The one-session model described in section 2 is a considerable simplification of the way buyers search for sellers. As is commonly observed in several markets with the sort of structure we are modelling here, customers that refuse a deal with one seller, usually shop around to find other offers. Indeed this is generally regarded as the principle motivation for refusal in standard search models. An alternative explanation is that customers refuse now in order to induce better offers in the future. In either case, to take this into account, we have to consider a model in which customers are given at least two occasions to purchase goods.
One further assumption to relax particularly in the case of perishable goods is the idea of a constant price for all sessions. In fact p is the price sellers would charge at each transaction if they were sure to sell exactly all the quantity they bring to the market. If they were able to predict precisely how many customers would visit their shop and accept this price, they would know exactly how much to supply. But, when their forecasts are not perfect they may not have the appropriate quantity, given the number of possible buyers they actually face at the close of the market. It might therefore, in this case, be better for them to sell at a lower price rather than to keep goods that they are not, by assumption, able to sell the next day. We ran the simulations with a constant afternoon price which is the morning price lowered by a factor . A more intelligent choice for the sellers would be to monitor previous fluctuations of the number of buyers and to decrease afternoon price in proportion.
To summarise then, we divide the day into two periods:
Sellers arrive in the morning with a quantity Q of the good corresponding to the number of customers they expect times q, plus some extra quantity of that good in case they have more customers than expected. The profit they expect from this additional amount is that obtained by satisfying new customers or unexpected former customers who might appear.
Buyers have to decide every morning whether to buy at the morning price or to wait for a better price in the afternoon. Of course waiting has a trade-off: they might not find anything to buy in the afternoon and thus make no profit. They choose an action according to their expectations of the average afternoon profit with respect to what they would get by buying in the morning, which they know from equation 2. Average afternoon profit is estimated from their past history of afternoon profits. We used in the simulations a simple quadratic fit of the afternoon profit as a function of morning prices. But for all reasonable choices of afternoon prices and extra supply by the sellers, expected afternoon profits for buyers are much smaller than morning profits, essentially because their chances of finding goods in the afternoon were smaller than in the morning. We discovered that even with their primitive prediction abilities, buyers soon (say after 50 time steps) realised that they would do better to accept the morning offers. For the present, we have not investigated further the issue of refusals by buyers.
Numerical simulations show that the introduction of a second session does not change the qualitative behaviour of the system: a low disordered regime and a high ordered regime still exist with the same characteristics as in the one session model. But the time to eventually reach the ordered regime and the width of the transition are increased. Estimated is 20 percent lower with two sessions than with one.
A change induced by the introduction of an afternoon session is that divorces are observed in the ordered regime for a much wider range of the learning parameter , for instance as soon as is larger than 0.1 . This is because on the occasion of an infidelity, since a buyer has a much better chance of making a higher afternoon profit with a new shop that has extra supplies, she then runs faster across the simplex.