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Duopoly and Game Theory: The Prisoners’ Dilemma and the Nash Equilibrium

Previously: some duopoly models led to eventual outcomes not the best for both firms.

Bertrand model, they get the worst result (same as perfect competition);

Cournot model: “in-between” result (but not as good as the monopoly outcome).

Cartel result (duplicate monopoly, and share monopoly profits)

- produced less than the Cournot equilibrium (i.e. done less work)

- charged a higher price than the Cournot equilibrium, and

- made a higher super-profit, equally sharing the monopoly profit.

Key fact: “the two firms got together” to agree on a joint strategy.

General question: if competing duopoly firms cannot get together (for whatever reason), is it still possible to arrive at some kind of “optimal” solution (Nash Equilibrium) which is better for both of them than the “worst case” scenario?

If they can, then there is a “Nash Equilibrium” available.

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eg Vodaphone and Digicel can compete on prices, or not.

In the pay-offs, (x,y): x represents the profit to Vodaphone (in $m)

y represents the profit to Digicel (in $m)

Explain the pay-offs in the 4 squares.

Intuitively- the best option would be the purple square: total profits = 10 equally shared.

But that is not the “Nash Equilibrium” ? Where each person’s “Best Response” is also the other person’s “Best Response”.

Profit matrix    Digicel

   Compete on

pricesDon’t Compete

on prices

 Vodaphone

Compete on prices (3,3) = 6 total (7,1) = 8 total 

Don’t Compete on prices (1,7) = 8 total  (5,5) = 10 total 

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The Prisoners’ Dilemma problem (in the Fiji context):

Take time to understand this, as the eventual solution does not seem to be particularly desirable or rational at first glance, but it is.

The police pick up Abdul and Bale (armed with screwdrivers) at 3 in the morning within a fenced private car-park of a bank.

Charged with intention to rob the bank. Typical sentence if convicted: 20 years.

But there is no direct evidence that they were going to rob the bank. The screwdriver? “Oh, officer, we were just cleaning our fingernails with it”. At 3 am??? Yeah.

But if found guilty of only trespassing, the gaol sentence is only 3 years.

The police would prefer a conviction for attempted bank robbery: looks better in their statistics than a conviction for trespassing.

How can the police encourage prisoners to confess?

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Police isolate each prisoner (no communications allowed)

Abdul is told: Listen mate- you can keep quiet, and if your friend also keeps quiet- you will both be convicted of “Trespassing” and we will give each of you 3 years in gaol.

But if you keep quiet, and Bale confesses and blames you for leading him astray - you are going to get 20 years in gaol, - and we will let him off with only 1 year in gaol.

Look, you seem like a nice guy. We will give you a deal: you confess, you implicate Bale, Bale will probably keep his trap shut, we will put Bale behind bars for 20 years, and reward you with only 1 year in gaol.

And (yawn yawn) if both of you confess to Attempted Robbery (and make it easy for us), you will both get 5 years in gaol.

Take your pick mate.

But honestly, remember: there is no honour among thieves. Look after yourself, mate.

Do you think Bale will care about you?

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And Jone is given the same options

Listen mate- you can keep quiet, and if your friend also keeps quiet- you will both be convicted of “Trespassing” and we will give each of you 3 years in gaol.

But if you keep quiet, and Abdul confesses and blames you for leading him astray - you are going to get 20 years in gaol, - and we will let him off with only 1 year in gaol.

Look, you seem like a nice guy. We will give you a deal: you confess, you implicate Abdul, he will probably keep his trap shut, we will put Abdul away for 20 years, and reward you with only 1 year in gaol.

And (yawn yawn) if both of you confess to Attempted Robbery (and make it easy for us), you will both get 5 years in gaol.

Take your pick mate.

But honestly, remember: there is no honour among thieves. Look after yourself, mate.

Do you think Abdul will care about you?

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What will Abdul and Jone do?

They both know that they are being given the same options.

Abdul and Bale are both 35 years old, not married yet, no children,

Unlike developed countries, Fiji prisons do not have “visitation rights” from girl-friends.

Twenty years behind bars is a long long long time.

By the time they come out they will be 55- no girl will look at a 55 year old ex-convict, who cannot even get a job in government now.

And life will have gone by.

Do they trust each other enough to choose the strategy that will benefit them both?

Is there an “optimal strategy” for each of them?

This is the central problem that the “Prisoners’ Dilemma” game tries to answer.

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Note: if Abdul chooses an option- that alone does not decide the result

The result depends on what both Abdul and Bale chooses. What are the possibilities?

Each of then can choose either “Confess” or be “Silent”: only 4 possible combinations. Neat graphical way of giving all the possible outcomes is through the table below:

For instance, if Abdul Confesses, (red box) what can Bale do?

If Bale also confesses, the “payoff” for both can be put into the the blue square

If Abdul confesses but Bale stays Silent, the “payoff” can be written in the pink square

    Bale 

    Confess Silent

 Abdul

Confess  ”Pay-off”   ”Pay-off”

Silent    

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How write the pay-offs?

If Abdul confesses, and Bale confesses - they both get 5 years in gaol.

Write the “payoff: as (5,5)

With the first number for Abdul and the second number for Bale.

    Bale 

    Confess Silent

 Abdul

Confess (5,5)   

Silent    

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If Abdul confesses and Bale remains silent

Abdul gets only 1 year, while Bale gets 20 years.

The payoff is written in the top right hand corner as (1, 20).

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent    

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If Abdul chooses to stay silent, but Bale confesses

Abdul gets 20 years in gaol, while Bale will only get 1 year.

Write the payoff in the bottom left hand corner as (20, 1)

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)   

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And if both Abdul and Bale remain silent

They each get only 3 years

Write the payoff in the bottom right hand corner as (3,3).

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (3,3) 

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What would seem to be the best option for both?

If both remained silent (green square) they would get only 3 years each (total of 6 years in gaol). Whereas the other options give a total of 10 years in gaol and 21 years in gaol.

Surely, being silent (the cartel result- green cell) ought to be seen by both as the best option for both of them?

But this is “not the market result”/this is not a “Nash Equilibrium”. Why not?

Both cannot be sure that the other will stay silent. Both worry that the other might confess in order to reduce their own sentence. Both worry they might end up with the worst case scenario.

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (3,3) 

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Why is (Silent, Silent) not a Nash Equilibrium?

Suppose that Abdul looks at the possibility of himself staying silent.

The two options for Bale are now shaded green.

Abdul now thinks: what might Bale do, if he thinks that I am going to stay silent: ie what would be in Bale’s interest if he thinks that I am going to be silent?

If Bale also stays silent then they will get only 3 years each (bottom right hand cell).

But if Bale confesses, then he will only get 1 year in gaol, and I will get 20 years.

Oh no, Abdul thinks. I just cannot take the risk involved with being silent.

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (3,3) 

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If Abdul chooses to Confess, what is the worst that can happen?

If Abdul chooses to confess, the options for Bale are in pink.

If Bale also confesses, then the worst that can happen to Abdul is he gets 5 years (while Bale also gets 5 years).

And who knows, if Bale stays silent, then the Police will give him 20 years, while, Abdul chuckles, he will get off with only 1 year!

So for Abdul, “Confess” is the “Best Response”- seems much safer than being “Silent”.

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (3,3) 

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Suppose that Bale also thinks about being silent

He wonders what might Abdul do? The two options for Abdul are in the pink column.

He thinks, yes, it is true that if Abdul also stays silent then they will get only 3 years each (bottom right hand cell).

But what if that tricky Abdul decides to confess, he will only get 1 year in gaol, and I will get 20 years.

Oh no. Being silent is very dangerous.

On the other hand, if he “Confesses”

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (3,3) 

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If Bale thinks about confessing

What can Abdul do? The two options for Abdul are then in the pink column.

If Abdul also confesses, they would both get 5 years- not too bad!

And who knows (chuckle, chuckle) if Abdul stays silent, then he will get 20 years, while I will get away with 1 year only.

It is a lot safer for me to choose “Confess”- that is the “Best Response”.

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (2,2) 

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i.e. for both of them, the “Best Response” is “Confess”

That means that the likely outcome is the top left hand corner where both get 5 years each- a total of 10 years- the “Nash equilibrium”

The “Nash Equilibrium” is the outcome where each person’s “Best Response” is also the other person’s “Best Response”.

This is a “stable equilibrium”

Even though it is worse than the option where both stay silent (3 years each and total of 6 years in gaol).

    Bale 

    Confess Silent

 Abdul

Confess (5,5)  (1,20) 

Silent (20,1)  (3,3) 

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Question: does this also involve an “attitude” towards risk?

Both Abdul and Bale choose “confess” as the Best Response- because it seems safer, or less risky.

Both reject “Silent” because of the potentially higher costs should the other person choose to confess.

Both are “risk averters”.

What if both Abdul and Bale were “risk takers”?

What if both choose “Being Silent” in the “hope” that the other also does so.

Both will get off with just 3 years each.

Better than the 5 years if they both Confess.

ie the outcome depends also on attitudes towards risk.

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Explain the following pay-off matrix for Voda and Diji

Explain the pay-offs in the 4 squares.

Which is the “Nash Equilibrium” ? Where each person’s “Best Response” is also the other person’s “Best Response”.

Profit matrix    Digicel

   Compete on

pricesDon’t Compete

on prices

 Vodaphone

Compete on prices (3,3)  (7,1) 

Don’t Compete on prices (1,7)  (5,5) 

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Compare with pricing options for Digicel and Vodaphone

Compare and contrast the two scenarios:

1. At the beginning of the battle when Digicel has an unacceptable share of the market.

2. When both have reached some “acceptable” balance in their shares of the market.

What are the best strategies for Digicel and Vodaphone

whether or not to use price competition

whether or not to use other avenues such as advertising, product differentiation etc

What does each firm stand to gain in terms of market share and profits

What does each firm stand to lose, in terms of market share and profits

.

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There are many other Game Theory models of great application to duopolies and cartels

Have a read in your text-books.