10(34)
further required to be symmetric, that is, participants are assumed to rely
on the same basic decision function.
An equilibrium for the game thus defined exists but will in general depend
on what is assumed about the valuations of participants. In the simplest
case, namely independent private values (IPVs), the valuations are
assumed to be independent of one another. There are also common value
(CV) situations, where the values can be assumed to be the same for all
participants, and affiliated value situations, which represent intermediates
between IPV and CV situations. Under IPV conditions, the best strategy
b*(v) is given by
b*(v) = (n – 1) ∫ x f(x) F(x)
(n-2)
dx / F(v)
(n-1)
,
where integration is between 0 and
v
. Another way of expressing this
strategy is
b*(v) = v – ∫ F(x)
(n-1)
dx / F(v)
(n-1)
.
It is possible to see the optimal bid as the actual valuation v minus a
correction term stemming from the interaction with other bidders. It is
clear to see that the optimal is bid is closer to the actual valuation as the
number of bidders rises and that the two will meet as the number of
bidders approaches infinity.
If the assumption of risk neutrality is relaxed and replaced by risk aversion,
bidding will become more aggressive and the price will be higher. Relaxing
the assumption on independency of valuations complicates the analysis
somewhat, but the deduction is similar.
In the standard model setting, procurement is simply assumed to be the
mirror image of auctions. Higher bids correspond to lower prices and so on.
Problems with the standard model
Even a quick comparison of the standard model with the list of factors
presented in the previous section indicates a serious mismatch. The
standard model has a fairly narrow focus on the number of bidders and
distribution of their valuations. Company policies and order intake are
virtually absent, although the order situation may enter indirectly via the
degree of risk aversion.
Even more troublesome is the fact that the assumptions made on
information available to participants are unrealistic or even demonstrably
false. Both the number of participants and the distribution of their
valuations are assumed to be known. It may be reasonable (although not
entirely unproblematic) to assume that the number of bidders is known in
a classical auction, but this is conspicuously inadequate in procurement,
where secrecy concerning the number and identity of tenderers is central to
the procedural design. A limited number of studies–for instance, Matthews
(1987), McAfee and McMillan (1987a, b), Levin and Smith (1994), Levin
and Ozdenoren (2004), and De Silva et al. (2009)–have been devoted to
the problem of tenders under uncertainty with respect to the number of
participants. They support qualitatively the common-sense hypothesis that
