  We introduce a novel axiom of co-loss aversion for a preference relation over
the space of acts, represented by measurable functions on a suitable measurable
space. This axiom means that the decision maker, facing the sum of two acts,
dislikes the situation where both acts realize as losses simultaneously. Our
main result is that, under strict monotonicity and continuity, the axiom of
co-loss aversion characterizes preference relations represented by a new class
of functionals, which we call the duet expectiles. A duet expectile involves
two endogenous probability measures, and it becomes a usual expectile, a
statistical quantity popular in regression and risk measures, when these two
probability measures coincide. We discuss properties of duet expectiles and
connections with fundamental concepts including probabilistic sophistication,
risk aversion, and uncertainty aversion.
