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Sony vs Microsoft • Assignment 1: Given the information in Exhibits 1, 2, 3 – Draw AVC and ATC – Calculate Sony’s profit over variable cost when Ps=Pm=399 – Assuming demand curves are linear, calculate ownprice elasticities of demand at price 399 – Calculate Sony’s and Microsoft’s profits for each combination of prices. • Would you predict that Sony and/or Microsoft will want to reduce console price by $100? Who are the players? Costumers Competitors (substitutes) Company Competitors (complements) Game Theory • Strategic interdependence – Each player’s best choice depends on what he expects other players to do – Outcome depends con choices by all players • Rational players – Maximise their payoffs • players have to reason strategically Elements of a game • Set of players I • Set of strategies for each player Si • Outcomes (s1, s2,…sI) • Payoffs ui (s1, s2,…sI) • Rules (timing and information) Suppliers 1 At the bar https://www.youtube.com/watch?v=2d_dtTZQyUM EN • The situation …. It is a game • Players: boys • Strategies: "go for the blonde" or "go for a brunette“ • Rules: Each boy has to decide what to do without knowing what the others will do. Types of games and solution concepts timing\ info Complete Incomplete Static Nash Equilibrium Bayesian NE Dynamic Prisoners’ dilemma Sony vs Microsoft S\ M $ 399 (in normal form) $ 299 $ 399 1032.5 960 $ 299 1186.8 637 767.3 978.8 Subgame Perfect Perfect Bayesian 1\2 1012.5 NC 920 C NC 2 C 2 0 3 3 0 1 1 2 definitions Nash equilibrium • Dominant strategy: maximises player’s payoff regardless of strategies chosen by the others. – In the PD C is the dominant strategy • Dominated strategy: there is always another choice taht gives higher payoffs it is never a good choice – In the PD NC is dominated by C Nash equilibrium: an example R\C L C • A strategy profile is a Nash equilibrium if each player’s strategy is a best response to the strategies chosen by other players • in equilibrium palyers no player can change his trategy and do better (noregret) Nash equilibrium: an example • (B, C) is a NE • To find a NE, write the BR functions R – For Row player T 4 0 2 2 2 3 M 1 2 1 0 4 0 B 2 1 3 3 2 2 • BRR(L)=T • BRR(C)=B • BRR(R)=M – For Column player • BRC(T)=R • BRC(M)=L • BRC(B)=C 3 Nash equilibrium Nash equilibrium: an example R\C L C R T 4 0 2 2 2 3 M 1 2 1 0 4 0 B 2 1 3 3 2 2 Problems • There may be multiple NE • An equilibrium may not exist • • • • • Define the best reply function: si=Bi(s-i) I= 2 …. A NE strategy profile is s*=B(s*) ∗ ∗ i.e. 𝜋𝑖 (𝑠𝑖∗ , 𝑠−𝑖 ) ≥ 𝜋𝑖 (𝑠𝑖′ , 𝑠−𝑖 ) To find a NE – Write BR functions – Find a fixed point, i.e. Solve simultaneously Multiple NE: a coordination game R\C L R T 3 3 0 0 B 0 0 1 1 4 Multiple NE: the «stag hunt» Stag Rabbit Stag 3 3 0 2 Rabbit 2 0 1 1 Matching pennies The Battle of the Sexes C S C 2 3 0 0 S 1 1 3 2 Mixed strategies H T H 1 -1 -1 1 T -1 1 1 -1 • Players randomise, i.e. choose probability distributions over pure strategies • Players must be indifferent between pure strategies: expected payoff from strategies must be equal • Matching pennies • Harsanyi’s interpretation of MS as uncertainty over opponent behaviour 5 Mixed strategy equilibrium: an example • Matching pennies • For player 1: prob (H)= p1; prob(T)=1-p1 • For player 2: prob (H)= p2; prob(T)=1-p2 • Randomization makes sense if players are indifferent between H and T • EU(H)=EU(T) …. p1= Nash equilibrium: existence • Nash’s theorem: Any finite game has at least an equilibrium, possibly in mixed strategies (Nash, 1950) – Proof by a fixed-point theorem p2=1/2 Quantity competition: an example (CW p. 246) Cournot Competition • Assumptions: – n=2 – strategy: output – Homogeneous product – Firms choose simultaneously – one shot • In what follows assume, – AC1=AC2=c – p=a-bQ 6 Nash-Cournot Equilibrium Graphically • Find best response functions – Firm 1 considers residual demand (total demand - q2 ) and maximises profits, which requires MR1=MC1 – Analogously for firm 2 MC q1* Nash-Cournot Equilibrium Firm 1 Best Response function q1*(q2) q2 qc qM qM N qc qM qc • A pair of strategies which are mutually best response • Graphically, it is where best response functions intersect • Analitically, one has to solve the system of equations given by the best response functions 7 Price competition Exercises P=a-bQ Q=q1+q2 MCi=cqi • In a wide variety of markets firms compete in prices – – – – ch. 8 CW Internet access Restaurants Consultants Financial services • With monopoly setting price or quantity first makes no difference • In oligopoly it matters a great deal – nature of price competition is much more aggressive the quantity competition 33 Price competition an example: Microsoft and Netscape • Product: browser (Netscape Navigator and Internet Explorer) were good substitutes • Navigator, since 1995; Explorer since 1996. • Price of N = $49; Price of E = $0 • Microsoft signed agreements with PC producers to incentivate use of E • Netscape in 1998 N free and open source Bertrand Competition • Assumptions: – n=2 – strategy: price – Homogeneous product – No capacity constraints – Firms choose simultaneously – one shot • In what follows assume, – AC1=AC2=c – p=a-bQ 8 Bertrand Competition Demand Bertrand Competition Best Response functions • No capacity constraints 0 se p1 p2 D1 p1 D p1 se p1 p2 1 D p1 se p1 p2 2 Bertrand’s paradox • Nash equilibrium p1=p2=c – (no firm wants to deviate) • “Proof”: – p1 p2 is not an equilibrium (firm setting lowe price wants to deviate) – p1=p2=p’>c is not an equilibrium (both firms want to deviate) • Result depends on assumptions: p M se p2 p M p1* p2 p2 se c p2 p M c se p c 2 Price competition with differentiated products • With differentiated products a firm setting a price higher than rival does not loose all costumers • Demand for each firm depends on prices set by both firms • Firms have market power ( P>MC) – No capacity constraints – Homogeneous product – One shot game 9 An example of product differentiation Exercise Coke and Pepsi are similar but not identical. As a result, the lower priced product does not win the entire market. • q1=130-1,5p1+0,5p2 • q2=130-1,5p2+0,5p1 Econometric estimation gives: QC = 63.42 - 3.98PC + 2.25PP • MC1=MC2=10 MCC = $4.96 QP = 49.52 - 5.48PP + 1.40PC MCP = $3.96 40 Price competition with capacity constraints • For the p = c equilibrium to arise, both firms need enough capacity to fill all demand at p = c – But when p = c they each get only half the market – So, at the p = c equilibrium, there is huge excess capacity • Firms are unlikely to choose sufficient capacity to serve the whole market when price equals marginal cost • since they get only a fraction in equilibrium – so capacity of each firm is less than needed to serve the whole market – but then there is no incentive to cut price to marginal cost Price competition with capacity constraints • With capacity constraints a firm setting a price higher than rival does not loose all costumers • Two-stage game: first firm invest in capacity and the compete over prices • Kreps and Sheinkman (1983) show that the solution to this two-stage game is Cournot equilibrium 10 Price competition in repeated interactions • If firms are in the market for many periods they may be able to reach a tacit agreement (collusion) to fix prices higher than NE. • In non-cooperative games this agreement must be self-enforcing (i.e. in the players’interest) Strategies vs Actions • Actions are the choices available to a player when it is his turn to move. • Strategies are complete plan of actions; tell what to do in every possible circumstance (if ...then ... else) Dynamic Games Game tree: sequential games • The Battle of the Sexes S 3 2 C 1 1 W S M S C W 0 0 C 2 3 Another example: entry deterrence E E NE I A 100 150 W -10 0 0 450 11 Backward Induction Capacity Expansion • Entry deterrence – Price war is not credible • Procede backward – Incumbent must choose between Accept entry (payoff=20) or Price War (payoff=-10) – Correctly anticipating this decision E enters • The credible equilibrium is: [E; A/E] • I invests in capacity at a cost equal 200. Capacity is only used if production is high. E E NE I A 100 -50 W -10 0 0 250 Subgame perfect equilibrium Evidence on predatory expansion • Some anecdotal evidence • Alcoa – evidence that consistently expanded capacity in advance of demand • Safeway in Edmonton – evidence that it aggressively expanded store locations in response to potential entry • DuPont in titanium oxide – rapidly expanded capacity in response to to changes in rivals’ costs – market share grew from 34% to 46% • Subgame: starting from any decision node, it include all subsequent choices • A strategy profile is a subgame perfect Nash equilibrium (SPNE) if the strategies are a Nash equilibrium in every subgame. • In a finite game the SPNE can be found by backward induction (example 9.4 in CW) 52 12 Subgame perfect equilibrium An example (9.4 in CW) Sequential quantity competition: Stackelberg • Firms choose outputs sequentially – leader sets output first, and visibly – follower then sets output • The firm moving first has a leadership advantage – can anticipate the follower’s actions – can therefore manipulate the follower • For this to work the leader must be able to commit to its choice of output • Strategic commitment has value 56 Sequential price competition • Firms choose prices sequentially • Price competition gives a second mover advantage. Repeated games • In many economic situations strategic interaction is repeaded over time. • A repeated game is a dynamic game in which the same basic game is repeated many times. The basic game can be – In normal form (simultaneous move) : G(Si,ui)iI e.g PD – In extensive form (sequential moves): game tree e.g. entry-deterrence • Players remeber history of play 13 Repeated games • A SPNE is a set of strategies optimal in every subgame. • In a repeated game a subgame starts each period. • SPNE adds sequential rationality to NE, thus refining NE to rule out non-credible threats. The collusion dilemma in Cournot model The collusion dilemma • Recognising that strategic interdependence may lead to inefficient outcome, players may try to reach an agreement to maximise joint profits (COLLUSION). • Collusion must be self-enforcing … • To mantain collusive agreements collusive strategies must contemplate Detection of deviation and Punishment of deviators The collusion dilemma in Bertrand model 1\2 Collude Deviate 1\2 Collude Deviate Collude 1800 1800 1350 2025 Collude 1800 1800 0 3600 Deviate 2025 1600 1600 Deviate 1350 3600 0 0 0 14 The collusion dilemma in the PD Finitely Repeated PD • The unique SPNE is (D, D …, D) 1\2 Collude Deviate Collude 2 2 0 3 Deviate 3 0 1 1 – …. • If history does not influence the strategies, then repeating the game does not change the equilibrium. Infinitely Repeated PD Infinitely repeated PD • Trigger strategy: – Play the collusive strategy in each period as long as all the others have done so in the past – If anyone has ever deviated from collusion, play the punishment strategy for ever. • If the opponent is playing the trigger strategy (i.e. is choosing Coll) – By choosing Coll., the stream of discounted payoffs is 2 2 2 2 2 ... 1 – By choosing Dev., the stream of discounted payoffs is – In the example, if δ>1/2 then efficient outcome can be sustained by the trigger strategy (play Coll. as long as the other has done so, play Dev. otherwise) 3 2 ... 3 – Thus if 1 2 1 3 cioè 1 1 2 – The trigger strategy is a SPNE 15 Folk Theorem Folk Theorem • Let a* be a static equilibrium of the stage game with payoffs ui . • For any feasible payoff v, with vi > ui for all i ∈ I, there exists some δ < 1, such that for all δ > δ, there exists a subgame perfect equilibrium of G ∞(δ) with payoffs v • • • • • The Great Salt Duopoly The evolution of cooperation (CW case study 10.4) http://www.cultureofdoubt.net/download/docs_cod/evolution%20of%20cooperation,%20axelrod.pdf Two firms: BS (55%) and WP (45%) Excess capacity: 25% (BS)and 35% (WP) Theory prediction: …. Observed behaviour: parallel pricing Rees: Firm behaviour is consistent with collusion supported by carrot-and-stick strategies • Under what conditions will cooperation emerge in a world of egoists without central authority? • To find a good strategy to play in the PD Axelrod invited experts in game theory to submit programs for a computer PD tournament. • The winner was tit for tat (reciprocity, provocability, forgiveness, simplicity) • Small number interacting for extended periods (indefinite) • Altruism nor trust are needed 16 Experiments • In lab experiments, there is more cooperation in prisoners’ dilemma games than predicted by theory. • More interestingly, cooperation increases as the game is repeated, even if there is only finite rounds of repetition. • Why? – Altruism – Fairness – revenge 17