The Prisoner’s Dilemma
The prisoner’s dilemma is a very famous problem in the area of Game Theory. It is an example of a situation where individuals tend to take decisions that give the maximum satisfaction for themselves as opposed to taking decisions that might benefit them as a group
But before that, let us get to know what exactly is Game Theory
What is Game Theory?
Game Theory is a field of study which examines real-life situations where players make decisions that are dependent on the pay-off available and the decisions taken by other players in the “game”
Since the theory models real-life situations as “Games” and the participants/people as “players” it is referred to as Game Theory
Game Theory finds applications in a lot of real-life situations and a variety of disciplines like analysing voting behavior, advertising campaigns, studying market dynamics in economics and also military decisions
What is the Prisoners’ Dilemma?
There are 2 thieves A&B who have been arrested for a petty theft. However, the police suspect they might have been involved in a larger crime that happened few days back. But, the police have no conclusive evidence linking them to the same.
In order to bring a confession out of them, the police interrogate them in separate rooms. Each thief has 2 options in front of them:
- Confess with the hope of a reduced sentence (also called Defect)
- Refuse to talk(Deny) (also called co-operate)
The police do not have sufficient evidence to charge them with the crime until atleast one of them confesses to the crime.
They put the following proposition in front of the thieves:
- If both confess to the crime, each gets a jail time of 10 years
- If both deny involvement in the crime, each gets a reduced jail time of 3 years for the minor crime
- However, if one of them confesses & the other denies, the one who confessed will get a reduced jail time of 1 year while the one who denied gets a jail time of 25 years
Since both are in separate rooms, there is no way the thieves can discuss amongst themselves. So, what decision do you think each thief will take to minimize their jail time?
The Pay-Off Matrix
One of the most basic things while dealing with situations in Game Theory is to construct the pay-off matrix. The pay-off matrix is a very concise representation of the pay-off that each player gets if they perform a particular action
A pay-off can be described as a reward/satisfaction a player gets on performing a particular action. It does not have any unit and is used more in a comparative sense. For example, if from playing table tennis, the pay-off for player A is 10 and for player B is 5, we can infer that player A derives more satisfaction from playing table tennis than player B. The higher the pay-off, the more the reward or satisfaction
However, in some cases, it could mean the opposite (like in the case of Prisoners’ Dilemma) i.e the lower the pay-off, more the reward.
The pay-off matrix for the Prisoners’ Dilemma can be constructed as follows:
There are 2 actions for each player:
Confess (or) Defect
Deny (or) Co-operate
This gives us totally 4 situations, whose pay-offs were described above
The numbers within the matrix represent the jail time for each player if he/she were to perform that particular action. If both thieves were to confess, the pay-off’s are (10,10) implying that both will serve a jail time of 10 years each. If A confesses & B denies, the pay-off’s are (1,25) implying that A will serve a reduced sentence of 1 year while B will serve 25 years
As already mentioned, in this case, a higher pay-off implies lower reward (more jail time). One way to avoid this confusion is to use negative numbers. But it might be confusing for first timers. So, I am leaving it as it is
Analyzing the situation
Let us begin by analyzing the individual pay-off’s for each player irrespective of the other player’s move
Let us think of it from Thief A’s point of view:
If Thief A thinks that Thief B is going to Confess, he would be better off Confessing since Confessing would get him a jail time of 10 years. However, If B confessed and A denied, A would end up in jail for 25 years
Similarly, if Thief A thought B would deny, A would be better off Confessing in this situation also since A also denying would get both a jail time of 3 years. Whereas, if A Confessed while B denied, A would be out in 1 year
We conclude that irrespective of whatever B does, the best strategy for A would be to “Confess”
Since the matrix is symmetrical, the pay-off’s are the same of Thief B also. Hence, the analysis we did for Thief A holds true for B also. You can try it out for yourself
Irrespective of whatever A does, the best strategy for B also would be to “Confess”
“Confess” is a Dominant Strategy for both the players
A Dominant Strategy is a strategy that gives a player the highest pay-off irrespective of the opponent’s strategy. In this case, confessing gives a better pay-off for each of the thieves irrespective of what the other thief decides
If both thieves were to play their Dominant Strategy, each would end up with a jail time of 10 years. Is it the optimal strategy? No
One look at the pay-off matrix would tell us that if both deny, they get a jail time of 3 years. Hence, (Deny, Deny) is the equilibrium strategy.
(Deny, Deny) is called a co-operative strategy. A co-operative strategy is a strategy which players decide after consulting with each other. If both the thieves were given an opportunity to interact with each other and decide, they would have definitely gone with the Deny strategy since it provides a better pay-off (lower jail time) for the BOTH OF THEM
But as already mentioned, the thieves are interrogated in separate rooms with no chance of communication. Hence, Prisoners’ Dilemma is called a non-cooperative game
Real-Life Example 1: Economics
One of the most widespread uses of Game Theory & in particular, Prisoners’ Dilemma, is in the field of economics.
Game Theory is used by economists to study the interaction of various players in the market and what kind of strategies they might employ. This is especially important in the analysis of oligopoly markets
Oligopoly is a market structure where the market is dominated by a small number of large firms. One of the most important aspects here is that the decisions(eg: pricing strategy) taken by one firm might cause the other firms to change their decisions
Members of a cartel might be involved in a multi-player prisoners’ dilemma. In this case, “Co-operating” would go on to mean that all the players keep their prices at a minimum level that has been collectively decided beforehand.
“Defecting” would mean that one of the member companies prices his products lower than the others, thereby taking away the customers from the other companies and making a profit
Another real-life example of Prisoners’ Dilemma is advertising
Example 2: Advertising
(Disclaimer: Smoking is injurious to health. Smoking causes cancer)
Suppose there are 2 rival cigarette companies (with equal market share) in the US and they decide to advertise their respective products in a bid to increase their profits
However, how effective their respective advertisement campaigns are, is also dependent on the advertising done by the rival company
If one firm advertises and the other doesn’t, the firm that didn’t advertise loses out on a significant portion of their market share to the rival
If both firms advertise, their advertising would cancel out each others, resulting in the same market share as before, but with an extra expense incurred (for the advertising)
You can attempt to draw a pay-off matrix for the above situation. You should have no trouble with it if you have understood my post till this point
Laws were based in US around the 1970’s forbidding cigarette companies from advertising their brand in televisions. However, this law faced little resistance from cigarette companies since they could observe that it made little difference to their market share and profits
Btw, the simple game of Rock, Papers and Scissors is one of the most trivial examples of Game Theory. Do not believe me, try constructing the pay-off matrix and find out for yourselves
There are lot more interesting real-life examples of Game theory which you can read about in this website: http://cassmba8.weebly.com/index.html
In Pop Culture
One of the most famous references to Prisoners’ Dilemma is in the Dark Knight(2008) movie.
The antagonist in the movie is the Joker (played by the late Heath Ledger), an antagonist whom all of us have admired because he uses his brains more than weapons.
In the movie, the Joker plants explosives in 2 ferries which contain civilians in one and prisoners in the other and places the detonator for each boat in the other one. He threatens to blow up both the ferries, but will let one of the ferries survive if its passengers blow up the other one
There is an interesting article in Medium on the above situation, which you can read here:
Breaking Down the Finale of Dark Knight via Game Theory
You can watch the scene here:
Thanks for reading!!
References
http://cassmba8.weebly.com/everything-is-a-prisoners-dilemma.html
Originally published at http://infinitesimallysmallcom.wordpress.com on December 20, 2021.
