Analysis

 

In this section, we present a game-theoretic analysis of the prescribed model viewed as a one-shot game. Assuming sellers are profit maximizers, we first show that there is no pure strategy Nash equilibrium, and we then compute the symmetric mixed strategy Nash equilibrium. A Nash equilibrium is a vector of prices at which sellers maximize their individual profits and from which they have no incentive to deviate [28]. Recall that ; in particular, the number of buyers is assumed to be very large, while the number of sellers is a good deal smaller. In accordance with this assumption, it is reasonable to study the strategic decision-making of the sellers alone, since their relatively small number suggests that the behavior of individual sellers indeed influences market dynamics, while the large number of buyers renders the effects of individual buyers' actions negligible.

Traditional economic models consider the case in which all buyers are bargain hunters: i.e., . In this case, prices are driven down to marginal cost; in particular, , for all sellers s (see, for example, Tirole [30]). In contrast, consider the case in which all buyers are of type A, meaning that they randomly select a potential seller: i.e., . In this situation, tacit collusion arises, in which all sellers charge the monopolistic price, in the absence of explicit coordination; in particular, , for all sellers s. Of particular interest in this study, however, is the dynamics of interaction among buyers of various types: i.e., . Knowing that buyers of type A alone results in all sellers charging the valuation price v, we investigate the impact of buyers of type B, or shopbots, on the marketplace.

Throughout this exposition, we adopt the standard notation , which distinguishes the price offered by seller s from the prices offered by the other sellers. Our analysis begins with the following observation: at equilibrium, at most one seller s charges . Suppose that two distinct sellers set their equilibrium prices to be , while all other sellers set their equilibrium prices at the buyers' valuation v. In this case, , for small, positive values of , which implies that is not an equilibrium price for seller s. Now suppose that two distinct sellers set their equilibrium prices to be , while all other sellers set their equilibrium prices at v. In this case, seller s prefers price v to , since , which again implies that is not an equilibrium price for seller s. In sum, no 2 (or more) sellers charge equal equilibrium prices strictly below v, and no 2 (or more) sellers charge unequal equilibrium prices strictly below v. Therefore, at most one seller charges .

On the other hand, at equilibrium, at least one seller s charges . Given that all sellers other than s set their equilibrium prices at v, seller s maximizes its profits by charging , since , for small, positive values of . Thus, v is not an equilibrium price for seller s. It follows from these two observations that at equilibrium, exactly one seller s sets its price below the buyers' valuation v, while all other sellers set their equilibrium prices . Note that , for all v' > v, since , implying that all other sellers s' maximize their profits by charging price v. The unique form of pure strategy equilibrium which arises in this setting thus requires that a single seller s set its price while all other sellers set their prices . The price vector , with , however, is not a Nash equilibrium. While v is in fact an optimal response to , since the profits of seller are maximized at v given that there exists low-priced seller s, is not an optimal response to v. On the contrary, , whenever . In particular, the low-priced seller s has incentive to deviate. It follows that there is no pure strategy Nash equilibrium in the proposed shopbot model.

There does, however, exist a symmetric mixed strategy Nash equilibrium. Let f (p) denote the probability density function according to which sellers set their equilibrium prices, and let F (p) be the corresponding cumulative distribution function. Following Varian [32], we note that in the range for which it is defined, F (p) has no mass points, since otherwise a seller could decrease its price by an arbitrarily small amount and experience a discontinuous increase in profits. Moreover, there are no gaps in the said distribution, since otherwise prices would not be optimal -- a seller charging a price at the low end of the gap could increase its price to fill the gap while retaining its market share, thereby increasing its profits. In this probabilistic setting, the event that seller s is the low-priced seller occurs with probability . Rewriting Eq. 2, we obtain the demand expected by seller s:

 

A Nash equilibrium in mixed strategies requires that (i) sellers maximize individual profits, given the other sellers' strategic profiles, so as there is no incentive to deviate, and (ii) all prices assigned positive probability yield equal profits, otherwise it would not be optimal to randomize. Following condition (ii), we define equilibrium profits , for all prices p. The precise value of can be derived by considering the maximum price that sellers are willing to charge, say . At this boundary, , which by Eq. 7 implies that . Moreover, the function attains its maximal value at price , yielding equilibrium profits . Now, by setting equal to this value and solving for F (p), we obtain:

 

which implicitly defines p and F (p) in terms of one another. Since F (p) is a cumulative probability distribution, it is only valid in the domain for which its valuation is between 0 and 1. As noted previously, the upper boundary is p = v; the lower boundary is computed by setting F (p) = 0 in Eq. 8, which yields:

 

Thus, Eq. 8 is valid in the range . A similar derivation of this mixed strategy equilibrium appears in Varian [32]. Greenwald, et al. [20] presents various generalizations of this model.

Figs 1 (a) and (b), respectively, exhibit plots of the functions F (p) and f (p) under varying distributions of buyer strategies -- in particular, the fraction of shopbot users -- with S = 5, v = 1, and c = 0.5. When exceeds a critical threshold (equal to 0.1071 for S = 5), f (p) is bimodal. In this regime, as either or S increases, the probability density concentrates either just below v, where sellers expect high margins but low volume, or just above , where they expect low margins but high volume, with the latter solution becoming increasingly probable. Since itself decreases under these conditions (see Eq. 9), it follows that both the average price paid by buyers and the average profit earned by sellers decrease. These relationships have a simple interpretation: buyers' use of shopbots catalyzes competition among sellers, and moreover, small fractions of shopbot users induce competition among large numbers of sellers.

  
Figure 1: Nash Equilibria for S = 5, v = 1, c = .5, and

Recall that the profit earned by each seller is , which is strictly positive so long as . It is as though only buyers of type A are contributing to sellers' profits, although the actual distribution of contributions from buyers of type A vs. buyers of type B is not as one-sided as it appears. In reality, buyers of type A are charged less than v on average, and buyers of type B are charged more than c on average, although total profits are equivalent to what they would be if the sellers practiced perfect price discrimination. Buyers of type A exert negative externalities on buyers of type B, by creating surplus profits for sellers.