Simultaneous online auctions, in which the auction of all items being sold starts at the same time and ends at the same time, are becoming popular especially in selling items such as collectables and art pieces. In this paper, we analyze the characteristics of bidders (Reactors) in simultaneous auctions who update their preauction value of an item in the presence of influencing bidders (Influencers). We represent an auction as a network of bidders where the nodes represent the bidders participating in the auction and the ties between them represent an Influencer-Reactor relationship. We further develop a random effects bilinear model that is capable of handling covariates of both bidder types at the same time and account for higher-order dependence among the bidders during the auction. Using the model and data from a Modern Indian Art auction, we find that Reactors tend to update their values on items that have high preauction estimates, bid on items created by high investment risk artists, bid selectively only on certain items, and are more active in the second half of the auction. Implications for the auction house managers are discussed.

1. Introduction

For the last three decades, simultaneous auctions have become one of the most popular auction settings for selling high-priced affiliated private value items, including FCC radio bandwidth spectrum [1], U.S. treasury bills [2], and timber [3]. In these auctions, all the items are sold simultaneously, meaning that their auctions start at the same time and end at the same time. The set of bidders who attend these auctions remain the same throughout the auction event. Such an auction setting is particularly successful in situations where the items are highly complementary and the buyers have a demand for more than one item. For example, broadcasting companies typically need to purchase more than one bandwidth of the radio spectrum; art collectors buy more than one item of their favorite artist and so on. Recently, this auction format has become popular in online auctions to sell collectables and fine arts [4].

In typical affiliated private value auctions such as art auctions, affiliation theory [5] suggests that some bidders will consult the bid amount of other bidders during the auction and update their preset valuation and their “willingness to pay” for the items they are interested in(Bidders’ valuations have some dependence with each other (loosely speaking, one bidder’s high value signal makes it more likely that other bidders will exhibit high values), implying that bidders can change valuation if others’ bidding behaviors are observable as in open bid format [5].). Particularly, if the bidders have limited information about the art items, they will consult bids of other bidders as additional item information [6, 7]. Furthermore, if the art items are sold in simultaneous auctions, bidders frequently encounter each other in more than one item. Therefore, it is possible for value-updating bidders to get influenced by the same bidder more than once. How does the characteristics of “known” bidders affect the propensity of value-updating behavior? In this paper, we intend to answer this question. In particular, we analyze the characteristics of such value-updating bidders (termed here as Reactors) in the presence of influencing bidders (termed here as Influencers). We examine three important questions pertaining to the Reactors. In what type of lots (art items) do bidders update their value? Do they bid more in the first half or in the second half of the auction? Are they selective in the lots on which they bid, or do they bid on many lots?

Traditional approaches such as general linear models and logistic regressions are useful in estimating aggregate effects of different characteristics of overbidding bidders and in identifying lots where such behavior can be witnessed but they fall short of fulfilling the goal of this study. Particularly, they are not capable of estimating effects of individual Reactors and Influencers characteristics that ultimately lead to overbidding of Reactors. Further, these models lack the capacity to capture higher-order bidder dependency that exists in simultaneous auctions. Particularly, with bidders competing for multiple items simultaneously, bidders “recognize” each other and engage in anticompetitive practices such as bidder collusion [3, 4]. To some degree, this limitation is due to the data structure available from the auction houses. To overcome this issue, we represent the auction data in the form of a network 𝑌𝑖𝑗 of bidders where bidder 𝑗 (the Reactor) has updated his valuation in reaction to the bids of bidder i (the Influencer). Using this rich and innovative framework, we examine the characteristics of Reactors using a randomeffects dyadic relation model [8]. This is based on a generalized regression framework and is capable of handling covariates of both bidder types. It builds on the social relations model [9, 10] and is capable of specifying random-effects between subsequent bidders. It is proficient in simultaneously considering regressor variables as well as correlation between Reactors having the same Influencers, between Reactors bidding on the same item, and reciprocity between Reactors and Influencers. In other words, this approach analyzes the bidder characteristics in consideration with the bidder dependence in these simultaneous auctions.

From the auction house manager’s perspective, understanding the behavior of bidders is important. Since the rivalry among auction houses has intensified in recent years, much more attention is now given to strengthen relationships with bidders. The underlying quest for all the managers is now to cultivate and promote a strong relationship with bidders and encourage them to participate more in future auctions. Reactors play a pivotal role in the success of the auction house. Their value updating behavior typically leads to overbidding and higher price [11] paid for the item, thus playing a critical part in the price formation process in the auction. Therefore, managers are interested in learning on what and when these bidders bid. (Auction house managers from multiple auction houses were interviewed during our investigation.)To this end, our paper attempts to investigate their characteristics in these simultaneous auctions.

Prior studies on this issue [12, 13], although limited, have mainly focused on understanding the underlying psychological factors that result in value updating behavior. They identified factors including bidder rivalry, social effects, and escalation of commitment that lead to a “win at any cost” mentality of the Reactors (when they start bidding irrationally in order to win the item [11]). (This emotional phenomenon is commonly known as “auction fever”.) Our paper is essentially an extension of this body of literature. We assume that bidders do update their value during auctions and such behavior is revealed when bidders place a high bid value after using a proxy bidding system earlier in the auction. Proxy bidding is a commonly available feature in most online auction houses where bidders set a maximum amount they are willing to pay, and then let the auction house place proxy bids on their behalf until that price. Along these lines, Ku et al. [11] performed a survey of bidders and found that most of them use proxies to set their maxima.

Another uniqueness of our paper from other online auction research is the context of our study, that is, auctions of high-end fine arts. Selling high-priced art items through online auctions has become a recent trend in the art market. As the demand for fine art has reached its all-time high [14], auction houses and art dealers have found Internet-based auctions as one of the reliable ways to sell art items to a wider group of art lovers. Established online auction houses such as SaffronArt (http://www.saffronart.com/), Attinghouse.com, and AspireArt (http://www.aspireart.com/), use the simultaneous auction setting in their auctions. Simultaneous online auctions are different from eBay auctions, which are frequently analyzed in academic research. In a simultaneous auction, all the items up for sale are sold concurrently to the same group of bidders over a certain period of time. This gives rise to a complex competitive environment, where there is a great level of interdependence between bidders leading to value-updating behavior by some of them. With the art market so hot and with so much at stake for the auction house managers, our paper is well focused in helping these managers develop a better relationship with their customers. From our analysis, we find that Reactors typically bid on fewer items, suggesting that they might be the collectors and not the art dealers [15]. They also bid on high-value items and bid more in the second half of the auction than in the first half. We further find that Reactors and Influencers rarely alter their role during the auction and there is more possibility for Reactors to encounter the same Influencers over other items than for Influencers meeting the same Reactors over other items.

In sum, the contribution of our study is threefold. First, we investigate characteristics of value-updating bidders in simultaneous online art auctions. Second, from the technical standpoint, we demonstrate an application of a random-effects dyadic relational model for complex human behavior and emphasize the importance of new and advanced statistical techniques available to the social sciences. And third, we introduce a new approach of considering bid history in the form of a network. This innovative and rich framework will not only allow us to examine the interbidder influence in our study but will also encourage future bidder behavior studies in other auction settings.

The rest of the paper is presented as follows. First, we describe the auction data of our research and discuss our unique approach of representing it as a social network. Second, we discuss the random-effects dyadic relation model and explain how we use it to determine the characteristics of Reactors. Third, we present the results of our investigation. Finally, we discuss the implications of our work and present directions for future research.

2. Auction Data and Bidder Network

Online auctions have become a hot research topic in economics, marketing, management, and statistics. (Laudon and Traver [16] estimate that online auction sales (C2C and B2C) will top $36 billion by 2007. Revenue exceeded $6 billion in 2006 at eBay, the pioneering online auction firm where everything from paperclips to private jets get sold. Even traditional auction houses like Christie’s (whose annual revenues are expected to top $4 billion in 2006) are adopting the online model. (http://www.iht.com/articles/2006/07/12/news/auction.php) (http://internet.seekingalpha.com/article/25034))Particularly, with the availability of detailed bidding data from online auctions, we are now able to investigate bidder behavior and auction characteristics in details that were not possible to explore earlier. For example, in the last decade, a wide range of new studies have looked at interesting auction issues such as price dynamics [17, 18], bidder surplus [17], importance of reference points in auctions [19, 20], herding behavior [21], and forward-looking behavior of bidders [22]. Interestingly, all these studies have focused exclusively on single-item auctions such as eBay, where bidders compete for one item at a time.

Unlike prior studies, this paper investigates simultaneous online auctions where multiple items are sold concurrently to a same group of bidders over a certain period of time. We have collected the data from an online auction house called SaffronArt (http://www.saffronart.com/). This auction house sells only Modern Indian Art and has become a prominent distribution channel of that genre in recent years. More specifically, the data come from a three-day auction where 199 art lots (a unique piece of art such as a painting, a drawing or a sculpture) were sold. Unlike eBay auctions, these auctions are in simultaneous first-price ascending format. The lots are open at a specific date and time, and they close simultaneously at a specific time. Moreover, to allow bidders to compete for multiple items, the lots are closed sequentially in a group of 20 to 25 lots. For example, lots 1–25 may close at 9:00 AM and lots 26–50 will close at 9:30 AM. Further, to discourage devious online bidder behavior such as sniping the auction has a soft closing time: the closing time extends by three minutes whenever a bid comes during the last three minutes of the auction. (Sniping is a strategic bidding activity where bids are submitted in the last moments of the auction to allow minimal time to other bidders to react to this bid. Such behavior is prominent in eBay auctions as the auction closes promptly at a specific time.),This time extension continues until no one bids during a span of three minutes.

2.1. Modern Indian Art

Modern Indian Art, with over $100 million in auction sales in 2006, is now one of the leading emerging art markets in the world. Although traditional auctions for Modern Indian Art have existed since 1995, it is only since 2000 that the market has exploded, with values realized at auctions growing at a brisk 68.7% annually (coincidentally, this is when SaffronArt (http://www.saffronart.com/), the source of our data, started its online auctions of Modern Indian Art). In 2006, online auction sales of Modern Indian Art from SaffronArt (http://www.saffronart.com/) ($36.76 million) had more sales (of Modern Indian Art) than the traditional auction houses like Sotheby’s ($35.29 million) and Christie’s ($33.08 million). Further, SaffronArt (http://www.saffronart.com/) sold more art items (537) compared to Sotheby’s (484) and Christie’s (329) in that year.(In 2005, online auction sales of Modern Indian Art by SaffronArt (http://www.saffronart.com/) were $18.06 million, more than that of Sotheby’s ($10.49 million) and Christie’s ($14.89 million). SaffronArt (http://www.saffronart.com/) also sold more art items (390) compared to Sotheby’s (276) and Christie’s (248) in 2005.) The top ten Indian artists sold 31% of the lots and contributed to 57% of the total value realized at auctions since 1995. Two of these artists are now ranked in the top 100 artists in the world based on their auction sales in 2005. A new set of emerging artists (the new trendsetters, typically born after 1955) have contributed 2% in value and 3% in lots and are becoming increasingly popular, commanding ever higher prices.

2.2. Bidder Influence Network

Identifying Reactors in an auction is challenging. First of all, no prior information is available about the private-value distribution of the bidders. These bidders do not reveal the maximum amount they are willing to spend, nor do they explicitly announce their value update at the end of the auction. Luckily, in online auctions, the auction house provides a proxy bidding service to the participants. Proxy bidding is a commonly available feature in most online auctions where bidders set the maximum amount they are willing to pay for the auctioned item, and then let the auction house place proxy bids on their behalf until that price. Bidders using this facility have a predetermined value for the item and use proxy bidding to stay within that value limit [23]. Ku and his colleagues [11] performed a survey of the bidders and found that most of them use proxies to set their maxima. The respondents used terms such as “maximum personal limit”, “what we were willing to spend”, “the most I was willing to bid”, and “by how I valued it” to explain their proxy bids. In the same lines, we considered a bidder to be updating his value if he/she reenters the bidding process and places a normal (nonproxy) bid that is higher than his/her earlier proxy bid.

One of the challenges with the available auction (see Figure 1) is how we capture the interbidder dependence during an auction. Particularly, as we frequently observe the same bidders competing for multiple items simultaneously, it is important that we include this auction feature in our analysis. That is, we should consider the influence of the Influencers over the Reactors in our modeling effort of their characteristics. In order to consider the inter-bidder dependencies, proper representation of the bidding data is necessary. We transform the bid history into an 𝑁×𝑁 bidder influence matrix where N denotes the number of bidders participating in the auction. The value 𝑦𝑖,𝑗 in each cell of the influence matrix indicates the influence of bidder i over bidder j. We considered the influence measure 𝑦𝑖,𝑗 as the total number of items in which the Influencer (bidder i) has bid between the proxy and nonproxy bid of the Reactor (bidder j). Since the auctions are held simultaneously, prior auction studies [3, 4] suggest that bidder-pair with multiple engagements tend to influence each other. Dass and Reddy [4] also showed that such engagements lead to dyadic level bidder effects, which has significant effect on the seller’s profit even after controlling for the aggregate competition in the auction. Figure 2 illustrates the network data structure with Reactors and Influencers as nodes. Here y1, 2 indicates the number of items in which bidder 1 (an Influencer) has placed a bid between the proxy and non-proxy bid of bidder 2 (a Reactor).

We also considered two other possible measures of influence for our bidder network Yi,j. The first approach considers the order in which the Influencers have placed their bids. We assumed that the bidders whose bid is close to that of the non-proxy bid of the Reactor will provide more value information than the bidders who have placed a bid earlier than that. We captured this notion by computing the difference between the bid ranks (order of the bids) of the Reactor and his/her Influencers. The second approach considers the reaction time of the Reactors. In other words, it considers the difference between the bids of the Influencers’ and the Reactors’ normal bid. The shorter the time difference is, the larger is the influence. The longer the time difference is, the weaker is the influence as the Reactor may have used other resources to update his value of the item. The second approach yielded the same results as our first approach, and thus their results are excluded from this paper. (The results from these approaches can be obtained from the auctions.)

3. Value Updating Behavior of Bidders

This study assumes that the characteristics of Influencers and Reactors are homogeneous within the bidder types. Since the bidder influence data are represented in the form of a sociomatrix 𝑌=[𝑦𝑖,𝑗], there are certain modeling constraints that need to be addressed. First, the model needs to accommodate bidder level covariates. The main goal of this paper is to characterize bidders who update their value during a simultaneous auction. Therefore, our modeling framework should be capable of investigating the characteristics that define as Reactors. And second, the model should allow second-order dependencies (such as reciprocity and common bidder effects in auctions where two interdependent bidders have common Influencers or common Reactors). Second-order dependencies are essential in identifying the stability of the bidder characteristics. For example, reciprocity will tell us whether bidders exchange their role as Reactors or Influencers during the auction. The effects of common Influencers and Reactors will emphasize how repeated encounters with the same bidders affect bidder decisions. In this paper, we develop a randomeffects dyadic relational model to investigate the characteristics of the bidders.

3.1. Random-Effects Dyadic Relation Model

This modeling approach is based on the works of Ho [24], and Hoff and Ward [8] that specify and analyze random-effects for the originator (Influencer) and the recipient (Reactors) in a social relations setting. It starts with the description of a simple linear model and builds the complexities around it sequentially. Consider modeling the dyadic influence data yi,j with a linear regression model of the following form:

𝑦𝑖,𝑗=𝛽𝑥𝑖,𝑗+𝜀𝑖,𝑗,(3.1) where yi,j represents the influence of bidder i over bidder j, and 𝑥𝑖,𝑗 represents the dyadic level covariates. The regressor 𝑥𝑖,𝑗is assumed to have enough individual level bidder information such that the distribution of the errors is invariant under permutations of the unit labels. This is similar to having an 𝑛×𝑛 matrix of errors with a distribution that is invariant under identical row and column permutations; thus {𝜀𝑖,𝑗𝑖𝑗} is equal to {𝜀𝜋(𝑖),𝜋(𝑗)𝑖𝑗} in terms of their distribution. This condition is called a weak row-and-column exchangeability [8] of an array. For undirected networks, such as our bidder influence network, it results into a random effects representation of the error term 𝜀𝑖,𝑗 such that it is equal in distribution to 𝑓(𝜇,𝑎𝑖,𝑎𝑗,𝛾𝑖,𝑗) where 𝜇,𝑎𝑖,𝑎𝑗,𝛾𝑖,𝑗 are independent random variables and f is a function that is specified [25, Theorem 14.11].

Therefore, individual bidders’ characteristics and higher-order dependence among the bidders’ effects during the auction are captured by an error component of the linear model where the model assumes a covariance structure that is exchangeable under identical permutations of i and j indices of the Influencers and Reactors, respectively. We model this error as Gaussian, and thus, the joint distribution of εi,j is as follows:

𝜀𝑖,𝑗=𝑎𝑖+𝑏𝑗+𝛾𝑖,𝑗,(3.2) where 𝑎𝑖 represents the effect of bidder characteristics of Influencer i, bj represents the effect of bidder level characteristics of Reactor j, and (𝑎𝑖,𝑏𝑗) multivariate normal [MVN] (0,Σab).


𝑎𝑖𝑏𝑗00,𝜎𝑁2𝑎𝜎𝑎𝑏𝜎𝑏𝑎𝜎2𝑏.(3.3) Additionally, (𝛾𝑖,𝑗𝛾𝑗,𝑖) also follows a multivariate normal distribution with MVN(0,Σ𝛾). Therefore,

𝛾𝑖,𝑗𝛾𝑗,𝑖00,𝜎𝑁2𝛾𝜌𝜎2𝛾𝜌𝜎2𝛾𝜎2𝛾.(3.4) The covariance structure of the error term is therefore

𝐸𝜀2𝑖,𝑗=𝜎2𝑎+2𝜎𝑎𝑏+𝜎2𝑏+𝜎2𝛾𝜀,𝐸𝑖,𝑗𝜀𝑗,𝑖=𝜌𝜎2𝛾+2𝜎𝑎𝑏,𝐸𝜀𝑖,𝑗𝜀𝑘,𝑙𝜀=0,𝐸𝑖,𝑗𝜀𝑖,𝑘=𝜎2𝑎,𝐸𝜀𝑖,𝑗𝜀𝑘,𝑗=𝜎2𝑏𝜀,𝐸𝑖,𝑗𝜀𝑘,𝑖=𝜎𝑎𝑏,(3.5) where 𝜎2𝑎 represents the variance in an observation due to the presence of common Influencer, 𝜎2𝑏 represents the variance in an observation due to the presence of common Reactors, and ρ represents the correlation of observations within a influencer-reactor pair and serves as a measure of reciprocity or mutuality in the bidder influence data [8]. (This model is also known as the “round-robin” model [9, 10].)To analyze the characteristics differences between Influencers and Reactors in a particular sample space, the error structure as shown above can be added to a linear predictor in a generalized linear model. Thus,

𝜃𝑖,𝑗=𝛽𝑥𝑖,𝑗+𝜀𝑖,𝑗,(3.6) where 𝜀𝑖,𝑗=𝑎𝑖+𝑏𝑗+𝛾𝑖,𝑗.

The above (3.6) is modeled such that the dyadic data are unconditionally dependent, but conditionally independent, given the random effects of Influencers and Reactors. Moreover, 𝛽𝑥𝑖,𝑗=𝛽0as the bidder influence is only modeled in terms of the individual bidder level covariates. Therefore, we consider it as a generalized linear mixed-model with inverse-link function 𝑔(𝜃). Thus,

𝐸𝑦𝑖,𝑗𝜃𝑖,𝑗𝜃=𝑔𝑖,𝑗,𝑝𝑦12,𝑦13,,𝑦𝑛,𝑛1𝜃12,𝜃13,,𝜃𝑛,𝑛1=𝑖𝑗𝑝𝑦𝑖,𝑗𝜃𝑖,𝑗.(3.7) In our case, a Poisson model with log-link is appropriate, given that we measure the level of influence 𝑦𝑖,𝑗 as the number of items where bidder 𝑖 has presumably influenced bidder j to update his/her value. Thus,


3.2. Incorporating the Effects of Higher-Order Dependence

In auctions, there are possibilities that the effect of intense rivalry initiated between two bidders in the early part of the auction may spillover to the later part of the auction and influence the Reactors to update their values. Consider a hypothetical scenario: say bidder A and bidder B have engaged in a bidding-war in the auction. This bidding phenomenon may influence the proxy bidder, say C to update his/her private value and place a higher non-proxy bid (similar to transitivity). Such second-order effects are common in art auctions [4, 15, 26].

In the social networks literature, researchers have used simple functions of latent characteristics vector 𝑧𝑖 for Influencers and 𝑧𝑗 for Reactors and added their inner product 𝑧𝑖𝑧𝑗 to the error model in a fixed effect setting to capture such higher-order effects [8]. They used models of the form 𝜃𝑖,𝑗=𝛽𝑥𝑖,𝑗+𝑓(𝑧𝑖,𝑧𝑗) where the latent characteristics function 𝑓(𝑧𝑖,𝑧𝑗) is considered either as a distance model {𝑓(𝑧𝑖,𝑧𝑗)=|𝑧𝑖𝑧𝑗|} or as a projection model 𝑓(𝑧𝑖,𝑧𝑗)=𝑧𝑖𝑧𝑗/|𝑧𝑗|. We followed a similar approach and added the inner product 𝑧𝑖𝑧𝑗 to the error model (3.2). Thus,

𝜀𝑖,𝑗=𝑎𝑖+𝑏𝑗+𝛾𝑖,𝑗+𝜁𝑖,𝑗,(3.9) where 𝜁𝑖,𝑗=𝑧𝑖𝑧𝑗 is a mean-zero random effect and the random effects 𝑎𝑖,𝑏𝑗, and 𝛾𝑖,𝑗 are modeled with the multivariate normal distributions described earlier. The moment properties of the distributions of 𝜁 are 𝐸(𝜁𝑖,𝑗)=0;𝐸(𝜁2𝑖,𝑗)=trace2𝑧;𝐸(𝜁𝑖,𝑗𝜁𝑗,𝑘𝜁𝑘,𝑖)=trace3𝑧 if the z’s are modeled as independent k-dimensional MVN random vectors with mean zero and covariance structure matrix as𝑧 and all other second-and third-order moments are equal to zero. Moreover, an orthogonal transformation of the z’s leaves 𝑧𝑖𝑧𝑗invariant, suggesting that 𝑧 is a diagonal matrix. For example, lets us consider the case where 𝑧=𝜎2𝑧𝐼𝑘×𝑘 and the above moments are 0, 𝑘𝜎4𝑧, and 𝑘𝜎6𝑧, respectively. When we add 𝜁𝑖,𝑗 to the error term, the nonzero second-and third-order moments become

𝐸𝜀2𝑖,𝑗=𝜎2𝑎+2𝜎𝑎𝑏+𝜎2𝑏+𝜎2𝛾+𝐾𝜎4𝑧,𝐸𝜀𝑖,𝑗,𝜀𝑗,𝑖=𝜌𝜎2𝛾+2𝜎𝑎𝑏+𝐾𝜎4𝑧,𝐸𝜀𝑖,𝑗𝜀𝑗,𝑘𝜀𝑘,𝑖=𝐾𝜎6𝑧𝜀,𝐸𝑖,𝑗𝜀𝑖,𝑘=𝜎2𝑎,𝐸𝜀𝑖,𝑗𝜀𝑘,𝑗=𝜎2𝑏𝜀,𝐸𝑖,𝑗𝜀𝑘,𝑖=𝜎𝑎𝑏.(3.10) The above shows that the effect 𝜁𝑖,𝑗=𝑧𝑖𝑧𝑗can be interpreted as a mean-zero random effect that is able to capture the higher-order dependencies found among the bidders in an auction. Marginally, if k increases, the distribution of 𝜁𝑖,𝑗 will converge to a normal distribution (due to central limit theorem). Jointly, if the 𝜁s of the Influencer and Reactor have at least one unit in common, the Markov dependence graph will plot the two bidders as neighbors. If 𝜁𝑠are considered as fixed effects, they can be viewed as interaction terms that are highly constrained due to the functional dependence on the 𝑧’s. For example, if the 𝑧𝑖 and 𝑧𝑗 are vectors of similar directions and magnitude, then 𝑧𝑖𝑧𝑘 and 𝑧𝑗𝑧𝑘 will not be different. This characteristic can be related to bidder transitivity as we described earlier. (For more information on how the moments are scaled, please refer to [8, 24, 27].)

Since the inner-product term 𝑧𝑖𝑧𝑗 is a fixed effect, it can be considered as a reduced-rank interaction term. This is typically known as the bilinear effect or multiplicative interaction. To include this inner product of the member’s latent characteristics in the random effect model, (3.6) is reparameterized as

𝜃𝑖,𝑗=inf𝑖+rea𝑗+𝛾𝑖,𝑗+𝑧𝑖𝑧𝑗,inf=𝛽inf𝑥inf,𝑖+𝑎𝑖,rea𝑖=𝛽𝑟𝑒𝑎𝑥rea,𝑖+𝑏𝑖,(3.11) where 𝑥inf,𝑖 are the Influencer specific covariates and xrea,i are the Reactor specific covariates in the model.

The above model is estimated using a Bayesian estimation process. A Markov Chain Monte Carlo (MCMC) algorithm is used to sample values of the Influencer and Reactor specific parameters from their posterior distribution. We estimate the parameters for the generalized bilinear regression model by constructing a Markov chain in {𝛽inf,𝛽rea,𝑎𝑏,𝑍,𝜎2𝑧,𝛾} (where 𝑍 is a 𝑘×𝑛 latent vector matrix) with 𝑝(𝛽inf,𝛽rea,𝑎𝑏,𝑍,𝜎2𝑧,𝛾𝑌)as the invariant distribution. A Gibbs sampling is used to obtain the chain and to sample θ.

The algorithm followed as per Hoff [8] is as follows.

(1)Sampling of linear effects in the model is as follows(a)Sample inf,rea𝛽inf,𝛽rea,𝑎𝑏,𝑍,𝜃,𝛾 with a linear regression.(b)Sample 𝛽inf,𝛽reainf,rea,𝑎𝑏 with a linear regression.(c)Sample 𝑎𝑏,𝛾 from their full conditionals.(2)Sampling of bilinear effects is as follows(a)For each bidder 𝑖=1,2,,𝑛, sample 𝑧𝑖{𝑧𝑗𝑗𝑖},𝜃,𝛽,𝑠,𝑟,Σ𝑧,Σ𝛾  with a linear regression.(b)Sample Σ𝑧 from its full conditional distribution.(3)Update {𝜃𝑖,𝑗,𝜃𝑗,𝑖} using Metropolis-Hastings step. (a)Propose [𝜃𝑖,𝑗𝜃𝑗,𝑖]𝑁([𝛽𝑥𝑖,𝑗+𝑎𝑖+𝑏𝑗+𝑧𝑖𝑧𝑗𝛽𝑥𝑗,𝑖+𝑎𝑗+𝑏𝑖+𝑧𝑗𝑧𝑖],Σ𝛾).(b)Accept [𝜃𝑖,𝑗𝜃𝑗,𝑖] with probability 𝑝(𝑦𝑖,𝑗𝜃𝑖,𝑗)𝑝(𝑦𝑗,𝑖𝜃𝑗,𝑖)/𝑝(𝑦𝑖,𝑗𝜃𝑖,𝑗)𝑝(𝑦𝑗,𝑖𝜃𝑗,𝑖)1. The prior distributions of the parameters are taken as follows:(i)𝛽MVN(0,80×𝐼4×4),(ii)Σ𝑎𝑏~ inverse Wishart Distribution (𝐼2×2,4),(iii)𝜎2𝑢,𝜎2𝑣 iid inverse gamma(1,1), 𝜎2𝛾=(𝜎2𝑎+𝜎2𝑏)/4,𝜌=(𝜎2𝑎𝜎2𝑏)/(𝜎2𝑎+𝜎2𝑏).

We used K = 2 dimensions in our analysis. (Five different values of K = 0 to K = 1 were tested with a fourfold cross-validation procedure as described by Ho in his seminal paper [8]. The predictive performances for all the K values were roughly the same. The biggest improvement in the marginal likelihood criterion was from K = 1 to K = 2. Therefore, K = 2 was selected.)

4. Bidder Covariates

Reactors are examined in the context of three types of characteristics: lot (item) characteristics, auction characteristics and bidder behavior characteristics.

4.1. Lot Characteristics

Prior work on auctions of fine arts [26, 28] found that both the preauction estimates of the items provided by the auction house and the art type (paper or nonpaper work) are to be significant drivers of auction prices. Before the auction starts, potential bidders are exposed to various types of information about the lots. Lot information and provenance provided by the auction house (in their printed catalogs and in their websites), and comments and suggestions of the art experts (in personal blogs, art magazines, etc.), provide information on the value of the art items such as the estimated price, artist information, and previous auction price history of similar paintings by the artists. The price estimates indicate the value of the items as suggested by the auction house experts (like curators, art specialists, etc.). Mei and Moses [29] found these value estimates to have high correlation with the final realized prices of the art items. Therefore, the higher the preauction estimates, the greater is the tendency for the item to fetch a higher price, and so higher stakes are associated with it. Such high stakes lead to higher bidder propensity to clarify and justify their future bids. Therefore, bidders look for other value signals from different sources and are inclined to change their value belief during the auction of these lots. Thus, we hypothesize that the bidders who act as Reactors tend to bid on lots that have high preauction estimates.

The media on which the art item is painted plays an important role in its maintenance and longevity [26]. For example, works on paper are typically of low price as compared to that of works on canvas since paper tends to be more fragile than canvas. Therefore, low financial risk is associated with the purchase of nonpaper items, and thus, we hypothesize that the Reactors will tend to update values more on nonpaper items.

4.2. Artist Characteristics

Artist characteristics such as reputation and previous auction history of the artists also play an important role in the valuation of the art items. Established artists are highly reputed and their works are well recognized in the art market. Most of their works have been resold many times in the market and, thus, their value commonly known to all. Therefore, the works of established artists present a low-risk purchase opportunity for the bidders [26]. On the other hand, emerging artists are new to the art market and their works are not well known. Further, not enough works of these artists are sold in the marketplace in order to estimate their values confidently. Therefore, the values of the works of these artists are highly uncertain, making them high-risk purchases. Thus, Reactors are hypothesized to react and change their values in lots created by the emerging artists as they tend to seek more information from other bidders.

Like artist reputation, historical market information such as the average price per square inch of the artists’ art works or the total number of items sold in the previous year provides a signal to the bidders about the market value of the artist [26]. If the value realized by the works of an artist is low in the previous year, stakes will be high for their art work. Since evaluators are unaware of how the market will react to the works of these artists, they are high-risk purchases for the bidders. Similarly, if fewer items of the artist are sold in the previous year’s auctions, less information is available about the artist’s present market value. Therefore, it is highly probable for bidders to rely upon other value signals in these lots. Thus, it is hypothesized that Reactors will tend to change their values in these lots.

4.3. Bidding Characteristics

During auctions, Reactors are assumed to wait and consult bids of others participants (Influencers) to reduce any uncertainty they may have about the value of a lot. Thus, we conjecture that Reactors will bid more in the second half of the auction than in the first half. Further, these bidders by the virtue of their behavior tend to be selective in the types of lots on which they bid. Their attachment to a particular item is an integral reason for them to update their private value for the lot [13]. Therefore, we hypothesize that they will participate in auctions of fewer lots than other bidders will.

5. Results

The description of the bid data is presented in Table 1. In this particular auction, 199 lots were sold, and 42 bidders were observed to change their value belief for 63 lots during a Modern Indian Art auction held in December 2005. Eighty bidders participated in the auctions of these 63 lots creating 947 bid instances. Thus, an 80×80 bidder matrix is created and only the influence on those forty-two bidders who changed their value belief is considered. Bidder level covariates are used in the model to determine the characteristics of those who overbid. Table 2 presents the correlation between the bidder level covariates.

A Markov Chain Monte Carlo algorithm was run for the influence data. Each chain was run for 200,000 iterations. However, since a large number of parameters (Influencer and Reactor specific covariates, and terms capturing higher-order dependencies) are analyzed in the model, only every 50th iteration is stored in order to keep the output file to a reasonable size, as suggested by MacEachernand and Berliner [30]. Outputs from first 20,000 iterations are considered as burn-ins and are not recorded. (A burn-in of 20,000 and capturing every 50th iteration suggests that the model should stabilize by the 400th row of the output. Examining the trace plot, we do find that the model has stabilized well before 400, suggesting that 20,000 burn-in was sufficient. Complete traces are available from the authors.) The posterior means and quantile-based 95% confidence intervals are presented in Tables 3 and 4 where the mean is highlighted with bolded fonts.

Here we will only discuss the results where the 95% confidence interval does not include a zero. From Table 3, we find that the pre-auction estimates have a positive effect for the Reactors. This suggests that bidders who typically change their value belief have bid on lots with high estimated values. We also examined the bidder characteristics of the Influencers and Reactors by comparing their bidding frequency during each half of the auction. Results show that the coefficient for the number of second half bids for the Reactors is positive, but there is no effect of the first half bid frequency. This suggests that Reactors tend to bid more in the second half of the auction than the first half. We also found that Influencers bid more in the first half as compared to the second half of the auction. In addition, we examined the coefficient for the total number of unique lots bid by the two types of bidder. We found that the coefficient for the Reactors to be negative (coefficient = − 6.4200) and no effect for the Influencers. This suggests that Reactors participate in auctions of fewer lots as compared to the Influencers. A summary of the results is presented in Table 5.

Estimates of other major parameters (Table 4) suggest that the common influence variance is smaller than the common Reactor variance, meaning that the Influencers maintain their role throughout the auction. Since these influencing bidders are frequently observed to bid on more lots, they tend to be the art dealers [15]. On the other hand, the common Reactor variance is large. This also supports the findings that the Reactors bid on fewer items and thus, the same Reactor is seldom encountered by the Influencers. Error variance was found to be small, suggesting a good fit for our model. Finally, we find that the reciprocity is negative, which further suggests that the Reactors and Influencers rarely switch roles during the auction.

6. Implications and Future Directions

This paper investigates the characteristics of bidders (termed here as Reactors) who update their private value of the items in the presence of the influencing bidders (termed here as Influencers). Traditional approaches such as linear models or logistic regressions, although useful in our context, lack the capacity to capture the inter-bidder effects and analyze Reactors and Influencers concurrently. To overcome this issue, we represent the auction data as networks of bidders where the nodes represent the bidders participating in the auction and the ties between them represent an Infuencer-Reactor relationship. We further develop a random-effects bilinear model capable of handling the covariates of both bidder types at the same time and account for higher-order dependence occurring during an auction.

From the auction house manager’s perspective, this study provides a way to identify different types of bidders in terms of how they manage and update their private value towards an item. Such information is vital for auction house managers in strengthening their relationship with their customers and helps them to determine how to attract them to future auctions. For example, our analysis suggests that Reactors mostly bid on fewer items, indicating that they might be the art collectors, who, unlike the art dealers, are very selective on the items they desire to purchase [15]. Therefore, there is great possibility and opportunity for the auction house managers to take part in the strategic decisions of the bidders and support in their decision making process. Suggestive decisions are shown in Table 6 to support the auction house managers.

As with most research work, this study also has some limitations that need to be acknowledged. First, it is difficult to determine why the Reactors have used the proxy bidding system. We used an identification system in the lines of work done by Ku and his colleagues [11], which is tested and proved using a bidder survey. Work by Bapna and his colleagues [23]) found that certain types of bidders in online auctions use proxy bidding as a part of their bidding strategy (referred to as Agent Bidders). They suggested that these bidders place more than 60% of their bids as proxy bids. Reactors in our study had less than 20% of their total bids as proxy bids, and thus can be assumed not to be Agent Bidders. We tried two different approaches to identify the level of influence of an Influencer over a Reactor. Although our measure is grounded on the auction theory, still, further study is needed to develop a better influence measure.

A bidder’s value change plays a fundamental role in determining the attractiveness of auctioned items. Therefore, sophisticated models may be developed using value updating behavior in the future to design more profitable auctions and also help in designing future auctions. Our study contributes to auction house strategy refinement by investigating competition among bidders at the dyadic bidder level. Further studies on this topic are essential to understand the underlying bidding dynamics in auctions.

We hope that this paper will encourage other researchers to start investigating bidder behavior and competition at a more microlevel. We also hope that our paper will promote future applications of bilinear effects models and other advanced techniques in other areas of social sciences.