Test Me If You Can—Providing Optimal Information for Consumers Through a Novel Certification Mechanism ^†

Abstract

Certifiers such as Stiftung Warentest (Germany), Which? (UK), and Consumer Reports (US) reduce asymmetric information between buyers and sellers by providing credible information about product quality. However, due to their limited testing capacities, they face a set selection problem and test only a subset of all available product models. We show theoretically that, under any current mechanism to select product models for testing, buyers always end up buying suboptimal product models, unless all product models which would be sold under complete information (or all but the overall cheapest one) happen to be tested. Instead, we propose a novel mechanism based on voluntary disclosure, but with the same testing capacity, which always yields the maximum possible consumer surplus and thus weakly dominates any current mechanism. Furthermore, we confirm in a controlled laboratory experiment that our mechanism significantly increases consumer surplus.

1. Introduction

From technical products such as smartphones to insurances or consumer products such as foodstuff, toothpaste, and strollers, sellers are often better informed about product quality than buyers. Unfortunately, this asymmetric information may lead to a fundamental economic problem: if sellers of high-quality products are not able to credibly convey the high quality of their products, not all buyers are able to choose the quality they would like to buy. Consequently, information asymmetry about vertical product quality1 can reduce consumer (and producer) surplus (Akerlof, 1970).2

Independent consumer organizations such as Stiftung Warentest (Germany), Which? (UK), and Consumer Reports (US) reduce asymmetric information between buyers and sellers by providing credible information about product quality. These organizations are third-party certifiers whose goal is to provide objective and comprehensive information about product quality. They are usually non-profit organizations, and neither require sellers to pay a fee for the rating service nor accept advertisements, in order to avoid conflicts of interest. Instead, sales of their own publications represent one of their main sources of financing (see Stiftung Warentest, 2019; Which?, 2023c; Consumer Reports, 2023; International Consumer Research & Testing, 2021).3 Usually, independent consumer organizations aim to provide a comprehensive rating of vertical product quality; i.e., they include ratings for a stroller’s weight, how waterproof the raincover is, and the level of toxic substances. However, they do not include ratings for horizontal quality, e.g., how tasteful a stroller’s color is. Independent consumer organizations often have employees whose task is to purchase products (for testing) while concealing that they are doing this on behalf of their organization. In order to obtain a comprehensive quality rating, e.g., very good, good, satisfactory, fair and poor, independent consumer organizations weight and add the ratings of all included quality dimensions. Consumers can access the testing results online or in print magazines. Moreover, in several European countries, certain sellers may print their own testing result on their product, so consumers are able to access this quality information directly while shopping online or in retail settings (see, for example, Stiftung Warentest, 2020; Which?, 2023b, and Appendix A.1 for screenshot examples). Independent consumer organizations are usually well-known and well-regarded. For instance, 96% (77%) of all German consumers know of (strongly trust) Stiftung Warentest (KantarEmnid & Verbraucherzentrale Bundesverband, 2018). In the US, Consumer Reports has more than 6 million paying members and their website receives an average of 11 million unique visits per month (Consumer Reports, 2023). Information provided by independent consumer organizations is thus very close to what Viscusi (1978) suggests in a reply to Akerlof (1970), namely to provide credible information to buyers. 4

While independent consumer organizations offer credible and precise information about product quality, they are hampered by limited testing capacities; i.e., they face a set selection problem as they need to select a subset of all available product models in the market for testing.5 Typically, they select product models based on which ones are perceived to be of greatest interest for consumers. Stiftung Warentest, for example, uses current sales numbers to select the bestselling product models for testing. Their subset of tested product models usually accounts for 2% to 33% of all available product models (as in the 09/2016 magazine; see GfK SE, Nuremberg, 2017). By contrast, Which? and Consumer Reports make their selections using a combination of sales numbers, price, and other criteria (see Appendix A.3 for details). To the best of our knowledge, no independent consumer organization currently allows sellers to directly suggest certain product models for testing.

The logic behind any of these current selection mechanisms is that consumers are likely to want information on the bestselling product models and/or on product models from different price ranges. However, our counterargument is that these are not necessarily the ones that buyers would have selected under complete information. It is not clear whether any of these current product model selection mechanisms do in fact provide optimal information for consumers, i.e., whether the selection of product models leads to optimal consumer surplus. In particular, there may be product models among the untested ones which dominate the tested product models, e.g., offer a higher quality at the same price, but have simply not been selected for testing (see Section 3.1 for formal definitions of dominated and non-dominated product models). Indeed, among a sample from Stiftung Warentest, there are many dominated product models (see Figure A4 and Figure A5 in Appendix A.2). Note that this observation is in line with numerous empirical studies which measure the correlation between product quality and price within the subsets of tested product models in different countries (see Note 4). Thus, the problem is not the limited testing capacity itself, but the consequences of this limited capacity for the provision of optimal information.

In this paper, we explore the mechanism by which a certifier solves the set selection problem, i.e., how it selects a subset of product models for testing. In particular, we propose a novel mechanism to select product models that yields greater buyer information. Importantly, our proposed mechanism uses the same number of testing slots. Only the selection process differs to take advantage of the sellers’ information (see Section 3.1.3 for a detailed description). To test the performance of our proposed selection mechanism, we develop a certification game to derive theoretical, testable predictions. Furthermore, we conduct a controlled laboratory experiment to test these theoretical predictions.

The rest of our paper proceeds as follows: Section 2 discusses the related literature. Section 3 introduces our theoretical framework and derives our theoretical, testable predictions. Section 4 presents our experimental design and hypotheses and reports our experimental results. Section 5 discusses our findings and concludes.

2. Related Literature

This paper is related to several strands of theoretical, empirical and experimental literature such as certification, information disclosure and unraveling, social choice under constraints and Bayesian persuasion.

Many papers on certification, including ours, analyze how certifiers influence market transparency (for an overview, see Dranove & Jin, 2010). The main contribution of our paper is that we focus on certifiers who face a set selection problem as they need to select a subset of product models for testing. To the best of our knowledge, we are the first to do so. For example, Stahl and Strausz (2017) explicitly mention independent consumer organizations in their paper, but they do not discuss any testing capacity constraints.

Our paper is, however, closely related to Stahl and Strausz (2017) as we also focus on different aspects of seller versus buyer certification. Following them, we define buyer (seller) certification as a mode of certification where buyers (sellers) (i) initiate and (ii) pay for certification. As to paying for certification, we view independent consumer organizations as containing mainly, but not only, elements of buyer certification because, in addition to selling their own publications to consumers, some consumer organizations also sell licenses to producers wishing to print their certification on their product (see, for example, Stiftung Warentest, 2020 and Which?, 2023b). While we do not suggest any changes in how consumer organizations finance themselves, we do suggest a change in who initiates certification. More specifically, we suggest including one aspect of seller certification by allowing sellers to apply for certification. However, we suggest to keep the aspect of buyer certification of letting the certifier have the final say in whom to test from the set of applicants. As we show below, this allows sellers to signal whether they are (non-)dominated, and it leads to more transparency as buyers are able to infer more about the quality of untested product models.

Our certification game represents a market with sellers, buyers, and a certifier (building on a model by Encaoua and Hollander (2007) without a certifier; see Section 3 for details). As our main research question is how a certifier can optimally solve its set selection problem, we need a game that allows for a certain number of sellers from whom the certifier then selects a subset for testing. Related papers such as Stahl and Strausz (2017) and Ispano and Schwardmann (2023) analyze models with one or two sellers, respectively. Importantly, our certification game allows for prices which may not be positively correlated with quality. As depicted in Figure A4 and Figure A5 in Appendix A.2, product models are represented by points in the quality–price space, and dominated product models may exist. This implies that, in our setting, it is not necessarily all the high-quality sellers who want to signal high quality, but it is all the sellers whose product models would be bought under complete information who want to signal that they are optimal.

Our proposed mechanism by which a certifier selects product models for testing allows sellers of non-dominated product models to voluntarily and credibly disclose their product quality which may lead to information unraveling (see Grossman, 1981; Milgrom, 1981). Note that, while our mechanism builds on Grossman (1981) and Milgrom (1981), it goes beyond their ideas by including, amongst other things, a certifier modeled as an algorithm which selects a subset of applicants for testing (see Section 3.1.3 for details). To the best of our knowledge, we are the first to do so.

Also, note that our study differs from Encaoua and Hollander (2007) in that we do not focus on sellers’ pricing and quality-setting choices. Instead, we investigate a situation in which the unknown variable has already been determined, similarly to Benndorf et al. (2015), and Jin et al. (2022). In other words, we take situations where qualities and prices have already been set, and focus on sellers’ decisions to apply for testing—and thus on the degree to which information unravels—as a first step in analyzing the performance of our mechanism.

Our proposed mechanism with unraveling and information disclosure is supported by previous theoretical, empirical, and experimental research (for overviews, see Dranove & Jin, 2010 and Brendel, 2021). On the theoretical side, several studies investigate the different conditions under which unraveling occurs (see, for instance, the seminal papers of Grossman, 1981 and Milgrom, 1981, as well as the overviews in Dranove & Jin, 2010 and Brendel, 2021) and find that complete unraveling requires several strong assumptions. By contrast, unraveling has been observed but to an incomplete degree in empirical studies (see, for instance, Mathios, 2000 and Jin & Leslie, 2003, amongst many others). The experimental papers also observe unraveling, sometimes to an incomplete degree (see, for example, Benndorf et al., 2015 and Benndorf, 2018), but sometimes to a complete degree when allowing for detailed feedback and learning (see, for example, Forsythe et al., 1989 and Jin et al., 2021). To the best of our knowledge, there is no study that investigates whether unraveling leads to improved information in contexts with limited information disclosure capacities. We contribute to this strand of literature by analyzing the first game with such limited information disclosure capacities.

While all third-party certifiers help to mitigate buyers’ information disadvantage, they may differ along several dimensions, e.g., seller versus buyer paid certification, for-profit or not-for-profit, limited versus unlimited testing capacities, and focus on subset (e.g., environmentally friendly product) versus comprehensiveness of quality information (overall high-quality product). As mentioned above, independent consumer organizations are usually not-for-profit, and they mainly rely on buyers paying for certification. By contrast, for-profit third-party certifiers such as Moody’s and PSA6 charge sellers fees for the rating service (see Dranove & Jin, 2010; List, 2006; Jin et al., 2010). Independent consumer organizations face limited testing capacities, and thus need to select subsets of product models testing. By contrast, both for-profit third-party certifiers such as Moody’s and PSA and not-for-profit third-party certifiers such as USDA organic or Blauer Engel are typically able to consider and, if applicable, certify each seller. (Yet, sellers may have to wait a certain amount of time for certification.) Thus, these certifiers do not need to select a subset of sellers for certification, which is the main focus of this paper. As for the comprehensiveness of quality information, independent consumer organizations such as Consumer Reports, Stiftung Warentest and Which? aim to provide comprehensive quality information including search, experience and credence characteristics. By contrast, certifiers such as USDA organic (US) or Blauer Engel (Germany) provide information about a distinct subset of quality dimensions, more specifically, credence characteristics. 7

Furthermore, this paper is also related to the literature on social choice under constraints (see, for example, Barberà et al., 2005; Bahel & Sprumont, 2020, 2021) as both our paper and the literature on social choice under constraints investigate mechanisms to solve certain set selection problems. Social choice under constraints analyzes mechanisms to solve social set selection problems when not all possible subsets of objects are feasible. We analyze mechanisms to solve seller set selection problems in which all subsets are, in principle, feasible, while the constraint is the number of sellers in the subset.

Finally, this paper is also related to the literature on Bayesian persuasion (Kamenica & Gentzkow, 2011) as both our paper and the literature on Bayesian persuasion tackle complex problems of information disclosure. Bayesian persuasion is concerned with optimal disclosure when a sender designs a disclosure signal to a receiver about the value of a prospect, sometimes through an intermediary (Bizzotto et al., 2021). In our paper, the intermediary (the certifier) reveals an always perfect signal, but has to select a subset of senders whose perfect signals it reveals. Senders may apply to be selected, but they do not design the disclosure signal.

3. Theory

We start by describing the theoretical framework (Section 3.1). Subsequently, we present theoretical results for three different versions of the game (Section 3.2). See

Table A2. List of symbols (in alphabetic order).

$b_h$	buyer, with $b_h\in B$, and $h\in \{1,\dots ,s\}$
B	set of buyers, with $\varnothing \ne B=\{b_1,\dots ,b_s\}$, and $s\in \mathbb{N}$
$c\left(q_t\right)$	seller $f_t$’s unit costs of production
D	set of globally dominated sellers
$D_Z$	set of locally dominated sellers within the set Z, with $Z\subseteq F$
$f_0$	non-existing seller with $q_0=0$ and $p_0=0$, denoting a buyer’s choice not to purchase a product model
$f_{F^c\wedge \underline{q}}$	seller offering the overall lowest quality and the overall lowest price
$f_t$	seller, with $f_t\in F$, and $t\in \{1,\dots ,n\}$
$f_t^{\mathit{false}}$	non-existing seller with $\left(p_t,q_t^{\mathit{false}}\right)$
$f_{Z^c}$	seller who offers the cheapest product model within the set Z, with $Z\subseteq F$
$f_{Z^{*}}^h$	seller who maximizes buyer $b_h$’s utility within the set Z, with $Z\subseteq F$
$f_{\tilde{Z}}^h$	seller who maximizes buyer $b_h$’s expected utility within the set Z, with $Z\subseteq F$
F	set of sellers, with $\varnothing \ne F=\{f_1,\dots ,f_n\}$, and $n\in \mathbb{N}$
h	index of buyer $b_h$ and ${\theta }_h$
k	certifier’s maximum testing capacity, with $k\in \mathbb{N}$
K	set of applicant sellers, with $K\subseteq F$
$K_{\mathit{false}}$	set of applicant sellers who state a false product model quality when applying
$K_{\mathit{true}}$	set of applicant sellers who state their true product model quality when applying
$K^{\prime }$	(final) set of tested sellers, with $K^{\prime }\subseteq F$
$K_i^{\prime }$	set of tested sellers in the $i^{\mathit{th}}$ algorithm iteration
n	number of sellers $f_t\in F$, with $t\in \{1,\dots ,n\}$
$ND$	set of globally dominated sellers
$N{D}_Z$	set of locally non-dominated sellers within the set Z, with $Z\subseteq F$
$\overline{ND}$	globally non-dominated sellers who have at least one buyer under complete information
$p_t$	seller $f_t$’s price, with $0\le p_t\in {\mathbb{R}}^{+}$
$q_t$	seller $f_t$’s quality, with $0\le q_t\in Q\subseteq {\mathbb{R}}^{+}$
$q_t^{\mathit{false}}$	seller $f_t$’s falsely stated quality
$\underline{q}$	overall lowest quality
Q	set of all quality levels $q_t$
$R_t$	set of dominating rivals of seller $f_t\in D$, with $R_t\subseteq F$
s	number of buyers $b_h\in B$, with $h\in \{1,\dots ,s\}$
t	index of seller $f_t$, quality $q_t$, and price $p_t$
$u_h$	buyer $b_h$’s utility
Z	subset of F
${\theta }_h$	buyer $b_h$’s valuation of quality, with $0<{\theta }_h\in {\mathbb{R}}^{+}$
${\pi }_t$	seller $f_t$’s profit

in Appendix A.6 for a list of symbols.

3.1. Theoretical Framework

In this subsection, we establish our theoretical framework. More specifically, we describe our certification game in Section 3.1.1. Subsequently, we establish a set of basic definitions in Section 3.1.2. Furthermore, we introduce three different versions of the game in Section 3.1.3 which later help us analyze different product model selection mechanisms. Last, we establish two assumptions about the distribution of market parameters in Section 3.1.4.

3.1.1. Certification Game

Our certification game represents a market with sellers, a certifier, and buyers. This one-shot game allows us to analyze how information about a limited subset of tested product models influences consumer surplus (and seller profits) in the short term. The sequence of the game occurs over three stages (see

Table 1. Sequence of the game.

	CompleteInformation	SellersMayNotApply	SellersMayApply
Stage 1	(Sellers are passive.)		Sellers may apply for testing.
Stage 2	(Certifier is passive.)	Certifier tests subset of product models.
Stage 3	Buyers decide which product model to buy (if any).

and Section 3.1.3).

Sellers (He/His)

We consider a market with a non-empty set of rational sellers F, with $\varnothing \ne F=\{f_1,\dots ,f_n\}$, and $n\in \mathbb{N}$. These sellers offer heterogeneous product models. For simplicity, we assume each seller offers exactly one product model, but can sell as many units of that product model as demanded. For seller $f_t\in F$, with $t\in \{1,\dots ,n\}$, we call $q_t$ the quality of the corresponding product model, with $0\le q_t\in Q\subseteq {\mathbb{R}}^{+}$, and with Q denoting the set of all quality levels. We call $p_t$ seller $f_t$’s price, with $0\le p_t\in {\mathbb{R}}^{+}$. Similarly to Benndorf et al. (2015), and Jin et al. (2022), we are mainly interested in product quality disclosure. Therefore, we analyze situations in which quality and price have already been set. Also, we assume the correlation between price and quality (when considering all product models) equals zero since certifiers are most helpful for buyers if buyers are unable to infer quality from price (see also Note 4).8 We further assume product quality comprises experience and credence characteristics, excluding search characteristics, according to Nelson (1970) and Darby and Karni (1973), and these quality characteristics are aggregated into the unidimensional, overall vertical quality measure $q_t$. Moreover, we assume there is no pair of sellers offering their product model at the same price, i.e., $\forall f_t,f_s\in F$ with $t\ne s$, we require that $p_t\ne p_s$. (Note that, in Appendix A.8, we present a generalized selection mechanism which allows us to relax, amongst other things, this assumption.) We define

$$f_{Z^c}:=\underset{f_k\in Z}{\arg \min }{p}_k$$

(1)

with $f_{Z^c}\in Z$, and with $Z\subseteq F$, as the seller offering the cheapest product model within the set Z.9 Furthermore, we define

$$f_{F^c\wedge \underline{q}}:=\left\{f_{F^c}|\exists f_{F^c}\mathit{with}{q}_{F^c}=\underline{q}\right\}$$

(2)

with $f_{Z^c\wedge \underline{q}}\in F$, and with $\underline{q}\in Q$ as the overall lowest quality, as the seller who offers the overall lowest quality and the overall lowest price. Note that $f_{F^c\wedge \underline{q}}$ may or may not exist depending on the market parameters, while $f_{F^c}$ always exists. If $f_{F^c\wedge \underline{q}}$ exists, he is always identical to $f_{F^c}$.

For seller $f_t$, we assume the function $c\left(q_t\right)$ denotes the unit costs of production. Thus, the unit costs of production $c\left(q_t\right)$ depend on only the quality level, are independent of the total number of produced units and are identical for a given quality level. Furthermore, the cost function is assumed to be continuously differentiable, strictly increasing and strictly convex in quality, i.e., $c^{\prime }\left(q_t\right)>0$ and $c^{″}\left(q_t\right)>0$. Since we are not interested in analyzing market entry or exit decisions and since positive fixed costs would thus not influence equilibrium predictions, we assume all sellers’ fixed costs equal zero. For simplicity, we assume there are no sellers providing the same utility for any buyer. Finally, we assume sellers have complete information. We write seller $f_t$’s profit function as

$${\pi }_t\left(p_t,q_t\right)=\left(p_t-c\right(q_{t}d\left(p_t,q_t\right)$$

(3)

where $d\left(p_t,q_t\right)$ represents the number of buyers buying seller $f_t$’s product model.

Certifier (It/Its)

Following the literature on honest certification, we assume the certifier provides credible information about product quality (see, e.g., Stahl & Strausz, 2017). However, it does so for a limited subset of product models which it selects according to its maximum testing capacity $k\in \mathbb{N}$, and according to its product model selection mechanism. Quality $q_t$ is perfectly revealed $\forall f_t\in K^{\prime }$, with $t\in \{1,\dots ,n\}$, and with $K^{\prime }\subseteq F$ as the set of sellers whose product models are tested.10 We assume the certifier has already determined its number of testing slots based on how precisely it wants to report quality. In particular, we assume the certifier provides at least as many certification slots as there are quality levels, i.e., $k\ge \#Q$. (Note that, in Appendix A.8, we present a generalized selection mechanism which allows us to relax, amongst other things, this assumption.) We model two different product model selection mechanisms: any current mechanism SellersMayNotApply and our new mechanism SellersMayApply (see Section 3.1.3 for details). We assume both mechanisms use the same number of testing slots.

We refrain from modeling the certifier’s surplus function since, as mentioned above (Section 1), it is a non-profit organization11 which does not rely on fees for the rating service. Since we model the certifier as an algorithm without its own surplus function, we do not call it a player.

Buyers (She/Hers)

We next identify a non-empty set of rational buyers B, with $\varnothing \ne B=\{b_1,\dots ,b_s\}$, and $s\in \mathbb{N}$.12 These buyers decide whether, and if so which, product model to buy (at most one). They are not able to resell. For buyer $b_h$, with $h\in \{1,\dots ,s\}$, we call ${\theta }_h$ her valuation of quality, with $0<{\theta }_h\in {\mathbb{R}}^{+}$. Before having received any information from the certifier, buyers are assumed to have complete information about prices and valuations of quality. As to quality, we assume buyers only know the quality distribution. Moreover, we assume buyers believe that the correlation between price and quality equals zero (once more, see Note 4). We assume

$$\mathbb{E}\left(u_h\left(p_t,q_t,{\theta }_h\right)\right)={\mathbb{1}}_{\left\{f_t\in K^{\prime }\right\}}\left({\theta }_h{q}_t-p_t\right)+{\mathbb{1}}_{\left\{f_t\in \left\{F\setminus K^{\prime }\right\}\right\}}\left({\theta }_h\mathbb{E}\left(q_t\right)-p_t\right)$$

(4)

is the expected utility function of buyer $b_h\in B$ buying seller $f_t$’s product model, with $K^{\prime }$ being the set of sellers whose product models have been tested.13 ${\theta }_h{q}_t$ is a buyer’s willingness to pay for $q_t$. If a buyer chooses a tested product model, her utility is deterministic since all buyers know the quality of a tested product model prior to purchase. Thus, if a buyer chooses to buy a tested product model, her utility simplifies to the first summand of Equation (4), i.e., to $u_h\left(p_t,q_t,{\theta }_h\right)={\theta }_h{q}_t-p_t$. In this case, her utility equals the product of her valuation of quality and the selected product model’s quality, minus the selected product model’s price. On the other hand, if a buyer chooses to buy an untested product model, her expected utility is probabilistic since she does not know the true quality of this product model prior to purchase. Thus, her expected utility simplifies to the second summand of Equation (4), i.e., to $\mathbb{E}\left(u_h\left(p_t,q_t,{\theta }_h\right)\right)={\theta }_h\mathbb{E}\left(q_t\right)-p_t$. In this case, her utility equals the product of her valuation of quality and the selected product model’s expected quality, minus the selected product model’s price. Finally, if a buyer chooses not to purchase a product model, her utility is zero. After having received information from the certifier, we assume buyers update the expected quality of all untested product models (also see Section 3.2.2 and Section 3.2.3).

3.1.2. Local and Global Dominance, and Dominating Rivals

Before describing the different product model selection mechanisms in more detail, we need to establish a set of basic definitions. We begin by defining local and global dominance to distinguish if a product model is dominated within the whole market (as in Section 3.2.1 when analyzing a world of complete information), or within a certain submarket like the set of tested product models (as in Section 3.2.2 and Section 3.2.3 when analyzing worlds of incomplete information with different product model selection mechanisms).

Definition 1 

((Non-)Dominated products). Let $\varnothing \ne Z\subseteq F$ be a non-empty set of sellers. A seller $f_t\in Z$ offers a dominated product in Z if $\exists f_j\in Z$ with $\left(\left(p_j\le p_t\right)\wedge \left(q_j>q_t\right)\right)\vee \left(\left(p_j<p_t\right)\wedge \left(q_j\ge q_t\right)\right)$. A seller $f_t\in Z$ offers a non-dominated product in Z if $\forall f_j\in Z$

• if $p_j<p_t$, then $q_j<q_t$,
• if $q_j>q_t$, then $p_j>p_t$.

When referring to a true submarket of F, i.e., when $Z\subset F$, we call a product locally (non-)dominated. When referring to the whole market F, i.e., when $Z=F$, we call a product globally (non-)dominated.

Essentially, a product is dominated in a set (or market) if at least one seller in this set offers a strictly higher product quality without being more expensive, or a strictly lower price without offering a lower product quality. By comparison, a product model is non-dominated in a set if every seller in this set offering a strictly higher product quality also has a strictly higher price, and every seller offering a strictly lower price also offers a strictly lower quality. In the following, we use the terms “seller with (non-)dominated product” and “(non-)dominated seller” equivalently.

To illustrate Definition 1, consider the following local market (see Figure 1): $Z=\{f_1,f_2,f_4,f_5,f_6\}$, with

• $q_1=2$, $p_1=5$,
• $q_2=3$, $p_2=10$,
• $q_4=1$, $p_4=11$,
• $q_5=2$, $p_5=9$,
• $q_6=4$, $p_6=28$.

Furthermore, consider the following global market: $F=Z\cup f_3$, with $q_3=5$ and $p_3=27$. While sellers $f_1,f_2$, and $f_6$ are locally non-dominated in market Z, sellers $f_1,f_2$, and $f_3$ are globally non-dominated in market F.

Having defined local and global dominance, we now partition any set of sellers Z into two sets of sellers: $N{D}_Z\subseteq Z$, the set of locally non-dominated sellers in Z, and $D_Z\subseteq Z$, the set of locally dominated sellers in Z, with $N{D}_Z\cup D_Z=Z$. If $Z=F$, we use the notation D instead of $D_F$, and $ND$ instead of $N{D}_F$. In the following, we assume all sellers are sorted according to these two disjoint sets. Without loss of generality, $F=\{f_1,\dots ,f_n\}=ND\cup D$, with $\varnothing \ne ND:=\{f_1,\dots ,f_m\}$⊂F as the set of globally non-dominated sellers, and with $\varnothing \ne D:=\{f_{m+1},\dots ,f_n\}$⊂F as the set of globally dominated sellers, with $m\in \mathbb{N}$. We denote the subset of globally non-dominated sellers who have at least one buyer under complete information with $\overline{ND}$.

Next, we define a seller’s dominating rivals.

Definition 2 

(Dominating rivals of a certain seller). We call the set

$$R_t:=\left\{f_j\in F|\right(\left((q_j>q_t\wedge p_j\le p_t)\right)\vee \left((q_j\ge q_t)\wedge (p_j<p_t)\right)\left)\right\}$$

(5)

the set of dominating rivals of seller $f_t\in D$, with $R_t\subseteq F$.

Definition 2 implies that seller $f_t$ is globally (and locally) dominated by every seller in $R_t$. We do not define sets of dominating rivals for globally non-dominated sellers as such sets would be empty.

3.1.3. Three Versions of the Game

This section introduces three different versions of the game which later help us analyze different product model selection mechanisms.

Benchmark CompleteInformation

As a benchmark, we model an ideal world of CompleteInformation about product quality (see Table 1). Since information is complete, there is no need for a certifier.14 Therefore, stages 1 and 2 of the game are excluded, and the game reduces to stage 3. In this stage 3, buyers observe the quality and price of all product models, and buyers decide which product model to buy (if any). The theoretical results can be found in Section 3.2.1. Note that, while we include this game version of CompleteInformation as a theoretical benchmark, the following incomplete information game versions provide the basis for our experimental treatments.

Any Current Selection Mechanism SellersMayNotApply

To represent any current product model selection mechanism in which sellers are not able to influence directly which product models are selected for testing, we model a mechanism in which sellers are passive. We name this mechanism SellersMayNotApply (see Table 1). Since sellers are passive, stage 1 of the game is excluded, and the game reduces to stages 2 and 3. In stage 2, the certifier tests the set of product models $K^{\prime }$, with $\varnothing \ne K^{\prime }⊊F$. This set of tested product models could, in principle, be determined by any combination of currently used selection criteria. Note that, when analyzing the performance of any current mechanism SellersMayNotApply in Section 3.2.2 and Section 3.2.4, we consider any possible scenario of which product models the set of tested product models $K^{\prime }$ may contain. In stage 3, buyers observe the quality of the tested product models as well as the prices of all product models, and then decide which product model to buy (if any).

New Selection Mechanism SellersMayApply

Under our new mechanism, sellers are able to influence whether a certifier will test their product model. Therefore, the game occurs over all three stages (see Table 1). In stage 1, sellers decide whether, and if so, with which quality, to apply for testing of their product model. Note that sellers apply by submitting their product model number.15 Sellers who apply incur positive $\mathit{application\_costs}$ which we assume to be lower than the profit margin of every globally non-dominated seller who has at least one buyer under CompleteInformation, i.e., $p_t-c_t>\mathit{application\_costs}>0 \forall f_t\in \overline{ND}$. Note that these $\mathit{application\_costs}$ could be very low, e.g., application paper work, and are not a fee for the testing service. After all sellers made their application decision, stage 2 starts. We assume the certifier has immediate access to each applicants’ price16 and uses a pre-specified algorithm to select product models from the set of applicants (see below). Eventually, all product models selected by the algorithm are tested, and the quality stated during the application may or may not be confirmed. If an applicant seller’s product model is chosen for testing and the submitted quality is found to be false, this seller must also pay a positive $\mathit{punishment\_fee}$ which is assumed to be higher than the maximum additional profits any globally non-dominated seller could make when being tested. (Note that, in Appendix A.8, we present a generalized version of SellersMayApply which uses a different method to deter lying, namely combining a lower $\mathit{punishment\_fee}$ with not revealing a liar’s quality. Furthermore, note that the $\mathit{punishment\_fee}$ exists under SellersMayApply, but not under SellersMayNotApply as, in our theoretical framework, sellers are only able to state a (true or false) quality when applying for testing. We abstract from quality statements made to buyers, e.g., marketing.) After the final test, the true qualities of all tested product models are published. In stage 3, buyers observe the quality of the tested product models as well as the prices of all product models, and then decide which product model to buy (if any).

The SellersMayApply mechanism is based on the algorithm described in Figure 2. See Figure A7 in Appendix A.7.1 for a formal description. Note that the algorithm stops after algorithm step 2 in its first iteration if all sellers submit true qualities when applying to be tested (which they are predicted to do in equilibrium according to Proposition 2 below). To illustrate our algorithm, consider once again global market F in Figure 1. Assuming all sellers applied stating their true qualities, the algorithm would pre-select seller $f_4$ ($f_1,f_2,f_6,f_3$) as the cheapest seller for quality level 1 (2, 3, 4, 5, respectively), and then exclude the locally dominated sellers $f_4$ and $f_6$ in algorithm step 1. In algorithm step 2, all untested, locally non-dominated product models selected at the end of the preceding algorithm step, i.e., sellers $f_1,f_2,f_3$, would be tested. Since we assumed all sellers applied stating their true qualities, no false quality statements would be detected, and the algorithm would stop. The theoretical results for our new mechanism SellersMayApply can be found in Section 3.2.3.

3.1.4. Two Distribution Assumptions

Last, we make two assumptions about the distribution of market parameters. (Note that, in Appendix A.8, we present a generalized version of SellersMayApply which does not require these distribution assumptions.) In the following, let $f_t$ be a seller from $\overline{ND}\setminus f_{F^c}$17, and let $b_h$ be any buyer who would select seller $f_t$ under CompleteInformation. Furthermore, let $f_k$ be the seller offering the next-lowest quality level in $\overline{ND}$, and let $f_g$ be any seller in $ND$ offering a cheaper product model than $f_t$. Moreover, let $\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayApply}\right]$ denote seller $f_t$’s expected quality under SellersMayApply.18

Assumption 1 

(Joint distribution of price and quality). Consider a hypothetical scenario in which seller $f_t$, seller $f_{F^c\wedge \underline{q}}$ (if any), and all globally dominated sellers are untested; therefore, their qualities are known in expectation only to buyers. Furthermore, seller $f_k$ and all sellers in $\overline{ND}$ offering quality levels higher than $f_t$ are tested; therefore, their qualities are known to buyers. (All other sellers (if any) may or may not be tested; therefore, their qualities may or may not be known to buyers.) We assume price and quality are distributed such that, given this scenario, $\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayApply}\right]\le q_k$.

To illustrate Assumption 1, consider seller $f_t=f_7$ in experimental market 1 (see Section 4.1 and Appendix A.10 for details). In this example, $f_7$ is the seller offering a quality of 3 in $\overline{ND}$. Furthermore, $f_k=f_4$ is the seller with the next lowest quality level in $\overline{ND}$ compared to $f_7$, i.e., $q_4=2$. Last, $f_{F^c\wedge \underline{q}}=f_1$. Therefore, we consider the following scenario. Sellers $f_7$ and $f_1$, and all globally dominated sellers, i.e., $f_2$, $f_3$, $f_5$, $f_6$, $f_8$, $f_9$, $f_{11}$, $f_{12}$, $f_{14}$, $f_{15}$, are untested; therefore, their qualities are known in expectation only to buyers. Furthermore, sellers $f_4$, $f_{10}$, and $f_{13}$ are tested; therefore, buyers know that $q_4=2$, $q_{10}=4$, and $q_{13}=5$. (There are no other sellers to be considered in this example.) In this scenario, $\mathbb{E}\left(q_7\right)\left[\mathit{SellersMayApply}\right]=(3\times 1+2\times 2+3\times 3)/8=2$.19 Since $q_4=2$, it follows that $\mathbb{E}\left(q_7\right)\left[\mathit{SellersMayApply}\right]\le q_4$, which implies that Assumption 1 holds.

Assumption 2 

(Joint distribution of price, quality, and valuation of quality). Consider a hypothetical scenario in which seller $f_g$, seller $f_{F^c\wedge \underline{q}}$ (if any), and all globally dominated sellers are untested; therefore, their qualities are known in expectation only to buyers. Furthermore, seller $f_t$ and all sellers in $\overline{ND}$ offering quality levels higher than $f_t$ (if any) are tested; therefore, their qualities are known to buyers. (All other sellers (if any) may or may not be tested; therefore, their qualities may or may not be known to buyers.) We assume price, quality, and buyers’ valuation of quality are distributed such that, given this scenario, $u_h\left(p_t,q_t,{\theta }_h\right)>\mathbb{E}\left(u_h\left(p_g,q_g,{\theta }_h\right)\right)\left[\mathit{SellersMayApply}\right]$.

To illustrate Assumption 2, consider once more seller $f_t=f_7$ in experimental market 1 (see Section 4.1 and Appendix A.10 for details). In this example, $f_g\in \left\{f_1,f_4\right\}$, and $f_{F^c\wedge \underline{q}}=f_1$. Furthermore, buyer $b_h=b_3$ (with ${\theta }_3=7$) would select seller $f_7$ (with $q_7=3$, and $p_7=10.9$) under CompleteInformation. Thus, we check Assumption 2 for two different cases: $f_g=f_1$ (with $q_1=1$, and $p_1=2.9$) and $f_g=f_4$ (with $q_2=2$, and $p_2=5$). In the former case, $u_3\left(p_1,q_1,{\theta }_3\right)>\mathbb{E}\left(u_3\left(p_4,q_4,{\theta }_3\right)\right)\left[\mathit{SellersMayApply}\right]$. In the latter case, $u_3\left(p_7,q_7,{\theta }_3\right)>\mathbb{E}\left(u_3\left(p_4,q_4,{\theta }_3\right)\right)\left[\mathit{SellersMayApply}\right]$.20 Combining both inequalities implies that Assumption 2 holds.

Both Assumptions 1 and 2 imply that we rule out “too left-skewed” distributions of quality. Note that we need them in steps 3 and 6 of Proposition 2’s proof. Furthermore, note that both assumptions hold for all our experimental markets in which quality is uniformly distributed, and in which $ND$ starts from the highest quality level and does not have any “gaps”.21

3.2. Theoretical Results

Having established our theoretical framework in the previous subsection, we are now able to analyze the three different versions of the game. We start with an ideal world that contains CompleteInformation about product quality (see Section 3.2.1) and then examine two worlds consisting of incomplete information about product quality that therefore require an certifier. Specifically, Section 3.2.2 examines SellersMayNotApply, any current product model selection mechanism. Section 3.2.3 examines SellersMayApply, our proposed product model selection mechanism.

Since sellers are passive in our benchmark case and in any current mechanism SellersMayNotApply, we analyze only buyer behavior. By contrast, since sellers are active in our proposed mechanism SellersMayApply, we analyze both buyer and seller behavior by applying backward induction. We calculate equilibrium payoffs for both buyers and sellers in all three versions of the game (Nash equilibria in the games under CompleteInformation and under SellersMayApply, Bayesian Nash equilbria under SellersMayNotApply).22

3.2.1. Benchmark CompleteInformation

In this subsection, we present our benchmark case of an ideal world of CompleteInformation about product quality (see Section 3.1.3). Buyers maximize their utility according to Equation (4). Because they have complete information about product quality, Equation (4) simplifies to $u_h\left(p_t,q_t,{\theta }_h\right)={\theta }_h{q}_t-p_t$. Hence, a buyer’s maximization condition is given by:

$$\underset{f_t\in \left\{F\cup f_0\right\}}{\arg \max }{\theta }_h{q}_t-p_t$$

(6)

with $f_0$ representing a non-existing seller with $q_0=0$ and $p_0=0$, denoting a buyer’s choice not to purchase a product model. Equation (6) implies that a buyer will choose the product model among all those yielding a non-negative utility which maximizes her utility. If all available product models yield a negative utility, she will refrain from buying. Therefore, buyer $b_h$ receives her maximum possible utility in equilibrium

$$u_h\left(p_{{\left\{F\cup f_0\right\}}^{*}}^h,q_{{\left\{F\cup f_0\right\}}^{*}}^h,{\theta }_h\right)={\theta }_h{q}_{{\left\{F\cup f_0\right\}}^{*}}^h-p_{{\left\{F\cup f_0\right\}}^{*}}^h={\theta }_h{q}_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h-p_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h$$

(7)

with $f_{Z^{*}}^h$ denoting the seller who maximizes buyer $b_h$’s utility within the set Z, and with $q_{Z^{*}}^h$ and $p_{Z^{*}}^h$ denoting the respective quality and price. Note that a globally dominated product model cannot maximize buyer $b_h$’s utility under CompleteInformation since each dominating rival product model would generate a higher surplus for buyer $b_h$. Therefore,

$$f_{{\left\{F\cup f_0\right\}}^{*}}^h=f_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h.$$

(8)

In addition, under CompleteInformation, buyer $b_h$’s utility is always non-negative in equilibrium. See Appendix A.7.3 for a formal definition of the corresponding market areas.

Equilibrium profits of globally dominated sellers equal zero. If seller $f_t$ offers a globally non-dominated product model, his equilibrium profit can be calculated as:

$${\pi }_t\left(p_t,q_t\right)=d\left(p_t,q_t\right)(p_t-c\left(q_t\right))=\sum _{h=1}^s{\mathbb{1}}_{\left\{f_t=f_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h\right\}}\left(p_t-c\left(q_t\right)\right).$$

(9)

The above equation implies that a product model maximizes buyer $b_h$’s utility by taking into account all sellers who would sell at least one product model under CompleteInformation and the option of not buying.

The following definition summarizes aggregate consumer surplus and seller profits.

Definition 3 

(Aggregate consumer surplus and seller profits under CompleteInformation). Aggregate consumer surplus equals

$$\sum _{h=1}^s{\theta }_h{q}_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h-p_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h.$$

(10)

Globally dominated seller profits equal zero. Aggregate profits of globally non-dominated sellers equal

$$\sum _{t=1}^m\sum _{h=1}^s{\mathbb{1}}_{\left\{f_t=f_{{\left\{\overline{ND}\cup f_0\right\}}^{*}}^h\right\}}\left(p_t-c\left(q_t\right)\right).$$

(11)

3.2.2. Any Current Selection Mechanism SellersMayNotApply

In this subsection, we analyze a world with incomplete information about product quality where a certifier perfectly reveals information about a subset of product models under any current mechanism SellersMayNotApply (see Section 3.1.3). Buyers maximize their utility according to Equation (4). Hence, buyer $b_h$’s maximization condition is given by:

$$\begin{array}{c}\underset{f_t\in \left\{F\cup f_0\right\}}{\arg \max }{\theta }_h\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayNotApply}\right]-p_t\hfill \\ \hfill =\underset{f_t\in \left\{F\cup f_0\right\}}{\arg \max }\left\{\underset{f_t\in \left\{K^{\prime }\cup f_0\right\}}{\max }{\theta }_h{q}_t-p_t,\underset{f_t\in \left\{\{F\setminus K^{\prime }\}\cup f_0\right\}}{\max }{\theta }_h\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayNotApply}\right]-p_t\right\}.\end{array}$$

(12)

This maximization condition implies that, among all product models yielding a non-negative expected utility, a buyer will choose either the optimal tested one, or the optimal untested one, depending on which of these two yields the higher expected utility. If all available product models yield a negative expected utility, a buyer will refrain from buying.

Regarding the expected quality of untested product models, we assume in Section 3.1.1 that buyers take the information from the certifier into account. Under any current mechanism SellersMayNotApply, this implies that buyers calculate the expected quality of untested product models by mentally removing the set of tested product models $K^{\prime }$ from the complete set of sellers F; i.e., they calculate $\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayNotApply}\right]$ based on the set of untested sellers $F\setminus K^{\prime }$. Note that, $\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayNotApply}\right]$ is the same for all untested product models. Since ${\theta }_h$ is given, it follows that price is the only decision parameter when choosing among all untested product models. Therefore, the cheapest untested product model maximizes the expected utility among all untested product models for all buyers.

Buyer and seller equilibrium payoffs vary, of course, depending on which product models are selected for testing. The respective equilibrium payoffs, given a certain set of tested product models $K^{\prime }$, can be found in Appendix A.7.4. Below, we compare aggregate buyer surplus and aggregate seller profits under SellersMayNotApply to those under CompleteInformation (see Definition 3) while considering all possible scenarios of which product models are selected for testing.

Proposition 1 

(Consumer surplus and seller profits under any current mechanism SellersMayNotApply). Any current mechanism SellersMayNotApply always leads to a lower consumer surplus (and higher (lower) profits of globally dominated (globally non-dominated) sellers) compared to a world of CompleteInformation except for two scenarios. In only two rare scenarios, any current mechanism SellersMayNotApply leads to the same consumer surplus (and same seller profits) as under CompleteInformation. These two scenarios are as follows:

(i) 

In the first scenario, all product models which would be sold under CompleteInformation are selected for testing. In other words, $\overline{ND}\subseteq K^{\prime }$.

(ii) 

In the second scenario, all product models which would be sold under CompleteInformation except for the overall cheapest one are selected for testing. In other words, $f_{F^c}\in \overline{ND}$, $f_{F^c}\notin K^{\prime }$ and $\left\{\overline{ND}\setminus f_{F^c}\right\}\subseteq K^{\prime }$. Moreover, the overall cheapest product model is selected by every buyer who would have selected it in under CompleteInformation. In other words, $\forall b_l\in \left\{b_1,\dots ,b_s\right\}$ with $f_{F^c}={\arg \max }_{f_k\in \left\{\overline{ND}\cup f_0\right\}}{u}_l\left(p_k,q_k,{\theta }_l\right)$, the following holds: $f_{F^c}={\arg \max }_{f_k\in \left\{{\left\{F\setminus K^{\prime }\right\}}^c\cup K^{\prime }\cup f_0\right\}}\mathbb{E}\left(u_l\left(p_k,q_k,{\theta }_l\right)\right)\left[\mathit{SellersMayNotApply}\right]$.

Proposition 1 implies that any current mechanism SellersMayNotApply always leads to a lower consumer surplus compared to a world of complete information unless all product models which would be sold under complete information (or all but the overall cheapest one) happen to be selected for testing. The proof consists of two steps. In the first step, we show that buyers select suboptimal product models in most scenarios of which product models are selected for testing which leads to a lower consumer surplus. In the second step, we show that buyers select their complete-information-optimal product models only in the two exception scenarios which leads to the optimal consumer surplus. The detailed Proof of Proposition 1 can be found in Appendix A.7.5.

3.2.3. New Selection Mechanism SellersMayApply

In this subsection, we analyze another world with incomplete information about product quality where a certifier perfectly reveals information about a subset of product models but with our new mechanism SellersMayApply (see Section 3.1.3). As in the previous subsection, buyers maximize their utility according to Equation (4), and buyer $b_h$’s maximization condition is given by Equation (12).

In contrast, to the previous subsection, however, buyers can infer more from the certifier’s information since they know that sellers have the option to apply for testing under our new mechanism SellersMayApply, while this option does not exist under any current mechanism SellersMayNotApply. Regarding untested product models, buyers calculate their expected quality by mentally removing not only the set of tested product models $K^{\prime }$ from the complete set of sellers F. In addition, buyers mentally remove all quality levels larger than or equal to the quality of the next most expensive tested product model. In other words, buyers calculate $\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayApply}\right]$ based on the set of untested sellers $F\setminus K^{\prime }$ while considering only the quality levels

$$q_t\in \left\{Q\setminus \left\{q_i|\exists f_k\in K^{\prime }\mathit{with}\left\{p_k>p_t\wedge q_k\le q_i\right\}\right\}\right\}$$

(13)

due to the following reasoning.

Rational buyers know that, in equilibrium, no seller will apply to be tested stating a false quality (see step 1 in the Proof of Proposition 2). Buyers can also conclude that, in equilibrium, any tested seller must be globally non-dominated (see step 2 in the Proof of Proposition 2). It follows that the quality of an untested product model must be lower than that of any tested product model offered at a higher price. Otherwise, i.e., if the quality of an untested product model was higher than or equal to the quality of any tested product model offered at a higher price, this cheaper untested product model would dominate the tested one. However, all tested products must be globally non-dominated (see step 2 in the Proof of Proposition 2) which would not be the case if the untested product model dominated any of the tested ones.

Note that $\mathbb{E}\left(q_t\right)\left[\mathit{SellersMayApply}\right]$ may differ among untested product models. However, it is identical for all product models within a certain price range23 between tested product models. Since ${\theta }_h$ is given, it follows that price is the only decision parameter when choosing among all untested product models within this price range. Therefore, the cheapest untested product model maximizes the expected utility among all untested product models within a certain price range for all buyers.

Sellers maximize their profits according to Equation (3). Since quality and price are set, sellers make decisions regarding only the product test. Unlike under any current mechanism SellersMayNotApply where we need to consider all possible scenarios of which product models are selected for testing, there is a unique Nash equilibrium under our new mechanism SellersMayApply which we describe in the following Proposition 2.

Proposition 2 

(Unique Nash equilibrium, consumer surplus and seller profits under our new mechanism SellersMayApply). A unique Nash equilibrium

$$\begin{array}{c}(\mathit{apply}\mathit{with}{\mathit{q}}_{\mathit{1}},\dots ,\mathit{apply}\mathit{with}{\mathit{q}}_{(\#\overline{\mathit{ND}})},\mathit{do}\mathit{not}\mathit{apply},\dots ,\mathit{do}\mathit{not}\mathit{apply},\hfill \\ \multicolumn{1}{c}{\mathit{buy}\mathit{product}\mathit{model}\mathit{of}\mathit{seller}{f}_{\tilde{F}}^1,\dots ,\mathit{buy}\mathit{product}\mathit{model}\mathit{of}\mathit{seller}{f}_{\tilde{F}}^s{)}^T}\end{array}$$

exists, with $f_{\tilde{F}}^1,\dots ,f_{\tilde{F}}^s\in \overline{ND}$.

In equilibrium, all sellers (or all but the overall cheapest one)24 of product models that buyers would have bought under CompleteInformation apply to be tested stating their true quality, and are tested. All other sellers do not apply to be tested. All buyers select the product model they would have selected under CompleteInformation. Consumer surplus and seller profits are identical to those occuring under CompleteInformation, except that profits of sellers who applied for testing are reduced by the $\mathit{application\_costs}$.

Proposition 2 implies that our new mechanism SellersMayApply always creates the maximum possible consumer surplus, equivalent to a world of complete information. The proof consists of six steps in which we sequentially eliminate strategy profiles either because they contain strictly dominated strategies, or because they cannot be a Nash equilibrium due to other reasons. In step 1, we show that all strategy profiles can be eliminated in which at least one seller applies to be tested stating a false quality. In step 2, we show that, given all remaining strategy profiles, all strategy profiles can be eliminated in which seller $f_{F^c\wedge \underline{q}}$ or at least one globally dominated seller applies to be tested stating his true quality. In step 3, we show that, given all remaining strategy profiles, all strategy profiles can be eliminated in which at least one seller from $\overline{ND}\setminus f_{F^c}$ does not apply for testing. In step 4, we show that, given all remaining strategy profiles, seller $f_{F^c}\ne f_{F^c\wedge \underline{q}}$ may apply to be tested stating his true quality, or he may not apply to be tested, depending on the market parameters. In step 5, we show that, given all remaining strategy profiles, all strategy profiles can be eliminated in which at least one of the sellers from $ND\setminus \left\{\overline{ND}\cup f_{F^c}\right\}$ (if any) applies to be tested stating his true quality. In step 6, we show that, in equilibrium, all buyers select the same product model as under CompleteInformation. The detailed Proof can be found in Appendix A.7.5.

3.2.4. Comparing Consumer Surplus Resulting from Any Current and Our New Mechanism

We end this section with a comparison of the consumer surplus generated from the two mechanisms.

Proposition 3 

(Comparing consumer surplus resulting from any current mechanism SellersMayNotApply and from our new mechanism SellersMayApply). SellersMayApply outperforms SellersMayNotApply by leading to the optimal, higher consumer surplus in all possible scenarios but two. In the two exceptions stated in Proposition 1, both SellersMayApply and SellersMayNotApply lead to the same optimal consumer surplus.

Proposition 3 implies that our new mechanism SellersMayApply weakly dominates any current mechanism SellersMayNotApply.

Proof Proposition 3. 

(Comparing consumer surplus resulting from any current mechanism SellersMayNotApply and from our new mechanism SellersMayApply). According to Proposition 1, any current mechanism SellersMayNotApply leads to a lower consumer surplus than a world of CompleteInformation for all but two rare scenarios of which product models are selected for testing. In those two exception scenarios, it leads to the same consumer surplus as a world of CompleteInformation. According to proposition 2, our new mechanism SellersMayApply always leads to the optimal consumer surplus of a world of CompleteInformation. It follows that, for all but two possible scenarios of which product models are selected for testing under any current mechanism, our new mechanism SellersMayApply outperforms any current mechanism SellersMayNotApply by leading to a higher consumer surplus. In the two exception scenarios stated in Proposition 1, both mechanisms lead to the same, optimal consumer surplus. □

4. Experiment

4.1. Experimental Design and Hypotheses

Based on the certification game introduced in the previous section, we design a laboratory experiment to test our theoretical predictions and ascertain the extent to which these predictions are observed with human decision makers. In particular, unraveling has been shown to decrease in more complex experimental settings (see Hagenbach & Perez-Richet, 2018; Jin et al., 2022). Recall that our mechanism includes two product model dimensions (quality and price) as well as the option for sellers to state a false quality when applying to be tested. We design four experimental treatments. The first two represent two scenarios of any current mechanism SellersMayNotApply in which sellers cannot directly influence whether their product model is tested, while the latter two represent two versions of our new mechanism SellersMayApply (see Section 3). Note that we do not include an experimental treatment for our benchmark version of the game CompleteInformation since it is comparatively simple, and since there is no real-world equivalent.

SellersMayNotApply-WorstCase 

To model a scenario under any current selection mechanism in which the market functions extremely poorly, we design a worst-case scenario regarding the set of tested product models. In this worst-case scenario, the tested product models are the ones vertically furthest away from the globally non-dominated ones.25

SellersMayNotApply-Random 

We also design an intermediate scenario under any current mechanism where the set of tested product models is chosen randomly among all available ones. In this random scenario, the share of globally non-dominated, CompleteInformation optimal product models among all tested product models is almost identical to the share of globally non-dominated product models in all markets.26 We include this treatment to investigate whether our new mechanism outperforms chance.

SellersMayApply-LyingPoss(ible) 

This treatment represents the version of our new mechanism where sellers may apply for testing and may provide a true or a false quality. While the option of providing a false quality does not change the equilibrium predictions (see Proposition 2), it makes the SellersMayApply mechanism more complex. Therefore, we consider it important to investigate this treatment in the lab.

SellersMayApply-Truth 

This treatment represents another version of our new mechanism where sellers may apply for testing and are not allowed to provide a false quality.

In our experiment, we use a between-subject design; i.e., per session, we conduct one treatment. Each session consists of twelve rounds/markets with different quality–price combinations. At the end of a session, one of the twelve rounds is chosen randomly for payment (4 ECU = 1 EUR). In each session, we include 15 sellers ($n=15$), 8 buyers ($s=8$) and one certifier which is implemented as a computer algorithm rather than a participant. Player roles are assigned randomly at the beginning of a session and remain constant afterwards. The certifier selects at most five product models to be tested ($k=5$).

During the experiment, sellers are assigned product models at one of five different quality levels, $q_t\in \{1,2,3,4,5\}$. This range allows for a balance between experimental simplicity and the ability to distinguish seller behavior across product model quality levels.27 Furthermore, we choose one of the simplest possible unit costs functions fulfilling $c^{\prime }\left(q_t\right)>0$ and $c^{″}\left(q_t\right)>0$, namely the quadratic unit costs of production, i.e., $c\left(q_t\right)=q_t^2$. In our experiment, buyers are assigned across four quality valuations, ${\theta }_h\in \{3,7,11,15\}$, with two subjects per ${\theta }_h$. While buyers know that there are three sellers per quality level, they do not know if price is related to quality. They learn a product model’s quality only if the certifier has revealed this or if they have purchased the product model. If applicable, sellers incur $\mathit{application\_costs}$ of 0.5 ECU and a $\mathit{punishment\_fee}$ of 24 ECU if a false quality statement is detected.

To ensure that our experiment yields only positive total payoffs, each subject receives an initial endowment of 100 ECU. Sellers earn 0.5 ECU per correct answer when we ask them their beliefs about other sellers’ behavior. For a variation of SellersMayApply-Truth, we ask sellers for their beliefs about both seller and buyer behavior.

Since our experiment is conducted in Germany, where Stiftung Warentest mainly selects bestselling product models for testing, we use a bestsellers frame for the tested product models under SellersMayNotApply. Therefore, in all treatments, participants are informed in the instructions that they should act on the assumption that there are five bestselling product models per round. These product models are chosen for testing in the two SellersMayNotApply treatments (see Appendix A.9 and Appendix A.10 for details).

Among the 12 different markets, there are four different types: markets with 5, 4, 3, or 2 globally non-dominated product models (markets 1–3, 4–6, 7–9, and 10–12, respectively; see Appendix A.9 and Appendix A.10 for details). Within a session, one type of market is played for 3 rounds in the same random order to exclude learning about market types. Moreover, all 12 markets differ in their quality–price combinations, which are assigned exogeneously. For all globally non-dominated product models in each market, prices are slightly higher than marginal costs. In addition, the price–quality combinations of these product models are chosen such that, under CompleteInformation, two buyers with the same theta would either buy the globally non-dominated product model with optimal quality if available (${\theta }_1={\theta }_2=3\to q^{*}=2,{\theta }_3={\theta }_4=7\to q^{*}=3$, ${\theta }_5={\theta }_6=11\to q^{*}=4$, ${\theta }_7={\theta }_8=15\to q^{*}=5$), or refrain from buying otherwise. Thus, we ensure that, under CompleteInformation, no buyer would choose a product model with $q=1$, which corresponds to Stiftung Warentest’s “poor” rating. This rating is given when a product model is considered unacceptable for all, as when it does not suit its claimed purpose and/or entails unacceptable risks such as high toxic material levels. Again to prevent learning across rounds, we choose quality–price combinations for globally dominated product models such that there are three product models per quality level and the correlation between quality and price is below 0.01.

We base our hypotheses on our theoretical results from Section 3.2. In particular, H1, H2, and H3 are consequences of Proposition 2, while H4 is a consequence of Proposition 3.

H1. 

Seller behavior. Under our new mechanisms SellersMayApply-LyingPoss and SellersMayApply-Truth

–

globally dominated sellers will not apply to be tested.

–

with the exception of $q_t=1$, globally non-dominated sellers will apply to be tested and will, if applicable, state their true quality.

H2. 

Content of the product test. The product test will contain the following:

–

no information on globally non-dominated product models under SellersMayNotApply-WorstCase,

–

information on 21.7% of globally non-dominated product models under SellersMayNotApply-Random, and

–

information on all globally non-dominated product models under both SellersMayApply-Truth and SellersMayApply-LyingPoss.

H3. 

Buyer behavior. Buyers will choose the least number of globally non-dominated product models under SellersMayNotApply-WorstCase, more under SellersMayNotApply-Random, and only globally non-dominated product models under both SellersMayApply-Truth and SellersMayApply-LyingPoss.

H4. 

Surplus and profits. Per capita, the following will hold:

–

consumer surplus as well as globally non-dominated seller profits are lowest under SellersMayNotApply-WorstCase, higher under SellersMayNotApply-Random, and highest under both SellersMayApply-Truth and SellersMayApply-LyingPoss.

–

globally dominated seller profits are highest under SellersMayNotApply-WorstCase, lower under SellersMayNotApply-Random, and lowest under both SellersMayApply-Truth and SellersMayApply-LyingPoss.

Our experiment comprised 25 sessions and was conducted between January 2017 and December 2018 at the Essen Laboratory for Experimental Economics (elfe), Germany. We conducted five sessions per treatment, with the exception of SellersMayApply-Truth, where we conducted an additional five sessions in which sellers were asked for their beliefs about buyer behavior. In total, 575 subjects participated in the experiment. On average, a session lasted two hours, and a subject earned 27.39 EUR. More details on the number of participants are displayed in

Table 2. Number of sellers and buyers per session, and number of sessions and participants per treatment.

Treatment	Sellers per Session	Buyers per Session	Number of Sessions/ Independent Observations	Participants
SellersMayNotApply-WorstCase	15	8	5	115
SellersMayNotApply-Random	15	8	5	115
SellersMayApply-LyingPoss	15	8	5	115
SellersMayApply-Truth	15	8	5	115
SellersMayApply-Truth (with beliefs about buyer behavior)	15	8	5	115
Total			25	575

. Participants were invited to participate in the experiment using ORSEE (Greiner, 2015). The experiment was programmed and conducted with zTree (Fischbacher, 2007). A translated version of the instructions can be found in Appendix A.11. Translated screenshots of the main decision situations in z-Tree can be found in Appendix A.12.

4.2. Experimental Results

We prepare and analyze the data with R 4.5.1 (R Core Team, 2023), including the R package zTree (Kirchkamp, 2019). Unless stated otherwise, we report the results of two-sided Mann–Whitney U tests for all treatment comparisons, conservatively counting one experimental session as one independent observation. Since we did not find any significant differences between SellersMayApply-Truth with and without asking for beliefs about buyer behavior, we pool these data in our subsequent analyses.

4.2.1. Seller Behavior

Figure 3 depicts the share of sellers who do and do not apply to be tested, split by sellers with globally dominated versus globally non-dominated product models.28 In line with our first hypothesis H1, Figure 3 shows that most globally dominated sellers do not apply to be tested (70.9% under SellersMayApply-LyingPoss, 83.1% under SellersMayApply-Truth). Also in line with H1, we see from Figure 3 that most globally non-dominated sellers do apply to be tested (70.5% under SellersMayApply-LyingPoss, 82.1% under SellersMayApply-Truth). Moreover, under SellersMayApply-LyingPoss, we see that only 7.4% (2.9%) of globally dominated (globally non-dominated) sellers apply stating their false quality, which is marginally significantly (not significantly) different from zero (sign test, p-values 0.06 and 0.12, respectively). Interestingly, we further see from Figure 3 that more globally dominated sellers apply to be tested under SellersMayApply-LyingPoss compared to SellersMayApply-Truth, but fewer globally non-dominated sellers do so. Thus, we see that SellersMayApply-LyingPoss reflects a greater degree of out-of-equilibrium behavior. We summarize our first main result as follows.

Result 1 

In line with H1, under SellersMayApply-LyingPoss and SellersMayApply-Truth:

–

most globally dominated sellers do not apply to be tested.

–

most globally non-dominated sellers do apply to be tested and, if applicable, state their true quality.

However, not in line with H1, there is more out-of-equilibrium behavior under SellersMayApply-LyingPoss than under SellersMayApply-Truth.

Result 1 is consistent with previous experimental findings (see, for example, Benndorf et al., 2015) which show that participants behave largely but not completely in line with theoretical predictions.

The finding that SellersMayApply-LyingPoss elicits greater out-of-equilibrium behavior means that the number of sellers with globally non-dominated product models who apply to be tested will impact the number that are tested. It further implies that if sellers with globally dominated product models apply stating false qualities to the extent that they create testing congestion, globally non-dominated product models may end up being squeezed out of the testing pool. The next subsection presents the degree to which this actually happens.

4.2.2. Content of the Product Test

Figure 4 shows the respective shares of globally dominated and non-dominated product models in the product test pool. In line with our second hypothesis H2, the product test yields the least globally non-dominated product model information under SellersMayNotApply-WorstCase (0%), more under SellersMayNotApply-Random (21.7%), more still under SellersMayApply-LyingPoss (79%), and most under SellersMayApply-Truth (96.1%). Note that the shares under SellersMayNotApply-Random and SellersMayNotApply-WorstCase are determined by design. In particular, under SellersMayNotApply-WorstCase, since the exogenously assigned bestsellers are all globally dominated, no globally non-dominated product models are tested. Also note that, by design, 21.7% of globally non-dominated product models are tested under SellersMayNotApply-Random as this reflects the share of globally non-dominated product models in all markets (23.3%).

Figure 4 further shows that the product test pool contains fewer globally non-dominated product models under SellersMayApply-LyingPoss than under SellersMayApply-Truth (difference: 17.1 percentage points), but more than under SellersMayNotApply-Random (difference: 57.3 percentage points). Thus, we find that although out-of-equilibrium behavior under SellersMayApply-LyingPoss decreases the share of globally non-dominated product models in the test, this share remains higher than that under SellersMayNotApply-Random. We summarize our second main result as follows.

Result 2 

In line with H2, the product test provides the least information on globally non-dominated product models under SellersMayNotApply-WorstCase, more under SellersMayNotApply-Random, and more still under SellersMayApply-LyingPoss. However, not in line with H2, the product test provides even more information on globally non-dominated product models under SellersMayApply-Truth.

4.2.3. Buyer Behavior

We next examine the relation between buyer information and behavior under the different mechanisms. Figure 5 shows that, consistent with H3, buyers choose the fewest globally non-dominated product models under SellersMayNotApply-WorstCase (48.1%), more under SellersMayNotApply-Random (65.4%), more still under SellersMayApply-LyingPoss (82.1%), and the most under SellersMayApply-Truth (88.2%). Note that the relatively large shares of buyers choosing globally non-dominated product models under SellersMayNotApply-Random and SellersMayNotApply-WorstCase can be explained by the large share of buyers who choose the cheapest product model, which is always globally non-dominated (53.5% of all buyers who choose globally non-dominated product models under SellersMayNotApply-Random, 90.9% under SellersMayNotApply-WorstCase). Figure 5 further shows that buyers choose fewer globally non-dominated product models under SellersMayApply-LyingPoss (82.1%) compared to SellersMayApply-Truth, but more than under SellersMayNotApply-Random. Figure 5 also shows a positive share of non-buyers for each treatment, but no significant difference across treatments. We do find that buyers choose the highest number of globally dominated product models under SellersMayNotApply-WorstCase (35.2%), fewer under SellersMayNotApply-Random (24.2%), and the fewest under SellersMayApply-LyingPoss and SellersMayApply-Truth (5% and 3.8%, respectively). Thus, we summarize our third main result as follows.

Result 3 

In line with H3, buyers choose the fewest globally non-dominated product models under SellersMayNotApply-WorstCase, more under SellersMayNotApply-Random, and more still under SellersMayApply-LyingPoss. However, not in line with H3, buyers choose an even greater number of globally non-dominated product models under SellersMayApply-Truth.

4.2.4. Surplus and Profits

In our final set of experimental results, we examine participant payoffs across different treatments. From Figure 6, we see that, consistent with H4, per capita consumer surplus is lowest under SellersMayNotApply-WorstCase (5.7 ECU), higher under SellersMayNotApply-Random (12.4 ECU), and highest under SellersMayApply-LyingPoss and SellersMayApply-Truth (17 ECU and 17.8 ECU, respectively). Figure 6 further shows that globally non-dominated seller profits are lowest under SellersMayNotApply-WorstCase (2.3 ECU), higher under SellersMayNotApply-Random (5.1 ECU), and highest under SellersMayApply-Truth (8.3 ECU). Figure 6 also shows that globally non-dominated seller profits are lower under SellersMayApply-LyingPoss (7.2 ECU) compared to SellersMayApply-Truth, but still higher than under SellersMayNotApply-Random. Again in line with H4, Figure 6 shows that globally dominated seller profits are highest under SellersMayNotApply-WorstCase (5.4 ECU), lower under SellersMayNotApply-Random (3 ECU), and lowest under SellersMayApply-LyingPoss and SellersMayApply-Truth (0.3 ECU in both treatments). Thus, we summarize our fourth main result as follows.

Result 4 

In line with H4, per capita

–

consumer surplus is lowest under SellersMayNotApply-WorstCase, higher under SellersMayNotApply-Random, and highest under SellersMayApply-LyingPoss and SellersMayApply-Truth.

–

globally non-dominated seller profits are lowest under SellersMayNotApply-WorstCase, higher under SellersMayNotApply-Random, and highest under SellersMayApply-Truth and SellersMayApply-LyingPoss.

–

globally dominated seller profits are highest under SellersMayNotApply-WorstCase, lower under SellersMayNotApply-Random, and lowest under SellersMayApply-Truth and SellersMayApply-LyingPoss.

5. Discussion and Conclusions

In this study, we propose a novel mechanism to solve a certifier’s set selection problem given its limited testing capacity in order to provide more valuable information to consumers. Our mechanism relies on the unraveling prediction as it allows sellers to indicate a product model’s quality when applying for testing. We first develop a certification game to derive testable predictions for different mechanisms to select product models for testing. We show theoretically that a unique Nash equilibrium exists in which our proposed mechanism yields optimal buyer information equivalent to a world of complete information. We also show that any current mechanism always leads to a lower consumer surplus unless all product models which would be sold under complete information (or all but the overall cheapest one) happen to be selected for testing. Therefore, our mechanism weakly dominates any current mechanism.

We then use an experimental setting to test the predictions derived from our game. The results of our experiment show that, under our new mechanism, most sellers with globally non-dominated product models apply for testing, suggesting that information unraveling is sufficient for increasing the information provided on globally non-dominated product models. Buyers benefit from the superior information as they buy more globally non-dominated product models. Thus, our experimental results confirm that our mechanism increases consumer surplus compared to current mechanisms with limited testing capacities. Our results further show that globally non-dominated seller profits increase while those of globally dominated sellers decrease under our mechanism. Our experimental results are consistent with those of previous studies that find that unraveling does occur, yet usually to an incomplete degree, in complex settings (see Hagenbach & Perez-Richet, 2018; Jin et al., 2021, 2022). Furthermore, we show that even in our two-dimensional context where price does not necessarily equal quality, and where false quality statements are allowed, our mechanism outperforms any current mechanism. Importantly, our mechanism can be applied to any certifier with limited testing capacities.

We acknowledge the following limitations of our theoretical framework. First, while all product tests are based on a partly arbitrary set of weights in order to create a unidimensional overall quality measure, we acknowledge that our mechanism would be particularly sensitive to the set of weights used as it might determine which product models are globally dominated or non-dominated. However, test results published by Consumer Reports show that more than half of all tested product models are globally dominated on all quality sub-dimensions (Hjorth-Andersen, 1984).29 This implies that more than half of these tested product models are, in fact, insensitive to the set of weights; i.e., they remain globally dominated no matter which set of weights is used. In addition, our mechanism could be extended to include more than one set of weights, which would be an interesting research question. For example, a certifier could issue different calls for applications based on different sets of weights if it identifies separate groups of buyers with different preferences. Thus, a certifier could provide customized information for each group of buyers. Note that, irrespective of whether a certifier wishes to include one or more sets of weights, we consider it important that a certifier announces precisely how it will measure product quality. In particular, sellers need to know which quality characteristics the certifier includes, and how these are weighted to calculate an overall, unidimensional quality measure.

Second, we limit our analysis to the short term. More specifically, in situations in which quality and price have already been set and remain constant, we show that buyers are able to select more globally non-dominated product models under our new mechanism. However, we acknowledge that globally non-dominated sellers may have an incentive to increase their price after the product test if the game allowed for a longer time horizon. Yet, this concern also exists for any current mechanism since sellers who are locally non-dominated within the set of tested sellers may also increase their price after the product test. Therefore, we consider this as a general problem, i.e., a problem that both any current mechanism and our new mechanism face, and not as a specific disadvantage of our mechanism. In relation to this, while we assume in our theoretical framework that each seller can sell as many units of his product model as demanded, we acknowledge that this may empirically not always be the case, especially in the short term. In particular, one may fear that globally non-dominated, but previously less well-known sellers may not be able to satisfy the increased demand in the short term. However, a similar problem also seems to exist with current mechanisms.30 Also, since our mechanism is more transparent than current mechanisms (see Appendix A.3 for details), it may actually allow sellers to estimate their future demand more precisely, which may mitigate the concern about the production capacities of less well-known sellers.

Third, we assume a high degree of precision when measuring product quality. More specifically, we assume in our theoretical framework that sellers are perfectly informed about their own and others’ product quality. While we acknowledge that this may not always be the case, we would still argue that sellers have an informational advantage over buyers. In relation to this, we assume that the certifier is able to measure product quality perfectly. We think that relaxing either assumption provides interesting avenues for future research: extending our certification game to a case where (i) sellers are less precisely informed about product quality, and (ii) where the certifier may not be able to measure product quality perfectly.

Fourth, our theoretical framework does not allow us to investigate how real-world complexities, such as strategic misreporting, market power, or collusion among sellers, might impact the effectiveness of our mechanism in other settings. Strategic misreporting, for example, should theoretically not occur in our setting where one seller sells one product model. Yet, it would be interesting to analyze settings in which one seller offers several product models. Here, a different way to deter lying might be more appropriate. Similarly, in a more complex setting in which one seller offers several product models, it would be interesting to investigate if and, if yes, how, our mechanism needed to be modified when sellers have different degrees of market power. As to analyzing collusion, a repeated setting in which sellers can set quality and price before the testing occurs would be an interesting extension.

Finally, we also acknowledge that there may be concerns about adopting our mechanism in the field for two reasons. First, the role of sellers in the application process in our mechanism may raise the concern that independent consumer organizations will incur a reduction in their perceived credibility. To mitigate this concern, they could provide a transparent explanation of the product model selection process to buyers. This information could emphasize that the role of sellers does not present a conflict of interest since the final decision which product models to test remains with the consumer organization. Moreover, they could emphasize that the testing process continues to include anonymous test buyers and objective testing methods. Note that other third-party certifiers, which do not face the same testing capacity constraints, do allow sellers to directly suggest certain product models to be tested. However, these other certifiers do not need to select a subset from their applicants which is the main focus of this paper (see end of Section 2 for more details).

Second, independent consumer organizations may be concerned about the effect of our mechanism on their publication sales. Low ratings of (previous) bestsellers under current mechanisms may elicit surprise and media attention, leading to increased interest in consumer organization publications. By contrast, under our new mechanism, none of the bestsellers may be included in the test. However, it is also possible that our mechanism would generate increased interest in consumer organization publications since it yields a more helpful portrait of what to purchase.

Importantly, it is also possible for independent consumer organizations to use a hybrid of our new mechanism and any current mechanism depending on their objectives. For example, consumer organizations could reserve a certain number of testing slots for bestselling product models to ensure that they are included in the test, while all other testing slots are filled using our new mechanism. Overall, our paper presents an alternative to current product model selection mechanisms that reduces information asymmetry between buyers and sellers in order to increase consumer surplus.