4.1.2. Definition 2 (Collision Score)

We use the resource experience results from Definition 1 to calculate the *collision scores*. This is defined from the view point of collision probability, that must be constrained under some threshold. Let


The *collision score* of the agent *j* is defined as the probability of how likely a collision occurs to the agent *j* on *at least* one of the resources given its resource experience *D<sup>j</sup>* :

$$P\left(\text{Col}\_{\hat{\jmath}}\middle|D^{\jmath}\right) = 1 - \prod\_{k \in \{1, \ldots, L\}} \left(1 - f\_k(D\_k^{\jmath})\right). \tag{8}$$

Note that *P Colj Dj* calculates the *complement* of the success probability—the joint probability of being tolerable at all resources.

Figure 2 shows two example designs of *f* : *f*1(*D*) = *sigmoid*(*D* − *δ*), with a discontinuity point *f*1(0) = 0, is a sigmoid-based CDF function, featuring a surge in the collision score (the derivative is bump-shaped) at the experience value around *δ*. This function is suitable to important resources that are sensitive to the agent; *f*2(*D*) = *min*(1, *D*/(4*δ*)) is a linear CDF with a shallow slope (the derivative is flat). This function can apply to trivial resources that are not very sensitive to the agent but still accumulate to contribute to the collision score. We use the offset parameter *δ* to adjust the *tolerance level* of the dissatisfying experience. With larger *δ*, the agent will tolerate a longer unpleasant experience before announcing a collision.

Although the definition of the collision score can be customized according to different practices, the probabilistic definition of collision score introduced here is a general one: Different types of resources may have different value ranges, and Equation (8) standardizes the resource ranges, mapping them to a value within [0, 1] and enabling an efficient integration of different types of resources to the framework.

**Figure 2.** Example designs of cumulative distribution functions (CDFs), mapping the resource experience *D* of an agent to a collision probability on certain resource. *f*1: sigmoid-based CDF for important (sensitive) resources. *f*2: linear CDF for trivial (insensitive) resource. *δ*: offset parameter adjusting the *tolerance level*.

#### 4.1.3. Definition 3 (Soft-Collision Function)

Now, according to the collision scores from Definition 2, we want to pick out the above-threshold agents and place them into the soft-collision set via the *soft-collision function* for the purpose of applying the sub-dimensional expansion.

Given a path *π* = *π* (*vs*, *vb*) and corresponding resource experience *D<sup>j</sup>* for the agent *j*, the *soft-collision function* of the agent *j* is

$$\left(\widetilde{\psi}^{j}\left(v\_{b}\right)\right) = \begin{cases} \ \left\,, \ for \, P\left(\text{Col}\_{j}|D^{j}\right) \ge T\\ \mathbb{O}\_{\prime} \qquad \text{otherwise} \end{cases}\tag{9}$$

where *T* is the *threshold of collision*. The definition of the *global soft-collision function* is then defined as

$$\tilde{\Psi}\left(\upsilon\_b\right) = \bigcup\_{j \in I} \tilde{\Psi}^j\left(\upsilon\_b\right). \tag{10}$$

Based on Definition 3, we can formally construct the soft-collision constraint on common resources and obtain the soft-collision constrained MAPP problem:

$$\begin{aligned} \min\_{\pi} & g\left(\pi\left(v\_{\mathbf{s}}, v\_{d}\right)\right) \\ \text{s.t.} & \\ & \check{\psi}\left(v\_{p}\right) = \bigcirc & \forall v\_{p} \in \pi. \end{aligned} \tag{11}$$

This problem setting is general and can be utilized to express the hard collision setting in Equation (2) by setting *T* = 0 or changing the condition inside the indicator function of Equation (6) to *Ak*(*e j pq*) <sup>=</sup> *<sup>A</sup><sup>j</sup> k*(*e j pq*) with infinite cost.
