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
A
COLLUSIVE
outcome
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 ‘cooperation’ [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 +
• Y = 300
5
1+0.05
+
5
1+0.05 2
+
5
1+0.05 3
+⋯
Payoff to co-operate
• 𝑃𝑎𝑦𝑜𝑓𝑓 =
𝜋𝐶
𝜋𝐶 +
1+𝛿
+
𝜋𝐶
1+𝛿 2
+
𝜋𝐶
1+𝛿 3
+⋯
• 𝜋𝐶 = profit under cooperative regime
• 𝛿= discount rate
• E.g.
• Y = 50 +
• Y = 1050
50
1+0.05
+
50
1+0.05 2
+
50
1+0.05 3
+⋯
Trigger strategy
• Payoff𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 > Payoff𝑐ℎ𝑒𝑎𝑡𝑖𝑛𝑔
• 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