# Indistinguishability Operators Applied to Task Allocation Problems in Multi-Agent Systems
## 1 Introduction  
The distribution of a determined number of tasks among a group of agents is a problem intensely studied in different fields, such as economics or robotics (we refer the reader, for instance, to [1; 2; 3] for a deeper treatment of the topic). It consists in allocating a collection of labors on a number of agents in the most efficient way, i.e., in such a way that the best agent is selected to perform each one of the labors to be carried out. This problem, commonly referred to as the task allocation problem, is still an open issue in real environments where the agents have a limited number of resources to obtain the optimal allocation. One of those challenging environments is the one formed by two or more autonomous robots that cooperatively perform a common mission, from now on referenced as multi-robot systems. Among all the methods proposed to address the task allocation problem, this paper focuses on swarm methods, which are inspired by insect colonies where an intelligent behavior emerges from the interaction of very simple skills running on each agent (see [4; 5]). Concretely, this work focuses on the so-called Response Threshold Method (RTM). In these methods each involved agenthas an associated task response threshold and task stimulus. The task stimulus value indicates how attractive the task is for the agent and its threshold is a parameter of the system. Thus, an agent starts the execution of a task following a probability function, referenced as response function and denoted as \(P\), that depends on both aforementioned values. As the probability of executing a task only depends on the current task, or state, the decision process can be modeled as a probabilistic Markov chain. Some examples of methods based on Response Threshold can be found in [6; 7; 8; 9]. This classical probabilistic approach presents some well known disadvantages (see [10]), for instance problems associated with the selection of the probability function (response function) when more than two tasks are considered, asymptotic converge to a system's stable state, and so on. Due to the inconveniences, in [10] new possibilistic theoretical formalism for a RTM was proposed. The RTM is implemented considering transition possibilities (response functions) instead of transition probabilities (response functions) and possibilistic Markov chains (also known as fuzzy Markov chains) instead of the classical probabilistic ones. The theoretical and empirical results demonstrated, among other advantages, that fuzzy Markov chains applied to task allocation problems require very few steps to converge to a stable state. As will be proved for the first time in this paper, the most widely used response functions are a specific kind of mathematical functions called indistinguishability operators. This work demonstrates that indistinguishability operators, in general, are useful in modeling the response probability function in response threshold task allocation problems in those cases in which the stimulus of each agent of the group only depends on the distance to each task that must be carried out. Hence, the aim of this paper from a theoretical point of view is twofold. On the one hand, we will use a well-known technique to induce indistinguishability operators in order to provide a few examples that could be useful in modeling response functions. On the other hand, we will show that two popular response functions exposed in [11] can be reproduced from indistinguishability operators. Thus, this paper provides a new systematic method to generate response functions. This paper extends the previous work in [10; 12] in order to apply the indistinguishability operators as possibility transition functions to allocate tasks in a multi-robot system with fuzzy Markov chains. As tested in the aforementioned references, each robot uses a fuzzy Markov chain to decide the next task to execute.