the entities and executes Ds (e.g., we will often write HA instead of H ( 5 ) A ). If time specifications are omitted, we assume the transfers happen at the optimal times, 4 i.e.,
the times that lead to highest expected utility. If we consider strategies with the same sequence of actions but different timings to be the same strategy, the agent has O( | E | k ) possible strategies to select from, where k is the maximum length of the strategy and | E | is the number of entities. Thus, the agent has a wide range of options, even if practical considerations lead to a reasonable upper bound on k and | E |. The agent must select the strategy that maximizes the overall expected utility of α .

In the rest of this section, we present a mathematical model of transfer-of-control strategies for AA and use that model to guide the search for a solution. Moreover, the model provides a tool for predicting the performance of various strategies, justifying their use and explaining observed phenomena of their use.  Section 3.1 presents the model of AA strategies in detail. Section 3.2 reveals key properties of complex strategies, including dominance relationships among strategies. Section 3.3 examines the E-Elves application in the light of the model, to make specific predictions about some properties that a successful AA approach reasoning for that application class will have. These predictions shape the operationalization of strategies in Section 4.

## A Mathematical Model Of Strategies 3.1

The transfer-of-control model presented in this section allows calculation of the expected utility (EU) of individual strategies, thus allowing strategies to be compared. The calculation of a strategy's EU considers four elements: the likely relative quality of different entities' decisions; the probability of getting a response from an entity at a particular time; the cost of delaying a decision; and the costs and benefits of changing coordination constraints. While other parameters might also be modeled in a similar manner, our experience with the E-Elves and other AA work suggests that these parameters are the critical ones across a wide range of joint activities.

The first element of the model is the expected quality of an entity's decision. In general, we capture the quality of an entity's decision at time t with the functions EQ = {EQf(t) :
R → R }.  The quality of a decision reflects both the probability that the entity will make an
"appropriate" decision and the costs incurred if the decision is wrong. The expected quality of a decision is calculated in a decision theoretic way, by multiplying the probability of each outcome, i.e., each decision, by the utility of that decision, i.e., the cost or benefit of that decision. For example, the higher the probability that the entity will make a mistake, the lower the quality, even lower if the mistakes might be very costly. The quality of decision an entity will make can vary over time as the information available to it changes or as it has more time to "think". The second element of the model is the probability that an entity will make a decision if control is transferred to it. The functions, P = {Pf(t) : R -> [0,1]},
represent continuous probability distributions over the time that the entity e will respond.

That is, the probability that ej will respond before time to is fto Pej (t)dt.

The third element of the model is a representation of the cost of inappropriate timing of a decision. In general, not making a decision until a particular point in time incurs some