In this essay,
I would like to tell the story of a minor discovery in mathematical
game theory that Freeman Dyson and I made in 2011. Dyson was a personal
friend and one of the great mathematical physicists of the twentieth
century. He died in 2020, at the age of ninety-six. He was famously
self-effacing, which is not to say that he lacked an accurate opinion of
his own abilities. Freeman would deny that he had done anything at all
and then allow friends—or even strangers—to vehemently contradict him.
Our discovery was not of that character. It really was very
minor. The reasons for telling the story now are less about the
discovery itself and more about the tendency of scientists to seek
lessons in moral philosophy in the least likely of places—high-school
algebra, for example.
Imagine that a group of scientists gather to play a kind of terror
game. They must propose scenarios that, should they eventuate, would
shake their belief in the foundations of their fields. The
mathematician’s proposed terror is that a long message, in English, is
found to be encoded—in excess of any plausible random
probability—somewhere in the first billion digits of pi.
The physicist’s terror is that the interaction cross-section of a
fundamental particle will have significantly different values when
measured in different places on earth, or in the same place at different
times.
The biologist’s terror is that some feature of the living world will be
unexplainable by the principle of natural selection. Within biology’s
subspecialty of evolution theory, there is a small area of study known
as evolution of cooperation. That study, some would say, lies closest to the biologist’s terror. That makes it worth poking at.
Cooperation and Defection
In biology,
a cooperator is an individual who pays a cost for another individual to
receive a benefit. When cooperation is mutually beneficial to two
individuals of the same or different species—a condition termed direct reciprocity—then
it is favored by natural selection. There are other possibilities. In
so-called kin selection, an individual’s self-sacrifice may be favored
if, on average, it helps another individual in the same gene pool to
survive. The unit of survival is understood in this case to be not the individual, but the gene that two individuals share.
It is harder to understand why individuals cooperate when defection
would be more favorable or when the reciprocity is only indirect.
Suppose that two microbe species, A and B, both need processed nutrients a and b. The cooperative state might be that A produces a, B produces b,
and each secretes a portion of its nutrient for the benefit of the
other. But this equilibrium is not evolutionarily stable: a defecting A with a mutation that halts its sharing of a becomes a free rider, benefitting from B
without paying the fare. Free riders, avoiding a cost, will tend to
take over the population. The evolutionarily stable endpoint is
noncooperation, even though cooperation would be better for both
species.
Cooperation among humans seems hardest of all to understand. “Humans
are the champions of cooperation,” Martin Nowak has remarked. “From
hunter-gatherer societies to nation-states, cooperation is the decisive
organizing principle of human society.”
In much, if not most, of our cooperation, reciprocity is indirect. To
be sure, some people give money to universities in the hope of getting
their own children admitted—kin selection—but many more give to
charities that are of no direct benefit to themselves or their kin. Many
billionaires become philanthropists, but from the standpoint of
evolution theory, why is this? A quirk of our culture, maybe? But
cultures, too, compete for dominance with other contemporaneous
cultures, and by a process akin to natural selection. Are we to
understand that generosity is selectively favored? Or are the generous
billionaires only transient?
Charles Darwin recognized that cooperation posed a challenge to his
theory of natural selection. He described an elegant experiment to
ferret out whether the aphid yields its excretion to the ant
voluntarily, or involuntarily with the ant as a parasite.
He provided a convincing argument that it was the former. Darwin, the
consummate naturalist, hated overgeneralized theory. Yet the significant
literature on the evolution of cooperation that has flourished in the
last fifty years is almost entirely theoretical. Much of it is cast in
the formalism of mathematical game theory, a subject that came into
existence more than half a century after Darwin’s death in the work of
John von Neumann and Oskar Morgenstern. Game theory describes how
competing, sentient players, in a well-defined universe of choices and
payoffs, may knowingly seek to optimize their own outcomes. Evolution is
the blind watchmaker,
optimizing only by trial and error. Exactly how the achievable outcomes
of evolution correspond to the mathematical optima of game theory is
not a settled question.
The Prisoner’s Dilemma
Go back to microbes A and B,
but now promote them to sentience. They become Alice and Bob, who are
arrested on suspicion of committing, together, a serious crime. Each has
sworn not to betray the other. They are questioned in separate rooms.
“We already have enough evidence to convict you both of a
misdemeanor,” the detective says to each, “that will put you away for
one year.” Each, separately, says nothing. “But if you defect, rat out
your partner and turn state’s evidence,” the detective continues, “we’ll
let you go, scot-free. Your partner will get a felony conviction, six
years in the state penitentiary.”
“What if we both turn state’s evidence?” Alice and Bob each ask.
“Well, I can’t let you both go free,” the detective says. “You’ll each get three years.”
Alice reasons as follows: there are only two possibilities. Either
Bob will rat me out, or else he won’t. If he rats, then I’ll get six
years—unless I rat also, in which case I’ll get just three years. So, if
he rats, I should too. But what if he doesn’t rat? What a chump! I can
rat on him and be out today. So, either way, I should defect.
Bob employs the same reasoning and defects on Alice. They each get three
years. The pair spend the time wishing that they had both kept their
promises not to betray each other and escaped with misdemeanor
convictions.
The prisoner’s dilemma (PD) game, played once, has no direct bearing on evolution. But consider the iterated
prisoner’s dilemma (IPD) game, first posited at the RAND Corporation in
the 1950s: Alice and Bob play many rounds of the same game with each
other. After following the same logic for a few games, Alice tentatively
tries a round of cooperation. In that round, Bob still defects and
Alice gets six years. But Bob has now seen Alice’s signal. He tries
cooperation himself. Alice reciprocates. And, for a string of games,
they are both cooperating, receiving only misdemeanor convictions. In
the IPD, there is information in the previous plays, and each player can
try to use that information to devise a superior strategy that remains
self-interested.
Is the best strategy to cooperate always? Certainly not. If Alice adopts
that strategy, Bob will always defect, getting off scot-free, while
Alice will always get six years. A good strategy would seem to be
something like, “Cooperate most of the time, but don’t be a chump if the
other player doesn’t follow suit.” Can this be formalized, or made
crisp, in some way?...
