Value of information example

As a concrete example of value of information, consider a problem of designing a kitchen assistant robot who must negotiate efficiently with a user about the various alternatives in jointly preparing a meal. Figure 5.1 illustrates the plan library for this problem, which is duplicated to populate each level of the belief model. The problem is one where agent 1 must decide whether to use some eggs in making an omelette. To make the decision it must find out whether agent 2 has fruit, since if agent 1 uses up the eggs, and agent 2 has fruit, his decision would cause agent 2's make-pavlova plan to fail.

Figure 5.1: Plan library for a value of information example

The utility function for the problem is defined by:

utility(make-omelette,200).
utility(make-fish,150).
utility(make-pavlova,100).

The game tree generated for this problem is illustrated in figure 5.2.

Figure 5.2: Game tree for the value of information example

This game tree matches the general form of a value of information problem, in that there are two alternatives, whose utility is a function of some belief, and there is a decision surface in the belief space corresponding to that belief. In particular, the utility for the upper branch of the game tree is given by:

$$200$$ (5.1)

For the lower branch, the utility function is

$$\begin{split}&150 + p.100 + (1-p).0 \\ \end{split}$$ (5.2)

At the true extreme of the belief, the agent chooses the lower branch. At the false extreme of the belief, the agent chooses the upper branch. A decision surface occurs where the two functions have an equal value, at 0.5. If the information about the belief is given to the agent, the expected utility value is a weighted sum of these two extremes, according to the prior value of the belief. This is:

$$p.200 + (1-p).250$$ (5.3)

If functions 5.1, 5.2 and 5.3 are plotted together, the result is as shown in figure 5.3. The area enclosed in the triangle indicates the utility gained from obtaining information. The gain is a maximum in the middle of the range, where the belief is least certain.

Figure 5.3: Comparison of expected utilities before and after information is obtained