Managerial Economics
Lecture Five:
What is competition?
The standard view (warts & all!)
Recap
• Last week
– Schumpeterian view of innovation process
• Pricing freed from constraints of “supply/demand” by
– Endogenous creation of new money to finance
entrepreneur
– Ability to undercut rivals/take over old markets with
new technology
• “creative destruction”
– Logistic process of
• product diffusion
• Price setting/change (high for early adopters, falling to
same input-output basis as for standard commodities)
• This week: just what is competition? First, the standard
view
Neoclassical models of strategic interaction
• Standard (Marshallian) model critiqued in first 2 lectures
has fundamental flaws
– “Horizontal firm demand curve” can’t co-exist with
downward sloping market demand
– Neoclassical “profit-maximisation” formula ignores
impact of other firms on profit in multi-firm industry
– Corrected equilibrium profit maximisation formula
predicts “competitive” industry will produce same
amount as monopoly for same price
– Assumption of upward sloping marginal cost doesn’t
apply in real world
• “Newer” Cournot-Nash-Game Theory “strategic
interaction” model not so (obviously) flawed…
Cournot-Game Theoretic Competition Models
• Game theory approach implicitly acknowledges firms
produce more than profit-maximising amount
• But then sees outcome as result of strategic interactions
between firms
– Key difference with “perfect competition”/Marshallian
analysis
• Marshallian presumes firms “atomistic”
– Too small (blah blah blah) to worry about what
competitors are doing; just tries to maximise profit
w.r.t. market
• Problem shown earlier: atomistic behavior means slope
of individual firm demand curve equals slope of market
demand curve
– End result: competitive outcome same as monopoly
Cournot-Game Theoretic Competition Models
• Game theory models don’t have this problem
– Firms presumed to react to expected strategies of
rivals
• In Marshallian theory dqj/dqi=0: one firm doesn’t react
to what other firms (might) do
• In Game theory dqj/dqi not zero: one firm does react to
what other firms (might) do
• Curiousity point: model predates neoclassical economics!
– French mathematician Augustin Cournot derived
mathematical model of how two competing spas might
price of access to spa in 1838!
• Neoclassical school didn’t begin till 1870s
– But one founder (Walras) corresponded with Cournot
Cournot-Game Theoretic Competition Models
• Cournot concluded
– Duopoly price lower than price if spas colluded
– Price converges from monopoly level to perfect
competition as number of competitors increases
• Cournot’s analysis: firms reach “Cournot equilibrium”
levels of output by independently pursuing “profit
maximising” strategies…
– Set “own-output” marginal revenue equal to marginal
cost
– NOT an interactive, strategic explanation—but
reaches same result as later game theory approach of
John von Neumann & “beautiful minds” John Nash
• with a little problem we’ll discuss after trudging through
the math…
Cournot on Monopoly & Oligopoly
• Cournot’s analysis: assume linear demand & cost functions
– (original paper actually assumed zero costs!)
P Q   a  bQ
C Q   c  dQ
• Profit=Revenue minus Cost:  Q   P Q Q  C Q 
• Maximum profit when gap between functions biggest
• Biggest gap when differential w.r.t. Q is zero:
d  Q 
max Q  :
0
dQ
d  Q 
 P Q   Q  P 'Q   MC Q   0
dx
d  Q 
  a  bQ   Q   b   d  0 Add these…
dx
Cournot on Monopoly & Oligopoly
• Monopoly profit-maximises by producing Q such that
a  2bQ   d
1 a d
QM 
2 b
• Cournot applied same analysis to two firms in duopoly
• Argued profit-maximise by equating marginal revenue
and marginal cost w.r.t. own outputs (q1 & q2)
• Each firm in a duopoly wants to maximise
1 q1   P Q  q1  C q1 
2 q2   P Q  q2  C q2 
where Q=q1+q2
• “Therefore” each firm in a duopoly solves
d  1 q1 
dq1
d  2 q2 
dq2
0
0
Cournot on Monopoly & Oligopoly
• Assume identical costs C(x)=c+dx
• Working out quantity for firm 1:
d 1 q1 
dq1
 a  b q1  q2   q1  b   d  0
b 2q1  q2   a  d
2bq1  bq2  a  d
a  d  bq2
q1 
2b
• Get symmetrical result for second firm:
a  d  bq1
q2 
2b
Cournot on Monopoly & Oligopoly
• To solve, substitute value for q2 into equation for q1:
 a  d  bq1 
a d b 
Cancel these…

a  d  bq2
2b


q1 

Multiply across
2b
2b
 a  d  bq1 
Multiply all terms
2bq1  a  d  

2


4bq1  2a  2d  a  d  bq1  Subtract
3bq1  a  d
a d
a d
q1 
Ditto for q2: q2 
3b
3b
2 a d
so QD 
3 b
Cournot on Monopoly & Oligopoly
• General rule is
– The more firms the higher the output & lower the
price
– If n is the number of firms in the industry, then
1 a d
qn 
n 1 b
n a d
Qn 
n 1 b
Output converges to perfect
competition level as n
• Perfect competition: price equals marginal cost:
P Q   a  bQ  MC  d
a d
Q
b
• But there’s a problem…
Cournot on Monopoly & Oligopoly
• Equating MC and MR not profit maximising behavior!
– Firm’s revenue a function not only of what it does, but
what other firms do (especially since can’t control
what they do!)
– So need to find maximum gap not w.r.t own output, but
w.r.t. industry output
d
d
  q1   not   q1  
• This is total derivative:
dQ
dq1
• What happens when firm really does maximise profits by
setting total derivative equal to zero?
d
d
d
 q1   
 q1  
 q1   0

dQ
d q1
d q2
This is MR-MC
This is impact of 2nd
firm on profit of 1st:
ignored by Cournot
Cournot on Monopoly & Oligopoly
• The real profit maximisation point for firm 1 is where
this equals zero:
d
d
 q1  
 q1   0
d q1
d q2
This we’ve already
worked out
d 1 q1 
dq1
What is this bit?
d
d
d
 q1  
P
Q

q

C q1 
  1

d q2
d q2
d q2
 a  b q1  q2   q1  b   d
This expands to This is zero
d
d
q1 
P Q    P Q  
q1 

d q2
d q2
For profit maximisation for firm 1,
the sum of these must equal zero: This equals q1  b
Cournot on Monopoly & Oligopoly
• Adding them together:
a  b q1  q2   q1  b   d  q1  b  0
a d
Adding these three together 3q1  q2 
b
a d
Symmetrical result for q2:
3q2 
 q1
b
1 a d
 a d
Substituting: 3q1  
 q1  
Multiply by 3:
3 b
b

a d
a d
a d
Cancel terms 8q1  2
9q1 
 q1  3
b
b
b
1 a d
1 a d
Ditto for q2: q2 
q1 
4 b
4 b
Cournot on Monopoly & Oligopoly
• So total output is:
1 a d 1 a d
1 a d
q1  q2 


4 b
4 b
2 b
• Oh dear… same as for monopoly!
• General formula is:
1 1 a d
qn  
n 2 b
1 a d
Qn 
2 b
• Oh dear… same as for monopoly again!
• Output independent of number of firms in industry…
• So if firms are profit maximisers, competition
makes no difference.
Cournot on Monopoly & Oligopoly
• Cournot oligopoly analysis therefore flawed
• But game theoretic version (which reaches same result) is
not
– Implicitly assumes strategic interactions force firms
not to profit-maximise but to produce more than
profit-maximising ouput…
• Based on analogy to “police interrogation trick”
– The “prisoners’ dilemma”…
Von Neumann: Games & Strategic Behavior
• Two people have
– Committed a crime together
– Been captured
– Suspected but only weak evidence
– Police offer deal to each:
• “rat on the other and we’ll let you go free”
• “if you don’t, you’ll get a year anyway”
– Prisoners face trade-off:
• Keep quiet and go to gaol for a year
• Rat on the other and go free
– But accomplice gets 10 years
• If both rat, the sentence is split 50:50
Von Neumann: Games & Strategic Behavior
• Prisoners’ Dilemma trade-off table:
Prisoner B
Prisoners’ Dilemma
“Truth table”
Defect
Cooperate
Prisoner A
Defect
Cooperate
A goes free/ B
gets 10 years
5 years each
B goes free/ A
gets 10 years
1 year each
• Dilemma:
– If both keep mum, 1 year each
– If one rats, gets off scot-free
– If not and other does, could get ten years
• Analysis:
– Both have incentive to “rat”; both get 5 years
Von Neumann: Games & Strategic Behavior
• Same analysis applied by analogy to duopolists:
– Maximum equilibrium profit if both follow “Keen”
profit-maximisation strategy (called “collusive” in
economic literature, “cooperate” in game lit.)
– If one “defects” to MC=MR strategy while other
sticks with profit-maximisation, gets:
• Large increase in sales ( 1/3 – 1/4 =1/12th increase)
• Smaller fall in price
– Reduces profit of other firm too
– So both have incentive to “defect”
• Profit-maximisation equilibrium “unstable”
– Each gains if “defects”
• Defect/Defect combination a “Nash equilibrium”
– Higher output/Lower profit outcome of strategic
interaction
Von Neumann: Games & Strategic Behavior
• With any price/cost combination
– Profit maximising strategy (MR-MC=[n-1]/n * [P-MC])
best for both
– “Defect” (increase output to where MR=MC) better
for one if other doesn’t change output
– MC=MR strategy “best response” of initial nondefector to defection
Firm 2
Duopoly
Dilemma
Firm 1
MC=MR
MC=MR
Profit
Max
q1  q2 
q1 
1 a d
3 b
1 a d
1 a d
, q2 
3 b
4 b
• E.g.: P(Q)=100- 1/100000000 Q
• C(q) = 100000000 + 50 q
Profit Max
q1 
1 a d
1 a d
, q2 
,
4 b
3 b
q1  q2 
1 a d
4 b
Von Neumann: Games & Strategic Behavior
• Firstly, quantities:
Firm 2
Duopoly
Dilemma
Firm 1
MC=MR
Profit Max
1666666667
MC=MR 1666666667
Profit
Max
3333333333
1250000000
1666666667
2916666667
1666666667
1250000000
2916666667
1250000000
1250000000
2500000000
• Then the price
Firm 2
Duopoly
Dilemma
Firm 1
MC=MR
Profit Max
MC=MR
67
71
Profit
Max
71
75
Von Neumann: Games & Strategic Behavior
• Finally profits
Firm 2
Duopoly
Dilemma
Firm 1
MC=MR
Profit Max
MC=MR
27677777778
27677777778
55355555556
25941666667
34622222222
60563888889
Profit
Max
34622222222
25941666667
60563888889
31150000000
31150000000
62300000000
• Aggregate profits higher if both profit maximise
• But both have incentive to increase output if other
doesn’t do so
• So both increase and get higher output, lower profit with
some output sold at a loss…
Von Neumann: Games & Strategic Behavior
• So game theory gives valid justification for Cournot
outcome: convergence to PC as number of firms rises
– Firm’s quantity not decided in isolation but with
respect to expected strategies of competitors
– Strategic interaction force non-profit-maximising
outcome on each firm
– Extrapolated to many firms, result gives convergence
to perfect competition
• Non-profit-maximising “strategy” of setting MR=MC
converges to P=MC as number of firms rises
• But there’s a problem (or four)…
Von Neumann: Games & Strategic Behavior
• “Nash equilibrium” unstable
– Both firms have incentive to reduce production
• “Prisoners’ Dilemma” solution also breaks down
– If firms don’t have complete information
– If game played more than once; and
• Experimental outcomes contradict theory
– Artificial firms
• Don’t follow Nash strategy in actual interactions
• Get higher profits by deviating from it
Von Neumann: Games & Strategic Behavior
• “Defect/Defect” strategy unstable
– One firm can increase profits if reduces output to
“Keen” level while other stays at “Cournot” level
• Both firms have incentive to increase production
Firm 2
Duopoly
Dilemma
Firm 1
MC=MR
Profit Max
MC=MR
27677777778
27677777778
55355555556
25941666667
34622222222
60563888889
Profit
Max
34622222222
25941666667
60563888889
31150000000
31150000000
62300000000
• More on instability later…
Von Neumann: Games & Strategic Behavior
• Information
– In true Prisoners’ Dilemma:
• Both suspects know they are guilty
• Both know that the other one knows
• Both know the consequences of ratting or not ratting
– Assumption of “perfect information” (except for
knowing whether accomplice will rat) justified
– In true industrial competition
• No firm knows any other firms costs
• No firm knows any other firms sales
– Assumption of perfect information unjustified
– Experiments with artificial firms result in no
equilibrium, or non-predicted equilibrium, once
information reduced from “perfect”:
Von Neumann: Games & Strategic Behavior
• “Multiple equilibria are the norm in multi-period games of
incomplete information, and Folk Theorems indicate that
a small amount of incomplete information can produce
almost any equilibrium payoffs when the discount rate
is low and the horizon is long.” Richard Schmalensee,
(1988), “Industrial Economics: An Overview”, Economic
Journal, 98, p. 447
• And repeated games…
– If prisoners know they will be made the same offer
many times, sensible thing is not to rat ever…
– If rat this time, one can “punish” other by ratting next
time—so best strategy is to keep quiet
– Ditto repeated “artificial economy” simulations:
• “Firms” converge to “cooperate” strategy and make more
profits as result
Von Neumann: Games & Strategic Behavior
• Some economists try to sidestep problem by “backward iteration”:
– “We know that if … the stage game is only played once … the
potential for cooperation is quite slim: playing D is a strictly
dominant strategy for each player…
– An extension of this logic suggests that we should not expect to
see cooperative behavior if the game is played twice, or 10 times,
or even 100 times. To see this, apply backward induction: in the
final game … playing D is a strictly dominant strategy for both
players, so both players should expect the outcome (D, D) in the
final game. We now look at the penultimate (second-to-last) game:
knowing that (D, D) will be the outcome in the final game, playing
D becomes a strictly dominant strategy in the penultimate game
as well! We continue working backwards all the way to the
beginning of the game, and conclude that the only rational
prediction is for the players to play (D, D) in each of the stage
games. So we cannot expect any cooperative behavior…” (Yoram
Bauman 2004, Quantum Microeconomics, p. 98)
Von Neumann: Games & Strategic Behavior
• But this depends on beliefs of other player’s behaviour:
– “Consider the prisoners' dilemma… Players believe that
the opponent cooperates in the first period. From
period 2 until period m-3, the opponent cooperates if
both players have always cooperated. Otherwise the
opponent defects. If both players have cooperated in
period m-3, then, in period m-2, the opponent
cooperates or defects with equal probability.
Otherwise the opponent defects. In the last two
periods, the opponent defects. Both players' best
response is to cooperate until period m-3 (included),
and to defect in the last three periods.” (Alvaro
Sandroni, 1998. “Does Rational Learning Lead to Nash
Equilibrium in Finitely Repeated Games?” Journal of
Economic Theory (78), p. 200)
Von Neumann: Games & Strategic Behavior
• “Much interest in recent years has attached to repeated
games … in which a relatively simple constituent or stage
game (such as the one-period Cournot or Prisoners’
Dilemma games) is played repeatedly by a fixed set of
players. Strategies of simply playing Nash equilibrium
strategies of the constituent game in each period form a
Nash equilibrium of the repeated game. But strategies in
the repeated game may involve taking actions conditional
on past history, and there are usually many other
equilibria when the horizon is infinite. In fact, many of
the main results in the supergame literature are variants
on the so-called Folk Theorem, which says that virtually
any set of payoffs can arise in a perfect equilibrium if
the horizon is long enough and the discount rate is low
enough.” Schmalensee p. 647.
Von Neumann: Games & Strategic Behavior
• So game theory looks like a sound way to justify perfect
competition…
– Until you go beyond one-off game
• “Perfect competition” only applies for “single shot” game
– Recommended economic strategy is “always defect”
• But “always defect” just one of many possible strategies
in repeated games
• Others include “tit for tat”:
– cooperate on first play
– cooperate if opponent cooperates
– if opponent defects, defect yourself next time:
“punish” non-cooperation
– Tends to win game contests
Economic modelling
• What is game theory supposed to do?
• Make economic theory more realistic by considering
interaction between agents
– Marshallian theory assumes “atomism”
• No reaction to what other firms might do
– Game theory allows for reactions to what other firms
might do
• While still assuming rational behavior
• Both Marshallian & game theory not supposed to be what
firms actually do;
– Instead, supposed to model “as if” behavior
– “Whatever they say they’re doing, they must really be
doing this if they are to be successful…”
– Friedman’s argument:
Economic modelling
• “Consider the problem of predicting the shots made by an
expert billiard player… excellent predictions would be
yielded by the hypothesis that the billiard player made
his shots as if he knew the complicated mathematical
formulas that would give the optimum directions of
travel, …, could make lightning calculations from the
formulas, and could then make the balls travel in the
direction indicated by the formulas. Our confidence in
this hypothesis is not based on the belief that billiard
players, even expert ones, can or do go through the
process described; it derives rather from the belief
that, unless in some way or other they were capable of
reaching essentially the same result, they would not in
fact be expert billiard players.” (Friedman 1953 “The
Methodology of Positive Economics”)
Economic modelling
• “It is only a short step from these examples to the economic
hypothesis that under a wide range of circumstances individual firms
behave as if they were seeking rationally to maximize ... "profits" ...
as if, that is, they knew the relevant cost and demand functions,
calculated marginal cost and marginal revenue from all actions open
to them, and pushed each line of action to the point at which the
relevant marginal cost and marginal revenue were equal. Now, of
course, businessmen do not actually and literally solve the system of
simultaneous equations ... The billiard player, if asked how he decides
where to hit the ball, may say that he "just figures it out" … the
businessman may well say that he prices at average cost, with of
course some minor deviations when the market makes it necessary.
The one statement is about as helpful as the other, and neither is a
relevant test of the associated hypothesis.” (Friedman 1953)
Multi-agent modelling
• Friedman’s purpose in “The Methodology of Positive
Economics”: to stop economists looking at empirical
research (Blinder’s predecessors)
– Economists prefer to try to prove what firms do
rather than find out…
• But fail… proofs either wrong (Marshallian) or
inconclusive, not robust as realism increases (game
theory)
– Computers offer new approach between proof and
questionnaires: simulate & see what happens
• Model large populations using computer simulations of
agents
• Carefully design behavior; see what well-defined agents
do
Multi-agent modelling
• Rather than predicting what profit-maximising firms will
do, let’s “find out” with simulation
– Define profit maximisers in terms of behavior rather
than calculus
• “instrumental profit maximisers”
– Try something (e.g., increase output)
– If profit increases, do same again
– If profit falls, reduce output
– Model single market, demand curve
– No assumptions about knowing/applying calculus,
etc.
• Just computer programming:
– Give computer precise instructions
– See what happens!
Multi-agent modelling
• Basic idea:
– Define demand curve & cost functions
– Create random list of initial outputs for n firms
– Work out initial price given sum of outputs
– Create random list of variations in output for n firms
– FOR a number of iterations
• Add variation to output of each firm
• Work out new price level
– FOR each firm
• Work out whether profit has risen or fallen
• IF rose, keep going the same way; ELSE
• IF fell, reverse direction
– See whether output converges to Cournot or “Keen”
prediction
• Trying it out with sample demand curve & ten firms…
Multi-agent modelling
• P(Q)=100- 1/100000000 Q
• C(q) = 100000000
+ 50 q
1
a  100 b 
100000000
P ( Q)  a  b  Q
MR ( Q) 
d
dQ
( P ( Q)  Q) MC  d
Demand Curve
Market Price
100
P ( Q)
MR ( Q)
• Cournot/Game theory
prediction: firms
equate MR & MC
Price
Marginal Revenue
Marginal Cost
50
MC
0
0
2 10
9
4 10
9
6 10
9
8 10
9
1 10
10
1 a d
qn 
n 1 b
Q
Industry Output
• “Keen” profit maximisation
prediction: firms produce where
MR–MC equals (n-1)/n times P-MC
1 1 a d
qn  
n 2 b
Multi-agent modelling
• Start with randomly allocated list of outputs by ten firms

• Random initial quantity
between Cournot & Keen
predictions for each firm
Q0 
• Initial outputs
• Next step: work out
initial profits:


Profit0  P 
Q0   Q0  TC  Q0


Profit0 
0
1
2
3
4
5
6
7
8
9

Q0  round runif Firms  qK ( Firms)  qC ( Firms)
0
4.365·10 8
3.988·10 8
3.866·10 8
3.144·10 8
3.126·10 8
2.722·10 8
4.241·10 8
2.817·10 8
2.662·10 8
3.811·10 8
0
1
2
3
4
5
6
7
8
9
QC ( Firms)  4.545  10
QK ( Firms)  2.5  10

Q0  3.474  10
8
x  0  10  10  10
9
9
9
9
Demand Curve
100
Demand curve
Marginal Revenue

0
6.559·10 9
5.985·10 9
5.799·10 9
4.698·10 9
4.669·10 9
4.053·10 9
6.371·10 9
4.198·10 9
3.962·10 9
5.715·10 9

Marginal Cost
Initial output level
50
Cournot prediction

P

0
0

Keen prediction

Q0   65.258

2 10
9
4 10
9
6 10
9
8 10
9
1 10
10
Multi-agent modelling
• Work out vector of changes in output (much smaller
amounts than the initial output so that firms won't end up
“producing” negative amounts)
• 6 firms reduce output and 4 increase:


qC ( Firms)  



dq  round rnorm Firms  0 



Firms  


dq 
0
1
2
3
4
5
6
7
8
9
0
-6.31·10 7
-9.766·10 7
-6.803·10 7
-1.368·10 8
-2.423·10 8
6.257·10 6
-1.734·10 7
7.998·10 7
3.15·10 8
1.162·10 8
dq  1.077  10
dq  Q


Q  augment Q0  Q0  dq
8
1
Q 
0  3.099 %
0
1
2
3
4
5
6
7
8
9
• Aggregate output drops a bit:
• Next what are new profit levels?
0
3.734·10 8
3.012·10 8
3.186·10 8
1.777·10 8
7.025·10 7
2.785·10 8
4.068·10 8
3.617·10 8
5.813·10 8
4.974·10 8
0
Q

0
1
2
3
4
5
6
7
8
9
0
4.365·10 8
3.988·10 8
3.866·10 8
3.144·10 8
3.126·10 8
2.722·10 8
4.241·10 8
2.817·10 8
2.662·10 8
3.811·10 8
Multi-agent modelling
• Curiousity point: some firms lose profit by reducing
output; others increase!

P


1 
Q   66.334


Profit1  P 

Profit1 
0
1
2
3
4
5
6
7
8
9

1  1
 1 
Q   Q  TC  Q 

0
5.999·10 9
4.82·10 9
5.104·10 9
2.802·10 9
1.048·10 9
4.449·10 9
6.544·10 9
5.808·10 9
9.395·10 9
8.024·10 9
Profit1  Profit0 
0
1
2
3
4
5
6
7
8
9
0
-5.608·10 8
-1.166·10 9
-6.95·10 8
-1.895·10 9
-3.621·10 9
3.953·10 8
1.733·10 8
1.61·10 9
5.433·10 9
2.309·10 9
dq 
0
1
2
3
4
5
6
7
8
9
0
-6.31·10 7
-9.766·10 7
-6.803·10 7
-1.368·10 8
-2.423·10 8
6.257·10 6
-1.734·10 7
7.998·10 7
3.15·10 8
1.162·10 8
• Firms 0-4 decreased output & saw profit fall
• Firm 6 decreased output & saw profit rise
• Cause: elasticity interactions between size of aggregate
price change & size of individual output change
Multi-agent modelling
• Firms 0-4 saw profit fall, so they will alter the direction
of their output changes
• Firms 4-9 increased profit, so they continue in the same
direction
• If profit rose, this function returns 1; if it fell -1:
 
sign P 
 

1   1 

Q    Q   P



0   0    1    0  
Q    Q   TC  Q    TC  Q   



0
1
2
3
4
5
6
7
8
9
0
-1
-1
-1
-1
-1
1
1
1
1
1
• This is multiplied by the dq amounts and added to second
period output to work out third period…
Multi-agent modelling
• The entire program:
Cts ( f  r  s) 
Seed ( s)
Random number generator


Q0  round runif f  qK ( f)  qC ( f)

Random initial outputs
Random change amounts


qC ( f)  



dq  round rnorm f  0 



f 


For r iterations
for i  1  r



PThen  P 

Calculate market price
Qi 1 

Change outputs
Qi  Qi1  dq



PNow  P 
Calculate new market price

Qi 

For each firm
for j  0  f  1
  j  PThen  Qi1 j  tc Qi j  tc Qi1 j  dq j
dq j  sign PNow  Qi

Q
f  400
r  250
Change direction if profit has fallen
Output  Cts ( f  r  10)
Multi-agent modelling
• Result for 400 firms:
5 10
i  0  r  1
9
Aggregate Output
Cournot
9
4.5 10
Quantity
Output  Cts ( f  r  10)
r  250
f  400
4 10
9
3.5 10
9
3 10
9
2.5 10
9
Keen
0
50
100
150
Iterations
200
250
• Converges towards Keen rather than Cournot…
• BUT doesn’t quite reach it; apparent complex interaction
effects between firms…
Multi-agent modelling
• But not ones predicted by game theory
• Reason: local instability of Cournot equilibrium, stability
of Keen
– If duopoly begins at Cournot or Keen level & firms
change output seeking higher profit, 1st mover will gain
at expense of 2nd
– 2nd will change behavior, affecting 1st mover
– Sustainable change only if both increase profit
– Local region around equilibria determines what happens
next
• Region around Cournot unstable: both firms gain if both
reduce output
• Region around Keen stable: no sustainable pattern of
output change possible:
Multi-agent modelling
• Cournot near-equilibrium dynamics
3
2
1
0
1
2
3
3
True
True
True
True
True
True
True
False
True
False
True
False
True
False
2
True
True
True
True
True
True
True
False
True
False
True
False
True
False
1
True
True
True
True
True
True
True
False
True
False
True
False
True
False
0
False
True
False
True
False
True
True
True
False
False
False
False
False
False
1
False
True
False
True
False
True
False
False
False
False
False
False
False
False
2
False
True
False
True
False
True
False
False
False
False
False
False
False
False
3
False
False
True
False
True
False
True
False
True
False
True
False
True
False
2
False
True
False
False
True
False
True
False
True
False
True
False
True
False
1
False
True
False
True
True
True
True
False
True
False
True
False
True
False
0
False
True
False
True
False
True
True
True
True
False
True
False
True
False
1
False
True
False
True
False
True
False
True
True
True
True
False
True
False
2
False
True
False
True
False
True
False
True
False
True
True
True
True
False
• Keen near-equilibrium dynamics
3
2
1
0
1
2
3
3
False
True
False
True
False
True
False
False
False
False
False
False
False
False
3
False
True
False
True
False
True
False
True
False
True
False
True
False
False
• “True” means
profit rise
from change in
output
• Only
sustainable
pattern where
both firms
have “True”
result
• Sustainable
change from
Cournot, not
from Keen…
Multi-agent modelling
• Outputs from 3 sample firms
k  round ( runif ( 3  0  Firms  1) )
2 10
Outputs of 3 randomly chosen firms
7
Firm 1
Firm 2
Output
1.5 10
7
Firm 3
Cournot
Keen
1 10
7
5 10
6
0
0
50
100
Iteration
• Many varying individual behaviors…
150
200
250
Multi-agent modelling
• Multi-agent results show
– Generally Keen formula better predictor of behavior
than Cournot
– Increasing number of firms reduces likelihood of
Cournot outcome
– Model still makes false assumption of rising marginal
cost
– But shows new analytic approach:
• Accurately define conditions of economy
• Simulate behavior to see whether theories describe
reality well
• Next week: sophisticated multi-agent model of
innovation & competition