Environmental Engineering Reference
In-Depth Information
a continuum ranging from no rain at all to a lot of rain, but to keep this
game simple, consider these two extremes. The two nodes for nature (N.1
and N.2) are chance decision nodes as there is uncertainty on what nature
will do. In chance decision nodes, all branches deriving from them have a
probability distribution equal to 1 (for example, b 1 1 − b 5 1). The agro-
pastoralist can have some information on these probabilities, such as what
the rainfall was last year, however, there is still uncertainty on what in fact
will happen.
The letters or numbers, in this case X 's at the terminal nodes, repre-
sent the payof s 2 of each decision path. These payof s will depend on the
utility functions of the players in the game. In this case only the actions
of nature will have an ef ect on the payof s for the agro-pastoralist and
not vice versa. Therefore, only one payof is sketched. In the case that
two or more players have a payof contingent on the i nal scenario,
these payof s will be written at the end of the terminal nodes separated
by a comma, the i rst letter or number being the payof for player 1, the
second letter or number the payof for player 2 and so forth. In the game
considered here, if the agro-pastoralist settles and it rains, she or he will
receive X 1, if he or she settles and there is drought, she will receive X 2
and so on.
In order to i nd a solution for this game, we have to compute the
expected payof s at each node, starting from the right and moving to the
left. This commonly used method is called solving by backward induction.
In this game we thus have to compute the expected payof s at N.1 and
N.2, that is, the expected payof s from settling and moving respectively.
To compute the expected payof s of a decision under uncertainty, one has
to multiply the payof by the probability of each payof and add them. 3
For example, the expected payof from settling in this game would be:
Ep ( S ) 5 b( X 1) 1 (1 − b)( X 2). The same would have to be done for the
other available strategies for player 1. These two payof s would have to
be computed and compared in order to derive a preferred strategy. 4 To
identify the preferred strategy, an arrow in the branch that represents the
preferred strategy can be drawn. To make this clearer, numerical payof s
to this example are given below.
Consider the hypothetical payof s for the agro-pastoralist from each
of her or his available strategies as those depicted in Figure 10.2. These
payof s would have to include all benei ts and costs (for example, mon-
etary costs from moving, opportunity costs, and so on) derived from each
of her or his strategies given the actions of nature. That is, the payof s
rel ect the net benei ts of all contingent scenarios.
Given the payof s from Figure 10.2, the expected payof for the
agro-pastoralist from settling would be:
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