<p>This is how I would try to model this. It needs some more detail, but I think this is the basic gist of it.</p>

<p>You need to allow firms to imperfectly observe other firms' prices. One way that I would do this is to assign some probability to the event that any given firms price is observed. Say, each firm flips a coin and if its head, the firm must reveal its price. Now, assume that the probability of a firms price being revealed is inversely proportional to the number of firms in the market. When the probability of having your price revealed becomes lower, a firm figures that it has a better chance of "cheating" the cartel agreement. Everybody knows this in a symmetric game. So if one firm thinks the other firm has a better chance of getting away with cheating, his/her best response is to also cheat. So, when the number of firms increases, the incentive for each firm to cheat gets larger and larger.</p>

<p>Just to note, I think Stigler has a paper ("A Theory Oligopoly") that outlines a model that gives an opposite result.</p>

CC BY-SA 3.0
2014-11-18T23:17:22.200Z
2014-11-18T23:17:22.200Z
jmbejara
<p>I think the question wishes you to refer to the so called "Bertrand Paradox" - the term Bertrand competition refers to price competition (i.e. firms compete by choosing prices, as opposed e.g. to quantities in so called "Cournot competition"). In the simplest case, with constant marginal costs equal to c, say, a single firm will set the monopoly price. Now if you consider the case with two firms competing on prices, with the same constant marginal costs and under the assumption that prices are measured on the real line, it is easy to show that there is a unique Nash equilibrium in which both firms (whose strategy consists in choosing a price) will charge a price equal to their marginal cost - that is, by adding a single firm you go from monopoly pricing to marginal cost pricing. </p>

<p>This is the simplest answer to your question that I can think of - now confess you were trying to solve an undergraduate assignment.... ;-)</p>

<p>p.s.
Osborne's ug Game Theory textbook is very clear on this, if you need to catch up with self study.</p>

CC BY-SA 3.0
2014-12-01T22:36:00.997Z
haha nearly, it was a postgrad industrial organisation exam question. Already sat. I thought it was an interesting question. That info was in the question prior to Foobars edit. The correct answer with respect to the course was very close to the one offered by @Lex.I was interested in other approaches as well.
Jamzy
CC BY-SA 3.0
2014-12-02T08:43:27.707Z
oh well, I gave you the intermediate micro answer :-)
loop
CC BY-SA 3.0
2014-12-01T22:28:56.137Z
2014-12-02T08:43:49.413Z
2014-12-02T08:43:49.413Z
loop
<p>Iet's say we have n identical firms and an infinite horizon of time.</p>

<p>The n firms sustaining the collusion, will find optimal to fix the same price $p_m$ where $p_m$ is the price of the monopoly level and we define $\frac{\Pi^m}{n}$  as the profits each firm is obtaining by sustaining the collusion in each moment t.</p>

<p>Now, of course each firm can betray the others by fixing a price lower than $p_m$, namely $p_m-ε$, where ε is small, and by doing so, the firm will capture the entire demand because in this market the firms are doing the Bertrand competition. In other words, the firm by betraying the others, will get almost π_m at the time T=t. We will also assume that in all t > T no firms will make any profits, because they will punish the firm, by fixing the price in Bertrand competition.</p>

<p>The firm will defect if:</p>

<p>$π_m/n + δπ_m/n + δ^2π_m/n.... &lt; π_m+0+0....$</p>

<p>Where δ is the discount factor.</p>

<p>This can rewritten as:</p>

<p>$(\frac{π_m}{n})(\frac{1}{(1-δ)}) &lt; π_m$</p>

<p>We can now see that if n, the number of firms, increases then the profits by sustaining the collusion will decrease, so the above inequality will be more likely to be true. This means that a firm has less incentives to sustain a collusion when there are too many participants, because the profits will be divided among too many firms and the punishment will be seen as less heavy.</p>

CC BY-SA 3.0
2014-12-11T22:35:21.847Z
It's worth noting that what is important here is not the number of firms per se, but rather the size of the smallest firm. One would get the same result if there were two firms with market shares $1/n$ and $(n-1)/n$ respectively.
Ubiquitous
CC BY-SA 3.0
2014-11-18T23:43:11.710Z
+1, was about to to write an answer precisely along these lines when you answer popped up. Do you mean "in all t>T" rather than in "t>0"? Also, shouldn't your condition for defection be (π_m/n + δπ_m/n + δ^2π_m/n + ...) = (π_m/n)*(1/(1-δ)) < π_m"?
Martin Van der Linden
CC BY-SA 3.0
2014-11-19T00:00:33.293Z
I edited my answer, it should be ok now.
Lex
CC BY-SA 3.0
2014-11-19T02:11:12.073Z
Yes.  Nearly identically what I got. Only addition I would have is to add a minimum delta value which would sustain collusion. In order to do that,  more needs to be said about the demand function.
Jamzy
CC BY-SA 3.0
2014-11-19T05:22:08.083Z
much clearer with your edits, thanks. If you have time, you may want to re-edit your question using mathjax now that it is available on this SE.
Martin Van der Linden
CC BY-SA 3.0
2014-11-19T10:36:02.820Z
Thank you for your suggestions. Anyway I actually don't know what mathjax is
Lex
CC BY-SA 3.0
2014-11-18T23:28:59.287Z
2014-11-19T22:23:30.777Z
2014-11-19T22:23:30.777Z
Lex
<p>How would you answer the following question?</p>

<p>You work for a CEO of a large firm. He says to you, <strong>"In my experience collusion is less likely to be sustained as the number of firms in the market increases. Demonstrate this using a model of Bertrand Competition.</strong>"</p>

CC BY-SA 3.0
2014-12-02T10:05:01.363Z
CEOs of large companies are unlikely to use GT models.
Deer Hunter
<industrial-organisation><game-theory>
Collusion and number of firms
Jamzy
# Collusion and number of firms

(Asked by: username_0 on 2014-11-18)

How would you answer the following question?

You work for a CEO of a large firm. He says to you, **"In my experience collusion is less likely to be sustained as the number of firms in the market increases. Demonstrate this using a model of Bertrand Competition.**"

**Tags:** industrial-organisation, game-theory

> **Comments:**
> **username_1**: CEOs of large companies are unlikely to use GT models.  

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## 3 answers

### Answer by username_4 on 2014-11-18. Score: 9 | Accepted answer

Iet's say we have n identical firms and an infinite horizon of time.

The n firms sustaining the collusion, will find optimal to fix the same price $p\_m$ where $p\_m$ is the price of the monopoly level and we define $\frac{\Pi^m}{n}$ as the profits each firm is obtaining by sustaining the collusion in each moment t.

Now, of course each firm can betray the others by fixing a price lower than $p\_m$, namely $p\_m-ε$, where ε is small, and by doing so, the firm will capture the entire demand because in this market the firms are doing the Bertrand competition. In other words, the firm by betraying the others, will get almost π\_m at the time T=t. We will also assume that in all t > T no firms will make any profits, because they will punish the firm, by fixing the price in Bertrand competition.

The firm will defect if:

$π\_m/n + δπ\_m/n + δ^2π\_m/n.... < π\_m+0+0....$

Where δ is the discount factor.

This can rewritten as:

$(\frac{π\_m}{n})(\frac{1}{(1-δ)}) < π\_m$

We can now see that if n, the number of firms, increases then the profits by sustaining the collusion will decrease, so the above inequality will be more likely to be true. This means that a firm has less incentives to sustain a collusion when there are too many participants, because the profits will be divided among too many firms and the punishment will be seen as less heavy.

> **Comments:**
> **username_6**: +1, was about to to write an answer precisely along these lines when you answer popped up. Do you mean "in all t>T" rather than in "t>0"? Also, shouldn't your condition for defection be (π_m/n + δπ_m/n + δ^2π_m/n + ...) = (π_m/n)*(1/(1-δ)) < π_m"?  
> **username_4**: I edited my answer, it should be ok now.  
> **username_0**: Yes.  Nearly identically what I got. Only addition I would have is to add a minimum delta value which would sustain collusion. In order to do that,  more needs to be said about the demand function.  
> **username_6**: much clearer with your edits, thanks. If you have time, you may want to re-edit your question using mathjax now that it is available on this SE.  
> **username_4**: Thank you for your suggestions. Anyway I actually don't know what mathjax is  
> **username_5**: It's worth noting that what is important here is not the number of firms per se, but rather the size of the smallest firm. One would get the same result if there were two firms with market shares $1/n$ and $(n-1)/n$ respectively.  

---

### Answer by username_2 on 2014-11-18. Score: 4

This is how I would try to model this. It needs some more detail, but I think this is the basic gist of it.

You need to allow firms to imperfectly observe other firms' prices. One way that I would do this is to assign some probability to the event that any given firms price is observed. Say, each firm flips a coin and if its head, the firm must reveal its price. Now, assume that the probability of a firms price being revealed is inversely proportional to the number of firms in the market. When the probability of having your price revealed becomes lower, a firm figures that it has a better chance of "cheating" the cartel agreement. Everybody knows this in a symmetric game. So if one firm thinks the other firm has a better chance of getting away with cheating, his/her best response is to also cheat. So, when the number of firms increases, the incentive for each firm to cheat gets larger and larger.

Just to note, I think Stigler has a paper ("A Theory Oligopoly") that outlines a model that gives an opposite result.


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### Answer by username_3 on 2014-12-01. Score: 3

I think the question wishes you to refer to the so called "Bertrand Paradox" - the term Bertrand competition refers to price competition (i.e. firms compete by choosing prices, as opposed e.g. to quantities in so called "Cournot competition"). In the simplest case, with constant marginal costs equal to c, say, a single firm will set the monopoly price. Now if you consider the case with two firms competing on prices, with the same constant marginal costs and under the assumption that prices are measured on the real line, it is easy to show that there is a unique Nash equilibrium in which both firms (whose strategy consists in choosing a price) will charge a price equal to their marginal cost - that is, by adding a single firm you go from monopoly pricing to marginal cost pricing.

This is the simplest answer to your question that I can think of - now confess you were trying to solve an undergraduate assignment.... ;-)

p.s.
Osborne's ug Game Theory textbook is very clear on this, if you need to catch up with self study.

> **Comments:**
> **username_0**: haha nearly, it was a postgrad industrial organisation exam question. Already sat. I thought it was an interesting question. That info was in the question prior to Foobars edit. The correct answer with respect to the course was very close to the one offered by @username_4.I was interested in other approaches as well.  
> **username_3**: oh well, I gave you the intermediate micro answer :-)  

---

