Selling mechanisms for perishable goods: An empirical analysis of an online resale market for event tickets

Abstract

This paper assesses the value of the availability of menus of different selling mechanisms to agents in an online platform in the context of scarce perishable goods. By analyzing the choice between auctions and posted prices in the context of National Football League tickets offered on eBay, it estimates a structural model in which heterogeneous, forward-looking sellers optimally choose which selling mechanism to use and its features. Counterfactual results suggest that sellers would experience an average 87.37% decrease in expected revenues if auctions were removed and just a 4.34% decrease if posted prices were. In turn, buyers would benefit from an auction-only platform since the expected number of transactions would increase and expected transaction prices would decrease. These results suggest that while sellers benefit from menus of different selling mechanisms, the same does not hold for buyers. Thus, the implications for a platform, which should take into account both sides of the market, are ambiguous.

Appendices

Appendix A: Sample construction

I describe here in detail the procedure I used to create the chains of listings described in Section 2. The key information used to create the chains consists of the number of tickets being offered, the game they corresponded to, and the section and row where the seats were located. This information is key for two reasons. First, the linking process to track the same set of tickets over time is based exclusively on them.¹² Second, these variables allow me to identify the price of these tickets on the primary market, which yields a measure of their quality.

The first step is obtaining information on the game corresponding to a given listing, which is often available in a standardized fashion. When this is not the case, I obtained this information from the title or subtitle of the listings. When these are not informative, the dates in which listings were created by the sellers are ordered to potentially fill in this missing information. I also make use of this procedure to correct listings for tickets that were created after the games they corresponded to had taken place, which were usually instances in which the seller corrected the information shortly after. When this was not the case, sellers just removed the listing within a few days indicating that they were erroneous and possibly the result of automatic re-listing.

Of the valid listings with information regarding the game to which the tickets corresponded, around 97% also had information on number of tickets and section and row where the tickets were located. To fill in the missing information I use the listing’s title or subtitle. I also verify whether sellers had offered multiple listings for the same team at the same location and whether the listings with missing information were created and terminated in between listings with complete information. This procedure was also useful to correct instances in which the information was erroneous, either because the section and row numbers were exchanged or because the information did not conform with what was reported on the title or subtitle.

With this information in hand I define as potential chains of listings combinations of different seller-game-section-row quadruples. I then identify instances in which listings within the same chain are created before the previous one was over. These cases are inspected and classified into five scenarios. First, multiple chains at the same location offered by the same seller. This is done based on information on the titles, complete sales, and other chains by the same seller. Cases in which the seller creates two identical listings across all dimensions at virtually the same time are assumed to be for different tickets. The second scenario is reorganization of quantities, or rebundling. For example, turning a single listing for four tickets into two listings for two tickets each. Third, listings which were removed within a day and recreated shortly after are assumed to be mistakes and deleted. The final two scenarios concern listing the same set of tickets more than once concurrently, which I call doublelisting.

I classify doublelisting scenarios into two cases. The first is separation across quantities: for example, having a listing for four tickets and, at the same time, two separate listings for two tickets each. This is again cross-checked with the seller’s history of listings and their outcomes. Returning to the example from the previous paragraph, if both a two-ticket and four-ticket listings are sold then they were for different sets of tickets, while if the four-ticket one is sold and the other two two-ticket listings are then removed from the website it suggests that the same tickets were listed twice. Finally, the second case consists of listing the same set of tickets through different mechanisms. I verified these cases according to the same procedure that I employed in the previous case.

At the end of this procedure I obtain a sample of 38,520 sets of tickets, which were offered across 78,863 listings. However, the analysis will be restricted to activity within four weeks of a game. This restriction is not extreme: in this period, almost 61% of tickets were introduced to the market, more than 70% of tickets were available at the website, and more than 67% of the transactions observed in the data took place. The resulting sample contains 27,047 sets of tickets across 43,215 listings. This subsample is used to construct the measures of market tightness and of potential demand on the platform at every point in time. To estimate the model, I drop the 293 listings, spread across 226 chains, that do not have information on the number of tickets, type of tickets, or section or row where tickets were located. I further restrict the sample to tickets that were always offered as a pair and were never rebundled or doublelisted. Therefore, the final sample is smaller, containing 19,174 pairs of tickets over 28,257 listings. Nevertheless, pairs are the most common bundle offered (more than 74% of listings).

Appendix B: More mechanisms

I now document patterns relative to mechanisms that were not explicitly considered in the main text: auctions with an immediate purchase option, also known as buy-it-now (BIN) or hybrid auctions, and bargaining-enabled posted prices. First, Fig. 8 shows which among the four options sellers choose across time. Hybrid auctions are not as popular as regular auctions until the week of the game, possibly because the additional flexibility they yield becomes more attractive closer to the deadline, when buyers seem to be relatively less willing to participate in auctions. In turn, despite being the default option for posted prices, bargaining-enabled listings only become slightly more prevalent within two weeks of the game, possibly due to their additional flexibility akin to hybrid auctions.

Fig. 8

Notes: Quantities displayed on the vertical axis are averaged over all listings in the sample

A different consideration is if and how prices are chosen differently across these mechanisms. Figure 9 displays the choice of start prices for regular and hybrid auctions, posted prices with and without the bargaining option, and buy prices for hybrid auctions. It is interesting to note that hybrid auctions consistently have higher start prices than regular ones, which could be a consequence of sellers with higher outside options self-selecting into the hybrid format. Moreover, posted prices with a bargaining option are consistently higher than those without it, possibly because sellers anticipate a potential negotiation that would likely reduce the final agreed upon price. Finally, it is interesting to note that buy prices in hybrid auctions are often higher than posted prices regardless of whether bargaining is available to buyers.

Fig. 9

Notes: Quantities displayed on the vertical axis are averaged over all listings in the sample

A final consideration is whether buyers make use of this richer set of mechanisms. First, it is interesting to note that despite being slightly more prevalent than regular auctions, hybrid auctions are less likely to convert, possibly because of higher start and buy prices, as displayed in Fig. 9. In addition, just a little more than 15% of successful hybrid auctions were actually sold via buy prices, in part because when the reserve price is met the buy-it-now option goes away. Similarly, bargaining-enabled posted prices are more commonly used than regular posted prices, but have a lower conversion rate. Furthermore, even though almost half of the bargaining-enabled listings were involved in negotiations at some point, less than 30% of these were sold at a negotiated price. These numbers show that abstracting from these more detailed, hybrid mechanisms is not a dramatic simplification.

Table 13 Distribution of listings across mechanisms II

Appendix C:: Derivation of expected utility and profit functions

I now derive the expected utility and profit functions from Section 3.

1.1 C.1 Expected utility from auctions

For ease of notation I will ignore the subscripts. Assume that a buyer with original valuation v is matched on day t with an auction with reserve price r, end date τ, Poisson arrival rate λ, and characteristics x. Conditional on v ≥ r and on the number of opposing bidders being n, the buyer’s expected utility from the auction is:

$$ \begin{array}{@{}rcl@{}} U_{A_{t}}(v,r,\tau,n,x)\!&=&\!\Pr(V^{(n:n)}\!<\!v| x,\tau) \left (v-\mathbb{E} \left [\max \left \{ V^{(n:n)},r \right \} \middle \vert V^{(n:n)}<v,x,\tau \right ] \right ) \\ &=&F_{V}(v|x,\tau)^{n} v-{{\int}_{0}^{v}}\max \left \{ u,r\right\}n F_{V}(u|x,\tau)^{n-1}f_{V}(u|x,\tau)du\\ &=&F_{V}(v|x,\tau)^{n} v-{{\int}_{0}^{r}}r n F_{V}(u|x,\tau)^{n-1}f_{V}(u|x,\tau)du\\ &&- {{\int}_{r}^{v}} u n F_{V}(u|x,\tau)^{n-1}f_{V}(u|x,\tau)du\\ &=&F_{V}(v|x,\tau)^{n} v-r F_{V}(r|x,\tau)^{n}\\&&- \left [ u F_{V}(u|x,\tau)^{n}{\vert_{r}^{v}} -{{\int}_{r}^{v}} F_{V}(u|x,\tau)^{n}du \right] \\ &=&{{\int}_{r}^{v}} F_{V}(u|x,\tau)^{n} du. \end{array} $$

The number of opposing bidders, n, is unknown to the buyer. As demonstrated by Myerson (1998), the Poisson distribution yields environmental equivalence: the number of opposing bidders from a buyer’s perspective will follow the same distribution as the one guiding the overall arrival of bidders. Hence, summing over all realizations of n:

$$ \begin{array}{@{}rcl@{}} U_{A_{t}}(v,r,\tau,\lambda,x)&=& \sum\limits_{n=0}^{\infty}\frac{\lambda^{n} e^{-\lambda}}{n!} {{\int}_{r}^{v}} F_{V}(u|x,\tau)^{n} du\\ &=& {{\int}_{r}^{v}} \sum\limits_{n=0}^{\infty}\frac{\lambda^{n} e^{-\lambda}}{n!} F_{V}(u|x,\tau)^{n} du \\ &=&{{\int}_{r}^{v}} e^{-\lambda\left [1-F_{V}(u|x,\tau)\right ]} du. \end{array} $$

1.2 C.2 Expected utility from posted prices

Now assume that buyer i with valuation v is matched with a posted price with price p such that v ≥ p and arrival rate λ. If the number of opposing buyers is zero, a purchase is made. If there is one opposing buyer, with 50% probability i is called first and makes a purchase and with 50% probability i is called second and purchases because the other buyer’s valuation is below p. Thus, applying this to all realizations of n yields:

$$ \begin{array}{@{}rcl@{}} U_{P_{t}}(v,p,\lambda,x) &=&(v-p)\sum\limits_{n=0}^{\infty} \frac{\lambda^{n} e^{-\lambda}}{n!}\frac{1}{n+1}\sum\limits_{l=0}^ n F_{V}(p|x,t)^{l} \\ &=&(v-p)\sum\limits_{n=0}^{\infty} \frac{\lambda^{n} e^{-\lambda}}{(n+1)!}\left (\frac{1-F_{V}(p|x,t)^{n+1}}{1-F_{V}(p|x,t)} \right ) \end{array} $$

$$ \begin{array}{@{}rcl@{}} \phantom{U_{P_{t}}(v,p,\lambda,x)}&=&\frac{(v-p)}{1-F_{V}(p|x,t)} \left \{ \frac{1}{\lambda} {\sum}_{n=1}^{\infty} \frac{\lambda^{n} e^{-\lambda}}{n!}- \frac{e^{-\lambda\left [ 1-F_{V}(p|x,t) \right ]}}{\lambda}\right.\\ &&\left. \times \sum\limits_{n=1}^{\infty} \frac{\left [ \lambda F_{V}(p|x,t)\right ]^{n} e^{-\lambda F_{V}(p|x,t)}}{n!} \right \}\\ &=&\frac{(v-p)}{\lambda \left [ 1-F_{V}(p|x,t) \right ]}\\&& \left \{ \left (1-e^{-\lambda}\right )-e^{-\lambda \left [ 1-F_{V}(p|x,t) \right ]} \left (1- e^{-\lambda F_{V}(p|x,t) }\right) \right \}\\ &=&\frac{(v-p)\left (1-e^{-\lambda \left [ 1-F_{V}(p|x,t) \right ]}\right)}{\lambda \left [ 1-F_{V}(p|x,t) \right ]}. \end{array} $$

1.3 C.3 Computing choice-specific value function for auctions

I now describe how the choice-specific value functions for auctions ($\tilde {\pi }_{jt}^{A_{\ell }}$) were computed. To ease notation I will drop the subscripts for seller (j), date when the auction started (t), and auction length (ℓ), as well as the superscript indicating that an auction was chosen (k = A).

To calculate $\mathbb {E}\left [\max \limits \{V^{(N-1:N)},r\}\right ]$, I first re-write the seller’s payoff using analogous equivalences from static auction theory. First, for any number of bidders n, any reserve price r, and any seller continuation value, π₀, it follows that revenue can be expressed as . Hence, the expected payoff from an auction is given by:

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\pi^{A} = \mathbb{E}_{N} \left [ \max\{V^{(n-1:n)},r\} \middle \vert N = n \right ] + \left( \pi_{0}-r\right)Pr_{N} \left (V^{(n:n)}<r \middle \vert N = n \right ). \end{array} $$

(C.1)

First, I compute the second term in the right-hand side of (C.1). It follows that, if bidders arrive according to a Poisson distribution with parameter λ,

$$ \begin{array}{@{}rcl@{}} \left( \pi_{0}-r\right)Pr_{N} \left (V^{(n:n)}<r \middle \vert N=n \right )&=&\left (\pi_{0}-r\right ) \sum\limits_{n=0}^{\infty} \frac{\lambda^{n} e^{-\lambda}}{n!}F_{V}(r)^{n} \\ &=& \left( \pi_{0}-r\right) e^{-[1-F_{V}(r)]\lambda }. \end{array} $$

(C.2)

To compute the first term in the right-hand side of (C.1) first note that if N ≤ 1, then r ≥ V^N− 1:N with probability one. For these two values the expression simply is:

$$ \begin{array}{@{}rcl@{}} \sum\limits_{n=0}^{1} \frac{ \lambda^{n} e^{-\lambda}}{n!} r=e^{-\lambda} r\left (1 + \lambda \right ). \end{array} $$

(C.3)

Finally, consider now the case when N > 1. In particular, for any n and r it follows that:

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\mathbb{E} \left [ \max \left \{ V^{n-1:n} ,r \right \}\middle\vert N=n \right ]&=&{\int}_{0}^{\infty}\max\{u,r\}f_{n-1:n}(u)du \\ &=&{{\int}_{0}^{r}} r f_{n-1:n}(u)du +{\int}_{r}^{\infty} u f_{n-1:n}(u)du. \end{array} $$

(C.4)

The two terms in (C.4) are now evaluated separately, beginning with the first in the right-hand side. Remember that due to properties of order statistics it follows that F_n− 1:n(r) = nF(r)^n− 1 − (n − 1)F(r)ⁿ. Thus,

$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}_{N} \left [r F_{n-1:n}(r)\middle \vert N=n \right ]\\&=&r\sum\limits_{n=2}^{\infty} \frac{\lambda^{n} e^{-\lambda}}{n!} \left [n F_{V}(r)^{n-1} - (n-1)F_{V}(r)^{n} \right ] \\ &=&r \left \{ \lambda \left [1-F_{V}(r)\right] \sum\limits_{n=2}^{\infty}\frac{\left [ \lambda F_{V}(r) \right]^{n-1}e^{-\lambda}}{(n-1)!} + \sum\limits_{n=2}^{\infty}\frac{\left [ \lambda F_{V}(r) \right]^{n}e^{-\lambda}}{n!}\right \} \\ &=&r e^{-\lambda\left[1-F_{V}(r)\right]} \left \{ \lambda \left [1-F_{V}(r)\right] \left (1-e^{-\lambda F_{V}(r)} \right ) \right. \\ &&\left.+ 1-\left [ 1+\lambda F_{V}(r)\right ] e^{-\lambda F_{V}(r)} \right \} \\ &=&r e^{-\lambda \left[1-F_{V}(r)\right]} \left\{1+\lambda\left[1-F_{V}(r)\right] \right \}-r e^{-\lambda} \left( 1+\lambda \right ). \end{array} $$

(C.5)

Now I compute the expectation over N of the second term in the RHS of (C.4):

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{n=2}^{\infty}\frac{\lambda^{n} e^{-\lambda}}{n!}{\int}_{r}^{\infty} y f_{n:1:n}(y)dy\\&=&{\int}_{r}^{\infty} y \left [ \sum\limits_{n=2}^{\infty}\frac{\lambda^{n} e^{-\lambda}}{n!} f_{n-1:n}(y) \right ] dy \\ &=&{\int}_{r}^{\infty} y \left [ \sum\limits_{n=2}^{\infty}\frac{\lambda^{n} e^{-\lambda}}{n!} \left [n f_{n-1:n-1}(y)-(n-1) f_{n:n}(y) \right ] \right ] dy \\ &=&{\int}_{r}^{\infty} y \left [ \sum\limits_{n=2}^{\infty}\frac{\lambda^{n} e^{-\lambda}}{n!} \left [n(n-1) F_{V}(y)^{n-2}f_{V}(y)-n(n-1) F_{V}(y)^{n-1}f_{V}(y) \right ] \right ] dy \\ &=&\lambda^{2} {\int}_{r}^{\infty}y e^{-\lambda[1-F_{V}(y)]} [1-F_{V}(y)]f_{V}(y) \left [ \sum\limits_{n=2}^{\infty}\frac{[\lambda F_{V}(y)]^{n-2} e^{-\lambda[1-F_{V}(y)]}}{(n-2)!} \right ] dy \\ &=&\lambda^{2} {\int}_{r}^{\infty} y e^{-\lambda[1-F_{V}(y)]} [1-F_{V}(y)]f_{V}(y) dy \\ &=& {\int}_{0}^{\lambda[1-F_{V}(r)]} x e^{-x} F_{V}^{-1}\left (1-\frac{x}{\lambda} \right )dx \\ &=&{\int}_{0}^{\lambda e^{-\frac{r^{2}}{2\sigma^{2}}}} x e^{-x} \sqrt{2\sigma^{2} \log \left (\frac{\lambda}{x} \right )} dx \equiv \xi, \end{array} $$

(C.6)

where the first equality follows from Fubini’s theorem, the second from properties of order statistics, the penultimate from a change of variables in which x = λ[1 − F_V(y)], and the last from the Rayleigh distribution which was assumed. The last integral, defined as ξ, is solved via Gauss-Chebyshev quadrature using ten nodes.

Plugging (C.5) and (C.6) in (C.4) yields:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}_{N} \left [\max \left \{V^{N-1:N},r\right\}\right ] &=& r e^{-\lambda \left[1-F_{V}(r)\right]} \left\{1+\lambda\left[1-F_{V}(r)\right] \right \}\\&&-r e^{-\lambda} \left( 1+\lambda \right ) + \xi, \end{array} $$

(C.7)

and plugging (C.2), (C.3), and (C.7) in (C.1) finally yields:

$$ \begin{array}{@{}rcl@{}} \pi^{A}=r \lambda [1-F_{V}(r)]e^{-\lambda [1-F_{V}(r)]}+\xi + e^{-\lambda [1-F_{V}(r)]}\pi_{0}. \end{array} $$

(C.8)