Impact of Asymmetry in Cost: Comparison of Special Cases of a

In the following section, I study two special cases when a = 1 and a = 2 in order to achieve some insights on how the asymmetry in firms’ costs affects their reward decisions and revenues. When a = 1, the game is symmetric. The case with a = 2

represents another special situation since, as I see in section 4, this point is a threshold for two different cases of a < 2 and a > 2, in which the firms’ reward decisions at equilibrium are significantly different. Recall that, when a < 2 and w ≤ ŵa, the equilibrium is in Case II, but when a > 2 and w ≤ ŵa, the equilibrium is in Case I. Moreover, in the case of a < 2, the game is more “intensive” than in the case of a > 2; as a result of asymmetry, when

a > 2, there is lower chance for the more favourable firm to have a better position in the competition.

Symmetric Duopoly Case (a = 1). A question arises regarding how Lemmas 4 and 5 can be simplified when firms are symmetric. Moreover, I would like to examine whether the equilibrium is symmetric (i.e., if r1* = r2*) and study the arrangement of users’ states at equilibrium. For example, which portion of users selects states AA, AP, PA or PP? For the symmetric case, ŵa = 1.5. I summarize the result in Proposition 2.

Proposition 2 The symmetric game (a = 1) has a unique and symmetric Nash equilibrium, which can be calculated as follows:

(a) If w ≤ 1.5, then r1* = r2* = w/3.

(b) If w > 1.5, then r1* = r2* = r*, where r* is the feasible unique solution to the

following equation:

(𝑤 − 3𝑟)𝑙𝑜𝑔(𝑟) + 2𝑟 − 1 = 0. (8)

Proposition 2 presents a simplified form of Lemma 4 and Lemma 5 for the symmetric game, which has a symmetric equilibrium and fixed threshold ŵa = 1.5. When w ≤ 1.5, both firmsshare one-third of w with their active users. (The equilibrium is on line OB in Panel A, Figure 1.) This result represents a case in which the revenue contribution of an active user is less than 1.5 times of the cost of a user with the highest effort (unit of cost). When w > 1.5, the equilibrium is derived based on equation (8), i.e., the equilibrium is on line BC in Panel A, Figure 1.

Note that, for a symmetric game, users decide whether to contribute or not to contribute to both firms; that is, users never choose states AP and PA. For example, if w = 1.5, at equilibrium, nearly 70 (30) percent of users decide to be active (passive) in both firms, and the total rewards they receive from firms equal the unit of cost (i.e., r1* = r2* = 0.5). Each firms share one-third of w and keeps two-thirds. Although the equal probabilities of being in states AP and PA seem natural because of the symmetric setting, selection of neither of these states may seem counterintuitive.

Asymmetric Duopoly Case (a = 2). When a = 2, one would presume an asymmetric equilibrium since the users’ cost of contribution to the less favourable firm is twice the cost of contribution to the more favourable firm. In this case, ŵa = 2 and the equilibrium solution is given in Proposition 3:

Proposition 3 When a = 2, the game has a unique Nash equilibrium, which can be calculated as follows:

(b) If 𝑤 > 2, then r1* = 1 and r2* is the feasible unique solution to this equation:

(𝑤 − 2𝑟2)𝑙𝑜𝑔 ( 2

𝑟₂) − 𝑟2+ 1 = 0. (9) An interesting result is that, when w ≤ ŵa = 2, the equilibrium is symmetric for the

asymmetric game. Both firms share half of w with their active users. When w > 2, the less favourable firm gives a higher reward than the more favourable firm; however, the less favourable firm always has a lower market share and revenue, even when it pays more than its competitor. For example, when a = 2 and w = 3, at equilibrium, nearly 83 percent of users decide to be active in both firms, and the remaining users exclusively contribute to the more favourable firm. Note that, when a = 2, the equilibrium is on line OA of Panel A in Figure 1 if w ≤ 2, and it is on line AB if w > 2.

Next, I compare the outcomes of the symmetric (a = 1) and asymmetric (a = 2) games, and I observe that, for both firms, payments in the asymmetric game are higher than those in the symmetric game. Also, the proportion of active users in the asymmetric game is greater (less) than the proportion of active users in the symmetric game for the more (less) favourable firm. Define πi a*as the revenue of firm i at equilibrium for a specific a. Corollary 4 compares the above cases.

Corollary 4The net revenue in the symmetric game is higher than the net revenue in the asymmetric game for both firms, i.e., πi1* > πi2*, for i = 1, 2.

Corollary 4 demonstrates that increasing asymmetry results in shrinkage of net revenues for both firms. Revenue shrinkage for the less favourable firm is not surprising due to the higher reward paid out and the loss in customers. However, the diminishing revenue for the more favourable firm is counterintuitive. This result can be explained by the fact that, as a increases, while φ1increases, the more favourable firm has to share a higher reward in order to better compete with its rival, which also increases its rewards. The negative impact of the higher reward is greater than the positive impact of the higher market share on the firm’s revenue. Therefore, the more favourable firm ultimately loses revenue.

Since the above results come from the comparison of just two special cases for a = 1 and 2, the question arises as to whether these results still hold for other general cases of a. In the next section, I aim to answer this question through a numerical study. I will

demonstrate whether increasing asymmetry always increases (reduces) the proportion of active users for the more (less) favourable firm and whether it shrinks the revenue for the more favourable firm. I further explore this effect in subsection 5.2.