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

2. 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

3. Reaction curves n prices
Bertrand-Nash equilibrium

4. Other outcomes B B
COLLUSIVE
outcome A

5. 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

6. Other outcomes C B A C
COMPETITIVE
outcome

7. 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

8. 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)

9. 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”

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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= …
Y = 300

16. 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 𝜋 𝐶 1+𝛿 3 +…
𝜋 𝐶 = profit under cooperative regime
𝛿= discount rate
E.g.
Y= …
Y = 1050

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

18. 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)

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

20. 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

21. 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 ???

22. 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

23. 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