When reason says ‘we’. Martin Hollis and the pitfalls of individualism
A simple game can throw a spanner in the works of a great idea. The game is straightforward. There are six coins on a table. Two players, Adam and Eve, take turns. On each turn, each player may take one or two coins. If they take one coin, the game continues and it is the other player’s turn. If they take two coins, the game ends immediately and the remaining coins disappear. Adam goes first. What should he do? The intuitive answer would seem obvious. If both take one coin at a time, they’ll play until the end and split the spoils: three coins each. If, on the other hand, Adam takes two coins straight away, the game ends immediately. In this case, Adam gets two and Eve gets none. If the game were to reach Adam’s final turn, however, when there are two coins left on the table, he would have a good reason to take them both. He would, in fact, end up with four coins in total, whilst Eve would be left with two. So, the final outcome, if the game were to actually reach the very last turn, would not be three each, but four to two. This is where the problems begin. Eve knows, in fact, that if she were to let Adam reach the final turn, he would choose four to two instead of three to three. Therefore, on the previous turn, Eve would have a very good reason to stop the game early. But Adam also knows that Eve knows this. So, on the turn before that, he would have a reason to pre-empt her. Using the same logic, we can work backwards, step by step, right back to the first decision point. The result is paradoxical. Adam should take two coins straight away and end the game. Two to nil. In their attempt to maximise their own gain, Adam and Eve condemn themselves to a worse outcome for both of them than the one they could have achieved had they simply been able to continue until the final decision node.
Game theorists call it the centipede game, the ‘centipede game’. Coined in 1981 by Boston University economist Robert Rosenthal, we have discussed it in the simplified version presented by Martin Hollis in his Trust within Reason (Cambridge University Press, 1998). The name derives from the shape of the game’s graphical representation: a sequence of nodes, like a succession of legs. Its importance lies not in the quirkiness of the example, but in the precision with which it forces the theory of rational choice to spell out its own assumptions. The players are perfectly rational. Their preferences are complete and consistent. Each knows that the other is also rational. Each knows that the other knows, and so on. This is the condition of common knowledge of rationality. In a world constructed in this way, the reasoning by ‘backward induction’ (backward induction) seems irrefutable, its logic absolutely compelling. Yet the conclusion it produces strikes us as absurd.
Martin Hollis emphasises precisely this point. The problem is not that Adam and Eve are selfish in a psychological sense. There is no need to imagine them as wicked, greedy or morally corrupt. The pay-offs in the game are in fact ‘utils’, that is, aggregate measures of preference that incorporate everything that matters to each player, including, where applicable, altruism, empathy, friendship, etc. If Adam cares about Eve’s satisfaction, this will already be included in his utility function. If he feels sympathy, guilt or a desire for approval, these elements too are already factored into the calculation. Therefore, simply adding a touch of altruism is not enough to make the problem go away. If, after incorporating all relevant motivations, Adam still prefers four to two over three to three, standard theory dictates that he should take the two coins at the final node. And from there, the backward induction begins again.
Hollis calls this conclusion the ‘sting’ of the centipede (the centipede’s sting). The game does not merely show that cooperation can be fragile. It reveals something even more troubling: namely, that a certain conception of rationality renders precisely the behaviour that appears most reasonable to be irrational. With a million coins on the table, writes Hollis, it would be scandalous if reason were to advise stopping the game immediately. And indeed, in real-world experiments, participants do not behave in this way. Some see it through to the end; many get almost all the way there. But to dismiss this finding by saying that human beings are less rational than the standard model assumes would be too easy. Perhaps, Hollis suggests, the problem is not that real people reason poorly. Perhaps it is the model that captures only part of what we consider to be rational.
The difficulty becomes even clearer if we imagine that Adam, in the first round, takes not two coins but just one. What should Eve conclude? If Adam were the perfect maximiser predicted by the model, he would have had to end the game immediately. The fact that he does not seem to indicate that he is not that sort of agent. But then Eve might think that if Adam is not a standard maximiser, perhaps she can continue down this path. However, if continuing is rational because Adam has shown a willingness to cooperate, then Adam’s initial move might have been rational. But if it was rational, then Adam is still a rational agent, and if he is a rational agent, the backward induction should hold. Hollis describes this short-circuit as follows: if Adam is rational and plays outside the predicted equilibrium, Eve’s rational choice is not to cooperate; but if Adam is not rational, it may become rational for Eve to cooperate; and then Adam’s move becomes rational once again. The reasoning does not come to a close. It spirals.
