Fairness & Altruism

FAIRNESS & ALTRUISM:

A BRIEF SURVEY OF SOME UNFORTUNATE APPARENT CONSEQUENCES OF EVOLUTIONARY PROCESSES, AND A DISCUSSION OF ONE SOLUTION

Prepared by Drew Schroeder for Phil-152, December 2006.

I. FAIRNESS1

Divide the Dollar. Two players are competing for a reward of $90. Each is told to secretly write down a fraction on a sheet of paper. If the fractions sum to more than 1, neither player gets anything. If they sum to 1 or less, each player gets a share of the $90 corresponding to the fraction she wrote down.

Define an evolutionarily stable strategy (ESS) as a strategy such that, if everyone followed it, an individual or small group of individuals who adopted any different strategy would do worse. The intuitive idea, in biological terms, is this: an ESS is a strategy such that if it’s dominant in a population, any mutant that arises will do worse. The strategy will therefore be very resistant to change by evolutionary processes. Now, what is an ESS in Divide the Dollar? Not surprisingly, writing down 1/2 is an ESS. If everyone is writing down 1/2, a mutant who writes more than 1/2 will never get anything, while those writing 1/2 will still usually get $45 (since they’ll usually be playing against others writing 1/2). A mutant who writes less than 1/2 will get something, but she’ll always get less than those writing 1/2. So, either way, a mutant does worse than those writing 1/2, so writing 1/2 is an ESS. There are other ESSs, however. Consider a population in which half write 2/3 and half write 1/3. (If you want to think of it as a strategy, you can either think of it as writing each half the time, or – to make things simpler – at birth flipping a coin and then subsequently writing either 2/3 or 1/3, depending on what comes up.) The expected payoff of writing 1/3 in this population is of course $30. (You’ll always get 1/3 of $90.) The expected payoff of writing 2/3 is also $30, since half the time you’ll get nothing (when you play a 2/3-er) and half the time you’ll get $60 (when you play a 1/3-er). Consider some mutant. Suppose she writes down a fraction less than 1/3. Then she’ll always get what she writes down times $90, but her expected payoff will be less than $30. So, she’ll do worse. Suppose that she writes down a fraction higher than 2/3. Then she’ll never get anything. Expected payoff=$0. Finally, suppose she writes down some fraction, N, between 1/3 and 2/3. She’ll get nothing when her opponent writes 2/3, and she’ll get N x $90 when her opponent writes 1/3. So her payoff will be N x 0.5 x $90. But since N is less than 2/3, this will be less than $30. (Do the math.) Thus, no matter what she does, a mutant will have an expected payoff of less than $30. She’ll do worse than a 1/3-er or a 2/3-er. It turns out that there are infinitely many ESSs: for any two fractions that add up to 1, there’s an ESS that has each played some fraction of the time. (For 4/5 and 1/5, for example, it’s writing 4/5 75% of the time, and 1/5 25% of the time.) The problem, of course, is that all of the non-1/2 ESSs are inefficient. That is, the average payoff is less than it could be. Part of the pie is being wasted. If everyone wrote 1/2, everyone would do better. This is crucial – we’re not talking about a case where some people could do better at the expense of others, or even a case where some people could do better without cost to others. This is a case where everyone would be better off with a different strategy. Since these are ESSs, though, unilateral change (e.g. by mutation) can’t get us out of the inefficient state. Any new strategy will do worse than the old ones, so evolution has gotten the population into a trap from which it can’t extricate it.

How big a problem is this, actually? We can run computer simulations. Start with a random assignment of strategies (of the sort “write N”). Have members of the population play against random other members of the population. Interpret the payoffs in terms of evolutionary fitness: if A has a higher payoff than B, then A’s strategy is represented in proportionally higher numbers in the next generation.

1 The material in this section very closely tracks the discussion in Skyrms 1996 (ch. 1).

1 Repeat the procedure through many iterations until an equilibrium is reached. What happens? It turns out that it depends on the initial distribution of strategies. If we allow members of the population three strategies, write-1/2, write-2/3, and write-1/3, then in about 74% of initial distributions, the population will evolve to the “write 1/2” ESS. In about 26% of cases, however, we’ll end up with half 2/3-ers and half 1/3-ers. (See the graph to the right.) This is not an insignificant fraction. If evolution worked this way, we should expect to find lots of inefficient, stable strategies out there – populations where everyone would be better off, if they could all just change their strategies, but where any change by a single individual will leave her worse off. What can we do to avoid these inefficient “traps”? Skyrms’ simulations suggest two solutions. First, if we admit more possible strategies, we don’t get rid of inefficiency, but we do minimize it. If, for example, we allow twenty strategies Each point in the triangle represents a particular distribution of strategies within the population. Points closer to a given vertex represent higher proportions of that strategy. All initial (write-1/20, 2/20...19/20, 20/20), the distributions evolve to one of the three labeled equilibria. A, though, is unstable, since a small efficient solution only comes about 57% increase or decrease in 1/2-ers will quickly lead the population to either B or C. B and C are of the time -- but an additional 36% of stable, since after any small change, the population will return to B or C. the time the population gets stuck in the 9/20-11/20 or 8/20-12/20 “traps”. These are indeed inefficient, but they’re much less inefficient than the 1/3-2/3 “trap”. 93% of the time, then, the population gets an average payoff of at least 80% of the pie. The more interesting solution, however, is through correlation. Notice that if we have a mixed population of 1/3-ers, 1/2-ers, and 2/3-ers, the 1/2-ers do well when playing against 1/2-ers or 1/3-ers. They do badly against 2/3-ers. The 2/3-ers do badly against 2/3-ers and 1/2-ers, but well against 1/3-ers. (1/3-ers don’t care who they play.) Suppose, then, that players are somewhat more likely to play against players who share their strategy than they are to play against players with different strategies. This won’t affect 1/3-ers at all. 2/3-ers will suffer, since they’ll be involved in 2/3 vs. 2/3 games more often. The 1/2-ers will benefit, since they’ll play 1/2-ers more often. We’d expect, then, that correlated interaction would help us get to the write-1/2 equilibrium. What’s surprising is how significant the effects are. If we introduce a correlation coefficient of 10%,2 the efficient outcome arises about 91% of the time (instead of 74%). If the coefficient is 20%, the inefficient trap is virtually gone. What’s the take-home lesson? Evolution can sometimes lead us into and get us stuck in inefficient situations. One way to break out of such a trap is to ensure that like interacts with like.

2 Formally, this means that for some player, A, 10% of the encounters she would have had with different strategies come instead with like strategies. That is, if As make up 20% of the population, given random pairing they’d meet non-As 80% of the time. With a correlation coefficient of 10%, they meet non-As only 72% of the time.

2 II. ALTRUISM

Define biologically altruistic behavior as behavior which increases the fitness of others, at cost to one’s own fitness. This is not the way we ordinarily use the term ‘altruistic’. When we call some act altruistic, we’re usually making some claim about the agent’s psychology: that she acted with the intention of helping someone else, at perceived cost to herself. Notice that a biologically altruistic act need not be psychologically altruistic: a worm presumably has no intentions at all, and so couldn’t possibly be a psychological altruist, though it could be a biological altruist. Also, a psychologically altruistic act need not be biologically altruistic: you might intend to spend your afternoons selflessly helping the homeless, but if society rewards you in some way that increases your fitness (e.g. by featuring you on a “most eligible bachelors” T.V. show), your act wasn’t altruistic in the biological sense. In what follows, we’ll be worrying only about biological altruism. Biological altruism seems like it should be ruled out by natural selection. Compare two organisms, the same in all respects except that one engages in biologically altruistic behavior. By definition, the fitness of the altruist will be lower, so she should be selected against – even if the results are catastrophic in the long term. A concrete example: consider a population of birds, who give warning calls when they see a cat. This is altruistic behavior, because making a warning call benefits others (they can fly away) but hurts the caller (since she calls attention to herself and is slightly more likely to be eaten). Now, suppose one mutant arises who doesn’t give warning calls. She’ll have higher fitness than the altruists: she benefits from their calls, but doesn’t suffer the costs of calling herself. So, she’ll have more offspring than the average altruistic bird. In the next generation, then, there will be a slightly higher proportion of selfish birds. Each of those selfish birds will have higher fitness than the altruists, and so in the third generation there will be still more selfish birds. This will continue until there are no altruists left. The new population of selfish birds will, however, do much worse than the original population of altruists – since there will be no warning calls at all, cats will catch a lot of birds. Despite the plausibility of this argument, there are many examples of apparently altruistic behavior in the animal kingdom. What are we to make of this? There are several possibilities: it might be that such acts are examples of imperfect evolutionary adaptation. Or maybe they can be explained by appeal to kin selection – organisms help their relatives, thereby increasing the fitness of their own genes. Or maybe, contrary to appearances, the acts aren’t really altruistic at all. Here, we’ll investigate more unusual solutions.

1. Sober and Wilson’s solution: group selection3

Suppose a population is divided into two groups, perhaps by some physical barrier. The first population is composed mostly of selfish individuals, the second mostly of altruists. As expected, everyone in the first group does poorly, since they don’t benefit from the presence of many altruists. Also as expected, altruists in each group do worse than their selfish comrades, since the selfish individuals gain the benefits of altruism without incurring the costs. Despite all this, it’s possible that in the whole population, altruists have a higher fitness than non-altruists. This is called Simpson’s Paradox. (See the table below.)4

Selfish Selfish Avg. Altruists Altruists Avg. before after fitness of before after fitness of selection selection selfish selection selection altruists Group #1 40 20 20÷40=0.5 5 0 0÷5=0.0 Group #2 5 8 8÷5=1.6 40 40 40÷40=1.0 Overall 45 28 28÷45=0.6 45 40 40÷45=0.9 Generation 1: Simpson ʼ s Paradox

3 The basic argument here comes from Sober and Wilson 1998. 4 The chart comes from Sterelney and Griffiths 1999, p. 162.

3 Across the whole population, altruists here have a fitness of 0.9, compared to 0.6 for selfish individuals. Could this explain the evolution of altruism? Not yet. If we continue the example another few generations into the future, the first group, devoid of altruists, will perhaps stabilize at a small size. (Although they no longer get the benefits afforded by having altruists in their midst, they also no longer need to compete for resources with so many other selfish individuals.) Within the second group, selfish individuals will continue to have higher fitness than altruists, so they’ll eventually take over. Here’s one way generation #2 might play out:

Selfish Selfish Avg. Altruists Altruists Avg. before after fitness of before after fitness of selection selection selfish selection selection altruists Group #1 20 15 14÷20=0.8 0 0 N/A Group #2 8 13 13÷8=1.6 40 37 37÷40=0.9 Overall 28 28 28÷28=1.0 40 37 37÷40=0.9 Generation 2: No more paradox

Here, the selfish individuals – both within group #2 and in the population as a whole – have higher fitness than the altruists. Eventually, the altruists in group #2 will be driven to extinction. While Simpson’s Paradox can therefore explain why altruists might for a short time have higher fitness than selfish individuals, so far we don’t have an explanation that could explain how altruism could persist in a population over a long period of time. Suppose, however, that after each generation the populations come together, then divide again into groups. If the groups form randomly, altruists will still eventually disappear, for the reasons we’ve already seen. But if the groups form non-randomly, with altruists tending to group with altruists, and selfish individuals with selfish individuals, then Simpson’s Paradox can appear again in generation #2 and beyond. (Exercise: take the output of the “Generation 1” chart above, and recombine the groups to create a 2nd generation where Simpson’s Paradox reappears. Convince yourself that this could be done indefinitely.) Altruists will have higher fitness and need not disappear. What’s the lesson? Non-random grouping can sustain altruism indefinitely.

2. A quick digression: the Prisoner’s Dilemma

Suppose you and a partner are caught by the police inside a jewelry store after hours. The police can certainly convict both of you of trespassing and vandalism (jail time: 1 year), though they aren’t able to prove you stole anything. You and your accomplice are taken to separate rooms and given the following offer: if you testify against your partner and he doesn’t testify against you, you’ll get off scot free (since they’ll need your testimony), and he’ll go to jail for ten years. If you both agree to testify, you’ll both get three years. You have five minutes to accept or decline the deal. Suppose you’re interested only in yourself. What should you do? If you think your partner will take the deal, then you should, too – it’ll reduce your jail time from ten years to three years. If you think your partner won’t take the deal, you should – it’ll reduce your jail time from one year to nothing. No matter what your partner does, then, you should take the deal. Assuming your partner reasons the same way, you’ll both take the deal, and you’ll both end up in jail for three years. If you’d both stayed quiet, you’d have only gotten one year. Lesson: self-interested reasoning leads to a situation where everyone is worse off. Now, let’s reinterpret the situation so that the payoff, instead of jail time, is given in terms of evolutionary fitness. Call the action analogous to staying quiet “cooperating” and the action analogous to taking the deal “defecting.” Any act of cooperation is biologically altruistic: by not defecting, you decrease your fitness, but raise the fitness of your partner. Therefore, showing how cooperators in the Prisoner’s Dilemma could evolve would also show how altruism could evolve.

Some possible real-world examples of Prisoner’s Dilemmas:

4 a. Arms Race. For a predator, it might be individually advantageous to devote resources to increasing running speed, to catch more prey. Similarly, it’d be advantageous to prey to devote resources to increasing running speed, to avoid predators. But if both get faster, neither gains any advantage. They’d both have been better off remaining slower and devoting the resources to other areas. Similar cases: trees growing taller and taller, increasing forest canopy height; actual human arms races.

b. Competing Gas Stations. You and I own the only gas The Prisonerʼs Dilemma stations in town. Each of us wants only to make money. I can lower my price from $3/gallon to $2.90, steal your Formally, we can deﬁne a Prisonerʼs Dilemma as customers, and increase my profits. Not wanting to be an ordered-quadruple, {T,R,P,S}, where T>R>P>S. run out of business, you’ll lower your price to $2.90, Itʼs often also required that 2R>T+S. taking back your share of the customers. Now we’re Cooperate Defect both making less money than we would have had we both charged $3. (Assume that the demand for gasoline Cooperate (R,R) (S,T) is relatively inelastic.) Defect (T,S) (P,P)

c. Pollution. Should I secretly dump my trash in the T=the temptation to defect. R=the reward for river? I would much rather have a clean river than one mutual cooperation. P=the punishment for mutual filled with everyone’s trash. But the river is a lot closer defection. S=the suckerʼs payment. than the dump, and of course my dumping will make virtually no difference to the river’s cleanliness; the convenience of dumping is well worth the marginal cost of my dumping, I think. Mathematically: suppose a clean river is “worth” 100 to each of us. Each dumper reduces that value by 1, but gains 5 in convenience. If n other people dump and I don’t, my payoff is 100-n. If I do dump, my payoff goes up by 5 (convenience) and down by 1 (pollution), for an overall gain of 4: 104-n. Therefore, no matter how many others dump, I should dump. If everyone else thinks that way, though, the river will be a cesspool, and everyone will get 5, rather than the 100 we’d each have gotten, had no one dumped. (There are obvious analogs in terms of fitness.)

3. Prisoner’s Dilemma on a square lattice

Consider the following game: a bunch of organisms are arranged on a large square lattice. Each round, every organism gets to play Prisoner’s Dilemma (with payoffs {T,R,P,S}, as defined in the last section) against each of its eight neighbors, adding up the payoffs from each encounter. Organisms can use only one of two strategies: always cooperate, or always defect. (If you like, assume that the “complexity cost” of playing multiple strategies is prohibitive.) At the end of the round, each square is occupied by whichever player in the 3x3 grid around it did the best in that round. (You may think of this either as reproduction – whichever neighboring organism had the most offspring colonizes the square – or else as imitation – organisms imitate whichever of their neighbors was most successful.) Now, there are several things we can quickly note about the lattice. First, any isolated defector can never be eliminated: an isolated defector will get to play against eight cooperators, getting payoff 8T – the maximum possible. He’ll therefore at least initially take over all the neighboring squares. An isolated cooperator can’t survive: she’ll get payoff 8S – the minimum possible. Beyond that, however, the dynamics are quite complicated, depending both on the exact values of {T,R,P,S} and on the initial distribution. Here are just a few examples, with diagrams of the relevant parts of the lattice.

a. Example #1: {10,8,1,0} Start with a population of defectors. A single cooperator can’t survive, nor can a cluster of four. But as soon as a 2x3 block appears, cooperation will quickly spread.

All the cooperators have D D D D D The highlighted cooperator has D D D D D neighborhood 3C+5D. Payoff= D D C C D neighborhood 5C+3D=40. The D C C C D 3x8+5x0=24. The highlighted D D C C D lightly-shaded defectors have 3C D C C C D defector has neighborhood 2C+6D. +5D=35 and 2C+6D=26. 2nd- row Payoff= 2x10+6x1=26. So the D D D D D Ds have payoff 8. So all lightly- D D D D D cooperators disappear. D D D D D shaded Ds will become Cs. D D D D D

5 b. Example #2: {10,6,1,0} Start with a population of cooperators. Suppose a single defector arises through mutation. He’ll expand in an “X” shape, but no further.

Generation 1: the defector has C C C C C Later on: corner D: 5C+3D=53. Black C D D D D neighborhood 8C=80, the C C C C C Cs: 8C=48. So lighly-shaded Cs will be C D D D D maximum possible. So it will C C D C C taken over by the corner D. Darkly- C D D D D expand to all neighboring shaded Ds: 3C+5D=35. Since 2nd-rank squares, eventually creating a C C C C C Cs get 48, C will hold onto the darkly- C D D D D 7x7 grid. C C C C C shaded Cs. Defectors can expand only at C C C C C C C C C C corners, leading to an “X” pattern. C C C C C

c. Example #3: {10,7,1,0} Start with a population of cooperators. A single defector will take over the neighboring squares, but go no further.

Generation 1: the defector C C C C C Generation 2: as in ex. #2, the corner C C D D D has neighborhood 8C=80, C C C C C defector is in the best position: 5C+3D =53. C C D D D the maximum possible. So it C C D C C The lighly-shaded Cs have at most 7C C C D D D will expand to all neighboring +1D=49. But the 2nd-row C has 8C=56. So squares. C C C C C the lighly-shaded squares will remain C, C C C C C C C C C C and Ds canʼt advance. C C C C C

What’s the lesson of all of this? Depending on the structure of the situation, cooperators can do quite well, so long as they stay in a group – that is, so long as they interact primarily with other cooperators.

III. SUMMARY

In Divide the Dollar, we saw that there is a significant chance that evolution can lead us to a state in which everyone is worse off than she would be following some alternate strategy. Since this state can be an ESS, it may be true that one individual can’t do anything to help: no matter what she does, she’ll be worse off, and in doing so she won’t make anyone better off. To reach the optimal outcome, a critical mass of players playing the optimal strategy is necessary. This threshold can be greatly diminished by increasing, even slightly, the chance that like strategies interact with like strategies. The evolutionary case (including Prisoner’s Dilemma-type cases) is, in a way, worse. Given a randomly-interacting population of altruists and selfish individuals, altruists will always have lower fitness than non-altruists. Evolution, therefore, will always lead to the proliferation of selfish types, to the eventual extinction of altruists. As in Divide the Dollar, this will leave everyone worse off than they would be in a population composed of altruists. Also, as in Divide the Dollar, one solution is for altruists to interact with each other preferentially. Sober and Wilson provide one model for how this might occur: if altruists segregate themselves non- randomly into groups of mostly altruists, they can survive. For example, if altruists are good at detecting selfish individuals and ostracize them, we might end up with the sort of preferential grouping Sober and Wilson suggest. The square lattice provides another, perhaps more concrete model for generating preferential interactions. If organisms tend to interact with those around them, then clusters of altruists (e.g. a family of altruists) will interact disproportionately with other altruists and can sustain themselves and even expand.

6 IV. BIBLIOGRAPHY

Axelrod, Robert. The Evolution of Cooperation (Basic Books, 1984). This is a famous book, and the inspiration for many of the issues discussed here. Axelrod solicited computer programs to play in an iterated (repeated) Prisoner’s Dilemma and had them play against one another. Famously, TIT FOR TAT (a simple program which cooperates first, then does whatever its opponent last did) won. Axelrod draws several conclusions from the data, generally optimistic for the evolution of cooperation.

Hauert, Christoph. http://www.univie.ac.at/virtuallabs/. This is one of the coolest sites on the internet. In addition to lots of tutorials and discussions of various games, there’s also a great Java program that lets you design your own games. You can set the payoff matrix however you like, decide whether to use a square or honeycomb lattice, decide how many neighbors each square interacts with, choose from different updating rules, etc. Go prepared to waste many hours.

Nowak, Martin. http://www.ped.fas.harvard.edu/people/faculty/index.html. Martin Nowak has done lots of work on the dynamics of various strategies within populations. For example, in a population playing iterated Prisoner’s Dilemmas, given certain reasonable assumptions about mutation, mistakes, and the like, a “war and peace” cycle will emerge: a random distribution is taken over by defectors, who will get taken over by tit-for-tat, which will move toward a slightly more forgiving strategy (called “generous tit-for-tat”), which will drift towards pure cooperation, which will quickly get taken over by defectors (and so on). Nowak has also done lots of work on the evolution of populations on a lattice. Many papers on both of these topics are available on his website. For basic discussions, see especially “Win-stay, lose-shift outperforms tit-for-tat,” and “Evolutionary Dynamics of Biological Games.”

Resnik, Michael D. Choices: an Introduction to Decision Theory (Minneapolis: Univ. of Minnesota Press, 1987). There are lots of texts out there on decision theory and game theory written by economists and mathematicians. They often end up either focusing on different results than philosophers tend to be interested in, or they require a substantial knowledge of mathematics. This text is written by a philosopher and is the best introduction I’ve come across for philosophers interested in the subjects.

Sober, Elliott and David Sloan Wilson. Unto Others (Cambridge, MA: Harvard Univ. Press, 1998). This book contains the argument for the evolution of altruism by way of group selection and non-random recombination. Sober and Wilson also give a number of interesting examples of actual cases of apparent biological altruism. The second half of the book talks about psychological altruism (i.e. behavior done with the intention of helping others) and its relationship to biological altruism.

Sterelny, Kim and Paul E. Griffiths. Sex and Death (Chicago: Univ. of Chicago Press, 1999). The discussion in Chapter 8 makes some helpful distinctions and gives a good, quick presentation of Sober and Wilson’s view.

Sykrms, Brian. Evolution of the Social Contract (Cambridge: Cambridge Univ. Press, 1996). This is where most of the information for the “fairness” section of this handout came from. Sykrms uses the lessons there to ground an argument for the evolution of fairness, justice, and (linguistic) meaning in human interactions.

Sykrms, Brian. The Stag Hunt and the Evolution of Social Structure (Cambridge: Cambridge Univ. Press, 2004). A “sequel” to Skyrms’ earlier book, focusing on the lessons to be learned from a game called the “Stag Hunt.” Suppose you and a friend can hunt deer or hare. It’s virtually impossible to bag a deer alone, but working together it’s likely to succeed, and the reward is huge. What should you do? If your friend is going to hunt deer, you should join her, since you’ll likely get a huge payoff. If your friend is going to hunt hare, though, you should hunt hare, since hunting deer by yourself would be ineffective. Considerations of risk, then, point to hare-hunting (since it’ll work no matter what your friend does), but the reward is potentially greater in deer-hunting. Formally, it’s set up with a payoff matrix like the Prisoner’s Dilemma, except that R>(T=P)>S.