There are three major sources of intractability in Equation 1--the alternation of the maximization and expectation operators (allowing decisions to be conditioned on an exponential number of possible sets of holdings), the large number of maximizations (forcing an exponential number of decisions to be considered), and the large number of expectations (resulting in sums over an exponential number of variable settings).

We attack the problem of interleaved operators by moving all but the
first of the maximizations inside the expectations, resulting in an
expression that approximates the value:

Thus, Equation 2 can be written:

where is identical to except the -th component reflects whether item is won--.

Note that there is a further approximation that can be made by
computing the expected prices (as point values) before solving the
optimization problem. This approach corresponds to further swapping
the expectations towards the core of the equation:

The technique of swapping maximization and expectation operators was previously used by hauskrecht97 hauskrecht97 to generate a bound for solving partially observable Markov decision processes. The decrease of uncertainty when decisions are made makes this approximation an upper bound on the true value of the auction: . The tightness of the approximations in Equations 2 and 5 depends on the true distributions of the expected prices. For example, if the prices were known in advance with certainty, then both approximations are exact.