A self-referential situation is one in which the forecasts made by the human agents involved serves to create the world they are trying to forecast. Such co-evolutionary systems have also been called reflexive and are generally self-reinforcing. They have attracted attention in several socio-economic contexts (Arthur 1994; Casti 1999; Batten 2000). A key feature of these systems is that the best thing to do (such as to go or not to go) depends on what everyone else is doing. As no individual agent knows that, the best thing that any agent can do is to apply the strategy or mental model that has worked best so far.

In self-referential systems, the mix of possible strategies is co-evolving incessantly over time, so much so that the choices made by individual agents seem to have little impact on collective outcomes. Sometimes, collective populations of agents exhibit emergent regularities. At other times, they display different types of complex behaviour. In the next section, we describe the El Farol bar problem, an early agent-based simulation of a complex adaptive system reported by Brian Arthur (1994).

Consider a system of N = 100 agents deciding independently each week whether or not to go to their favourite bar (called the El Farol) next Thursday. Space is limited, so the evening is enjoyable only if the bar is not too crowded (say N_max = 60). There is no collusion or prior communication among agents. Knowing the bar attendance over the past few weeks, each bar-loving agent simply decides independently to go if he expects less than N_max to attend or stay home if he expects more than N_max to go.

This problem is a metaphor for a broad class of self-referential situations: e.g. urban traffic congestion, canteen crowding or queue lengths at big sporting events. It has some interesting properties. First, if a decision model existed that agents could rely upon to forecast attendance, then a deductive solution would be possible. No such model exists. Irrespective of past attendance figures, many plausible hypotheses could be adopted to predict future attendance. Because agents’ rationality is bounded, they are forced to reason inductively. Second, any shared expectations will be broken up. If all agents believe most will go, then nobody will go. By staying home, that common belief will be destroyed. If all agents believe few will go, then all will go, thus undermining that belief. Because any mental model that is shared by most of the agents will be self-defeating, agents’ expectations must always differ.

Perplexed by the intractability of this problem, Arthur created a computer simulation in which his agents were given attendance figures over the past few months. He also created an alphabetic soup of several dozen predictors replicated many times. After randomly ladling out k of these to each agent, each kept track of his k different predictors and decided whether to go or not according to a preferred predictor in his set. This preferred predictor could be chosen in a variety of ways, although Arthur adopted the most accurate current predictor for each agent in his simulations.

Once decisions have been made in Arthur’s simulated bar, agents learn the new attendance figure, updating the accuracy of their own set of predictors. Then decisions are made for the following week. In this kind of problem, the set of predictors acted upon by agents—called the set of preferred predictors—determines the attendance. But the attendance history also determines the set of preferred predictors. We can think of this set as forming a kind of ecology (John Holland’s term). Of interest is how this ecology evolves over time.

The simulations show that weekly attendance fluctuates unpredictably, but mean attendance always converges to 60 in the long run. The predictors self-organise into an equilibrium pattern or ecology in which (on average) 40 per cent of the preferred predictors forecast above 60, and 60 per cent below 60. This 40/60 split remains although the population of preferred predictors keeps changing in membership. The emergent ecology is rather like a forest whose contours do not change, but whose individual trees do. Similar results appeared throughout Arthur’s experiments, robust to changes in the types of hypotheses.

There is another intriguing result. Although the computer-generated attendance results look more like the outcome of a random process than a deterministic one (see Figure 4.1), there is no inherently random factor governing how many people attend. Weekly attendance is a deterministic function of the individual predictions, themselves being deterministic functions of the past attendance figures.

Thus the bar problem is a relatively simple example of an emergent, self-defeating system. It is a situation in which a system of interacting agents can develop collective properties that are not at all obvious from our knowledge of the agents themselves. Even if we knew all agents’ individual idiosyncrasies, we are no closer to anticipating the emergent outcome. Under the influence of a sufficiently strong attractor, individually subjective, boundedly rational expectations self-organise to produce a kind of collectively rational behaviour (Arthur 1994).

Given that agents do communicate with each other in the real world, we wonder what may happen if they are permitted to exchange information? Bruce Edmonds’ work allows communication among agents before they make their final decisions whether or not to go to the bar (Edmonds 1999). Using a genetic programming algorithm to simulate adaptive learning, he allows each agent to competitively develop its models of what the other agents are going to do. Although the beliefs and goals of other agents are not known or represented by each agent, heterogeneity among the agents emerges in the form of non-uniform tactics and role-playing identities. These collective properties are features that emerge purely from the micro-dynamics.

Like the Prisoners’ Dilemma, the bar problem is currently receiving more attention outside economics—as a metaphor for learning and bounded rationality. It has inspired a new literature in statistical physics on a closely related problem known as the minority game. A minority game is a repeated game in which N (odd) players have to choose one out of two alternatives (say A and B) at each time step. Those who happen to be in the minority win.

Seemingly simple at first glance, the game is subtle in the sense that if all players analyse the situation in the same way, all will choose the same alternative and therefore all will lose. Thus, players behave heterogeneously over time. Moreover, there is a frustration since not all players can win at the same time (this is an essential mechanism for modelling competition).

Since its introduction, this game has generated incredible interest. As there are only three parameters, the game is suitable for detailed numerical studies and analytical descriptions. Although each alternative is unspecified, for A or B one could read ‘I am going to the bar’, or, ‘I am staying home’; ‘I am choosing the motorway’, or, ‘I am going by the scenic route’; ‘I am going to fish at the usual place’, or, ‘I am going to find a better spot’. Perhaps the most striking properties of the minority game are that:

The minority game is an abstraction of the bar problem. Like simulations of the bar problem, numerical simulations of this game have displayed a remarkably rich set of emergent, collective behaviours (Challet and Zhang 1998).

Both the bar problem and the minority game contain the key elements of a complex adaptive system: firstly, a medium number of agents—a number too large for hand-calculation or intuition, but too small to use statistical methods applicable to very large populations; secondly, these agents are intelligent and adaptive, making decisions on the basis of rules of thumb or heuristics, like the bar predictors (needing to modify these rules or come up with new ones if necessary, they reason intuitively); and, finally, no single agent knows what all the others are (thinking of) doing, because each only has access to limited information.