Self-Fulfilling Prophecies, Quasi Non-Ergodicity and Wealth Inequality

Abstract

We construct a model where people trade assets contingent on an observable signal that reflects public opinion. The agents in our model are replaced occasionally and each person updates beliefs in response to observed outcomes. We show that the distribution of the observed signal is described by a quasi non-ergodic process and that people continue to disagree with each other forever. Our model generates large wealth inequalities that arise from the multiplicative nature of wealth dynamics which makes successful bold bets highly profitable. The flip side of this statement is that unsuccessful bold bets are ruinous and lead the person who makes such bets into poverty. People who agree with the market belief have a low expected subjective gain from trading. People who disagree may either become spectacularly rich, or spectacularly poor.

Exact time-dependent dynamics of discrete binary choice models

“…There remains, however, to see how such memory effects could arise naturally from interactions. It should be noted, as highlighted in [19], that under a timescale of order 1/2ε the model we have described here is not ergodic: in the high imitation regime one may think that one of the two choices is optimal because it has been made all the time so far, but this may be only because under that timescale the ants are self-consistently 'trapped' in one given choice, and one has not had the time to observe a full collective switch.…”

“…Although inspired by an example coming from behavioural biology, this model, and others that are very similar, has been used to explain behaviour in financial markets [14][15][16], firm agglomeration [17], the dynamics of fishing boats [18] and even wealth inequality [19]. The model is in fact also identical to the Moran model in genetics [20], and is also closely related to the Pólya urn model reviewed in [21].…”

Decision-making with distorted memory: Escaping the trap of past experience

“…The probability of escaping such a trapping state can obviously depend on the value of noise in the decision. (19). Parameters as in Fig.…”

“…11, in a large part of the highnoise regime (19) provides a good long-time approximation to the numerics. This observation is supported by close agreement of fitted parameters from the numerics and the corresponding theoretical approximation; both the power-law exponent and the prefactor are well predicted by (19), as shown in the insets of Fig. 11.…”

“…Indeed, there is a link between a simplified version of our model and the well-known elephant random walk [17]; as we explain later, both are mathematically Pólya urns [18] with the distinction that in our case there is a non-linear dependence on previous steps. Other work exploring the connections between Pólya urns and reinforcement mechanisms in economics can be found in [19]. In particular, such models help highlight the importance of ergodicity-breaking, which is related to the distinction between ensemble averages and averages over trajectories.…”