Bid cost: the eighth invitation is the first that loses money

A bid costs 20,000 to prepare and 150,000 if it wins. One chance in seven still pays 1,429; one in eight loses 1,250. Seven holds and the eighth invitation is the first that loses money. The page names no number to invite.

Live result

Invitation 8 is the first loss; 7 still pays.

Expected profit at 7$1,428.57
Expected profit at 8−$1,250.00

At your count of 7, preparing a bid pays.

Givens

Try a scenario

Bid preparation cost: Bid preparation cost
$; From 0.01 to 10,000,000, in one currency.
Expected margin if won: Expected margin if won
$; After costs, from 0.01 to 1,000,000,000 in the same currency.
Count invited: Count invited; A whole count from 1 to 30.
Chance of winning: one in the count
Preparation cost: paid win or lose
Margin: same whoever wins

Expected profit from bidding at each count invited

Expected profitLast paying countFirst losing count

Profit falls as the invited count grows; a loss carries a minus sign in the same currency.

Count invited

invited

Frominvited

Toinvited

In steps ofinvited

It does not check the values in between.

Across the values sampled, the verdict on preparing a bid stays a profit from 1 invited, the lowest value sampled to 7 invited. The nearest value where it changes is 8 invited, where it becomes a loss.

The answer	a profit
Values sampled	1 invited to 20 invited, 20 values, step 1 invited
Holds from	1 invited, lowest sampled
Holds to	7 invited
Nearest value that changes it	8 invited
The answer there	a loss

What an unchanged region does and does not prove is set out in a short lesson.

Every admissible integer in this range was checked. Values outside the range remain unchecked.

The same count, read from two seats

For the bidder, each extra invitation spreads the same margin across another chance while preparation still costs the same.

For the buyer, a larger invitation list can leave fewer bidders expecting a profit.

Where this stops

The page calculates expected profit from your values, names no number to invite and predicts nobody's decision. It shows what a mechanism does under stated rules and says nothing about any public procurement regulation.

Here is the maths

Scale: Linear on both axes, with zero drawn
Basis: One equal chance of winning among those invited, independent of who else bids
Assumptions: One margin and one preparation cost, the same for every count read
Model version: 1.0.0

The formula, in words

The margin is shared across equal chances of winning. Subtract the preparation cost, paid regardless of the outcome, to find expected profit from bidding.

E(n)=\frac{m}{n}-c

where $E(n)$ is the expected profit from bidding, $m$ the margin if the bid wins, $n$ the count invited, and $c$ the cost of preparing the bid.

The count that stops it

The last paying count keeps expected profit above zero:

n^{*}=\max\left\{\,n\ge 1:\ \frac{m}{n}>c\,\right\}

The next count is $n^{*}+1$ . An exact whole ratio breaks even there; the last paying count is one below it.

A worked example

The extra invitation turns a profit into a loss with preparation costing 20,000 and winning worth 150,000.

The ratio marks where profit reaches zero:

\frac{150{,}000}{20{,}000}=7.50

The last whole count below it still pays:

n^{*}=7

Profit at that count, in the same currency:

E(7)=\frac{150{,}000}{7}-20{,}000=1{,}428.57

At the next count, profit becomes a loss:

E(8)=\frac{150{,}000}{8}-20{,}000=-1{,}250.00

Show the full maths

Weight both outcomes before counting entrants

Winning leaves the margin less preparation cost; losing leaves only the preparation cost to pay. Weight each result by its chance:

E(n)=\frac1n(m-c)+\left(1-\frac1n\right)(-c)=\frac mn-c

This uses the same cost whether the bid wins or loses.

Preparation pays only when expected profit is strictly positive. Both the cost and the invited count are positive, so rearranging keeps the inequality’s direction:

E(n)>0\iff\frac mn>c\iff n<\frac mc

The last paying count is therefore the largest whole number strictly below the margin divided by cost:

n^*=\max\!\left(0,\left\lceil\frac mc\right\rceil-1\right)

where

n^{*}

is the last paying count. Rounding the ratio up and subtracting one handles both whole and fractional ratios. Zero means that no positive invited count pays.

Break even is a different condition:

E(n)=0\iff n=\frac mc

It occurs at an invited count only when the ratio is a whole number. At that count there is no profit, so it is excluded from the paying counts.

The worked example has a fractional threshold, so seven pays and the next whole count loses:

\frac{150{,}000}{20{,}000}=7.5\qquad n^*=\lceil7.5\rceil-1=7

\begin{gathered}\frac{150{,}000}{20{,}000}=7.5 \\ n^*=\lceil7.5\rceil-1=7\end{gathered}

With the same preparation cost and a margin of 140,000, the threshold is whole; six is the last paying count and seven breaks even:

\frac{140{,}000}{20{,}000}=7\qquad n^*=\lceil7\rceil-1=6\qquad E(7)=0

\begin{gathered}\frac{140{,}000}{20{,}000}=7 \\ n^*=\lceil7\rceil-1=6 \\ E(7)=0\end{gathered}

The same count, in the other lesson

One more bidder thins expected profit here and deepens the gap between the lowest estimate and true cost in the linked lesson. More competition can discourage entry while making the lowest bid more likely to underestimate cost.

Why the lowest bid tends to be an underestimate

What this does not model

No unequal bidders or changing preparation costs. No learning, reserve price or bid subsidy. No prediction of entry.

What the sweep checks

Only the count changes, in whole steps; each verdict uses the same margin and preparation cost as the finding.

Source basis

Nicola Dimitri, Gustavo Piga and Giancarlo Spagnolo, Handbook of Procurement: costly preparation changes the participation a buyer can expect; Paul Milgrom, Putting Auction Theory to Work: entry costs meet diminishing gains from another entrant. Paul Klemperer, Auctions: Theory and Practice: entry responds to incentives and can lower common value auction revenue.

Calculation version

1.0.0

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Your margin, preparation cost and count determine the figures; the page neither assesses bidders nor recommends inviting, bidding or staying away.

Bid cost and the count invited