Superiority-Seeking and the Preference for Exclusion

4.2 Study 2: Auctions

We now proceed to test the predictions of our framework in the competitive setting of first-price auctions. Our model makes predictions on the effects of exclusion that run counter to the classic results on competing for a private good. This facilitates a test of superiority-seeking in a setting that is both conceptually interesting and important for applications. Additionally, because the environment always involves the allocation of a single good, a positive effect of exclusion also allows us to rule out a host of alternative explanations related to changes in supply, such as scarcity being a signal of quality, as well as explanations based on consumption externalities or on scarcity of the opportunity to participate per se.

4.2 Method

Participants (⁠ $N = 274$ ⁠) were recruited from a university-wide pool to take part in experiments on decision-making. As in the first study, sessions comprised groups of $M \in {4, 6, 8}$ and participants were assigned lab stations 1–12. Participants earned $15 as part of the same unrelated study and then told that they may get the opportunity to participate in an auction. Conditional on having the opportunity, participants could use up to $15 to bid on a good through a first price, sealed-bid auction. The good was a custom T-shirt as in Study 1. Participants would write down their bid privately on a sheet of paper. The highest bidder would receive the T-shirt and pay their bid. Everyone else would not receive the T-shirt and not pay anything.

Participants took part in one of three treatments. The Random Exclusion treatment was conducted in a similar way as in Study 1. Participants whose lab stations matched the outcome of the die roll would not have the opportunity to bid for the good and relinquished their bid sheets. The rest of the group reported their bids, which the experimenter then collected. The number of active bidders per session thus corresponded to $N = M - K$ ⁠.

In the Non-Random Exclusion treatment, participants arriving to the lab were told about the T-shirt as in the other conditions but not about the auction. Each was then asked to indicate the extent to which she would want to own the good on a scale of 1–10. Once these scores were collected, participants learned about the auction and that $K = \frac{M}{2} - 1$ people from the group would not have the opportunity to participate. However, unlike in the Random Exclusion treatment, exclusion was based on participant’s ex ante liking of the good: the K individuals who least wanted to own the T-shirt were not given the opportunity to bid for it and this was made common knowledge. The actual liking scores of the participants were never revealed, only that those with the lowest were excluded. The number of active bidders here was thus exactly the same as in the Random Exclusion treatment, $N = M - K$ ⁠.

We ran two versions of the Baseline treatment (⁠ $K = 0$ ⁠). In both versions, the experimenter announced that everyone in this session would have the opportunity to bid on the shirt, i.e. $K = 0$ ⁠. Participants then wrote down their bids, which were collected by the experimenter. The number of active bidders in this treatment was equal to the group size, $N = M$ ⁠. The only difference between the two versions of the Baseline treatment was whether or not participants first indicated the extent to which they would want to own the good, matching the initial procedures of both the Random and Non-Random Exclusion treatments. This allowed us to test whether reporting this measure affected bids orthogonally to the treatment variation.

It is important to stress that the nature of exclusion—whether it was random or depended on intrinsic taste—was emphasized and made common knowledge as part of the experiment. Additionally, both the group size M and the number of excluded participants K were always emphasized in both written and verbal instructions. Care was also taken to make sure that participants left the lab one at a time and were away from the facility before the next participant departed. At the end of the experimental session, the highest bidder was paid $15 minus her bid and received the shirt. All others were paid $15.

4.2 Results

The average bid size was $1.41 (SE = $0.12$ ⁠). There were no significant differences in bids between the two versions of the Baseline treatment (⁠ $p > 0.4$ ⁠), indicating that reporting one’s ex ante taste for the good did not meaningfully affect behaviour. We thus pool data from the two versions for the analysis that follows.³⁵

We begin by examining the simple comparison of average bids in the Random Exclusion and Baseline treatments. A OLS regression with standard errors clustered at the session level reveals a sizable effect of exclusion (Random Exclusion=1) on bids (⁠ $β = 0.78$ ⁠, $p < 0.01$ ⁠).

Table 1 presents results from regressions that further test our predictions. We first regress bids on dummy variables corresponding to the number of active bidders N in the Baseline treatment. As shown in Column 1, the coefficients on both N dummies are positive. This provides evidence for the standard prediction outlined in Proposition 2 that, in the sale of private goods, increased competition leads to more aggressive bidding and higher prices.

Table 1.

Effect of group size and exclusion on bids

	(1)	(2)	(3)	(4)
$N = 6$	0.29
	(0.22)
$N = 8$	0.95^***
	(0.19)
$K = 1$		0.67	−0.01	−0.08
		(0.50)	(0.48)	(0.48)
$K = 2$		0.73^**	−0.26^**	−0.33^**
		(0.35)	(0.12)	(0.15)
$K = 3$		1.28^***	0.88	0.81
		(0.18)	(0.53)	(0.54)
Random				−0.18
				(0.24)
Random* $(K = 1)$				0.85
				(0.71)
Random* $(K = 2)$				1.17^***
				(0.41)
Random* $(K = 3)$				0.58
				(0.59)
Constant	0.77^***	1.12^***	1.12^***	1.19^**
	(0.17)	(0.11)	(0.11)	(0.14)
N	142	210	206	274

	(1)	(2)	(3)	(4)
$N = 6$	0.29
	(0.22)
$N = 8$	0.95^***
	(0.19)
$K = 1$		0.67	−0.01	−0.08
		(0.50)	(0.48)	(0.48)
$K = 2$		0.73^**	−0.26^**	−0.33^**
		(0.35)	(0.12)	(0.15)
$K = 3$		1.28^***	0.88	0.81
		(0.18)	(0.53)	(0.54)
Random				−0.18
				(0.24)
Random* $(K = 1)$				0.85
				(0.71)
Random* $(K = 2)$				1.17^***
				(0.41)
Random* $(K = 3)$				0.58
				(0.59)
Constant	0.77^***	1.12^***	1.12^***	1.19^**
	(0.17)	(0.11)	(0.11)	(0.14)
N	142	210	206	274

Notes: $^{* * *} : p \leq 0.01$ ⁠, $^{* *} : p \leq 0.05$ ⁠, $^{*} : p \leq 0.1$ ⁠. Standard errors clustered at the session level are reported in parentheses below each estimate. Column 1 reports the effect of group size N on bids in the Baseline treatments; Column 2 reports results comparing Random Exclusion to Baseline; Column 3 reports results comparing Non-Random Exclusion to Baseline; Column 4 compares the relative effects of Random versus Non-Random Exclusion.

Table 1.

Effect of group size and exclusion on bids

	(1)	(2)	(3)	(4)
$N = 6$	0.29
	(0.22)
$N = 8$	0.95^***
	(0.19)
$K = 1$		0.67	−0.01	−0.08
		(0.50)	(0.48)	(0.48)
$K = 2$		0.73^**	−0.26^**	−0.33^**
		(0.35)	(0.12)	(0.15)
$K = 3$		1.28^***	0.88	0.81
		(0.18)	(0.53)	(0.54)
Random				−0.18
				(0.24)
Random* $(K = 1)$				0.85
				(0.71)
Random* $(K = 2)$				1.17^***
				(0.41)
Random* $(K = 3)$				0.58
				(0.59)
Constant	0.77^***	1.12^***	1.12^***	1.19^**
	(0.17)	(0.11)	(0.11)	(0.14)
N	142	210	206	274

	(1)	(2)	(3)	(4)
$N = 6$	0.29
	(0.22)
$N = 8$	0.95^***
	(0.19)
$K = 1$		0.67	−0.01	−0.08
		(0.50)	(0.48)	(0.48)
$K = 2$		0.73^**	−0.26^**	−0.33^**
		(0.35)	(0.12)	(0.15)
$K = 3$		1.28^***	0.88	0.81
		(0.18)	(0.53)	(0.54)
Random				−0.18
				(0.24)
Random* $(K = 1)$				0.85
				(0.71)
Random* $(K = 2)$				1.17^***
				(0.41)
Random* $(K = 3)$				0.58
				(0.59)
Constant	0.77^***	1.12^***	1.12^***	1.19^**
	(0.17)	(0.11)	(0.11)	(0.14)
N	142	210	206	274

In Column 2, we regress bids on dummy variables corresponding to the number of excluded K in the Random Exclusion treatment. The exclusion effect is economically significant: for example, the effect of excluding three out of eight people from the auction corresponds to 0.65 standard deviations of the mean bids in the study. Furthermore, the effect of excluding two people out of six from the auction (⁠ $M = 6, K = 0$ versus $M = 6, K = 2$ ⁠) is roughly double the impact of adding two people in the Baseline treatment (⁠ $M = 4, K = 0$ versus $M = 6, K = 0$ ⁠), i.e. the competition channel.³⁶ These results are consistent with the predictions of Proposition 2.

As outlined in Section 3, the superiority-seeking channel is muted in the Non-Random Exclusion treatment. When exclusion targets those with the lowest liking scores, in equilibrium the winner still has the highest realized private value in the group. At the same time, the classic effect, holding N constant, now predicts somewhat higher bidding due to positive selection on consumption utility. Comparing the relative effects of the Random versus Non-Random Exclusion treatments on average bids to that in the Baseline is thus a conservative test of the superiority-seeking motive. If $α = 0$ ⁠, then bids should be lower in Random Exclusion than in the Non-Random Exclusion treatment. A positive α alone does not imply a greater effect in the former than in the latter—only a sufficiently high α would.

Column 3 of Table 1 presents results from the Non-Random Exclusion treatment. The impact of exclusion here is much more muted, with smaller coefficients on K than in the Random Exclusion treatment. Column 4 compares the relative impact of the two treatments to Baseline. The coefficients on the interactions between treatment and number excluded are all positive, suggesting that Random Exclusion has a larger impact on bids than Non-Random Exclusion. Since $α = 0$ predicts a smaller effect in the former than in the latter, these results provide further evidence for superiority-seeking. Our third study, described in Section 5.1, provides more direct evidence for the role of beliefs about the desires of others in the superiority-seeking motive.

Above, we focused on average bids between treatments. But, we can also use our data to examine the predictions on expected revenue. To compute expected revenue, we ran a series of Monte Carlo simulations to generate bid distributions using the measured average bid and standard deviations for each group of active bidders N by treatment. We draw N number of bids from these distributions to reproduce the type of data collected in the study and take the maximum bid from each set of draws. This process is repeated 10,000 times for each combination of treatment and group size.

Expected revenues by treatment and group size M are presented in Figure 2. We find that the expected revenues from the Random Exclusion versus the Baseline treatment are $4.68$ versus $3.84$ for $M = 8$ ⁠; $4, 14$ versus $3.03$ for $M = 6$ ⁠; and $4.04$ versus $1.7$ for $M = 4$ ⁠. In line with Proposition 3, both in the Baseline treatment and in the Random Exclusion treatment, the seller’s expected revenue is increasing in the number of active bidders. Exclusion, however, increases the seller’s expected revenue by more than the increased competition effect of increasing the number of bidders with full inclusion.³⁷ Furthermore, in sharp contrast to the standard prediction absent superiority-seeking, the seller’s expected revenue is considerably higher under random exclusion than under full inclusion both for each group size M and also when holding the number of active bidders N constant (⁠ $M = 6$ and $K = 2$ versus $M = 4$ and $K = 0$ ⁠).

Figure. 2.

Expected revenue by group size M across treatments.

In contrast, expected revenues in the Non-Random Exclusion treatment are much closer to those in the Baseline treatment. For any fixed M and K these revenues are always below revenues in the Random Exclusion treatment, consistent with $α > 0$ ⁠. In line with Propositions 3 and 4, expected revenues in Non-Random Exclusion are roughly similar to those in the Baseline treatment (revenue equivalence follows for any α) and lower than under Random Exclusion for any fixed M and K (which follows for α sufficiently large; the opposite follows for $α = 0$ ⁠).

4.3 Discussion

The preceding studies provide support for our model and rule out a number of alternative channels. First, there was always a single good being auctioned off across all treatments in Study 2, which assuages concerns about consumption externalities driving our results since these are held constant. Second, behaviour in Study 2 also allows us to rule out mechanisms driven by variation in group size. Specifically, we can compare scenarios where the number of potential active bidders N is the same but there is presence versus absence of exclusion, i.e. $M = 6$ and $K = 2$ in Random Exclusion versus $M = 4$ and $K = 0$ in Baseline. The number of people excluded was also held constant across the Random and Non-Random Exclusion treatments. Since bids were higher in Random Exclusion than in both Baseline or Non-Random Exclusion, keeping the number of active bidders N constant, this allows us to also rule out differences in “scarcity of opportunity” per se or social pressure as driving the observed effect.³⁸ Motives relating to joy of winning or signalling one’s income are either held constant across treatments or presumably imply more aggressive bidding in the Baseline than in the Random Exclusion treatment, predicting the opposite of the observed effect.

Another potential alternative is that our exclusion effect was driven by beliefs about the opportunities for resale. We can rule out a pure resale motive since the external resale opportunity of the winner is constant across treatments. One possibility is that the winner can resell only internally, that is, amongst the randomly excluded bidders. This seems unlikely to occur in practice because bidders were anonymous and care was taken that individuals left the lab one by one; moreover, we explicitly shut down the resale channel in the studies described in Section 5, which are conducted anonymously online. However, even if this unlikely scenario was the case in Study 2, given independent private values, random exclusion should not increase the seller’s revenue—in sharp contrast to what we find empirically. Given the standard regularity condition, whatever resale opportunities there may be, as long as $α = 0$ ⁠, it would still need to follow that the seller’s revenue satisfied $Π (M, K) < Π (M + 1, 0)$ ⁠, e.g. Klemperer and Bulow (1996)—a prediction that is violated both in our setup and in the data. Instead, our results are consistent with the prediction of superiority-seeking, as described in Corollary 5, whereby randomly excluding bidders leads to a greater increase in expected revenue than expanding the number of bidders under full inclusion.

5 ROBUSTNESS

The effects observed in the first two studies may still be impacted by aspects of the design, such as exclusion occurring face-to-face, exclusion being perceived as a signal of actual scarcity (Study 1), or other factors related to how exclusion was implemented. We designed our third and fourth studies to account for these potential confounds, to test the model’s predictions even more directly, and to further explore the economic significance of the superiority-seeking motive.

First, the new studies are conducted online in a fully anonymous setting. Any exclusion is thus completely anonymous, which should rule out image concerns that may have been present in the lab. Second, we employ a within-subject design to elicit valuations under different scenarios. Specifically, participants reported their valuations for the same good under scenarios that differed only in the number of others who would later be excluded from the opportunity to obtain the good. Since all exclusion scenarios were equally salient at the time that valuations were elicited, this allowed us to rule out concerns regarding differential attention as a driver of value (Krajbich et al., 2010; Towal et al., 2013; Li and Camerer, 2022). Third, we set up an online store exclusively for the study in order to credibly deliver unique goods to the participants remotely. Participants bid on a unique good—a print of a painting made by one of the authors. This setting highlights both the unique aspect of the product and alleviates concerns of exclusion being some function of actual scarcity.

5.1 Study 3: Exclusion on the intensive margin

The first two studies compared no exclusion to some exclusion as the main source of exogenous variation. This may have potentially introduced confounds, such as exclusion being a signal of scarcity, the feeling of “luck” at not being excluded, or differences in group size at the time that valuations were elicited. The third study circumvents these issues by comparing valuations as the level of exclusion increases, conditional on some baseline level of exclusion. Participants reported their valuations for the same good under different levels of potential exclusion; decisions were incentivized by telling participants that one scenario would be selected at random and that decision would be played out “for real.” This design choice makes transparent that the degree of exclusion is independent from scarcity or other participants’ valuations. Valuations are elicited before the actual scenario was realized, which precludes ex post emotional factors such as “luck” and the “joy of facing lower competition” from impacting bids. Moreover, because the group size M was held constant at the time when valuations were elicited, this should further assuage concerns regarding differential social pressure as a driver of our results. The design also allows us to explore the superiority-seeking motive on the intensive margin. Finally, we introduce treatment variation in information about whether the desire of those potentially excluded is higher or lower than one’s own in order to directly test the proposition that superiority-seeking increases with the excess desires of others.

5.1 Method

A group of participants (⁠ $N = 446$ ⁠) were recruited from the Academic Prolific crowdsourcing platform to take part in experiments on decision-making.^39,40 We refer to this group as “Active” participants, which is in contrast to the “Passive” participants whose role will be described below. Each Active participant was paid a base fee of $1.00 for completing the study and was told that 1 in 10 would be randomly chosen to have their decisions played out for real.⁴¹

Participants were first presented with an exclusive, unique good—a print of a painting made by one of the authors—and asked the extent to which they desired to own this good using the same method as in Study 2. Each was then told that they would have the opportunity to potentially purchase the good from an endowment of $10 by reporting their maximum WTP, which was incentivized using the same common-price BDM mechanism as in Study 1. Further, these Active participants would be matched with three other Passive participants in the study.⁴²

Active participants were told that the Passive participants may have the opportunity to acquire the good and similarly report both their desire and their maximum WTP for it. Additionally, each of the Passive participants would have a separate coin assigned to each of them. The Active participants were then asked to report their WTP under three scenarios, with each scenario being equally likely to be selected and played out for real:

•
Scenario 1: One of the coins would be flipped by the computer. If the coin flip landed on Tails, the assigned Passive participant would have the opportunity to purchase the good, depending on her WTP; if the coin flip landed on Heads, the assigned participant would not have the opportunity to purchase the good regardless of her WTP.
•
Scenario 2: Two of the coins would be flipped by the computer and a second assigned Passive participant would be similarly affected by the outcome.
•
Scenario 3: All three coins would be flipped by the computer and the third assigned Passive participant would be similarly affected by the outcome.

Additionally, Active participants reported their WTP in one of three randomly assigned treatments. In the High Information treatment, each was told that the matched Passive participants had a desire for the good that was as high or higher than their own; in the Low Information treatment, the desires of matched participants would be as low or lower than the Active participant’s. We also included a No Information treatment for robustness, where Active participants were not given any information about the desires of others in the group.

This design has several key features. First, the total size of the group remains constant across all conditions at $M = 4$ ⁠. Second, as the number of coin flips increases, the greater the potential number of others excluded from the opportunity to own the good, i.e . K.⁴³ This allows us to test the model’s predictions on the intensive margin. Third, our Informational treatments allow us to directly compare valuations when participants know that others have greater desire than their own (High Information) or know that they do not (Low Information). Specifically, the model predicts that Active participants will monotonically increase their WTP for the good as the number of coin flips increases, but only in the High and No Information treatments; the number of coin flips should not affect valuations in the Low Information treatment.

If an Active participant was chosen to have their decisions played out for real, they would be matched with a group of Passive participants who were recruited separately for the study. The group’s choices would be realized as stated in the instructions; all Passive participants entered their bids and would have the opportunity to obtain the good depending on the realized scenario and the outcome of the coin toss. Payments were delivered digitally through the Prolific system. If a participant had purchased the art print, they received a code to be redeemed at an online store set up exclusively for the study. The online store was run through a third party that shipped the goods to participants upon redemption.

5.1 Results

Our analyses will focus on the High and Low Information treatments (where the relevant beliefs are directly controlled for) as these comprise the cleanest test of the framework, which predicts a boost in valuations as a function of beliefs about the excess valuations of others. The results from the No Information treatment were similar to those in the High Information treatment. Consistent with the prediction of the model, all of the comparisons were significant but a bit less pronounced (see Supplementary Appendix 1.3).

Figure 3 illustrates the results and Table 2 reports analyses from an OLS regression with standard errors clustered at the individual level. In Column 1, we regress participants’ bids on the number of coin flips in the High Information treatment. Increasing the number of those potentially excluded increases bids: compared to flipping only one coin (Scenario 1), flipping two coins increases bids by 32 cents (⁠ $p < 0.01$ ⁠), and flipping three coins increases them by 68 cents (⁠ $p < 0.01$ ⁠). This represents 0.12 and 0.25 standard deviations of the average bids in the study, respectively.⁴⁴

Figure. 3.

Average bids by information and level of exclusion

Table 2.

Effect of exclusion on WTP

	(1)	(2)	(3)
Two-coin (=1)	0.32^***	−0.13	−0.13
	(0.09)	(0.09)	(0.09)
Three-coin (=1)	0.68^***	−0.09	−0.09
	(0.16)	(0.14)	(0.14)
High Information (=1)			0.34
			(0.30)
Two-coin*High Information			0.46^***
			(0.12)
Three-coin*High Information			0.78^***
			(0.21)
Constant	3.67^***	3.33^***	3.33^***
	(0.22)	(0.21)	(0.21)
N	444	453	897

	(1)	(2)	(3)
Two-coin (=1)	0.32^***	−0.13	−0.13
	(0.09)	(0.09)	(0.09)
Three-coin (=1)	0.68^***	−0.09	−0.09
	(0.16)	(0.14)	(0.14)
High Information (=1)			0.34
			(0.30)
Two-coin*High Information			0.46^***
			(0.12)
Three-coin*High Information			0.78^***
			(0.21)
Constant	3.67^***	3.33^***	3.33^***
	(0.22)	(0.21)	(0.21)
N	444	453	897

Notes: $^{* * *} : p \leq 0.01$ ⁠, $^{* *} : p \leq 0.05$ ⁠, $^{*} : p \leq 0.1$ ⁠. Standard errors clustered at the individual level are reported in parentheses below each estimate. Column 1 reports the relationship between the number of coin flips and WTP in the High Information treatment. Column 2 reports the relationship between the number of coin flips and WTP in the Low Information treatment. Column 3 compares the High and Low Information treatments.

Table 2.

Effect of exclusion on WTP

	(1)	(2)	(3)
Two-coin (=1)	0.32^***	−0.13	−0.13
	(0.09)	(0.09)	(0.09)
Three-coin (=1)	0.68^***	−0.09	−0.09
	(0.16)	(0.14)	(0.14)
High Information (=1)			0.34
			(0.30)
Two-coin*High Information			0.46^***
			(0.12)
Three-coin*High Information			0.78^***
			(0.21)
Constant	3.67^***	3.33^***	3.33^***
	(0.22)	(0.21)	(0.21)
N	444	453	897

	(1)	(2)	(3)
Two-coin (=1)	0.32^***	−0.13	−0.13
	(0.09)	(0.09)	(0.09)
Three-coin (=1)	0.68^***	−0.09	−0.09
	(0.16)	(0.14)	(0.14)
High Information (=1)			0.34
			(0.30)
Two-coin*High Information			0.46^***
			(0.12)
Three-coin*High Information			0.78^***
			(0.21)
Constant	3.67^***	3.33^***	3.33^***
	(0.22)	(0.21)	(0.21)
N	444	453	897

Column 2 presents the same results from the Low Information treatment. In contrast to the High Information treatment, exclusion has no significant effect when people think that the excluded have lower valuations than their own.

Column 3 presents results comparing the High and Low information treatments directly. Interacting both the two-coin and three-coin scenarios with the information treatment generates significant and sizable coefficients, showing that excess desire is necessary for superiority-seeking to emerge. Note that this comparison also allows us to rule out that the increase in valuations in the High Information treatment was driven by some artefact of the within-subject design; the only difference between the High and Low information conditions was whether or not the excluded had excess desire vis-à-vis the Active participant.

We chose to pair multiple Passive participants with each Active participant in order to explore the superiority-seeking motive on the intensive margin, holding the group size constant. This design also allowed us to examine both the existence and strength of superiority-seeking between-subjects. By looking at whether an individual increased her bids with the number of potentially excluded, and measuring the extent of this increase, we can capture both how many participants exhibited exclusionary preferences and the size of this effect on an individual level. To do so, we classify people who increased (decreased) their bids as the number of coin flips increased as exhibiting exclusionary (inclusionary) preferences. We classify those who did not change their WTP with the number of coin flips as neutral.⁴⁵

In the High Information treatment, 51% of people can be classified as exhibiting strict exclusionary preferences. This is the largest group, with 29% exhibiting neutral preferences and only 20% exhibit the opposite tendency.⁴⁶ Importantly, these proportions do not reflect noise or artefacts of the experimental design: consistent with the theory, the proportion of people with exclusionary preferences is substantially lower—nearly half the size—in the Low Information treatment (26%; $p < 0.01$ ⁠).

This classification also allows us to examine the strength of the superiority-seeking motive. To do so, we replicate Column 1 of Table 2 conditional on exhibiting exclusionary preferences. Active participants increased their WTP for the good by $1.10 when going from one to two coins, and by $2.44 when going from one to three. These are sizable effects, representing 0.40 and 0.89 standard deviation, respectively, from the mean WTP in the study.

5.2 Study 4: Shifting demand through exclusion

We designed our last study to mirror the setting of Proposition 1 via a within-subject design and explore whether, in line with our prediction, a seller may increase profits through random exclusion compared to the classic optimal mechanism of price-based exclusion which targets only those with the lowest valuations.

5.2 Method

The paradigm builds on the third study with several notable differences. Participants (⁠ $N = 100$ ⁠) were recruited from the Academic Prolific crowdsourcing platform to take part in experiments on decision-making.⁴⁷ Every participant was paid a base fee of $1.00 for completion. Each also had a $10 endowment to use for decisions in the study, and all participants had their decisions carried out for real.

Participants were informed that 100 individuals had been recruited for this study. Each then reported her WTP for the good—incentivized in the same way as in Study 3—in two (within participant) scenarios. In the No Exclusion scenario, all participants who finished the study would be able to purchase the good depending on their WTP. In the Exclusion scenario, 60% of participants would be randomly selected to have the opportunity to purchase the good; the rest would not have this opportunity. A computer would then flip a coin to determine which scenario would be played out for real, with Heads corresponding to Exclusion and Tails corresponding to No Exclusion.⁴⁸ Note that, like in Study 3, the within-subject design makes it transparent that exclusion is not a function of actual scarcity.

5.2 Results

We first look at whether the prospect of exclusion affected average WTP for the good. Regressing WTP on a treatment dummy (Exclusion=1) with clustered standard errors reveals a significant effect: participants increased their WTP for the good by nearly a dollar (⁠ $β = 0.91; p < 0.01$ ⁠) for the Exclusion case relative to the No Exclusion case. This represents 0.27 standard deviations from the mean WTP.

We can then calculate the demand curves facing a monopolist both with and without exclusion. Below, we report the number of participants who would be willing to purchase the good at each price for both scenarios. The two demand curves are presented in Figure 4 below.

Figure. 4.

Demand as a function of exclusion

The results support our theoretical predictions, Propositions 1 and 5, and lend credence to revenue-generating non-price-based exclusion. Demand shifted to the right as a function of exclusion.⁴⁹ Participants’ median WTP increased by nearly 50%—from $4 to $6—as a function of others being excluded from the opportunity to purchase the good. Furthermore, as long as the marginal cost of supplying one good is $2 or more—which is true in our setting—the seller’s optimal profit under exclusion is higher than under no exclusion. This, despite getting rid of 40% of potential demand and thus having a non-selectively 40% smaller customer base.

We can also perform a similar classification exercise as in Study 3, tagging those whose bids in the Exclusion scenario are strictly higher (lower) than in the No Exclusion scenario as having exclusionary (inclusionary) preferences. Those with no differences in bids are classified as neutral. We observe a similar breakdown as Study 3, with 53% exhibiting exclusionary, 25% exhibiting neutral, and 22% exhibiting inclusionary preferences. Looking at the size of the superiority-seeking motive, we find that conditional on exhibiting exclusionary preferences, people’s median WTP for the good increases by nearly 80%—from $5 to $9—when others are barred from acquiring the good. We believe that this striking empirical demonstration provides evidence for the potential effectiveness of exclusion as a rent-generating tool for firms.

6 DISCUSSION

6.1 Related literature

A rich literature emphasizes the presence of social motives in people’s preferences. A first generation of social preference models focuses on direct consumption externalities, e.g. Becker (1974). The next wave of models attempted to explain empirical evidence on costly punishment behaviour, by considering preferences over relative allocations. Models of inequity aversion (Fehr and Schmidt, 1999) and competitive preferences (Charness and Rabin, 2002) over money assume that people dislike unequal allocations or actively prefer allocations that put them ahead, respectively. A third wave of theory considers whether prosocial or antisocial choices may be a function of imperfect information and signalling rather than underlying preferences, e.g. Bénabou and Tirole (2006, 2011).

In these models, utility is defined over the consumption, money, or the beliefs of others. This is in contrast to our approach which defines utility over others’ unmet desires. This distinction is important because it generates novel predictions on the effects of exclusion. For example, in the auction setting, only the winner receives the item while others do not. As a result, there are no changes in consumption externalities and no differences in the relevant informational asymmetries, as the allocation mechanism is common knowledge. Hence, these alternative models will not predict our findings, while superiority-seeking does.

In a line of work closer to our own, Frank (1985) models “keeping up with the Jonses” effects through a demand for positional, or status, goods. The existence of such goods is taken as a primitive and people not only care about their personal consumption utility from positional goods, but also the hierarchy of observable consumption amongst others. In particular, people’s utility from a positional good is a function of the percentile ranking of their own consumption of the good relative to the overall population’s consumption of it. If people make choices non-cooperatively, this may then lead to overconsumption of positional goods compared to non-positional goods.⁵⁰

Our framework is conceptually distinct from this literature. In contrast to a direct consumption externality, the superiority-seeking motive is defined over the unmet desires of others for what one has rather than over the consumption space independent of those desires. Importantly, our mechanism makes distinct empirical predictions. As noted above, consumption externalities such as Frank (1985) would predict a null effect in Study 2. At the same time, positional externalities are consistent with the implications of superiority-seeking, which imply an incentive to surpass the consumption of others, in quality or kind, if this allows a person to increase the unmet desires of others for what she has. Our framework may thus be viewed as a psychological foundation for the demand for positional or status goods, while making a host of additional predictions for individual decision-making and market behaviour. We expand on this further below.

6.2 Broader implications

In this section, we explore some economic consequences implied by our framework for markets and political economy.

Exclusive access and artificial scarcity. As captured by Proposition 1 and Corollary 4, firms and marketers can extract greater rents by artificially rationing the actual or perceived supply of goods and services even while employing a uniform price. Examples of such practices abound. Established venues face persistent excess demand at the going price in lieu of expanding capacity or raising the price despite ample opportunities to do so. Nightclubs grant random access to some but not others despite there being enough room for all.⁵¹ Firms regularly introduce new consumer products at artificially low quantities despite overwhelming demand, e.g. Gmail and Facebook rationing invitations (Wortham, 2011). As another example, the major Chinese cell-phone manufacturer OnePlus distributed their flagship OnePlus One phone by first soliciting a large list of people who were interested in buying it and then randomly allocating the opportunity to a select few. As discussed in Section 1, marketers work hard to emphasize such “artificial” scarcity when advertising products. Techniques include limited time offers, highlighting that there are “only” a few goods left (common in online marketplaces), or that a price discount is offered to a select few and only allowing people with last names beginning with certain letters of the alphabet to purchase a good (a classic infomercial practice). Practitioner guides note that such scarcity techniques are effective in increasing demand by making the eventual owners “feel powerful” over those who want the same product but cannot have it.⁵²

As discussed in Becker (1991), these phenomena are puzzling from the perspective of standard economics since a firm should either raise the price and/or expand capacity; firms should not choose to restrict supply while profit opportunities are seemingly positive. His paper highlights the widespread prevalence of such artificial scarcity but ultimately cannot explain it. In contrast, our model rationalizes the use of artificial scarcity and excess demand at the posted price as a way to increase profits.

Price-based exclusion, luxury goods, and Veblen effects. Economists have long considered the prevalence of status goods and so-called “Veblen effects,” where the demand for certain goods increases as its price increases. The goods that display this apparent violation of the law of demand are often characterized by their exclusivity and luxury. A line of work relates “Veblen effects” to the propensity to signal one’s income status via observable consumption for downstream non-pecuniary benefits. For example, Bagwell and Douglas Bernheim (1996) consider a setting where people allocate their income across two types of goods: observable “conspicuous” goods, such as a sports car, and less observable non-conspicuous goods, such as private home decor or art displayed at home. A “Veblen effect” is said to arise when rich people consume a higher-priced but otherwise equivalent version of the “conspicuous” good. However, the conditions which generate such effects also generate fairly counter-intuitive predictions. For example, a more efficient method of signalling income would be for richer people to destroy resources publicly, or simply just post their incomes. The framework suggests that firms have an incentive to produce perfect substitutes that only differ in price, creating a “budget” product and a “luxury” product of the same quality. Finally, the purely signalling explanation for “Veblen effects” and luxury goods has a difficult time rationalizing the often private consumption of the associated goods and the persistence of high prices despite observationally similar alternatives, e.g. real versus fake diamonds.

Our framework provides an alternative and perhaps complementary channel to signalling. In the presence of heterogeneous income, people’s demand for a luxury good may increase with its price irrespective of observability with regards to one’s own income status or consumption, as a high price creates exclusion. Increasing prices lead consumers who may very much desire the good to no longer be able to obtain it due to tighter liquidity constraints, which generates a superiority-seeking boost amongst those who may like the good less but can still afford it. This can lead to an increase in total demand. A simple stylized example provides intuition.

Example 3

Let there be M consumers, each is rich with probability γ and poor otherwise. A monopolist produces a good at zero marginal cost. Each person has a unit demand and her consumption utility is an i.i.d. draw from $F (v)$ over $[0, \bar{v}]$ ⁠. The only ex ante difference between rich and poor is that poor consumers face a tighter budget constraint. For simplicity, suppose that the poor cannot spend more on the good than some number $Z < \bar{v}$ ⁠.

If the price is less than Z, total expected demand is $M [1 - F (p)]$ ⁠. If it is greater, there is no demand by the poor, but the expected demand is now $γ M [1 - F (p - α g (p))]$ for some function $g (p) > 0$ as long as $p < \bar{v}$ reflecting the utility from superiority-seeking; rich consumers with an intrinsic taste well below p will now also want to buy the product. It is straightforward to see that, depending on F and γ, demand may indeed be higher than when the price is below Z, emblematic of the standard “Veblen effect.”⁵³

The logic outlined in this example can explain many phenomena that have thus far been discussed through the lens of positional externalities (Frank, 1985). For example, Bursztyn et al. (2018) find that when the income threshold for a particular credit card is lowered, this increases demand for a more expensive—and exclusive—alternative among existing customers. The change in demand had previously been attributed to lower income consumers crowding out the signalling value of owning the original card, imposing a positional externality that leads existing customers to switch to alternatives which retain the signalling value. However, our framework can microfound this result through superiority-seeking. Lowering the income threshold decreases the superiority-seeking boost of existing consumers because those who had previously been unable to afford the card can now obtain it—the budget constraint is less likely to bind. Switching to a more expensive alternative card restores this boost, which generates the observed increase in demand. This logic can explain a wide range of other examples involving exclusive or luxury goods, e.g. the decision to burn product instead of selling it at a discount (Burberry).

The stylized example also illustrates why, in contrast to the prediction of the signalling account, a firm may not want to engage in third-degree price discrimination—even if the company could sell to the verifiably rich and poor at different prices and it was clear at which price a particular good was bought. Instead, the firm would be better off shutting down the market for poorer individuals. Producing low-priced “budget” substitutes introduces the prospect of low-income individuals having access to the same goods as richer individuals, which would diminish the impact of superiority-seeking and crowd out demand in the latter group. This rationalizes why luxury firms rarely advertise budget products under the same umbrella; when a firm does offer both low and high priced goods, the former is typically advertised under a different brand and stresses accessibility, while the latter stresses quality (e.g. Armani Exchange versus Giorgio Armani).

Importantly, our model also makes distinct predictions on the types of objects that are more likely to display “Veblen effects.” Unlike in the signalling account, intrinsic quality now plays an important role—a Ferrari is more likely to be a Veblen good than a Nissan Altima because it is a better quality car. A higher quality good is associated with a greater intrinsic taste for it and will be linked to higher levels of superiority boost when this desire is unmet. This is in contrast to pure signalling motives where perceptions of quality do not factor into “Veblen effects” per se. Additionally, our framework allows for exclusive goods to be enjoyed in private, such as intimate dinners at an expensive restaurant, art collections housed in a private gallery, or luxury amenities for a gated community. Perceptions of exclusivity rather than observability of one’s own consumption is the key driver of increased demand in our framework.

We do not claim that the observability of consumption is not an important factor in the demand for luxury goods. Work by, for example, Heffetz (2011) and Bursztyn et al. (2018), have shown that visibility plays a significant role in the consumption of some premium goods. In fact, visibility may amplify the utility boost in our model by ramping up desire amongst those who cannot afford the good or by shaping the social context. We also do not aim to minimize signalling as a potential motive. Rather, we argue that the utility boost from unmet excess desires provides a distinct and potentially complementary channel which further increases the predictive power for which goods are more or less likely to generate such “Veblen effects” or become status goods.

Redistribution. Redistributive policies such as progressive taxation are common tools for mitigating income inequality. However, political scientists and economists are often puzzled by opposition to these policies by the U.S. electorate, particularly amongst the poor and lower middle-class.⁵⁴ Prior work has argued that such opposition may stem from the “prospect of upward mobility” (e.g. Bénabou and Ok, 2001), whereby people would prefer to avoid higher taxes given their prospects of moving up higher on the income ladder, or motivated beliefs about the potential of upward mobility which function to counteract limited willpower (Bénabou and Tirole, 2006). In the presence of income heterogeneity, superiority-seeking offers a potential complementary and distinct motive for the opposition to some re-distributive policies—even by those who may benefit from it, e.g. amongst the poor-but-not-poorest individuals of a community. These policies may decrease the overall utility associated with the respective group’s consumption despite providing greater social insurance.

Consider the prospect of increasing the minimum wage. A higher minimum wage may have positive implications for those currently earning below the minimum wage; they can purchase goods that previously only those who are richer could afford. Our framework predicts why opposition may be strongest amongst those who earn just one bracket above the current minimum wage. To see the intuition, consider the group earning just one income notch above the current minimum wage. Similar to the Veblen logic described above, this group is currently deriving a utility boost from consuming products that the lower income group—who earn at the current minimum wage—cannot afford. Increasing the minimum wage to the level of their income would eliminate this boost, which leads the poor-but-not-poorest earning group to oppose the policy. In fact, the decrease in the superiority boost may well be the largest for those who are currently just one income notch from the bottom. Given heterogeneity in tastes, others closer to the top of the income distribution will still continue to enjoy various goods that those below them like more but cannot afford.⁵⁵ This prediction is consistent with the findings of Kuziemko et al. (2014), who present survey evidence collected by a marketing group that people just one income notch above the minimum wage threshold oppose a minimum wage increase the most even after controlling for demographics.

Immigration, nationalism, and barriers to trade. The mechanism described in the paper is also broadly consistent with the phenomenon whereby “natives” may want to limit the rights of “immigrants,” even if such exclusion comes at a material cost to the former. By restricting access to certain rights and institutions that some immigrants may desire more, natives can derive extra utility through the superiority-seeking motive. This also helps explain the familiar notion of “pulling up the ladder,” whereby people who have recently immigrated oppose further immigration.⁵⁶ Even if further immigration were to increase their own material well-being and productive social networks, recent immigrants who know others who would like to immigrate as well—thereby gaining access to opportunities and institutions available in the host country—may oppose additional immigration because this would lead to a drop in their own utility (which is derived from exclusive access to this set of benefits).

Politicians appear aware of these fears, describing access to national institutions through the perspective and desires of those outside of the nation. Often this rhetoric takes the form of highlighting outsider desire, such as in the U.K. where the National Health Service is often referred to as the “envy of the world,” or in the U.S., where the U.S. economy is celebrated as “being the envy of the entire world.”⁵⁷

Although our analysis has focused on the individual’s superiority-seeking vis-à-vis her social context, it can also apply to the joy of identifying with a group and the rivalry between groups. In particular, people may enjoy identifying with a group they can belong to and derive pleasures from the goods and attributes this group as a whole possesses. The value of in-group identification is then amplified if that group possesses attributes or consumes goods that members of another group would like more, but do not have. Indeed, given the superiority-seeking motive, it is the exclusion of out-of-group members from such goods or attributes which causes a form of pride and generates a utility boost from one’s group identity. To protect the value of such group identities, maintaining exclusion is important, which generates psychological barriers to inter-group trade.

Discrimination and social stratification. Similarly, the above logic can help rationalize aspects of social exclusion and “taste-based” discrimination. In a framework termed stratification economics, members of social groups compete over relative positions in exogenous social hierarchies (Darity Jr et al., 2015, 2017). Higher positions provide members with a number of privileges including exclusive access to a broad category of club goods. Here, discrimination is a “rational” response by dominant groups to maintain access to these privileges, serving as a tool for exclusion so that their own supply is protected. This is consistent with classic examples of club goods such as country clubs or exclusive residential communities being historically marked by discrimination based on socially constructed groups.⁵⁸

Our framework provides a distinct and complementary account through the psychological motive of superiority-seeking. The model predicts exclusionary and discriminatory policies even in the absence of exogenous hierarchies or the club goods being rivalrous on the relevant margin. Specifically, majority groups may employ discriminatory policies in order to boost their own private utility from consuming private goods or possessing certain attributes. That is, the unmet desires of the excluded or persecuted minority yield additional utility benefits associated with consumption of those goods, which increases incentives for discrimination. Exclusion based on salient (or easily observable) characteristics such as race or ethnicity further facilitates superiority-seeking by members of the majority group. In this account, the preferences that lead to disparate treatment do not arise from some innate disutility from interacting with others of a certain physical or ethnic characteristic; rather, they arise due to superiority-seeking, whereby external and observable aspects of others serve as a coordination device or as modes of “valuable” identification as described before. The logic of how superiority-seeking reinforces social discrimination and stratification is similar to the other cases of non-price-based exclusion in the preceding discussion.

7 CONCLUSION

Our article proposes that an individual’s valuation of consuming an item or possessing an attribute is boosted by a comparative term reflecting others’ unmet desire for it. Our model of imitative superiority-seeking helps explain a host of market anomalies and generates novel predictions for competition and political economy. We present experimental evidence that supports the predictions of the framework across several economically important environments. Our model rationalizes the use of artificial restrictions to supply (non-price-based exclusion) by firms, scarcity marketing, and naturally generates “Veblen effects.” The framework also provides a novel motive for attitudes against redistribution and immigration, and points to a distinct psychological mechanism for inter-group discrimination that has broad implications for social rivalry in a variety of key economic and political contexts.

Future research can greatly refine and expand the predictions of the motive introduced in our article. From a theoretical perspective, the preference for exclusion may have important implications for a variety of pricing and allocation problems. It may also be useful to consider factors that shape one’s social context and determine the domains in which superiority-seeking is most pronounced. Similarly, further studies could consider social institutions, such as systems of honour or moral exclusion that may both amplify and channel superiority-seeking from one domain to another. We believe that this motive may well be key for understanding a host of issues in political economy.

A natural question emerges regarding how other aspects of social behaviour are related to the superiority-seeking motive. As an initial exploration, we re-ran the No Information treatment of Study 3 (⁠ $N = 150$ ⁠) and included measures of other social behaviour that have been studied in the past, such as altruism, cooperation, and prosocial and anti-social punishment.⁵⁹ We classified people as having exclusionary, neutral, or inclusionary preferences based on the method used in Study 3. Superiority-seeking was not correlated with any of the “prosocial” measures such as altruism, cooperation, or prosocial punishment. We did find a significant, albeit small, correlation between superiority-seeking and anti-social punishment.⁶⁰ This suggests that preferences for exclusion are conceptually distinct from other social motives but may share some underlying psychology with anti-social behaviour.

Our model has a number of limitations. An important ingredient of our model that we do not endogenize is one’s social context. This social context can be affected by attention, memory, and salience (see e.g. work by Bordalo et al., 2013; Gabaix, 2019; Smith and Krajbich, 2019; Bordalo et al., 2020). Furthermore, strategic agents, such as firms or politicians, may try to directly influence the social context by the narratives they promote and the rhetoric they engage in. Additionally, in many contexts superiority-seeking may be veiled by factors outside of our model, such as the desire to conform to norms and forms of behaviour that would, for example, prevent people for explicitly paying for social exclusion. Future research should also explore the psychological factors that determine how superiority-seeking interacts with broader economic conditions.

Appendix: Proofs

Proof of Corollary 1

Proof

Suppose person i has the object. If $v_{i} < v_{j}$ ⁠, trade takes place if and only if (iff) $(1 - α) v_{i} + α v_{j} + ε < v_{j} - ε$ ⁠. The probability that this inequality is satisfied is strictly decreasing in α and becomes zero for any α sufficiently large, but still smaller than one. If $v_{i} > v_{j}$ ⁠, trade never occurs for any $α \leq 1$ ⁠.

Proof of Corollary 2

Proof

Consider $p \in (0, \bar{v})$ ⁠. If $α = 0$ ⁠, the seller accepts iff $p \geq v_{s}$ and the buyer accepts iff $p \leq v_{b}$ ⁠. Consider $α > 0$ ⁠. We first show that equilibrium is in cutoff strategies. Consider the buyer’s strategy and fix any strategy for the seller. Suppose the seller says yes. If the buyer says no, her payoff is zero. If the buyer says yes, her payoff from buying is $v_{b} + α E_{v_{s}} [{v_{s} - v_{b}}^{+} |$ seller yes $] - p$ which is increasing in $v_{b}$ given any $α < 1$ ⁠. In turn, the buyer’s strategy is given by some cutoff $v_{b}^{*}$ and the buyer says yes iff $v_{b} \geq v_{b}^{*}$ ⁠. Consider the seller’s strategy. If the buyer says no, this affects the seller’s utility, but whether the seller says yes or no, has no bearing on the seller’s payoff. Suppose the buyer says yes. The seller’s payoff from keeping the object is $v_{s} + α E [{v_{b} - v_{s}}^{+} | v_{b} \geq v_{b}^{*}]$ while his payoff when selling is p. The former is increasing in $v_{s}$ ⁠, since $α < 1$ ⁠, while the latter is independent from it. In turn, the seller’s strategy is also given by some cutoff $v_{s}^{*}$ ⁠. We now proceed by contradiction. Suppose that $v_{b}^{*} \leq v_{s}^{*}$ ⁠. Note first that $v_{s}^{*} < p$ must hold since if the seller values the object more than the price, he will never sell (the superiority boost is positive in expectation). Hence, $v_{b}^{*} < p$ ⁠. Consider now buyer type $v_{b}^{*}$ ⁠. This type’s utility from purchase would be $v_{b}^{*} + α z - p$ for some equilibrium superiority boost z bounded from above by $v_{s}^{*} - v_{b}^{*}$ ⁠. Hence, $v_{b}^{*} + α z - p \leq (1 - α) v_{b}^{*} + α v_{s}^{*} - p < (1 - α) (v_{b}^{*} - v_{s}^{*}) \leq 0$ —and this buyer type would lose from trade.

Proof of Corollary 3

Proof

An increase in C decreases the utility boost from holding the cup and leaves the utility from holding a pen unaffected. The logic with respect to a decrease in P is analogous.

Proof of Proposition 1

Proof

Let the seller-optimal posted price, without exclusion, be $p^{*}$ ⁠, and the associated expected profit, per buyer, be V. If $α = 0$ ⁠, the seller-optimal mechanism is charging $p^{*}$ to all buyers with an expected overall profit of MV, e.g. Skreta (2006). Consider $α > 0$ ⁠. Excluding a single buyer leads to an expected loss of V. Holding $p^{*}$ constant, the probability that any given remaining buyer buys is now raised by some $q > 0$ ⁠, increasing in α, since $p^{*} < \bar{v}$ must hold. The gain in the expected profit is $(M - 1) q p^{*}$ ⁠. In turn, there exists $M_{α}^{*}$ such that if $M \geq M_{α}^{*}$ ⁠, then $(M - 1) q p^{*} > V$ ⁠. By revealed preference, re-optimizing the uniform price to some $p^{'}$ can only further raise the seller’s expected profit. The same argument holds if exclusion per buyer binds only probabilistically with a strictly positive probability for any given buyer type v.

Proof of Corollary 4

Proof

Suppose that $v^{h} > p^{*}$ ⁠. The expected loss from exclusion is still $V$ ⁠, the expected gain is $(M - 1) q^{'} p^{*}$ where $q^{'} > 0$ still holds since there is a strictly positive measure of types below $p^{*}$ who are now willing to buy at $p^{*}$ ⁠, but did not buy without exclusion, and the previous argument applies. Suppose that $p^{*} \geq v^{h}$ ⁠. Then, there is no loss from $v^{h}$ -constrained random exclusion, since only types who were excluded by the price $p^{*}$ are effectively targeted, but the WTP of types strictly below $v^{h}$ increases.

Proof of Proposition 2

Proof

Let

b (v)

be the symmetric monotone bidding strategy with

b (0) = 0

⁠. Let’s denote the payoff, maintaining equilibrium behaviour by others, when type v pretends to be type z by

E U (v, z)

and let

G (z)

be the cdf of the highest intrinsic taste amongst the remaining

N - 1

active bidders. While G depends on N, for simplicity, we suppress this from the notation. In case of a downward deviation,

z < v

⁠,

E U (v, z) = G (z) [v - b (z) + α \int_{v}^{\bar{v}} (x - v) h (x) d x],

where

H (y)

is the cdf of the highest intrinsic taste of the excluded bidders. Note that H depends on K, but for simplicity, we also suppress this from the notation. In turn,

E U_{z} (v, z) = g (z) v - g (z) b (z) - G (z) b^{'} (z) + g (z) α \int_{v}^{\bar{v}} (x - v) h (x) d x .

Evaluating at

z = v

implies the optimality condition of

\frac{d}{d v} [G (v) b (v)] = g (v) [v + α K (v)],

where, with a slight abuse of notation,

K (x) = \int_{x}^{\bar{v}} (y - x) h (y) d y

⁠, with

K (x) \equiv 0

K = 0.

Via integration we get that:

b (v) = \frac{1}{G (v)} \int_{0}^{v} g (x) [x + α K (x)] d x,

where we assumed the boundary condition of the lowest type making zero rent, Milgrom and Weber (1982). To show that downward deviations are not profitable, consider

E U (v, z) - E U (v, v)

given

z < v

⁠. Note first that this difference can be written as:

(G (z) - G (v)) (v + α K (v)) + \int_{z}^{v} g (x) (x + α K (x)) d x .

Substituting terms and integrating by parts, the difference is:

(1 - α) G (z) (v - z) - (1 - α) \int_{z}^{v} G (x) d x + α [\int_{z}^{v} G (z) H (x) d x - \int_{z}^{v} G (x) H (x) d x] < 0.

Consider now an upward deviation

z > v

⁠. Here,

\begin{aligned} E U (v, z) & = G (z) (v - b (z)) \\ + α \int_{v}^{z} (x - v) (g (x) H (x) + G (x) h (x)) d x + α G (z) \int_{z}^{\bar{v}} (x - v) h (x) d x], \end{aligned}

because the highest unsatisfied consumption utility may now be vis-a-vis an included bidder and valuations are i.i.d and exclusion is random. Then

E U_{z} (v, z)

⁠, evaluated at

z = v

⁠, is again:

E U_{z} (v, v) = g (v) v - g (v) b (v) - G (v) b^{'} (z) + α g (v) \int_{v}^{\bar{v}} (x - v) h (x) d x .

Hence, the local optimality condition is the same.

Consider now the difference

E U (v, z) - E U (v, v)

⁠:

\begin{aligned} G (z) (v - z) + \int_{v}^{z} G (x) d x - α \int_{v}^{z} K (x) g (x) d x \\ + α [\int_{v}^{z} (x - v) (h (x) G (x) + H (x) g (x)) d x \\ + G (z) \int_{v}^{\bar{v}} (x - v) h (x) d x - G (v) \int_{v}^{\bar{v}} (x - v) h (x) d x] . \end{aligned}

Note that

\int_{v}^{z} K (x) g (x) d x = K (z) G (z) - K (v) G (v) + \int_{v}^{z} (1 - H (x)) G (x) d x

⁠, since

K^{'} (x) = - (1 - H (x))

⁠. Hence, the above can be further written as:

\begin{aligned} (1 - α) [G (z) (v - z) + \int_{v}^{z} G (x) d x] \\ + α [- K (z) G (z) + K (v) G (v) + \int_{v}^{z} H (x) G (x) d x \\ + \int_{v}^{z} (x - v) (h (x) G (x) + H (x) g (x)) d x \\ + G (z) \int_{z}^{\bar{v}} (x - v) h (x) d x - G (v) \int_{v}^{\bar{v}} (x - v) h (x) d x - G (z) (z - v)] . \end{aligned}

Simplifying terms,

K (v) G (v) = G (v) \int_{v}^{\bar{v}} (x - v) h (x) d x

and

G (z) \int_{z}^{\bar{v}} (x - v) h (x) d x = G (z) K (z) + G (z) (z - v) (1 - H (z))

⁠, the part inside the second square brackets can then be written as:

\begin{aligned} \int_{v}^{z} H (x) G (x) d x + \int_{v}^{z} (x - v) (h (x) G (x) + H (x) g (x)) d x \\ + G (z) (z - v) (1 - H (z)) - G (z) (z - v)] \end{aligned}

which is non-positive, and the overall difference is negative.

It follows that if

α = 0

⁠, bidding is independent of K. Note also that for

K > 0

⁠, given integration by parts and using the Leibniz rule,

b (v)

can be written as

v + α K (v) - \int_{0}^{v} \frac{G (x)}{G (v)} [1 - α (1 - H (x))] d x,

where

1 - α (1 - H (x)) > 0

which is independent of N. Recall that G depends on N and H on K. Holding K constant, bids are increasing in N since

\frac{G (x)}{G (v)} = {[\frac{F (x)}{F (v)}]}^{N - 1}

and

F (x) < F (v)

for

x < v

⁠. Holding N constant, bids are increasing in K since

K (v)

increases in K and

H (x)

is decreasing in K.

Proof of Proposition 3

Proof

From the proof of Proposition 2 it follows that:

b (v) = (1 - α) \frac{N - 1}{N} v + α \frac{K}{K + 1} + α (1 - \frac{K}{K + 1}) (\frac{N - 1}{K + N}) v^{K + 1},

and

Π (M, K)

is thus:

(1 - α) \frac{M - K - 1}{M - K + 1} + α [\frac{K}{K + 1} + (1 - \frac{K}{K + 1}) (\frac{M - K - 1}{M}) \frac{M - K}{M + 1}] .

Ignoring integer constraints, consider

Π_{K} (M, K)

⁠. Note that

Π_{K} (M, K) < 0

α = 0

and

lim_{α \to 1} Π_{K} (M, K) > 0

⁠, with

Π_{K} (M, K)

increasing in α and

Π_{K, K} (M, K) < 0

⁠. It follows, that there exists

α_{M}

such that iff

α > α_{M}

⁠, then

Π (M, K)

is globally increasing in K. Simple calculations show that

Π (M, 1) > Π (M, 0)

iff

α > α^{*}

⁠. If

α > α^{*}

⁠, there then exists

K_{M, α}

such that

Π (M, K) > Π (M, 0)

K \leq K_{M, α}

where

K_{M, α}

is increasing in α. In turn, for

α \in (α^{*}, α_{M}),

Π (M, K)

is inverse U-shaped in

K .

Finally, to show that

K_{M, α}

is increasing in M, note that

s i g n {Π (M, K) - Π (M, 0)} = s i g n {α (1 - K) + M (3 α - 2)}

⁠. In turn, if for a given α and K, this difference is positive at some M, it is also positive for any

M^{'} > M

⁠. Corollary 5 follows from the above.

Proof of Proposition 4

Proof

Under lowest exclusion, the winner of the auction is the person with the highest consumption utility and her expectation of the second highest active valuation is unaffected. In turn, for any fixed M, the bidding strategy, hence, the expected revenue is the same as under no exclusion, $Π (M, 0) = Π_{Low} (M, K)$ for any α and any K. Fix M and K. Let $α = 0$ ⁠. Under random exclusion the bidding strategy is $b (v) = v - F {(v)}^{1 - M + K} \int_{0}^{v} F {(x)}^{M - K - 1} d x$ ⁠, under lowest exclusion it is $b_{Low} (v) = v - F {(v)}^{1 - M} \int_{0}^{v} F {(x)}^{M - 1} d x$ ⁠.

Proof of Proposition 5

Proof

If $α = 0$ ⁠, then $b_{i} (v_{i}) = v_{i}$ ⁠, is the unique equilibrium since a person’s payoff is not affected by the strategies of others. Note that for any given p, holding the other players’ strategies constant, the payoff from “winning” the object is strictly increasing in $v_{i}$ as long as $α < 1$ ⁠. Hence, $b_{i} (v)$ cannot be decreasing in $v_{i}$ since, holding the strategies of the other players constant, this would violate incentive compatibility. Hence, $b_{i}$ must be monotone increasing since the bid does not directly affect the price paid only the probability of winning. In turn, given symmetric strategies, the superiority boost can only come from the excluded ones and in equilibrium $b_{i} = v_{i} + α E max_{j \in K} {v_{j} - v_{i}, 0}$ holds.

Acknowledgments

We are grateful to Sulllivan Anderson, Roland Benabou, Sylvain Chassang, James Choi, Stefano DellaVigna, Dan Gilbert, Shengwu Li, George Loewenstein, Ryan Oprea, Pietro Ortoleva, Wolfgang Pesendorfer, Klara Steinitz, Adam Szeidl, Leeat Yariv for discussions and the Editor, Nicola Gennaioli, and four anonymous referees for excellent comments. We also thank seminar audiences at Arizona, Ashoka, Berkeley, Chicago, CEU, CMU, Harvard, Hebrew U, Hong Kong, LSE, Princeton, Vienna, Zurich, Peking University Business School, VIBES June 8, 2020, and the 2019 Warwick-Princeton-Utah Conference on Political Economy in Venice. All errors are our own. An earlier version of the article was titled Mimetic Dominance and the Economics of Exclusion. Contact: [email protected] or [email protected].

The data and code underlying this research is available on Zenodo at: https://doi.org/10.5281/zenodo.8104170.

Supplementary Data

Supplementary data are available at Review of Economic Studies online.

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