given the specific model in Appendix I, is:

$$f(K)<\omega(\exp^{-\mathcal{D}_{\varepsilon\,\mathrm{div}\,\varepsilon}\omega}-1)\times(\frac{\rho}{\delta}\exp^{-\delta T}-\frac{\omega}{\delta}\exp^{-\delta\Delta}-\exp^{\omega\Delta-\rho T})\tag{8}$$

Depending on the nature of f ( K ), Equation 8 can hold for any number of Ds, so, for any K t there will be some conditions for which a strategy with K D s is optimal. For instance, in Section 5.3, we show that the maximum length of the optimal strategy for a random configuration of up to 25 entities is usually less than eight actions.

Equation 8 illustrates how the value of an additional D can be limited by changing D cost :
but Lemma 3 also shows us that other factors can affect the value of an additional D . For example, even with a constant Dcost, the value of an additional D depends on how many other D actions the agent performs. Figure 4 shows that the value of the D depends on the rate at which wait costs accrue. If the rate of wait cost accrual accelerates over time (e.g., for the exponential model), a D action slows that acceleration, rendering a second D action less useful (since the wait costs are now accruing more slowly). Notice also that Ds become valueless after the deadline, when wait costs stop accruing.

Taken together, Lemmas 1-3 show that no particular transfer-of-control strategy dominates all others across all domains. Moreover, very different strategies, from single-shot strategies to arbitrarily complex strategies, are appropriate for different situations, although the range of situations where a particular transfer-of-control action provides benefit can be quite narrow. Since a strategy might have very low EU for some set of parameters, choosing the wrong strategy can lead to very poor results. On the other hand, once we understand the parameter configuration of an intended application domain, Lemmas 1-3 provide useful tools for focusing the search for an optimal transfer-of-control strategy. The Lemmas can be used off-line to substantially reduce the space of strategies that need to be searched to find the optimal strategy. However, in general there may be many strategies and finding the optimal strategy may not be possible or feasible.

## 3.3  Model Predictions For The E-Elves

In this section, we use the model to predict properties of a successful approach to AA in the E-Elves.  Using approximate functions for the probability of response, wait cost, and expected decision quality, we can calculate the EU of various strategies and determine the types of strategies that are going to be useful. Armed with this knowledge, we can predict some key properties of a successful implementation.

A key feature of the E-Elves is that the user is mobile. As she moves around the environment, her probability of responding to requests for decisions changes drastically, e.g., she is most likely to respond when at her workstation. To calculate the EU of different strategies, we need to know P + (t), which means that we need to estimate the response probabilities and model how they change as the user moves around. When Friday communicates via a workstation dialog box, the user will respond, on average, in five minutes. However, when Friday communicates via a Palm pilot the average user response time is an hour. Users generally take longer to decide whether they want to present at a research meeting, taking approximately two days on average. So, the function P T ( t ) should have an average value of 5 minutes when the user in her office, an average of one hour when the user is contacted via a Palm pilot and an average of two days when the decision is whether to present at a