  Work extraction in quantum finite systems is an important issue in quantum
thermodynamics. The optimal work extracted is called ergotropy, and it is
achieved by maximizing the average work extracted over all the unitary cycles.
However, an agent that is non-neutral to risk is affected by fluctuations and
should extract work by following the expected utility hypothesis. Thus, we
investigate the optimal work extraction performed by a risk non-neutral agent
by maximizing the average utility function over all the unitary cycles. We
mainly focus on initial states that are incoherent with respect to the energy
basis, achieving a probability distribution of work. In this case we show how
the optimal work extraction will be performed with an incoherent unitary
transformation, namely a permutation of the energy basis, which depends on the
risk aversion of the agent. We give several examples, in particular also the
work extraction from an ensemble of quantum batteries is examined. Furthermore,
we also investigate how work extraction is affected by the presence of initial
quantum coherence in the energy basis by considering a quasiprobability
distribution of work.
