All Pay Quality-Bids in Score Procurement Auctions

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 effort 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’ payoffs, but reduce total surplus and the procurer’s payoff. 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 payoff 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.