All Pay Quality-Bids in Score Procurement Auctions

Abstract

We study score procurement auctions with all-pay quality bids, in which a supplier's
score is the difference between his quality and price bids. Equilibrium quality and
price bids are solved without first obtaining the corresponding equilibrium scores. In
particular, our approach accommodates the case with a minimum score or quality
requirement. When the convex e¤ort cost function takes a power form, a higher all-
pay component of the quality bid reduces quality provision, total surplus, and suppliers'
payoffs, but may increase or decrease the procurer's payoff. If the procurer reimburses
the all-pay components of losing suppliers or all suppliers, this would increase quality
provision and suppliers’ payo¤s, but reduce total surplus and the procurer's payo¤.
Finally, we rely on our approach to identify the procurer-optimal score rule, which
is quasi-linear in quality and price. Relative to the procurer's payff function, the
optimal score rule values quality less, and the score rule function increases in quality
at a lower rate than the procurer's payoff function. When suppliers' type distribution
has a weakly convex reverse hazard rate, the optimal score rule is more responsive to
quality or values the quality more when placing the quality bid gets more costly or
there are more suppliers.

Keywords: All-pay quality bids, Equilibrium analysis, Score auctions, Score procurements, Winner-pay quality bids

JEL Classification: C70, D44, D89, L12, O32