Bidder Strategies 679steadily with its bid. Looking at the last column of the table, we see that the
firm’s expected profit is minimal (1) for very low bids, which have little chance
of winning, and (2) for bids near its reservation price, since these generate lit-
tle profit. Expected profit is maximized at a bid of $328 thousand, which the
firm predicts has a 49 percent winning chance. This is firm 1’s optimal bid.
The firm’s probability assessment of its winning chances usually is based on
its past bidding experience: how often its bids have won auctions against vary-
ing numbers of competitors in the past. A useful way to think about the firm’s
winning chances is in terms of the distribution of the best (i.e., highest) com-
peting bid. The probability of the firm’s winning is simply the probability that
the best competing bid (BCB) is smaller than the firm’s own bid. Figure 16.1
shows the graph of the cumulative distribution of BCB (labeled H). The curve’s
height indicates the probability that the best competing bid is smaller than the
value shown on the horizontal axis. For instance, at $320 thousand the height
of the curve is .25, meaning that there is a .25 chance that the highest com-
peting bid will be lower than $320 thousand (and, of course, a .75 chance that
it will be higher than this value). The median of the BCB distribution is about
$328 thousand (actually, very slightly higher). There is a 50 percent probabil-
ity of BCB being lower than the median value.
The BCB distribution curve is important because it precisely measures the
firm’s winning chances for its various bids. Thus, a bid at the distribution median
has a .5 chance of winning the auction because half the time the best compet-
ing bid will be below this value. According to the BCB curve, a $320 thousand
bid has a .25 chance of winning, and so on. Using the BCB curve, there is a simple
geometric means of describing the firm’s optimal bid. In Figure 16.1, a vertical line
has been drawn at the firm’s reservation price, $342 thousand. Suppose the firm
chooses bid b. Then the firm’s profit, if the bid wins, is $342 thousand b. This
profit is measured by the horizontal distance between $342 thousand and b.
The probability that the bid wins the auction is given by the height of the curve
H(b). It follows that the firm’s expected profit, (342 b)H(b), is measured by
the area of the rectangle inscribed under the BCB curve.
Maximizing the firm’s expected profit is equivalent to choosing a bid that
maximizes this rectangle’s area.^6 The figure shows the firm’s optimal bid, $328
thousand, and the associated inscribed rectangle. This rectangle has a larger
area than any other possibility. In general, the same geometric procedure can
be used to determine an optimal bid for any reservation price the firm might
hold. The right side of the expected-profit rectangle simply is set at the reser-
vation price. (For instance, if the firm’s value is $330 thousand, we can find
the best bid to be $320 thousand after some experimentation.)(^6) The interested reader may wish to apply calculus to find the optimal bid. A bidder with value v that
submits bid b earns an expected profit of E() (v b)H(b). Marginal profit is MdE()/db
(v b)h(b) H(b), where h(b) dH/db is the density function of the BCB distribution. Setting
Mequal to zero, we find b v H(b)/h(b). The optimal bid is below the firm’s value, v, and
the size of this discount is given by H(b)/h(b).
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