The Prisoner’s Dilemma and the problem of deciding whether to take an umbrella are both instances of the same general pattern of cause and effect:
a particular outcome happens if I do a certain action and the world is in a particular state.
Similarly:
I will be rich if I buy a lottery ticket and my number is chosen. I will be famous if I write a book and it receives critical acclaim. It will rain tomorrow if I do a rain dance and the gods are pleased. In all of these cases, you can control your own actions, but you cannot completely control the actions of others or the state of the world. At best, you might be able to judge the exact probability that the world will be in a particular state. At worst, you might just assume that the odds of its being or not being in the state are simply equal.
However, suppose that in the case of the Prisoner’s Dilemma, you decide to do a little high school algebra. Let:
the utility of your getting N years in jail be–N. the probability that John turns witness be P.
Therefore the probability that John refuses to turn witness is (1– P). These utilities and probabilities can be added to the decision table: Action State of the world
John turns witness with probability P
John refuses with probability (1–P)
Expected utility P × utility1+ (1–P) × utility2 I turn witness I get 3 years
with utility1=–3
I get 0 years with utility2= 0
–3P
I refuse I get 6 years with utility1=–6
I get 1 year with utility2=–1
–6P –(1–P) = –5P – 1
But the expected utility–3P of turning witness is greater than the expected utility−5P–1 of refusing to turn witness, for all values of P between 0 and 1. So no matter what the probability P that John turns witness against you, you are always better off turning witness against him.
Unfortunately, if John has the same beliefs, goals and utilities as you, then he will similarly decide to turn witness against you, in which case both of you will get a certain 3 years in jail. You would have been better off if both of you had
ignored decision theory, taken a chance and refused to turn witness against the other, in which case you would both have got only 1 year in jail.
But there is a different moral you could draw from the story: that the fault lies, not with decision theory, but with your own selfish judgement of utility. You have placed no value at all on the consequences of your actions for the time that John will spend in jail.
Suppose, for example, that you assign equal value to the time that both of you will spend in jail. The corresponding new judgements of utility can be incorpo- rated into a revised decision table:
Action State of the world John turns witness with probability P
John refuses with probability (1–P)
Expected utility P × utility1+ (1–P) × utility2 I turn witness I get 3 years
John gets 3 years with utility1=–6
I get 0 years John gets 6 years with utility2=–6
–6P –6(1–P) = –6
I refuse I get 6 years John gets 0 years with utility1=–6
I get 1 year John gets 1 year with utility2=–2
–6P–2(1–P) = –4P–2
But–6 ≥ –4P –2, for all values of P between 0 and 1. Therefore, no matter what the probability P that John turns witness against you, there is never any advantage in your turning witness against him. Moreover, if John has the same beliefs, goals and utilities as you, then he will similarly decide not to turn witness against you, in which case you will both get a certain 1 year in jail. But it is probably unrealistic to expect you to value equally both what happens to John and what happens to yourself. To be more realistic, suppose instead that you value what happens to John only half as much as you value what happens to yourself:
Action State of the world John turns witness with probability P
John refuses with probability (1–P) Expected utility P × utility1+ (1–P) × utility2 I turn witness I get 3 years
John gets 3 years with utility1=–4.5
I get 0 years
John gets 6 years with utility2=–3
–4.5P–3(1–P) = –1.5P–3
I refuse I get 6 years
John gets 0 years with utility1=–6
I get 1 year
John gets 1 years with utility2=–1.5
–6P –1.5(1–P) = –4.5P–1.5
The expected utilities of the two alternatives are the same when: 1:5P 3 ¼ 4:5P 1:5
i:e: 3P ¼ 1:5
i:e: P¼ 0:50
Therefore, if you judge that the probability of John turning witness is less than 50%, then you should not turn witness. But if you judge that the probability is greater than 50%, then you should turn witness. Tit for tat.
Just as in the case of deciding whether to take an umbrella when you leave home, these calculations are a normative ideal. But in real life, we more normally compile our decisions into rules (or heuristics), which approximate the decision-theoretic ideal, but which can be applied more simply and more efficiently. For example:
Goals: if an agent requests me to perform an action, and the action does not harm another person then I perform the action.
if an agent requests me to perform an action, and the action harms another person then I refuse to perform the action.
These rules are not very subtle, but clearly they can be refined, both by adding extra rules to deal with other cases, and by adding extra conditions to accom- modate extra qualifications.