by Haim Shapira
Incidentally, if German philosopher Immanuel Kant (1724–1804) could speak today, he’d suggest that we resolve the dilemma with the following categorical imperative (which I’ve adapted from Kant’s words): Before you act, think about this question: Would you like your move to become a universal law? Kant would have expected the condo tenants to say: ‘Of course, we don’t want the idea of avoiding the fees to become universally accepted. Since that might turn out very unpleasant, perhaps we should pay our fees after all.’ That’s very nice, but instead of waiting for all the tenants to familiarize themselves with Kant’s writings, we’d better introduce by-laws regarding those fees. When it comes to fees and taxes, people generally don’t pay of their own free will … even if they have read Kant.
Referring to a similar issue, Spanish philosopher José Ortega y Gasset (1883–1955) said that ‘Law is born from despair of human nature.’
What would be the best strategy for the iterated multi-player Prisoner’s Dilemma? Well, things become more complicated than before. Tit for tat, for example, cannot apply here. When I play against a single player, I know what he’s done and I react accordingly; but when I play against 20 tenants – eight who did not pay their fees and 12 who did – what would be my tit-fortat strategy? Follow the majority? Pay only after everyone else has paid? Could one tenant who pays be enough to convince me to pay too? This is very complicated, mathematically and intuitively, and so we let it go for now.
Chapter 7
PENGUIN MATHEMATICS
This chapter is devoted to animals – experts in playing games and stars of a field called Evolutionary Game Theory. We’ll discuss the seemingly strange behaviour of the Thomson’s gazelle as related to altruism, join a gang of penguins in their search for a volunteer, and learn about a nice definition from Evolutionary Game Theory that expands the Nash Equilibrium.
One of the Game Theory branches that I find most fascinating is known as Evolutionary Game Theory, which attempts to study and understand animal behaviour.
I was attracted to this field of study because, among other things, animals tend to be almost completely rational. Now, rationality is the very thing that encourages mathematicians to produce models that attempt to predict behaviours. And it’s nice to see how such models fit with natural phenomena.
One of the fascinating issues I addressed when I first started studying the application of Games Theory to animal behaviour was altruism.
In The Selfish Gene (1976) Richard Dawkins offers this definition: ‘An entity … is said to be altruistic if it behaves in such a way as to increase another such entity’s welfare at the expense of its own.’ That is, an act is considered altruistic when its results lower the altruist’s chances of survival. Dawkins is in effect trying to offer possible explanations of altruism, because the phenomenon seems to conflict with his own fundamental concept of the ‘selfish gene’. He argues that living organisms are mere survival machines for genes that want to move on to the next generation in a competitive world where egocentricity is advantageous. After all, if the living organisms’ only interest was to send their genes forward in time (we could say that self-replication is the only thing that genes care about), altruism should not have survived evolution and natural selection. Nevertheless, nature provides us with many examples of altruistic behaviour, such as the lioness that fights to protect its cubs. Dawkins spoke of a Thomson’s gazelle that leaps up and down (‘stots’) instead of running for dear life when a predator comes knocking: ‘This vigorous and conspicuous leaping in front of a predator is analogous to bird alarm calls, in that it seems to warn companions of danger while apparently calling the predator’s attention to the stotter himself.’ The gazelle’s behaviour could be viewed as self-sacrificing, or extremely risky; its only motivation is a wish to warn the pack. These are only two examples. Nature provides us with many more, from bees to monkeys.
As noted, at first glance altruism seems to contradict Dawkins’s selfish-gene theory, but in reality there’s no contradiction, since there’s no true altruism in the wild.
The lioness that fights for her cubs may be altruistic on the individual level, but genetically speaking her action is extremely egotistical – the creature is trying not so much to save its cubs as to protect its genes (or, rather, their carriers).
THE THOMSON SHOW
But how can the behaviour of the Thomson’s gazelle be explained? When a gazelle spots a prowling cheetah headed towards its herd, it sometimes leaps up and down, makes strange noises, and generally seems to attract the predator. Is that a good idea? Should it not just flee as other (apparently wiser) Thomson’s gazelles do? How can that be explained?
A while ago, zoologists believed that the ‘stotter’ is warning its group, but later they changed their minds. Professor Amotz Zahavi, another animal altruism researcher, believes that the bouncing Thomson’s gazelle isn’t trying to warn the pack but is actually sending a message (or a ‘signal’ in Game Theory language) to the predator, no less. Translated into human language, the message is: ‘Dear predator, look here. I’m a young and strong Thomson’s gazelle. Did you see how high I just jumped? Did you notice my graceful movements and agile body? If you’re really hungry, dear predator, you’d better chase another gazelle (or preferably a zebra), because you won’t catch me, and you’ll remain hungry. Listen to me: find yourself some easier prey, because I won’t be on your plate today. Sincerely, Stotter.’
So which is it? Is it true that the gazelle leaps to warn the group, as pre-selfish-gene theorists believed, or is it merely looking out for Number One?
There are two possible answers here. One would be a mathematical solution – the application of a potential and plausible model that attempts to describe a given situation to see where mathematics would take us. In most cases, this is quite complicated. The other solution is much simpler: see what the predator does in real life. Observations have shown that predators rarely go after prancing gazelles. Apparently, they get the message.
Once, when I lectured on mathematical models in the animal kingdom, a man in the audience stood up and said: ‘You got it all wrong, sir. The models you’re showing may be very nice, but they’re extremely complicated. I’ve never heard of a Thomson’s gazelle that’s familiar with differential equations or evolutionary games theory, and only very few lions took classes in functional optimization and analysis. They couldn’t possibly understand your lecture.’
I answered that all Thomson’s gazelles, and indeed all predators in existence, actually know quite a lot about Game Theory, differential equations and other mathematical models – only they don’t understand them the way we humans do. For example, though I’ve never heard of a snail who attended a class on logarithmic spirals, clearly all snails are quite skilled in making them, and do so beautifully. Bees build their hives optimally, though they probably don’t have a master’s degree in applied mathematics. Animals in nature attend a different school and have a wonderful teacher named Evolution. It’s a fantastic educator, but very tough too: if you fail, even once, you’re eliminated – not from school, mind you, but from nature as a whole. Though harsh, this school has the advantage of keeping the best students.
Suppose an uneducated rabbit wakes up feeling it must tap a wolf on the shoulder one day as a dare to himself. Evolution doesn’t think twice before eliminating that rabbit, because although he did surprise the wolf (and even enjoyed his prank), the naughty rabbit made a horrible strategic mistake. As a result, the rabbit’s genes, which made that mistake (if such behaviour on the rabbit’s part was indeed determined by its genes – this assumption is controversial), nicely line the wolf’s stomach and fail to reach the next rabbit generation.
I sometimes wonder what would happen in universities if students were kicked out for making one big mistake or several small ones. That would leave only a few students, but they would be the absolute best. Perhaps that’s not such a bad idea.
GAZELLE’S SWAN-SONG LEAP
All this
made me wonder. If the leaping strategy is so good, why don’t all Thomson’s gazelles ‘stot’ habitually? If they did, the cheetah that came to dinner would merely feast its eyes on an amazing sight: dozens of rejoicing Thomson’s gazelles jumping for joy because Mr Cheetah dropped by. Why are there no such shows in nature? The answer is simple. You’re at liberty to show off only if you can back it up. Yes, it’s easy for young Thomson’s gazelles to jump, but an older guy, who might jump high for his age, is not as agile as he used to be. He might injure his back in the most inconvenient moment, or land hard, sprain an ankle or even break a leg. The cheetah might be surprised by the Thomson’s gazelle’s incompetence, but soon the elderly stotter would turn into a snack.
PENGUINS AND THE VOLUNTEER’S DILEMMA
A wonderful documentary I watched many years ago on a TV nature channel showed a group of penguins arriving at the shore in search of food. Their diet exclusively comprises fish that, naturally enough, swim in the ocean. Penguins can swim there too. The problem is that so can seals, and penguins are their favourite dish. The best thing in such a case is to have a volunteer, a penguin who jumps in the water first to make sure the coast is clear – literally. It’s a very simple sink-or-swim test: if the volunteer comes out of the water and calls on his friends to join him, all will be fine; if the water turns red, no lunch today, at least for the penguins. Naturally, no penguin in his right mind would volunteer, so they all just stand around, waiting.
The mathematical model of that situation is an n-players game known as the Volunteer’s Dilemma. Strategically, this situation doesn’t yield the Nash Equilibrium, because if (you are a penguin and) one or more other volunteers present themselves, you shouldn’t step up. On the other hand, waiting around is neither the Nash Equilibrium nor a good option: how long can you and all the other penguins wait before starving to death? Now, if all the penguins choose the strategy of waiting forever, you’d be wise to volunteer, because you can only gain. If you stand with the gang on the shore, you’ll surely die; but if you jump in, either a seal will get you or, if there are no seals around, you’ll eat and live. Thus, volunteering actually gives you some chance of survival. At the same time, as we’ve already seen, all penguins would rather someone else jumped in first.
Note that the volunteering strategy is not the Nash strategy, because if everyone just dives in, the penguin that does so last takes no risk at all, since the seal is no longer hungry, having eaten a prompter penguin.
So should you jump in or not? The answer was quite simple, and all I had to do was wait to the end of the documentary. As it turned out, the penguins had several interesting strategies for such situations.
Strategy 1
War of Attrition
The penguins’ first strategy was to simply wait on the shore in an Arctic version of the ‘chicken’ game. They just stand there and wait for someone to go first. This is a war of attrition that the penguins wage among themselves. Eventually, someone dives in. It’s hard to tell how long they have waited. It could be seven hours, but the documentary editors kept only seven seconds of the original footage. At the end of all that suspense, realizing he’s going to remain hungry, one penguin decides to dive in. We can’t call that first diver a ‘volunteer’, because if he wanted to volunteer for his mates, he would have done so right from the start, and not strain everybody’s nerves like that. We could mathematically examine if and when a penguin should volunteer – it’s a question of probabilities, known as the ‘mixed Nash strategies’. It turns out that mathematics and reality sometimes go hand in hand, because the mathematical model predicts that someone always steps up, as they do in real life.
Strategy 2
The Slow Race
Another strategy is popular when the group of penguins is rather large: they all run into the water together. Let me try to explain this, even though I’ve never been a penguin, and thinking like one doesn’t come naturally to me. Here goes. Why would 500 penguins run into the ocean at the same time? What’s their guiding logic? Well, they may be telling each other (in genes language) that perhaps there’s no seal out there, which is wonderful. However, even if there is a hungry seal lurking, the odds of getting eaten are 1:500. That’s not so bad. The risk is reasonable, and penguins are willing to take it.
When I first saw that documentary, I remember thinking that this icy stampede is not a Nash Equilibrium, because if everyone runs into the water, particularly if there’s plenty of fish there, the penguin that plays the famous ‘shoelaces trick’ and hangs back, will gain. After all, on the odd chance that a hungry seal is lurking out there, by the time the unruly laces are tied again, the seal will have been satiated and the slow penguin is no longer at risk. Indeed, the film clearly showed certain penguins not running as fast as others, but there was no way of knowing if they were brilliant mathematicians or just poor athletes. After all, even if all penguins were created equal, some run faster than others. But if the penguins should start thinking about running ever slower, and all of them keep slowing down, in the end they’ll stand still, which would take us back to square one: all the penguins stand on the shore, no one volunteers, and the war of attrition begins again.
Strategy 3
Hey, Don’t Push!
The third penguin strategy in that film was the nicest and most amusing, at least to my taste. To explain this strategy I’d like to draw an analogy with a human situation involving soldiers.
After a month of intensive training, a company is about to go on home leave. As they line up for final inspection, their commander suddenly appears with grim news. One of the company soldiers must stay on base for guard duty. ‘I’ll be back in five,’ the officer says. ‘When I return, I want a volunteer to stay behind. If no one volunteers, no one goes on leave.’
The displeased soldiers and the penguins are now in a similar bind. Everyone wants someone to volunteer for the rest of them, and if no one does, no one eats – at sea or at mom’s table. The soldiers could draw straws or something, but penguins can’t draw anything, not to mention the problem of finding enough straws in Antarctica. Yet both the soldiers and the penguins find a solution.
One of the soldiers lined up for inspection is Max, who, like his comrades, is upset about the situation. A few seconds later, however, he recovers, slaps Little Joe on the shoulder and says: ‘Hey Joe, I say you volunteer.’ This is a very surprising move for Max. Clearly, it’s risky for both Joe and Max. I hope you can see that. I mean, as soon as Max volunteers Joe, the other soldiers might turn to him and gently suggest that he, not Joe, should sacrifice himself. Max’s move would be daring indeed, if it were not for a single fact: Max is the biggest guy in the company. He’s tall and broad-shouldered, and very strong too. All the soldiers know that all too well, which is why they now politely surround Joe. ‘What’s your problem, Joe? Max is telling you like it is. You’ll stay behind for the rest of us, and that’s that!’ Everyone wants big bad Max on their side, and Little Joe will probably volunteer, despite himself.
Strategy 3 has the penguins doing quite the same thing. After they’ve stood around on the shore for a few minutes, Penguin Max walks up to one of the smallest guys and slaps him on the back, hard. I don’t generally appreciate the humanization of animals, but I could actually see the surprised look on the small penguin’s face as it flies into the ocean. It was a most impressive concluding scene, up there with the endings of Casablanca and Some Like It Hot. In any event, the penguins produce a volunteer. It’s also important that we remember that Max was no ordinary penguin. Shoving another like that would be very risky for an average penguin, because when you raise your wing and push someone, you might lose your balance and another, stronger penguin could then give you a shove.
Considering the penguins’ predicament a little longer, we can see that they are playing a game within a game. On top of the volunteer-choosing game, they also play ‘who should I stand next to?’ The pushed penguin volunteers because he chose the wrong spot to stand on, too
close to Max. So remember: when playing a pushing game, stay away from the big guys.
It’s reasonable to assume that there’s almost always a strategic explanation for an allegedly altruistic behaviour in animals. Using mathematical tools from Evolutionary Game Theory, I once constructed a model that explains certain penguin realities without resorting to altruism. In truth, none of the penguin strategies included altruism. The penguin who lost the war of attrition, the one who came first in the slow race and the one who was pushed in – none of these ended up in the water for altruistic reasons. The pushing penguin took a chance because he could have lost his own balance, but he too doesn’t deserve to be called an altruist. By the same logic, the penguin who found himself swimming all alone doesn’t deserve a medal for volunteering under fire (or water, in this case) because he never intended to volunteer in the first place.
