Strategy -1- Strategy

Transcription

1 Strategy -- Strategy A Duopoly, Cournot equilibrium 2 B Mixed strategies: Rock, Scissors, Paper, Nash equilibrium 5 C Games with private information 8 D Additional exercises pages

2 Strategy -2- A Cournot Duopoly Two firms produce related products Demands are linear q p 0 p2, q p 20 p2 The demand price functions are as follows: p 9 q q, p 2 q q The cost of production in firm is C ( q ) 4q and in firm 2 is C2( q2) 7q2 Cournot (842) Firm : Computes its marginal revenue and adjusts its output until MR ( q, q2) MC 4 R ( q) p ( q) q 9q q q q R MR 9 q q q

3 Strategy -3- Firm then chooses q to satisfy MR MC 9 q q q q Therefore q ( q ) q 2 2 Best response function q( q 2) is firm s best (ie profit-maximizing) response for any choice by firm 2 We can similarly derive firm 2 s best response function q2( q ) Exercise: Show that q2( q ) q

4 Strategy -4- The two best response lines are depicted Cournot visualized an adjustment process With each firm adjusting to the other as depicted Note that the adjustment process converges to a q ( q, q ) where * * * 2 q ( q ) q and * * 2 q ( q ) q * * 2 2 At this point the strategy if the firms are said to be mutual best responses Then the stable outcome of this adjustment process satisfies the two first order conditions q ( q, q ) and ( q, q2) 0 q 2

5 Strategy -5- B Rock Scissors Paper Suppose that the game is to be played 00 times for $20 per round A pure strategy is a choice from the set of possible actions A { a, a2, a3} { R, S, P} A mixed strategy p ( p, p2, p3) places positive probability on at least two of these actions The probabilities are positive and sum to one Suppose that player 2 adopts the strategy 2 p ( p, p2, p3) ***

6 Strategy -6- B Rock Scissors Paper Suppose that the game is to be played 00 times for $20 per round A pure strategy is a choice from the set of possible actions A { a, a2, a3} { R, S, P} A mixed strategy p ( p, p2, p3) places positive probability on at least two of these actions The probabilities are positive and sum to one Suppose that player 2 adopts the strategy 2 p ( p, p2, p3) Player s expected payoffs are as follows: U ( a ) 20( p p ) 2 3 U ( a ) 20( p p ) 2 3 U ( a ) 20( p p ) 3 2 **

7 Strategy -7- B Rock Scissors Paper Suppose that the game is to be played 00 times for $20 per round A pure strategy is a choice from the set of possible actions A { a, a2, a3} { R, S, P} A mixed strategy p ( p, p2, p3) places positive probability on at least two of these actions The probabilities are positive and sum to one Suppose that player 2 adopts the strategy 2 p ( p, p2, p3) Player s expected payoffs are as follows: U ( a ) 20( p p ) 2 3 U ( a ) 20( p p ) 2 3 U ( a ) 20( p p ) 3 2 It is easy to see that if p (,, ), then player s payoff is the same (zero) for all three strategies Then * p (,, ) is a best response to p (,, )

8 Strategy -8- B Rock Scissors Paper Suppose that the game is to be played 00 times for $20 per round A pure strategy is a choice from the set of possible actions A { a, a2, a3} { R, S, P} A mixed strategy p ( p, p2, p3) places positive probability on at least two of these actions The probabilities are positive and sum to one Suppose that player 2 adopts the strategy 2 p ( p, p2, p3) Player s expected payoffs are as follows: U ( a ) 20( p p ) 2 3 U ( a ) 20( p p ) 2 3 U ( a ) 20( p p ) 3 2 It is easy to see that if p (,, ), then player s payoff is the same (zero) for all three strategies Then p (,, ) is a best response to p (,, ) By a symmetric argument, mutual best responses p (,, ) is a best response to p (,, ) So the strategies are 3 3 3

9 Strategy -9- Suppose instead that p (,, ) Then U ( a ) 20( p p ) 20( ) 2 3 U ( a ) 20( p p ) 20( ) 2 3 U ( a ) 20( p p ) 20(2 ) 3 2 **

10 Strategy -0- Suppose instead that p (,, ) Then U ( a ) 20( p p ) 20( ) 2 3 U ( a ) 20( p p ) 20( ) 2 3 U ( a ) 20( p p ) 20(2 ) 3 2 So player s best response is p for all 0 (0,0,) Thus the dynamic process underlying Cournot s solution concept does not converge to the mutual best response outcome *

11 Strategy -- Suppose instead that p (,, ) Then U ( a ) 20( p p ) 20( ) 2 3 U ( a ) 20( p p ) 20( ) 2 3 U ( a ) 20( p p ) 20(2 ) 3 2 So player s best response is p for all 0 (0,0,) Thus the dynamic process underlying Cournot s solution concept does not converge to the mutual best response outcome Despite this is does seem a very reasonable outcome In modern economic modelling of strategy, economists solve for the mutual best response outcome and remain silent as to how it might be learned/discovered

12 Strategy -2- Nash equilibrium Let S i be the set of feasible strategies (pure or mixed) for player i I {,, I} The strategies s i S i and Nash Equilibrium strategies if they are mutual best response strategies That is, for each i and strategies of the other players, s ( s,, s, s,, s ), i i i I s i is a best response for player i

13 Strategy -3- C A game with private information Sealed bid auction There are I bidders Each of the buyers may submit a non-negative sealed bid **

14 Strategy -4- C A game with private information Sealed bid auction There are I bidders Each of the buyers may submit a non-negative sealed bid Allocation rule Bidder i with bid b i loses if another bid is higher If there are m bidders who submit the tying high bid, the winner is selected randomly from one of these high bidders so that win probability of each such bidder is /m *

15 Strategy -5- C A game with private information Sealed bid auction There are I bidders Each of the buyers may submit a non-negative sealed bid Allocation rule Bidder i with bid b i loses if another bid is higher If there are m bidders who submit the tying high bid, the winner is selected randomly from one of these high bidders so that win probability of each such bidder is /m Payment rule (i) Sealed high-bid auction The winner pays his or her bid Losers pay nothing (ii) Sealed second-bid auction The winner pays the highest of the losing bids (the second highest bid)

16 Strategy -6- The model Private information: Each buyer s value is private information ****

17 Strategy -7- The model Private information: Each buyer s value is private information Common knowledge: It is common knowledge that buyer i s value is an independent random draw from a continuous distribution We define F( ) Pr{ v i } This is called the cumulative distribution function (cdf) ***

18 Strategy -8- The model Private information: Each buyer s value is private information Common knowledge: It is common knowledge that buyer i s value is an independent random draw from a continuous distribution We define F( ) Pr{ v i } This is called the cumulative distribution function (cdf) The derivative, f( ) F ( ), is called the probability density function (pdf) **

19 Strategy -9- The model Private information: Each buyer s value is private information Common knowledge: It is common knowledge that buyer i s value is an independent random draw from a continuous distribution We define F( ) Pr{ v i } This is called the cumulative distribution function (cdf) The derivative, f( ) F ( ), is called the probability density function (pdf) The values: The values lie on an interval [0, ] *

20 Strategy -20- The model Private information: Each buyer s value is private information Common knowledge: It is common knowledge that buyer i s value is an independent random draw from a continuous distribution We define F( ) Pr{ v i } This is called the cumulative distribution function (cdf) The derivative, f( ) F ( ), is called the probability density function (pdf) The values: The values lie on an interval [, ] Pure strategy With private information a pure strategy is a mapping b B( ) from every possible value (ie every i into a non-negative bid i i i Bayesian Nash Equilibrium (BNE) strategies: With private information mutual best response strategies are called Bayesian Nash Equilibrium strategies

21 Strategy -2- An example: Sealed high-bid auction bidders We first examine the special case of a uniform distribution in which the values lie in the interval [0, ] i F( ) Pr( v i ) 00 As we shall see, there is an equilibrium in which the bidding strategies are linear B( ) k, i I {,, I} i i With two bidders the equilibrium bid function is B( ) Method: To check whether any proposed pair of strategies are mutual best response (BNE) strategies all we need to do is show that they are mutual best responses i 2 i

22 Strategy -22- Step : Equilibrium win probability W() b We use the proposed equilibrium strategy of buyer 2 to obtain the equilibrium win probability for buyer If buyer bids b he wins if b b and loses if b b Since the tie occurs with probability zero, the win probability is W ( b) Pr{ b b} 2 Since b B( ) the win probability is W ( b) Pr { b} Pr{ 2b} F( 2b) For the uniform case, F( ) 00 Therefore 2b W ( b) F(2 b) 00

23 Strategy -23- Step 2: Solve for buyer s best response Buyer s expected payoff is the win probability Wb () multiplied by the profit if she wins, b 2b 2 2 u(, b) W ( b)( b) ( b) ( b b ) u 2 (, b) ( 2 b) b 00 0, if b 2 Therefore buyer s best response to buyer 2 s proposed strategy B ( ) is the strategy B ( ) 2 By an identical argument, B ( ) is a best response to B ( ) Therefore the pair of strategies { B ( )} { ( )} 2 2 i i i 2 i i are mutual best responses

24 Strategy -24- D Exercises: Pricing game If a firm cannot change its capacity quickly then the quantity setting model makes sense But what if capacity can be easily changed Then a firm can lower a price and still guarantee delivery, or raise a price and sell off unused capacity C ( q ) c q where c 4 and c2 7 Demands are f f f f q p 0 p2, q p 20 p2 Then the profit of firm f is ( p, q ( p)) p q ( p) C ( q ( p)) ( p c ) q ( p) f f f f f f f f f f (a) For each firm solve for the best response function p b ( q2) and p2 b2 ( p ) (b) Depict these in a neat figure What are the equilibrium prices in this game? (c) If firm 2 produces nothing what will firm do? Plot a price vector p ( p, p ) indicating the monopoly outcome with only firm producing Hint: The price of commodity 2 must be chosen so that demand for firm 2 s output is zero (d) Starting from this price pair, examine the adjustment process proposed by Cournot

25 Strategy -25- (e) Compare the equilibrium profits with those in the quantity setting game Exercise 2: Dominant strategy equilibrium in a pricing game with two firms producing the same product Firm has a constant marginal cost c and firm 2 a constant marginal cost c 2 Let q f ( p, p 2) be the demand for the goods produced by firm f And let q( p) q( p) q2( p) be the market demand If one sets a lower price then the other it sells to the entire market If the firms sets equal prices consumers are indifferent so any demands q ˆ ˆ ( p, p ) and q ˆ ˆ 2( p, p ) satisfying q ( pˆ, pˆ ) q ( pˆ, pˆ ) q( pˆ, pˆ ) 2 are best responses by consumers (a) Suppose that c c2 c Show that the unique equilibrium pair of prices is ( p, p2) ( c, c) (b) Explain why there is a continuum of equilibrium demands (c) Suppose next that c c2 Show that there is a unique equilibrium pair of prices (d) Is there more than one pair of equilibrium demands?