Another important qualitative result of the mean field approach is the existence of hysteresis effects: buyers might still have a strong preference for one shop that offered good deals in the past, even though the current deals they offer are less interesting than those now offered by other shops.
Let us come back once more to the case of two shops 1 and 2, and now suppose that they offer different prices and hence different profits and . Replacing profit in equations 11 and 12 by respectively and , equation 13 becomes:
When is above , the three intersections remain as long as the difference in profits is not too large. Which of the two extreme intersections is actually reached by the learning dynamics depends on initial conditions.
Figure 1: Hysteresis of fidelities. Plot of both fidelities versus , the profit to be obtained from shop number 1 when the profit to be obtained from shop number 2 is held equal to 1. ( and ). The thick lines correspond to stable equilibria for both fidelities, and , and the thin lines, existing when is around , to unstable equilibria. In the three solutions region, the larger value of is reached from initial conditions when is already large. Thus if is decreased from above one, is kept large (and is kept small) even when becomes less than . The stability of this metastable attractor is lost when . In a symmetrical manner, the high attractor existing at low can be maintained up to . (the figure was drawn using GRIND software, De Boer 1983).
Thus, as illustrated on figure 1, buyers can remain faithful to a shop asking for a higher price (which results in a lower profit for the buyer), provided that they became attached to this shop when it practiced a lower price. When the most often frequented shop changes its prices, the fidelity to that shop describes the upper branch of the fidelity versus profit curve (figure 1). The fidelity remains on the upper branch as long as it exists, i.e. until the point where the slope is vertical. When profit decreases beyond that level, a sudden and discontinuous transition to the lower branch occurs. This is the point when customers change their policy and visit the other shop. But, if the first shop reverses its high price/low buyer profit policy when fidelity is on the lower branch, the transition to the higher branch only occurs when the slope of the lower branch becomes vertical, i.e at a higher profit than for the downward transition.
A consequence of this phenomenon, is that in order to attract customers who are faithful to another shop, a challenger has to offer a profit significantly greater than the profit offered by the well established shop: when preference coefficients have reached equilibrium in the ordered regime, customers switch only for differences in profits corresponding to those where the slopes of the curves in figure 3 are vertical (i.e. not when profits are equalised!!). In other words, economic rationality (i.e. choosing the shop offering the best deal) is not ensured in the region where hysteresis occurs.