# Indistinguishability Operators Applied to Task Allocation Problems in Multi-Agent Systems
## 2 Swarm Task Allocation: Response Threshold  
In this section we will introduce the main concepts of classical (probabilistic) Response Threshold Methods (RTMs) and we will motivate that their response functions can be assimilated to indistinguishability operators. It must be recalled that classical RTMs are modeled using probabilistic Markov chains.  
As was mentioned in Section 1, one way to model the probability transition function of the aforementioned Markov chain is by means of the so-called stimulus and response thresholds. Concretely, the stimulus expresses the need perceived by the agent to develop a task and the threshold determines the tendency of an agent to respond to stimulus intensity and, therefore, to make the task. In [11], a method was proposed, based on response functions, to model the aforesaid probability when the response threshold is fixed over time. In the aforesaid reference, the probability response function \(P(s,\theta)\) can be defined by  
\[P(s,\theta)=\frac{s^{n}}{s^{n}+\theta^{n}}, \tag{1}\]  
where \(s\) denotes, for each agent, the intensity of a stimulus to carry out a particular task and \(\theta\) denotes the threshold for each agent and task. Notice that, according to [11], \(n>1\) (with \(n\mathbb{N}\)) determines the steepness of the threshold. Of course, the numerical value \(P(s,\theta)\) can be interpreted as follows. On one hand, values of the stimulus intensity much smaller than threshold (denoted by \(s<<\theta\)) implies response values (probabilities of engaging task performance) close to 0. On the other hand, stimulus intensity much greater than the threshold (denoted by \(s>>\theta\)) means probability of engaging task performance close to 1.  
Other authors have used response functions of type (1) in order to model the probability of engaging tasks performance in multi-robot task allocation. We can find an instance in [14] where a mathematical model was proposed to assign particular events to individual robots in such a way that each robot is limited to one task at time. Concretely, they assume that each robot senses the need to handle the closest task. In this direction, the stimulus produced by a task \(e\) for a robot \(r\) was taken as the inverse of the distance between the task and the robot, i.e., \(s=\sigma(r,e)=\frac{1}{d(r,e)}\). Then, the probability response function is formulated as follows.  
\[P(s,\theta)=\frac{\sigma(r,e)^{n}}{\sigma(r,e)^{n}+\theta^{n}}. \tag{2}\]  
As was pointed out in [14], the best performance was achieved with the inverse of the expected distance between tasks \(D\), i.e., \(\theta=\frac{1}{D}\). In this case, \(s<<\theta\), or equivalently \(d(r,a)>>D\), implies low response to engage the task, and \(s>>\theta\), or equivalently \(d(r,e)<<D\), implies high motivation to take on the task.  
A straightforward computation yields that (2) can be transformed into the response function  
\[P(s,D)=\frac{D^{n}}{D^{n}+d(r,e)^{n}}, \tag{3}\]  
Notice that the expression (3) maintains the essential properties of the response function (2). It must be stressed that these kind of response functions have been recently applied to possibilistic multi-robot task allocation problems (see [10] for more details).  
Expression (3) has motivated this paper, since, as we will show in Section 4, \(P(s,D)\) is an indistinguishability operator (see Section 3).