  We consider a new variant of the multi-robot task allocation problem -
Inverse Risk-sensitive Multi-Robot Task Allocation (IR-MRTA).
  "Forward" MRTA - the process of deciding which robot should perform a task
given the reward (cost)-related parameters, is widely studied in the
multi-robot literature. In this setting, the reward (cost)-related parameters
are assumed to be already known: parameters are first fixed offline by domain
experts, followed by coordinating robots online. What if we need these
parameters to be adjusted by non-expert human supervisors who oversee the
robots during tasks to adapt to new situations? We are interested in the case
where the human supervisor's perception of the allocation risk may change and
suggest different allocations for robots compared to that from the MRTA
algorithm. In such cases, the robots need to change the parameters of the
allocation problem based on evolving human preferences. We study such problems
through the lens of inverse task allocation, i.e., the process of finding
parameters given solutions to the problem. Specifically, we propose a new
formulation IR-MRTA, in which we aim to find a new set of parameters of the
human behavioral risk model that minimally deviates from the current MRTA
parameters and can make a greedy task allocation algorithm allocate robot
resources in line with those suggested by humans. We show that even in the
simple case such a problem is a non-convex optimization problem. We propose a
Branch $\&$ Bound algorithm (BB-IR-MRTA) to solve such problems. In numerical
simulations of a case study on multi-robot target capture, we demonstrate how
to use BB-IR-MRTA and we show that the proposed algorithm achieves significant
advantages in running time and peak memory usage compared to a brute-force
baseline.
