Bidding rules that guarantee procedural fairness may induce more equilibrium bidding and moderate other-regarding concerns. Here, we investigate procedural fairness as in Güth (2011). In our experiment, we assume commonly known true values and only two bidders to implement a best-case scenario for other-regarding concerns. The two-by-two factorial design varies ownership of the single indivisible commodity and the price rule. We find more equilibrium behavior under the procedurally fairer price rule, although this does not completely crowd out other-regarding concerns. In addition, the limits of procedural fairness are much more important than expected.