In order for society to operate, we need people to follow the rules, to work together, to cooperate. In this post, Bridget Welch begins a series on how we make (or fail to make) people cooperate. Up for today — the Prisoner’s Dilemma and Nash’s Equilibrium.
It goes a little something like this:
James and Daryl are arrested (separately) for some petty crime which the prosecutor can easily make the case and give them 2 years. During questioning, it becomes evident to the prosecutor that this is the team that robbed a bank a few weeks back. Unfortunately, the prosecutor has no evidence to back this up. So she schemes and tells each, while they are kept in separate rooms and not able to communicate, that:
“You have two choices. You can confess to the crime or remain silent. I have enough to put you both away now for 2 years on this other case. However, if you confess to the bank robbery, I’ll give you 1 year while your partner will get 10. He gets the same offer. If you both confess, you’ll both get 3 years.”
Pretend you’re James:
- If neither of you confess, you’ll get the two years for the petty crime.
- If you confess (fink), and Daryl doesn’t, you get 1 year and he gets 10.
- If Daryl confesses, and you don’t, you get 10 years and he gets 1.
- If both confess, you both get 3 years.
These options can be shown in what is called a payoff matrix. Obviously, the optimal scenario here is for both of them to deny they had anything to do with the armed robbery. In game theoretic speak, denying is to talk of them both cooperating — that is going along with each other so they both get the lowest possible cost. Confessing, in game theoretic speak, is defecting because, in effect, it is selling out your partner in an attempt to get the lowest sentence for yourself. In determining whether people cooperate or defect, as yourself this: How much would you trust Daryl?
Unless you have a strong trust in Daryl (remember, you can’t talk to him), you will be sitting in your room thinking that Daryl is going to try to get the best outcome he can possibly get. Looking at the payoff matrix, you’ll see the scenario that offers that is for Daryl to confess to the armed robbery. But, if he does that, and you don’t, then you’ll end up with TEN YEARS! Clearly, if he confesses you should also confess. Well, what if Daryl denies? In this case it’s your best interest still to confess because you will end up with one year. Thus, regardless of what Daryl does, your best (most rational choice) is to confess (by the way, this is the same for Daryl!).
Confession by both is a Nash equilibrium (yeah, that Nash, from Beautiful Mind played by Russel Crow)– or “a stable state of a system that involves several interacting participants in which no participant can gain by a change of strategy as long as all the other participants remain unchanged.” Looking at the payoff matrix, once both have decided to confess there is no possible benefit for either of them to change their answer to denying (they would be moving from 3 years to 10 years in prison if they changed their answer).
The optimum state, however, is not stable. If they both are denying in this example, at any moment one of them may decide to start cooperating in order to try to move from 2 years to only 1 in prison. The same is true of the other two boxes. For example, in the box where James confesses and Daryl denies, Daryl can improve his outcome by now confessing — meaning it is not a Nash equilibrium.
This social dilemma is one in which it is in our own best RATIONAL interest to defect. From governments refusing to curb pollution, to athletes choosing to dope, to students choosing whether to slack or work hard on a group project — prisoner’s dilemmas form the basis for many our decisions (some of these are special kinds of social dilemmas like the tragedy of the commons and freeriders that I will discuss in subsequent posts). The question becomes: How do we change this scenario to make people more likely to cooperate?
One possibility is to change outcomes by adding incentives to reinforce cooperative choices. This is a frequent social control solution utilized in our society. We positively (reward) and negatively (punish) sanction behavior based on what is desirable. Unfortunately, as powerful as this can be, it doesn’t always insure cooperation when the benefit of defecting outweighs any potential social cost (for example, watch the clip in dig deeper question 2). Thus, it becomes important that we make any social sanctions powerful enough to make cooperation likely when it really counts.
- Another game often played is evens and odds (also called “chasing pennies” or “chooses”). Watch this Seinfeld clip if you are unfamiliar. Draw the payoff matrix for this game. Is there a Nash Equilibrium here? If so, where?
- Watch this clip from the game show Gold Balls where an altered prisoner’s dilemma (here they can communicate) is presented. Draw the payoff matrix from the show. What should the outcome be? How does Nick’s behavior change this?
- Many people argue that the world would be better off if all nations got rid of their nuclear arms. Assuming the US and Russia as two nations trying to come to a disarmament agreement, draw the payoff matrix showing each nations’ two options (cooperating through disarming or defecting by keeping weapons). Explain what these two nations are likely to chose and why. A common prisoner’s dilemma students face is whether to slack or fully contribute to a group project. What types of incentives/punishments could be added to reduce the likelihood of defection?