Strategic Interaction of and pricing: II Economics for Business Enterprise Lecture 7

Bertrand competition Bertrand Competition in prices Differentiated products Firm 1’s demand depends on its own and firm 2’s prices; vice versa Firm 2 Reaction curves in prices Equivalence to Cournot game – where reaction curves are in quantity

Reaction curves n prices Bertrand-Nash equilibrium

Other outcomes B B COLLUSIVE outcome A

Incentive to cheat Collusive outcome like to be unstable Firm 2 realise that their ‘best reaction’ to firm 1 charging a high price is to cut prices Firm 1 vice versa See reaction curves Reversion to Nash Equilibrium in dynamic (or repeated) game – i.e. one that is played in more than one period

Other outcomes C B A C COMPETITIVE outcome

Incentives to cheat Still possibility to cheat at Nash Equilibrium Firms can grab market share at any time by “cheating” Under-cutting rivals on price Equivalent of a price war Is this wise? given that retaliation is inevitable

Bertrand Paradox Incentive to undercut rivals is always there whilstever price (P) is above marginal cost of production (MC) Will end up in mutual throat-cutting until P=MC Competitive outcome ‘Bertrand Paradox’ Mutually Assured Destruction (MAD)

Tit for Tat Tit for tat :“The infliction of an injury or insult in return for one that one has suffered” Thesaurus: revenge; retaliation In situations of dynamic interaction retaliation type strategies are described as “Tit for Tat”

Tit for Tat Ex-ante: Credible announcement of ‘Tit for tat’ intentions; OR Proven track record of ‘Tit for tat’ behaviour Sufficient to enforce Nash type ‘co-operation’ [sic][ex post] Price wars averted

Alexrod tournament In 1980, Robert Axelrod, professor of political science at the University of Michigan, held a tournament of various strategies of a simple game Players could invent a strategy of their liking based on possibilities of Cooperating; Cheating; or Randomising With payoffs each period Repeated interactions based on 200 iterations of the game

Alexrod tournament Game theorists submitted strategies to be run by computers Strategies are prescribed in advance Strategies were many and complex! Each game had a winner based on total payoff over 200 iterations

Alexrod tournament The tournament winner was: TIT FOR TAT strategy cooperates on the first move, then does whatever its opponent has done on the previous move benefit of cooperating with a friendly opponent benefit of punishing an opponent who cheats when matched against itself, the TIT FOR TAT strategy always cooperates forever

Trigger strategy A harsher strategy than Tit for tat Firm 2 announces that if firm 1 cuts price it will retaliate and will be unforgiving Firm 1 is “punished” forever thereafter by keeping prices low Both firms are then locked into grim episode

Payoff to cheat Payoff= 𝜋 ∗ + 𝜋 1+𝛿 + 𝜋 1+𝛿 2 + 𝜋 1+𝛿 3 +… 𝜋 ∗ = windfall profit (short run) 𝜋 = grim profit (long run) 𝛿= discount rate E.g. Y=200+ 5 1+0.05 + 5 1+0.05 2 + 5 1+0.05 3 +… Y = 300

Payoff to co-operate 𝑃𝑎𝑦𝑜𝑓𝑓= 𝜋 𝐶 + 𝜋 𝐶 1+𝛿 + 𝜋 𝐶 1+𝛿 2 + 𝜋 𝐶 1+𝛿 3 +… 𝑃𝑎𝑦𝑜𝑓𝑓= 𝜋 𝐶 + 𝜋 𝐶 1+𝛿 + 𝜋 𝐶 1+𝛿 2 + 𝜋 𝐶 1+𝛿 3 +… 𝜋 𝐶 = profit under cooperative regime 𝛿= discount rate E.g. Y=50+ 50 1+0.05 + 50 1+0.05 2 + 50 1+0.05 3 +… Y = 1050

Trigger strategy Payoff 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 > Payoff 𝑐ℎ𝑒𝑎𝑡𝑖𝑛𝑔 Cheating doesn’t pay in most reasonable parameterisations If trigger strategy applies

Trigger strategy Cheating only worthwhile if: Massively high rate of discounting “here today gone tomorrow” Weak punishment (grim episode not so punitive) Short lived punishment (not forever; grim period short lived)

Price wars So why price wars? Main theoretical ideas: Noisy signals Weak market concentration Firm asymmetries (more in seminar paper)

Noisy signals Noise means that other firms’ actions cannot be properly understood A firm observes temporary falling demand Firm doesn’t know if this is due to: Rivals undercutting on price (best response is to retaliate) An economic downturn (best response is to do nothing) A market blip (false signal to which the best response is to do nothing) Price war could be sparked inadvertently

Weak market concentration Weak market concentration refers to many rather than few firms Gains to cooperation (versus cheating) may be weakened Achieving best behaviour all round (collusion) is going to be easier when markets are concentrated Coordination problem 2 player – collusive outcome ? 20 players – degeneration to competition ???

Firm asymmetries Different costs, capacities and/or products Small firms may have incentive to cheat: grabbing market share might win consumer loyalty Small firms may go unpunished – small firm impact may be limited so large firms more reluctant to pull the trigger Large firm may have incentive to cheat: deeper pockets than smaller firms, so more able to survive grim episode

Conclusion Some form of ex-post cooperation is likely to emerge in a MAD world Simple ‘Tit for tat’ or ‘Trigger’ strategies sufficient Price wars unlikely in theory Price wars do happen from time to time in the real world Some theoretical explanations