Elicited bid functions in (a)symmetric first-price auctions

“joy of winning” as studied by Goeree et al. (2002) in the context of QRE models. In the next
section, we check whether any of the EBR, the RNNE and the RANE outperforms the others
in explaining the shapes of the submitted bid functions.

3.2.3. Shapes of Bid Functions

As subjects submitted complete bid functions in every round, the classification of bid
functions in different shapes is straightforward. We categorize them into four possible shapes:
Concave, Convex, Linear, and Humped. We assume a bid function to be concave if (s₁ - s₂)∕s₁
> .05 and s2 ≥ 0, where s1 and s2 stand for the slopes of the first and second segments of the
piecewise linear bid function. We define a bid function as convex if (s1 - s2)/s1 < -.05; as
linear if |(s1 - s2)/s1| ≤ .05 and as humped if (s1 - s2)/s1 > .05 and s2 < 0. A comparison of the
relative frequencies of each shape to the relative frequencies of EBR shapes allows an
additional qualitative assessment of the extent of strategic behavior.

3.2.3.1. Asymmetric Treatments

Figure 3 reports the relative frequencies of each shape in the asymmetric treatments, together
with the proportion of submitted bid functions that have a shape matching the one of the EBR
in a particular round t. In both treatments, the shapes of observed and EBR functions are
mostly convex for Strong bidders (about 49% of all their bid functions) and concave for Weak
bidders (about 49% of all their bid functions). Convex-shaped bid functions for Strong
bidders match the Nash equilibrium predictions and represent 82% of all EBR functions. In
both treatments, such matched cases with convex bid functions accounted for about 40% of
the Strong bidders’ observed strategies. The upper panels of Figure 4 report the plots of
average convex bid functions and indicate that Strong bidders did not low-ball enough when

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