Bias and Size Effects of Price-Comparison Platforms: Theory and Experimental Evidence

Appendix A. Proofs

A Nash equilibrium is a n-tuple of cumulative distribution functions over prices, {F₁(·), …, F_n (·)}, such that for some Πj∗ on ℝ0+, and j=1, …, n: (i) Πj(p)=Πj∗, for all p on the support of F_j (·), and (ii) Πj(p)≤Πj∗, for all p.

When vendors are identical we focus on symmetric equilibria, in which case: F_j (·)=F(·), p_j =p, p̅_j =p̅ and Πj∗=Π∗, for all j.

Denote by p^−j the minimum price charged by any indexed vendor other than vendor j, and denote by m^−j the number of indexed vendors that charge p^−j. The profit function of vendor j when it charges price p_j is:

πj(pj; τ)={pj[λn+(1−λ)ϕjτ]if pj<p^−j≤1pj[λn+1−λm^−jϕjτ]if pj=p^−j≤1pjλnif p^−j<pj≤10if 1<pj.

The next Lemma states some auxiliary results.

Lemma 1For all j: (i) ljτ≤p_j≤p¯j≤1; (ii) Fjτis continuous on[ljτ, 1]; (iii) p̅_j =1; (iv) Πj∗=λn; (v) p_j=ljτ; (vi) Fjτhas a connected support.

Proof of Lemma 1. For τ=s and j=k+1, …, n the proofs are obvious, so consider: (a) τ=s and j=1, …, k, and (b) τ=c, u.

• For any j, any price pj<ljτ or p_j >1 is strictly dominated by p_j =1;

• Suppose not, i.e. suppose that Fjτ has a mass point at price p. Let ε>0 be arbitrarily small and such that no mass point exists at price p–ε. The expected profits of firm j are:

  Πj(p−ε)=(p−ε)λn+(p−ε)(1−λ)ϕjτProb[p−ε<p^−j]+(p−ε)(1−λ)ϕjτProb[p−ε≤p=p^−j],

  and

  Πj(p)=pλn+p(1−λ)ϕjτProb[p<p^−j]+p(1−λ)ϕjτm^−jProb[p=p^−ij].

  Subtracting the second expression from the first and taking the limit as ε approached zero, one obtains

  limε → 0[Πj(p−ε)−Πj(p)]=p(1−λ)ϕjτ(m^−j−1m^−j)Prob[p=p^−j]>0.

  Hence, vendor j would increase profit by shifting mass from p to an ε neighborhood below p. But this implies that it cannot be an equilibrium strategy to maintain a mass point at p;

• Suppose not, i.e. suppose p̅_j <1. Then

  Πj(p¯j)=p¯jλn+p¯j(1−λ)ϕjτ[1−F(p¯j)]k − 1=p¯jλn,

  since from (ii) there are no mass points at p¯jλn. However, the payoff from setting a price equal to 1 is λn>p¯jλn;

• Follows from (ii) and (iii);

• Parts (ii) and (iv) imply that p_jλn+p_j(1−λ)ϕjτ=Πj(p_j)=λn. Hence p_j=ljτ;

• Suppose not, i.e. suppose there is an interval [p_l , p_h ] satisfying ljτ≤pl<ph≤1 such that F(p_l )=F(p_h ). Suppose also that p_l is the infimum of all prices p, ljτ≤p≤1. Then p_l is in the support of F(·) and, from (ii) Πj∗=Πj(pl)=plλn+pl(1−λ)ϕjτ[1−F(pl)]k − 1<phλn+ph(1−λ)ϕjτ[1−F(ph)]k − 1=Πj(ph), a contradiction.□

Proof of Proposition 1. We show constructively that equilibrium exists. Alternatively, existence follows from theorem 5 of Dasgupta and Maskin (1986). (i) Use Lemma 1(iv) to set Πj(p)=pλn+p(1−λ)ϕjτ[1−F(p)]k − 1=λn. Solving for F(p) the result follows; (ii) Obvious.□

Proof of Remark 1. (i) Follows from the fact that all firms are indifferent between any equilibrium price and the monopoly price. (ii) Follows directly from the definition of μc=lc+∫lc1(1−Fc)ndp and εc=lc+∫lc1(1−Fc)dp.□

Theorem 1 (i) ε^c (n)<ε^c (n+1); (ii) μ^c (n)>μ^c (n+1).◆

Proof of Theorem 1. (i) See Morgan et al. (2006); (ii) Follows from (i) and Remark 1(i).      □

Proof of Corollary 1. Obvious.      □

Proof of Proposition 2. (i) Obvious; (ii) Follows from Corollary 1 and the Theorem 1; (iii) Obvious.      □

Proof of Corollary 2. (i) Obvious; (ii) Follows from Corollary 1 and the Theorem 1.      □

Proof of Proposition 3. (i) Obvious; (ii) Follows from (i); (iii) Obvious; (iv) Follows from (iii).      □

Proof of Corollary 3. (i) Obvious; (ii) Obvious; (iii) Follows from (ii); (iv) Follows from (ii).      □

Appendix B. Instructions for Subjects (Translated From Spanish)

• The purpose of this experiment is to study how subjects take decisions in specific economic contexts. This project has received financial support by public funds. Your decision making in this session is going to be of great importance for the success of this research. At the end of the session you will receive a quantity of money in cash which will depend on your performance during the session.

• The environment in which the experiment takes place is an industry. This industry has the following characteristics:

  • a price comparison search platform like the ones on the Internet,

  • 3 firms, (Treatments 4–8: 6 firms),

  • 1200 consumers.

  Each firm in the industry produces a homogeneous product, and this product is the same for all firms.

• Transactions will take place in UMEX (our lab’s Experimental Monetary Units).

• This session will consist of 50 rounds.

• You are one of the 3 firms (Treatments 4–8: 6 firms) in the industry. Your production costs are zero. Therefore, your profits are equal to your revenue.

• Each round, you and the rest of the firms in the industry have to decide the price at which you want to sell the product. Price is your only decision variable.

• (Treatments 1 and 4) Each period, a Price search platform lists the prices of all firms in the industry.

• (Treatments 2, 5 and 7) Each period, a Price search platform lists the prices of 2 firms (Treatment 5: 4 firms) in the industry. More precisely, each round, the price comparison search platform randomly chooses 2 firms (Treatment 5: 4 firms), whose price will be included in its price list. The identity of the firms whose price will be included in the list of the price search platform, will be announced publicly to the members of the industry after the firms’ prices are posted.

• (Treatments 3, 6 and 8) Each period, a Price search platform lists the prices of 2 firms (Treatment 6: 4 firms) in the industry. More precisely, each round, the price comparison search platform randomly chooses 2 firms (Treatment 6: 4 firms), whose price will be included in its price list. The identity of the firms whose price will be included in the list of the price search platform, will be announced publicly to the members of the industry before the firms’ prices are posted.

• Each consumer wants to buy one unit of the product per round. The maximum willingness to pay of each consumer for a unit of the product is 1000 UMEX. That is, if the price you fix is higher than 1000 UMEX, nobody will buy from you.

• There are two types of consumers. Half of them, i.e. 600 consumers, will read the list of price created by the search platform. The other half do not actually read the list of prices of the search engine (maybe because they are not able to do so).

• The consumers who read the price list of the search platform will buy, each period, from the firm whose price for that period is the lowest among all prices included in the price list, if such price does not exceed 1000 UMEX. In case of a “tie” (i.e. several firms fix the same minimum price) the consumers are distributed evenly among the firms with the same minimum price.

• The consumers who do not read the search platform’s price list will buy “randomly” from any vendor, so that this group of consumers will be distributed evenly among all firms in the industry.

• In each round, 3 firms (Treatments 4–8: 6 firms) forming (together with you) the same industry, will be randomly drawn among the 18 participants of this session. Therefore, the probability of competing with the same 2 firms (Treatments 4–8: 5 firms) in 2 different periods is very low (less than 10%).

• Once the participants have been assigned to the industries, you must set your price. The master program in the computer will simulate the consumers’ reactions. At the end of each round, you will see on your screen the information about your own sales, your earnings and the prices fixed by your competitors in the market.

• At the end of the session you will be paid in cash. Your reward will be determined taking into account the earnings you accumulate over 10 (randomly selected) out of the total 50 periods. The exchange rate will be: 1,000,000 UMEX=10 €.

Thank you very much for your participation. Good luck!