Demonstration 1: Lower branch

This demonstration explores the form of the utility function in the lower branch of the overall tree, from the second agent's perspective. There is in fact only one choice node in the subtree that has more than one alternative, and for it a decision surface is expected between ask and chat. The only source of variation in each of the alternatives is the have-seat node, and so the following solution is found for the decision surface:

$$\begin{split}&u(ask) = u(chat) \\ \Rightarrow &-5 + 85 + 10q = 87 \\ \Rightarrow &q = 0.6 \end{split}$$ (4.13)

Along the intend(book-flight-window) axis, the utility function is expected to be linear since there is a weighted sum over the two invariant sub-branches. The plot from the planner's output is shown in figure 4.16. The decision surface is just visible.

Figure 4.16: Utility of strategies against P(intend(book-flight-window) and P(bel(have-seat))

To further verify the results, the corner points of the belief space were each checked, and were found to be consistent with the planner's output. intend(book-flight-window) is in the x position, and bel(have-seat) is in the y position.

• At 0,0, the user doesn't intend to have a seat, chat is expected to be chosen with an outcome of $100$ for the reward plus a negative cost of $1 + 1$ for the chats, totalling $102$

• At 0,1 the user doesn't intend to have a seat, so again chat is chosen giving $102$

• At 1,0, the user intends to have a seat but doesn't ask since he believes there is none available, giving $85 + 1 + 1 = 87$

• At 1,1 the user intends to have a seat and does ask, giving $100$ for success $+ 1$ for chat $- 5$ for ask and $- 5$ for give, totalling $91$