  We study a participatory budgeting problem of aggregating the preferences of
agents and dividing a budget over the projects. A budget division solution is a
probability distribution over the projects. The main purpose of our study
concerns the comparison between the system optimum solution and a fair
solution. We are interested in assessing the quality of fair solutions, i.e.,
in measuring the system efficiency loss under a fair allocation compared to the
one that maximizes (egalitarian) social welfare. This indicator is called the
price of fairness. We are also interested in the performance of several
aggregation rules. Asymptotically tight bounds are provided both for the price
of fairness and the efficiency guarantee of aggregation rules.
