Decision table for two-period model

In practice, it is very difficult to draw the decision tree for multiple-period models due to the large number of actions available to the decision maker. In the case of the two-period model (the simplest multiple-period case), the players can choose from the following three kinds of option for each city that has not yet started its fibre roll out—finishing the first stage of construction for this city, finishing the whole construction for this city or waiting.

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decision tree, which is very messy (the reader may compare it with the decision tree of our basic model (which has only 16 states) to imagine how messy the decision tree will be). To illustrate the possibilities, we illustrate the actions available once the constructions of two cities have been completed, meaning that the firm only needs to consider two remaining cities when it is making the decisions (There are only city HH (high demand city with high cost) and city LH (low demand city with high cost) left, which dramatically reduce the number of states from 81 to 9).46 Since the reduction of city number only exists to

demonstrate the firm‟s decision marking, it does not affect the valuation and results of the general model.47 We summarize players‟ decisions in table 5.2.

Table 5.2: Decision table for two-period two cities model

{0,0} {0,1} {0,2} {1,0} {2,0} {1,1} {1,2} {2,1} {2,2} {0,0}          {0,1}       {0,2}    {1,0}       {2,0}    {1,1}     {1,2}   {2,1}   {2,2}

In Table 5.2, each column represents the possible future states in the next period, while each row represents the possible current states. The first number in the bracket indicates the stage of construction for city HH, while the second number indicates the stage of construction for city LH. For example, {0, 0} in a row indicates that the player has started neither city at the current stage; {0, 0} in a column indicates that neither city will be started even in the next period. Ticks have been used to indicate the possible transitions between current states (row states) to corresponding future states (column states). Consider the following example. If the

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Since there are three choices that can be made for each city and two types of city left, the number of total states equals .

47 _{Although we only provide the decision tree for the two-period two cities case since it is complicated enough,} the decision tree for any more complicated models (Such as two-period four cities and three-period four cities, etc.) could be derived by the same method.

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player is in the first row at date 0 (which is {0, 0}), he has not started any of these two cities, therefore one action he may take is waiting, which is the first tick; Or he may start to build up one of these two cities but not go full speed, which corresponds to the second tick if he is doing city LH or the fourth tick if he is doing city HH; Or he may start to build up one of these two cities with the full speed, which corresponds to the third tick if he is doing city LH or the fifth tick if he is doing city HH; Or he may start to build up both cities, which

corresponds to the last four ticks (depending on different levels of speed). All above possible behaviours are motivated by the player‟s economics consideration. That is, by waiting, the player has a chance to observer whether the demand goes up or down; by investing in two stages at once (full speed), although the player incurs the higher capital expenditure, he can earn the revenues earlier. The player chooses its optimal behaviour according to the trade-off between capital expenditures and revenues. All his future states are determined by which transitions are taken from date 0 to date 1.

Now assume the player already took the action {0, 1} at a previous period, hence he is in row 2 now. Once again, he can choose to wait to see whether demand goes up or down, which corresponds to the suspension option (the first tick in this column); or he can earn the revenues earlier by finishing city LH, which corresponds to the second tick in this row; or he may start city HH, which corresponds to the third tick if he only starts the first stage or the fifth tick if he is trying to finish both stages (which the player is exercising a speed-up option); or he may finish city LH and start city HH simultaneously, which corresponds to the fourth tick if he just starts the first stage of city HH or the sixth tick if he is working through both stages of city HH (once again, the player is choosing to exercise a speed-up option).As above, the optimal behaviour of the player depends on the trade-off between its capital expenditures and revenues.

Similarly, we can derive all possible transactions in row three, row four…, and row nine. The process continues until either the player complete all constructions (that is the ninth row {2, 2}) or the development right expires. At these points no more decisions need to be made by the firm and all future cash flows can be estimated without the decision table.