Unformatted text preview: GAME THEORY Lecture 1011 Stable Play: Nash Equilibria in continuous games Instructor: Nuno Limão slides_414_10_11_2011_no_solution 1 MOTIVATION Games analyzed thus far feature discrete strategies Sometimes reasonable (enter or not, invest in machine or not) but not applicable in other: e.g. bids, price, quantities, quality are continuous variables can simplify analysis in certain cases but may also lead to misleading results Definition: A continuous game is one where the strategy set is composed of intervals of real numbers, e.g. price is between 0 and 1000. Implications of continuous game We have an infinite number of strategies. We can use calculus to find solution slides_414_10_11_2011_no_solution 2 Recall example of a continuous game played as experiment: outguessing game Players x,y choose number between 0 and 100 as close as possible to ½((x+y)/2) Best reply for each: y= (2/3) x and x=(2/3) y Nash equilibrium: y*= (2/3) x* and x*=(2/3) y*, only possible if x*= y*=0 x y 2
y 3 y 2
x 3 x slides_414_10_11_2011_no_solution 3 APPLICATIONS I: PRICE COMPETITION WITH SIMILAR GOODS Basic question I: how is the price determined if a finite number of sellers of similar products compete in prices (aka as Bertrand competition, after its initial developer: Joseph Bertrand, 1833) Setup I: two shops (1,2) selling exactly same product Unit cost of the good for each shop is the same, e.g. $10 Total demand is D=100‐p Strategies for each shop: a price, p i [0,100] since if >100 sell zero. Buyers buy from lowest price or “split” demand if price is same, e.g. demand for 1 D1(p1,p2) = 100‐ p1 if p1 <p2 D1(p1,p2) = ½ (100‐ p1) if p1 = p2 D1(p1,p2) = 0 p1 >p2 slides_414_10_11_2011_no_solution if 4 Payoff for shops: Vi (p1,p2) = (p i ‐c)Di(p1,p2), e.g. for i=1 (p1 ‐10) (100‐ p1) if p1 <p2 ½ (p1 ‐10) (100‐ p1) if p1 = p2 0 if p1 >p2 Illustrating i=1 payoff for special case where p2=$40 slides_414_10_11_2011_no_solution 5 Nash equilibria? Candidates: any price p i [0,100] Consider p i [0,10[ : not an equilibrium since it yields V<0 and can always get V=0 if pi=10 Consider pi 10 and focus on symmetric equilibria (p* = p1 = p2) payoff for each firm is then ½ (p’ ‐10) (100‐ p’) For any candidate price p’>10 each firm has an incentive to deviate to a price p’‐ε because for a small enough ε we have (p’‐ε ‐10) [100‐ (p’‐ε)] > ½ (p’ ‐10) (100‐ p’) Consider p i 10 and focus on symmetric equilibrium so payoff=0 for both pi>p’ implies a profit of 0 (no sales) so no incentive to deviate pi<p’ implies a profit of <0 (can’t cover cost) so no incentive to deviate Conclusion: under Bertrand competition between similar firms and products there is a unique symmetric Nash equilibrium with prices exactly equal to marginal cost slides_414_10_11_2011_no_solution 6 Questions for discussion How does this compare to the monopolist and the perfectly competitive outcome? If there are n>2 sellers, how would this affect the result above? Would marginal cost pricing by all still be a NE? If the goods are imperfect substitutes will we still obtain MC pricing? Are there simple pricing schemes that prevent such intense competition? slides_414_10_11_2011_no_solution 7 Setup II: Same as I but with a “Price matching guarantee” If pi > pi then consumer who bought from i can get a refund , e.g. pi pi so the effective price is min{ p1 , p2} Implication of matching price guarantee is that both stores end up selling at lowest posted price so the demand is split and payoffs are ½ (min{ p1, p2} ‐10)(100‐ min{ p1, p2}) slides_414_10_11_2011_no_solution 8 Nash Equilibria An intermediate step: optimal price for a monopolist: V =(p ‐10) (100‐ p) so FOC yields 100‐2p =10 →p=55 slides_414_10_11_2011_no_solution 9 Consider symmetric equilibria candidates p ' [10,55] . If p2=p’ is posted by firm 2 then the red line depicts firm 1’s payoff with mutual price matching guarantees If 1 posts p1 > p’ then still sells at p’ and so payoff is independent of whether posted price is higher If 1 posts p1 < p’ then other firm also sells at p1 and payoffs for each are half the monopolist and so decreasing for p<55. Thus the best reply of 1 to p’ is p’ for any p' [10,55] Infinite symmetric Nash equilibria but payoff dominant one is p’=55 slides_414_10_11_2011_no_solution 10 Notes on price matching guarantees While at first these may seem to create downward pressure on prices they can actually generate higher prices and even sustain the monopoly price! Thus a simple pricing scheme can neutralize the intense competition Hess and Gerstner (1991) find evidence that this scheme raised prices in supermarkets. slides_414_10_11_2011_no_solution 11 APPLICATIONS II: PRICE COMPETITION W/ DIFFERENTIATED GOODS Using calculus to define best‐reply and finding NE in continuous games Recall: s*i is a best response to s‐i if and only if Vi(s*i, s‐i) Vi(si, s‐i) for all si Si. If the payoff function is concave for any s‐i (so it has a unique maximum, BR(s‐i)) then the best response for i, BRi is defined by the FOC: ∂Vi(BRi, s‐i)/ ∂si = 0 Since a strategy profile (s*i, s*‐i) is a Nash equilibrium if s*i is a best response to s*‐i for all i = 1, 2, …, n, we can find it as the solution to the system of n equations ∂Vi(BRi, s‐i)/ ∂si = 0 for all i. slides_414_10_11_2011_no_solution 12 Application of calculus approach to pricing with differentiated goods Two suppliers, of PCs: i= Dell and HP Symmetric demand functions: Di = 100 – 2pi + p‐i Asymmetric unit production costs: Dell is $10 per unit; HP is $30 per unit Payoffs Dell: V_dell = (p_dell – 10)(100 – 2p_dell + p_hp) HP: V_hp = (p_hp – 30)(100 – 2p_hp + p_dell) Each V concave in own price for any given price of the other (verify yourselves) slides_414_10_11_2011_no_solution 13 Exercise Find the best response for Dell and HP What are the Nash equilibria in this game? slides_414_10_11_2011_no_solution 14 Solution slides_414_10_11_2011_no_solution 15 Solution (ctd) slides_414_10_11_2011_no_solution 16 OTHER APPLICATIONS Basic Question: How is price competition affected in the presence of trade costs? Setup: Bertrand competition with spatial element (borrowed from Schwab lecture) Firms (A, B) choose prices simultaneously on goods that are perfect substitutes. Costs are symmetric: MC = AC = .25. Spatial dimension Firms are located 1 mile apart each at the edge of a linear market m consumers uniformly distributed between firms (assume m is large so do not worry about integer problems) slides_414_10_11_2011_no_solution 17 Setup (ctd) Demand Each consumer demands 1 unit of the good from either Firm A or Firm B. When a consumer buys the good, he incurs transportation costs of 0.5 per mile. A consumer who lives x miles from A, and therefore 1 – x miles from B Will buy from A if pA + .5x < pB + .5(1 – x) Will buy from B if pA + .5x > pB + .5(1 – x) Will be indifferent between buying from A or B if pA + .5x = pB + .5(1 – x) Questions Which market does each firm capture and how does it depend on their price? What are the sales for each firm? What is the equilibrium price strategy for these firms slides_414_10_11_2011_no_solution 18 Solution slides_414_10_11_2011_no_solution 19 ...
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