In this section we depart from the assumption that bidders are symmetric and have independent private costs. We discuss how asymmetries and cost interdependence will affect the outcome. This is of relevance as bidders are often asymmetric in the sense that one has a better cost structure or is an incumbent with more information about the market (less uncertainty). Also, there are situations where bidders’ costs are interdependent which means that ex-post, bidders have the same cost realization but ex-ante this realization is unknown to each bidder. It is interesting to see how limited liability will affect the bidding behavior in such environments.
Asymmetries
It is a well known result in auction theory that if asymmetric bidders compete in an FPSB-auction, the weakness of a disadvantaged bidder leads to more aggressive bidding. This is bad news for the agency if bidders have limited liability, as more competition increases the risk of non-fulfillment. The logic behind this result is the following: assume that two bidders with different cost functions compete in an FPSB-auction. Maskin and Riley (2000a) show that in this case, an increasing equilibrium bidding function exists under certain conditions. If a strong bidder’s (indexed by S) cost term distribution dominates the weak bidder’s (indexed by W) in the sense of reverse hazard rate dominance (which implies first order stochastic dominance), the bidders’ bids are distributed the same way. The stronger bidder’s bid distribution will be lower than the weaker bidder’s. Knowing this, the weaker bidder will bid more aggressively, i.e. a weaker bidder will bid less than a stronger
bidder for each cost realization. Maskin and Riley (2000b) show that βW(c) < βS(c)
bidder has a lower equilibrium bid distribution but his bidding strategy is such that
he bids more than the weaker bidder for each cost term.49 Hence, the stronger
bidder can, even if he has a lower cost term, sometimes lose the auction as the weaker bidder bids more aggressively. The results for the SPSB-auction are not affected by asymmetries, it is still a weakly dominant strategy to bid the cost term and—in contrast to the FPSB-auction—this format is efficient in the sense that the bidder with the lowest cost term always wins.
Transferred into our framework, this would mean that the agency should not use the FPSB-auction as it does not allocate the contract to the most efficient bidder and it increases competition which in turn leads to a higher non-fulfillment rate. The result that the SBSP-auction might be preferred to the FPSB-auction is in contrast to most of the empiric and theoretic results where the FPSB-auction— without limited liability—is preferred as it gives the right bias towards the weak bidder.
Common costs
A crucial point of our analysis so far is the independent private cost assumption. Private means that each bidder’s costs depend only on his own type and independent means that there is no statistical dependence between the types. The other extreme are pure common costs where all bidders have the same ex-post cost realization but
different ex-ante signals about the true realization.50 For instance, the cost to finish
a project is identical for the bidders but their estimates (signals) differ, i.e. their information is of different quality.
To give an example how cost interdependence will affect the outcome in our framework, we follow the analysis of Bulow and Klemperer (2002) and investigate the almost common cost case for 3 symmetric (or asymmetric) bidders in a single-unit
49A good discussion of this topic is given in Milgrom (2004), pp. 149-155.
50In many situations the costs of the bidders are interdependent, i.e. bidders have common or
(or multi-unit) English auction.51 In a common-cost environment, the winner is the
bidder with the most optimistic signal. Conditional on winning without updating the belief about the true costs, this signal would be too low on average and the winning bidder will lose money on average. To avoid this winner’s curse, bidders will use any information to update their beliefs about the true cost realization of the project and bid more cautious. For example, in the English auction an exit of a bidder means that the beliefs of the remaining bidders about the true cost realization are lower. This is bad news as these might be too optimistic. Hence, the remaining bidders bid more cautiously. Does this affect the outcome in our framework? Note first that the possibility to declare bankruptcy can be regarded as an insurance against the winner’s curse. However, also bidders with limited liability will take account of the winner’s curse—although maybe not as much as without limited liability—as they have no interest in lowering their survival rate too far.
Assume that bidders’ costs have a common-cost part (e.g., market factors like exchange rates) and a private-cost part (e.g., efficiency levels) and the error term ∆ is part of the private costs. Also with common costs bidders ignore the error term, i.e. bid below possible cost realizations. If bidders are symmetric, then bidders bid
cautiously in respect to their common cost signal to avoid the winner’s curse.52 If
the agency procures two units, the expected payment will be higher than with one unit as there is less competition and the probability of bankruptcy is reduced.
But this is no longer true for the asymmetric case. If a bidder has a very large private cost advantage (e.g., a better production technology), the other bidders have to bid very cautiously. Because if a disadvantaged bidder wins against the advantaged bidder, then his signal about the common costs must have been very optimistic. Thus, the advantaged bidder almost always wins. If bidders are restricted to bid only for one unit, selling two units reduces the winner’s curse for the second unit as only bidders 2 and 3 compete for this unit. And as bidders 2 and 3 bid more
51The results of Bulow and Klemperer (2002) and of the example in appendix A.2 hold if hazard
rates are increasing.
aggressively due to a reduction of the winner’s curse, bidder 1 is now not much more likely than bidder 2 or 3 to be the winner of the auction. Therefore, bidder 1 will also have to bid lower (to win the first unit) and the expected payment will decrease. As procuring two units instead of one leads to a reduction of the winner’s curse and more aggressive bidding, reducing the winners curse might lead to an increase in risk.