Proof of Proposition 1: (i) Let y≡δσ2. It follows that E(χS)=2λ3>(2+0.5y)λ3+y=E(χN(y)) for all y>0. On the other hand, var(χS)=σ29<σ24=var(χN). (ii) E(χS22)=2λ29+σ218 and E(χN22)=(2+0.5y)2λ22(3+y)2+18σ2, thus, information sharing increases expected consumer surplus if

Moreover, ∂λ∗∂δ<0, and limδ→∞λ∗=σ57. This implies that for all λ guaranteeing positive equilibrium quantities there exists large enough δ∗ such that λ∗(δ∗,σ2)<λ, since, σ<λ2 is required for QN to be positive in the sharing equilibrium. Thus, δ>δ∗ implies that sharing increases expected consumer surplus. (iii) Differentiating λ∗ with respect to σ reveals that ∂λ∗∂σ>0.

Proof of Proposition 2: Let G denote the difference between the sum of consumers’ expected utilities under each regime. Assuming f is continuous with support [−z,z],^[5] it follows from (11) that

As noted in the proof of proposition 1, D(0)>0, thus, for sufficiently small α, it follows that G(α)>0.

Proof of Proposition 3: (i) Let Uij denote firm i’s expected utility where j∈{S,N} denotes whether information is shared, with S indicating shared, and N indicating not shared. Plugging in QA∗ and QN∗ into (1) reveals that

and

Thus, UNS−UNN>0 if

or, equivalently:

It follows that dλNdδ<0, limδ→0λN(δ)=∞, and limδ→∞λN(δ)=σ75. Noting that 2σ<λ for positive equilibrium quantities, it follows that for all λ that induce positive equilibrium quantities, there exists δN such that information sharing is beneficial to N if and only if δ<δN. Moreover, (17) immediately implies that for all δ information sharing is beneficial for N if and only if the market is sufficiently small.

On the other hand, E(πAS)=λ29+σ29 and var(πAS)=var((λ+ε3)2). f’s symmetry implies that E(ε3)=0, and assuming that ɛ has a symmetric binomial distribution implies that E(ε4)=σ4. Thus, var(πAS)=481λ2σ2, and, therefore

Table 1 also reveals that E(πAN)=1+0.5δσ2(3+δσ2)2λ2 and var(πAN)=λ2σ24(3+δσ2)2. Therefore,

Thus,

Re-arranging this expression reveals that UAS>UAN if:

Note that dλA(δ)dδ<0, limδ→0λA(δ)=∞, and limδ→∞λA(δ)=0. Therefore, for all λ>0, there exists δA>0, such that A prefers to share information if and only if δ<δA. Finally, note that comparing λA(δ) and λN(δ) reveals that λA(δ)>λN(δ) for all δ>0, implying that δA<δN for all λ>0.

These observations together imply that, for any given λ, both firms increase their expected utilities by agreeing to share information if δ<δA. Assuming indifferent firms are unwilling to share information, it follows that if δ≥δA, firm A is unwilling to share information, and, thus, information sharing does not take place. Similarly, for any δ, it follows that information is shared if and only if λ<λA.

(ii) When δ≥δA, firms do not share information even when it is allowed. Thus, allowing information sharing has no effect on welfare. When δ<δA, firms share information, because this makes them better off. As shown in proposition 2, information sharing also makes consumers better off when α is sufficiently small. Thus, allowing information sharing may make some parties worse-off compared to banning information sharing only if both δ<δA, and 1−α is sufficiently small.

(iii) Comparing λ∗ and λA reveals that λ∗<λA for sufficiently small δ. Thus, when δ is sufficiently small, there exists a range (λ∗,λA), such that when λ∈(λ∗,λA), per part (i) of this proposition and the proof of proposition 1, it follows that information sharing takes place when allowed, and there is an increase in consumer surplus.

(iv) When λ<λA, both parties are willing to share information as noted in the proof of part (i) of this proposition. However, this reduces expected consumer surplus if λ<λ∗ as noted in the proof of proposition 1.