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dependent (Liebowitz and Margolis 1995). First is an irreversible path because to be left require some cost, but at the same time it appear optimal and so doesn't provoke a market failure. Second and third paths occur when there is imperfect information. Second path happens when a non optimal decision is taken in the beginning. This decision is taken because information aren't known and when they are discovered is too late to change the path. In this case we can identify inferiority of path and results are unpleasant, hard and expensive to change. Third case introduce the same hypothesis of the second path but the path should be remediable while not done. Besides in this case outcomes cannot be predicted and we can find the Lock-in effect, because we have a market failure indeed there is the persistence of certain choices. Like in the paper from Liebowitz and Margolis (1995), path dependences can be easily explain as a mathematical probability. We need to consider two different choice (A1 and A2). Those are mutually exclusive and are taken like a sequence. We have two probabilities indicate with Pn and (1-Pn) where letter n indicate the number of attempt. At the attempt number i correspond an event Ei. The probability P(n+1)= f(Pn,En,En-1,a€|,E1). The response probability is said to be 'd-trial path dependent' where the probability function can be written f=f(Pn,En,En-1,a€|,En-d). In the special case where d=0, the process is 'path independent'. All of this concept are connected with the concept of memory; in fact today's decision are constrain by past decision.

递增收益-Increasing return

As Brian Arthur (1989) said, path inefficiency and market failure can occur only when there are some kind of increasing returns. Increasing returns and lock-in effect can be easily explain by an example. Like in the Table 1, we can see benefits for adopters according to the total number of adoption of technology. In this example there are two alternative technology: A and B. The first adopters will have more benefits adopting the technology A, but on the long term technology B lead more benefits than A. if we choose technology A we can't return or switch to Technology B thanks to externalities and network effects, and this is Lock-in effect.

Table 1: Adoption payoffs, Source adapted from Arthur Table 2 (1989).

This is a banal example where we have agents at the same level and events can make no different. First person choose best and most profitable technology and this cause the increasing of adoption this technology (in our example Tech. A). This can continue and all agents prefer Tech. A to Tech. B that lead to a dominance of Tech. A and an failure of Tech B. This result can be easily predictable, but we can see in table 1, if returns increase at the same rate, that adoption process can become path-inefficient. We can see that at thirty choice of adoption Tech. A have a similar adoption of Tech. B. Then if the process proceeded and the number of adoption increase we can see that market became more 'locked-in'.

The effect of increasing returns to lock-in can be also explained by the use of the graph in Figure 1 taken from the paper of Unruh (2000): the horizontal axis represent the degrees of adoption while the vertical measure the increasing of relative benefits for adopters. The curve in the graph show that returns of scale are not linear with the increasing of t