Key Arguments

• He generalizes from two agents to many: 'Though we can seldom know that we face a Two-Person Prisoner’s Dilemma, we can very often know that we face Many-Person Versions. And these have great practical importance.' He then defines a Many-Person Dilemma as a case 'when it is certain that, if each rather than none of us does what will be better for himself, this will be worse for everyone.'
• He illustrates a Many‑Person Case with the Samaritan’s Dilemma: in large communities, 'It may here be better for each if he never helps. But it will be worse for each if no one ever helps. Each might gain from never helping, but he would lose, and lose more, from never being helped.'
• He introduces the general Positive Conditions: '(i) each of us could, at some cost to himself, give to others a greater total sum of benefits, or expected benefits; (ii) if each rather than none gives this greater benefit to others, each would receive a greater benefit, or expected benefit; and (The Negative Condition) there would be no indirect effects cancelling out these direct effects.' These conditions cover a wide range of cooperation problems.
• He explains that the Positive Conditions include extremes where each benefits just one other (like the Samaritan’s Dilemma) and where each benefits all others, as well as intermediate cases where benefits go to some sub‑group; he notes that in some of these cases condition (ii) is implied by (i), while in others it holds probabilistically (e.g. if 'the benefits were randomly spread').
• He introduces uncertainty by distinguishing between certain and probabilistic benefits: 'At one extreme, each could certainly give to the others a greater total sum of benefits. At the other extreme, each would have a very small chance of giving to the others a very much greater benefit. In this range of cases each could give to the others a greater sum of expected benefits.'
• To handle uncertainty, he revises the dilemma definition and introduces 'Risky Dilemmas': 'We face a Risky Dilemma when it is certain that, if each rather than none gives himself an expected benefit, this will either reduce the expected benefit to everyone, or will impose on everyone an expected harm or cost.' Thus even when results are probabilistic, similar structural conflicts between individual and group interest arise.
• He also notes that in some many‑person cases only the Positive Conditions hold because 'the numbers involved are sufficiently small' that 'what each does might affect what most others do.' These are important real‑world cases involving 'nations, or business corporations, or trade unions' that resemble but do not strictly satisfy his definition of a true Prisoner’s Dilemma, since it is not clear which option is better for each.
• He argues that Many‑Person Dilemmas are 'extremely common,' and explains why the Negative Condition, hard to satisfy in two‑person cases, often holds naturally in large‑number contexts: 'if we are very numerous, what each does would be most unlikely to affect what most others do. It may affect what a few others do; but this would seldom make enough difference.'

Source Quotes

The problem is that, if each rather than neither does what is certain to be better for himself, this will be worse for both of them. Though we can seldom know that we face a Two-Person Prisoner’s Dilemma, we can very often know that we face Many-Person Versions. And these have great practical importance. The rare Two-Person Case is important only as a model for the Many-Person Versions.

The rare Two-Person Case is important only as a model for the Many-Person Versions. We face a Many-Person Dilemma when it is certain that, if each rather than none of us does what will be better for himself, this will be worse for everyone. This definition covers only the simplest cases. As before ‘everyone’ means ‘all the people in some group’.

As before ‘everyone’ means ‘all the people in some group’. One Many-Person Case is the Samaritan’s Dilemma. Each of us could sometimes help a stranger at some lesser cost to himself. Each could about as often be similarly helped. In small communities, the cost of helping might be indirectly met.

But in large communities this is unlikely. It may here be better for each if he never helps. But it will be worse for each if no one ever helps. Each might gain from never helping, but he would lose, and lose more, from never being helped. Many cases occur when (The Positive Conditions) (i) each of us could, at some cost to himself, give to others a greater total sum of benefits, or expected benefits; (ii) if each rather than none gives this greater benefit to others, each would receive a greater benefit, or expected benefit; and (The Negative Condition) there would be no indirect effects cancelling out these direct effects.

Each might gain from never helping, but he would lose, and lose more, from never being helped. Many cases occur when (The Positive Conditions) (i) each of us could, at some cost to himself, give to others a greater total sum of benefits, or expected benefits; (ii) if each rather than none gives this greater benefit to others, each would receive a greater benefit, or expected benefit; and (The Negative Condition) there would be no indirect effects cancelling out these direct effects. The Positive Conditions cover many kinds of case.

At the other extreme, each would have a very small chance of giving to the others a very much greater benefit. In this range of cases each could give to the others a greater sum of expected benefits. This is the value of the possible benefits multiplied by the chance that the act will produce them. When the effects of our acts are uncertain, my definition of the Dilemma needs to be revised.

In these cases it is not certain that, if each rather than none does what will be better for himself, this will be worse for everyone. We face a Risky Dilemma when it is certain that, if each rather than none gives himself an expected benefit, this will either reduce the expected benefit to everyone, or will impose on everyone an expected harm or cost. In some Many-Person Cases, only the Positive Conditions hold.

The questions raised by true Dilemmas are quite different. Many-Person Dilemmas are, I have said, extremely common. One reason is this.

One reason is this. In a Two-Person Case, it is unlikely that the Negative Condition holds. This may need to be specially ensured, by prison-officers, or game-theorists. But in cases that involve very many people, the Negative Condition naturally holds. It need not be true that each must act before learning what the others do.

Key Concepts

• Though we can seldom know that we face a Two-Person Prisoner’s Dilemma, we can very often know that we face Many-Person Versions. And these have great practical importance.
• We face a Many-Person Dilemma when it is certain that, if each rather than none of us does what will be better for himself, this will be worse for everyone. This definition covers only the simplest cases.
• One Many-Person Case is the Samaritan’s Dilemma. Each of us could sometimes help a stranger at some lesser cost to himself. Each could about as often be similarly helped.
• It may here be better for each if he never helps. But it will be worse for each if no one ever helps. Each might gain from never helping, but he would lose, and lose more, from never being helped.
• (The Positive Conditions) (i) each of us could, at some cost to himself, give to others a greater total sum of benefits, or expected benefits; (ii) if each rather than none gives this greater benefit to others, each would receive a greater benefit, or expected benefit; and (The Negative Condition) there would be no indirect effects cancelling out these direct effects.
• In this range of cases each could give to the others a greater sum of expected benefits. This is the value of the possible benefits multiplied by the chance that the act will produce them.
• We face a Risky Dilemma when it is certain that, if each rather than none gives himself an expected benefit, this will either reduce the expected benefit to everyone, or will impose on everyone an expected harm or cost.
• Many-Person Dilemmas are, I have said, extremely common.
• In a Two-Person Case, it is unlikely that the Negative Condition holds. This may need to be specially ensured, by prison-officers, or game-theorists. But in cases that involve very many people, the Negative Condition naturally holds.

Context

Middle of section 23, where Parfit generalizes the two‑person dilemma framework to many‑person and uncertain ('risky') situations, introduces the Samaritan’s Dilemma and formal Positive/Negative Conditions, and explains why many‑person and risky dilemmas are pervasive.