Prisoner’s Dilemma

Game theory is the science of strategy. It attempts to determine mathematically and logically the actions that players should take to secure the best outcomes for themselves in a wide array of games. Prisoner’s Dilemma is the oldest and most studied model in game theory.

In the book  Game Theory 101 – William Spaniel gives the following scenario

Two thieves plan to rob an electronics store. As they approach the backdoor, the police arrest them for trespassing. The cops suspect that the pair planned to break in but lack the evidence to support such an accusation. They therefore require a confession to charge the suspects with the greater crime. Having studied game theory in college, the interrogator throws them into the prisoner’s dilemma. He individually sequesters both robbers and tells each of them the following:

We are currently charging you with trespassing, which implies a one month jail sentence. I know you were planning on robbing the store, but right now I cannot prove it – I need your testimony. In exchange for your cooperation, I will dismiss your trespassing charge, and your partner will be charged to the fullest extent of the law; a twelve month jail sentence. I am offering your partner the same deal. If both of you confess, your individual testimony is no longer as valuable, and your jail sentence will be eight months each.

Let us assume the 2 thieves are Andy and Bob. I have captured all the 4 outcomes in a tabular format. Make sure you really understand the contents of the table.

Bob
Quiet Confess
Andy Quiet -1, -1 -12, 0
Confess 0, -12 -8, -8

Let us now take a look at how Andy will think about this situation. He will clearly see that Bob can either be quiet or confess. If Bob is quiet then Andy has the following options.

Bob
Quiet
Andy Quiet -1, ?
Confess 0, ?

For deciding between the 2 choices Andy will only look at number of months he will go to jail and not care about how many months Bob will be in jail. Hence I have put a ? for Bob.

If Andy is quiet then he will go to jail for 1 month. This is represented by -1. If Andy confesses then he will be set free. This is represented by 0. Since 0 > -1 Andy will confess.

If Bob confess then Andy will have the following options.

Bob
Confess
Andy Quiet -12, ?
Confess -8, ?

If Andy is quiet then he will go to jail for 12 months. This is represented by -12. If Andy confesses then he will go to jail for 8 months. This is represented by -8. Since -8 > -12 Andy will confess.

Hence in both the cases Andy will confess as his outcome is better if he confesses.

Let us now take a look at how Bob will think about this situation. He will clearly see that Andy can either be quiet or confess. If Andy is quiet then Bob has the following options.

Bob
Quiet Confess
Andy Quiet ?, -1 ?, 0

If Bob is quiet then he will go to jail for 1 month. This is represented by -1. If Bob confesses then he will be set free. This is represented by 0. Since 0 > -1 Bob will confess.

If Andy confess then Bob will have the following options.

Bob
Quiet Confess
Andy Confess 0, -12 -8, -8

If Bob is quiet then he will go to jail for 12 months. This is represented by -12. If Bob confesses then he will go to jail for 8 months. This is represented by -8. Since -8 > -12 Bob will confess.

Hence in both the cases Bob will confess as his outcome is better if he confesses.

Hence both Andy and Bob will confess and will go to jail for 8 months. If both of them cooperated and kept quiet then they would have been in jail for only 1 month. This is known as prisoner’s dilemma.

Two individuals acting in their own best interest pursue a course of action that does not result in the ideal outcome. The typical prisoner’s dilemma is set up in such a way that both parties choose to protect themselves at the expense of the other participant. As a result of following a purely logical thought process to help oneself, both participants find themselves in a worse state than if they had cooperated with each other in the decision-making process.

The image given below should summarize the whole concept.

PrisonersDilemma

Watch the video which explains this concept in detail.

Let us look at some real life examples.

1. Coca-Cola and Pepsi

I came across this article on how Coca-Cola and Pepsi will decide on their pricing strategy.

Consider two firms, say Coca-Cola and Pepsi, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of ten million dollars per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to twelve million dollars, and that of the rival falls to seven million. If both set low prices, the profit of each is nine million dollars. Here, the low-price strategy is akin to the prisoner’s confession, and the high-price akin to keeping silent. Call the former cheating, and the latter cooperation. Then cheating is each firm’s dominant strategy, but the result when both “cheat” is worse for each than that of both cooperating.

I have represented this information in a tabular format.

Coca-Cola
High Low
Pepsi High 10, 10 7, 12
Low 12, 7 9, 9

Because of prisoner’s dilemma both of them will reduce the price and it will result in worst outcome for both of them.

2. Buffett explains Prisoner’s Dilemma

In the 1985 letter to shareholders Buffett explains how prisoner’s dilemma affects the textile business.

Over the years, we had the option of making large capital expenditures in the textile operation that would have allowed us to somewhat reduce variable costs. Each proposal to do so looked like an immediate winner. Measured by standard return-on-investment tests, in fact, these proposals usually promised greater economic benefits than would have resulted from comparable expenditures in our highly-profitable candy and newspaper businesses.

But the promised benefits from these textile investments were illusory. Many of our competitors, both domestic and foreign, were stepping up to the same kind of expenditures and, once enough companies did so, their reduced costs became the baseline for reduced prices industry-wide. Viewed individually, each company’s capital investment decision appeared cost-effective and rational; viewed collectively, the decisions neutralized each other and were irrational (just as happens when each person watching a parade decides he can see a little better if he stands on tiptoes). After each round of investment, all the players had more money in the game and returns remained anemic. 

3. Why countries manufactures weapons

India and Pakistan are considering to develop a new military technology. For this they need to manufacture weapons. Manufacturing weapons is expensive but provides greater security against each other. Both these countries have 2 choices; either to build or not to build (pass). I have constructed the payoff’s for each of the scenario in a tabular format.

Pakistan
Pass Build
India Pass 3, 3 1, 4
Build 4, 1 2, 2

Because of prisoner’s dilemma both of them will manufacture weapons and it will result in worst outcome for both of them.

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2 thoughts on “Prisoner’s Dilemma

  1. Sir

    That was extremely insightful. I could not understand the build and pass matrix in the weapon manufacturing example you mentioned. Could you kindly elaborate it a little further ?

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