MC

  6. The Cournot Game

  7. Nash Equilibrium

  8. The fixed point of the BR functions This is the point where the BR functions have a fixed point.

  9. Review Cournot solution Cournot best response functions have other player’s quantities in equation In equilibrium, Strategies are Functions of Parameters Once we get strategies as functions of parameters then we can do comparative statics

  10. Cournot model for N firms

  11. Bertrand Competition Seemingly innocent changes in assumptions about strategy sets can change predictions Consider the exact same case as above, but assume firms have different choice variables 2 firms, identical products, constant mc, but firms simultaneously choose prices not quantities

  12. Proof Claim: the unique Nash Equilibrium is each firm sets price equal to marginal costs We cannot use calculus to prove this because the demand function is discontinuous We use a “suppose not” argument; to show that something is not a NE we need to show that at least one player is not best responding.

  13. Proof

  14. Weakly dominated strategies

  15. Chairman’s Paradox Consider a voting game – Outcomes {a, b, c} Players {1, 2, 3} The player’s vote, and the outcome is chosen by majority rule. -if there is a majority ex: (a, a, b) then a wins -the Chair (player 1) is allowed to break ties ex: (a, b, c) then a wins Note: no matter what player’s preferences are, we get the following NE (there are others). (a, a, a) (b, b, b) (c, c, c) a wins b wins c wins There is no clear prediction with the best response logic -if the other 2 players vote for a, then it doesn’t matter what the remaining player does -so voting for a is a BR One way to proceed is to eliminate weakly dominated strategies

  16. Elimination of strategies Suppose preferences are as follows: (1) a>b>c (2) b>a>c (3) c>b>a For (2) c is weakly dominated by b -if (1) and (3) choose the same thing, (2) is indifferent between voting for a, b, or c -if (1) and (3) choose different things then (2) might be the deciding vote - but he would choose b over c in order to ensure that c didn’t win Similarly, for (3) a is a weakly dominate by c For (1) both b and c are weakly dominated by a -if (2) and (3) choose the same thing, (1) is indifferent between voting for a, b, c -if (2) and (3) choose different things, then since (1) breaks ties, (1) will vote for a The remaining NE: (1) must play a. We get (a, a, c ) : a wins or (a, b, b) : b wins After another round of eliminating weakly dominated strategies, since c can never win The prediction is (a, b, b). The Chairman’s Paradox is that even with more power, his preferred outcome does not win.

  17. Horizontal Merger Among Firms Producing Complementary Products Demand for systems -consider a market for computer systems. -a computer system is defined as a combination of two complementary products - computers and monitors Px is the price of a computer; Py is the price of a monitor The price of the system is Ps= Px+Py Suppose the demand for systems is Q= –Ps= – (Px+Py) Denote the number of computer and monitors sold by X and Y respectively Assume the two goods are always sold as a system: Q=X=Y

  18. Complementary Products Consider 2 separate firms – one makes computers – the other makes monitors Suppose firms choose pricesand for simplicity assume that costs are 0

  19. Systems Monopolist Note: the price of a system is cheaper, more systems are sold, and firm profits are higher - social welfare unambiguously improves - integration removes the externality each firm imposes on the other - a rise in the price of one component lowers the demand for both Think in terms of externalities With two firms if firm (2) raises its price, then firm (1)’s demand falls -firm (2) does not consider how increasing Py affects firm (1)’s profits. The monopolist takes into account both markets and internalizes the externality

  20. Mixed Strategies

  21. Randomization

  22. Generalized definition All of the definitions – strictly dominant, strictly dominated, weakly dominant, and weakly dominated can be generalized in a straightforward way.

  23. Pure Strategy A pure strategy might be dominated by a mixed strategy Playing a mixed strategy is like spinning a roulette wheel or flipping a coin to determine which pure strategy to play This proposition says: -to test whether a pure strategy si is dominated when randomized play is possible we just need to check against all possible profiles of opponents’ pure strategies -not their mixed ones.