Detroit Mayor’s Prisoner Dilemma – Why in reality people choose not to confess
In class we have discussed the classic Game Theory Problem – The Prisoner’s Dilemma. We also analyzed why in a lot of cases the dominant strategy (the strategy a player will use no matter which strategy the opposing player plays) is to confess. In today’s blog, however, I will discuss why in reality, in some cases players will chose to not confess.
Here is the article I will be discussing http://www.usatoday.com/news/nation/story/2012/09/21/ousted-detroit-mayors-corruption-trial-gets-underway/57819054/1
In short it is about the corruption trial of a governor, his father, his best friend and a water-boss. They are accused of rigging water contracts. All the defendants plead not guilty.
Now, let’s set up a simple game theory matrix. If they are convicted of the crime they will get 30 years in prison. Let’s say that if one of them confesses and agrees to testify against the others in court they can get their sentence reduced (which is common practice). (Here I just put 5 years because it is easier to calculate, but it would probably be more) If both of the confess then the judge can think about giving them only 20 years because they came clean (again 20 is easier to calculate). If both do not confess, it would be hard for the plaintiff to get evidence for their crimes, so let’s say they only face 10 years in prison for a lesser crime than the one they actually committed. Here is the matrix:
| Confess | NotConfess | |
| Confess | -20,-20 | -5,-30 |
| NotConfess | -30,-5 | -10,-10 |
As you can see, the dominant strategy for both Player 1 and Player 2 would be to confess, since it yields better outcome for them no matter what the other person plays. The pure Nash equilibrium (where no player wants to change their strategy) would be Confess and Confess (C,C). Hence it would be likely to think that the governor and everyone else would confess to their crimes.
However, given the fact that the governor and the other people charged have strong positive relationships (they are friends and they have a close bond) with each other, the above matrix is not a practical approximation of the Game Theory at hand. It’s easy to see why these people would have strong positive bonds to each other: 1) they all know each other and 2) would have to trust each other a lot to commit a crime together. Also, three of the people have familial ties to each other.
Hence we can safely assume that the other player’s prison sentence factors into the player’s utility. Let’s say his utility is the negative of his prison sentence minus half of the other player’s (just so it is easy for calculation). Then the payoff matrix would be following:
| Confess | NotConfess | |
| Confess | -30,-30 | -20,-32.5 |
| NotConfess | -32.5,-20 | -15,-15 |
Hence we can see that the best outcome is both not confessing, with total welfare -30 (of the accused). There is no dominant strategy now, players will not choose to confess no matter what. There is a mixed Nash equilibrium. So now there is a probability that the players will choose not to confess. This probability is heightened by the fact that before the trial, the respective players can communicate with each other, so they can agree among themselves that they will all not confess. In addition, because they all know each other, they can assert influence on each other to come to a consensus to not confess.
Related concepts: prisoner’s dilemma, game theory, pure/mixed nash equilibrium, strong/weak ties, positive/negative ties
-butterknife