Verification of results

In this section, the plots obtained in the last section are explained. First, the pass and tell alternative (see figure 5.9) is examined. Pass is followed by a chance node, with each branch of the chance node giving agent 2 a choice between ending the dialogue and telling. There is a third alternative, pass, but it is always dominated by end. In the true branch, agent 2 is expected to use tell-true only when the value for have-fruit is likely to be false at level 4. Similarly, tell-false is used when this value is likely to be true.

Figure 5.11: Utility of (a) end (b) pass

Figure 5.12: Difference image between pass and end

Figure 5.11 plots the utility of pass and end, with respect to the have-fruit belief at levels 2 and 4, while figure 5.12 plots the difference in utility between them. A quadrant-by-quadrant analysis of this difference plot is now used to explain the results.

• level 2 low, level 4 low This is where agent 2 most likely prefers omelette, because the level 2 belief is low and agent 2 believes that agent 1 chooses omelette, because the level 4 belief is low. Since both beliefs are low, there is generally agreement between the agents, but as the value of the second level belief increases, there is a greater chance that agent 2 actually prefers fish, therefore it tells agent 1 and obtains a small gain in utility in figure 5.11 (b) over the corresponding quadrant in figure a. If this value increases over the 0.5 mark, there is a sudden change where agent 1 begins to choose fish without being told, and so the tell act is wasted.

• level 2 high level 4 high This is where agent 2 most likely prefers fish, and agent 2 believes that agent 1 chooses fish. Since both beliefs are high, there is generally agreement between the agents, but as the value of the second level belief decreases, there is a greater chance that agent 2 prefers omelette, therefore it tells agent 1 and obtains a gain in utility. Once the 0.5 line is crossed, agent 1 chooses omelette without being told and so the tell is wasted.

• level 2 low level 4 high This is where agent 2 most likely prefers omelette, but agent 2 believes that agent 1 chooses fish. agent 2 tells him so that it will change to omelette. However, since it generally chooses omelette whether it is told or not, there is a slight loss in this quadrant. The loss eases as the level 2 belief increases since agent 2 tends towards choosing fish.

• level 2 high level 4 low

  This is where agent 2 most likely prefers fish, and agent 2 believes that agent 1 chooses omelette. agent 2 tells him so that it will change to fish. Since it would have chosen fish anyway, there is a slight loss in this quadrant. This eases off slightly as the second level belief decreases.

An easier but incomplete way of verifying these results is to analyse the corner points. At (0,0) agent 2 should reason that since agent 1 already has the same beliefs about have-fruit, the dialogue should end, and make-omelette be chosen at 200. By the same token, at (1,1) the dialogue ends with make-fish and make-fruit-salad chosen at 250. At (0,1), agent 2 needs to always correct agent 1's mistaken belief, resulting in make-omelette chosen at 200, with 10 taken away for the cost of always telling. Otherwise, make-fish would have been chosen, giving 150 and no fruit salad. Similarly at (1,0 ), the belief is corrected, resulting in make-fish and make-fruit-salad at 250 with 10 taken away, instead of a mistaken choice of make-omelette at 200.

Apart from the [pass, tell] strategy, agent 1 can use an [ask, tell] strategy ( see figure 5.9, and 5.10 ). The difference between the two strategies is the cost incurred in using an ask instead of a pass, and a small belief revision change at level 4 to the have-fruit belief that is caused by the dry-land algorithm. This belief revision change makes do difference to the second agent's decision to tell. Therefore, the ask plot has a similar form to the pass plot.

This demonstration has shown how using initiative in holding the floor can depend on a probabilistic model of belief. Figure 5.10 shows how the decision depends on the probability value in the belief model. It is interesting as well to note from these figures the sensitivity of the decision of whether to continue with an ask or pass or whether to end the negotiation. Another note of interest is the deep nesting required to decide whether an ask is efficient. Notice in figure 5.10 that for the agent to decide between asking and ending the negotiation, the belief model must be examined to a depth of level 4.

Figure 5.13: Difference image between (a) request(tell(P)) and end, and (b) propose(tell(P)) and end