Analysis

 

We now present a game-theoretic analysis of the prescribed model viewed as a one-shot game.  Assuming sellers are utility maximizers, we derive 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 [13]. There are no pure strategy Nash equilibria in our economic model whenever  [8, 9]. There does, however, exist a symmetric mixed strategy Nash equilibrium, which we derive presently.

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 [17], 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. The cumulative distribution function F(p) is computed in terms of the quantity .

Recall that represents the probability that buyers select seller s as their potential seller. This function is expressed in terms of the probabilistic demand for seller s by buyers of type i, namely , as follows:

The first component . Consider the next component, . Buyers of type 1 select sellers at random; thus, the probability that seller s is selected by such buyers is simply . Now consider buyers of type 2. In order for seller s to be selected by a buyer of type 2, s must be included within the pair of sellers being sampled, which occurs with probability , and s must be lower in price than the other seller in the pair. Since, by the assumption of symmetry, the other seller's price is drawn from the same distribution, this occurs with probability 1 - F(p). Hence, . In general, seller s is selected by a buyer of type i with probability = i / S, and seller s is the lowest-priced among the i sellers selected with probability , since these are i - 1 independent events. Therefore, we have derived , and

 

A Nash equilibrium in mixed strategies requires that all prices assigned positive probability yield equal payoffs -- otherwise, it would not be optimal to randomize. Thus, assuming for all sellers s, the equilibrium payoff , 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 price, , which by Eq. 3 implies that . Identifying the expression v (p - r) g (p) as the profit function of a monopolist, this function attains its maximal value (the monopolist's profit) at price . Therefore, for all sellers s,

 

and hence,

 

implicitly defines p and F(p) in terms of one another, and in terms of g (p), for all p such that .

In previous work, all buyer valuations were taken to be equal (i.e., for all buyers b, ), and hence (see [9] and [10]). In this paper, we assume is a uniform distribution over interval [0, v], with v > 0, in which case the integral yields a step function as follows:

For this uniform distribution of buyer valuations, the monopolist's profit function is simply (p - r) (v - p), for , which is maximized at the price . At this price, the monopolist's profit . Inserting these values into Eq. 5 and solving for p in terms of F yields:

 

Eq. 7 has several important implications. First of all, in a population in which there are no buyers of type 1 (i.e., ) the sellers charge the production cost r and earn zero profits; this is the traditional Bertrand equilibrium. On the other hand, if the population consists of just two buyer types, 1 and some , then it is possible to invert p (F) to obtain:

 

The case in which i = S and for all buyers b was studied previously by Varian [17]; in this model, buyers either choose a single seller at random (type 1) or search all sellers and choose the lowest-priced among all sellers (type S) and all buyers have equal valuations.

Since F(p) is a cumulative probability distribution, it is only valid in the domain for which its valuation is between 0 and 1. The upper boundary is , since prices above this threshold leads to decreases in market share that exceed the benefits of increased profits per unit. The lower boundary can be computed by setting in Eq. 7, which yields:

 

In general, Eq. 7 cannot be inverted to obtain an analytic expression for F(p). It is possible, however, to plot F(p) without resorting to numerical root finding techniques. We use Eq. 7 to evaluate p at equally spaced intervals in ; this produces unequally spaced values of p ranging from to .

Fig. 1(a) and 1(b) depict the CDFs in the prescribed model under varying distributions of buyer strategies -- in particular, -- when S = 2 and S = 5, respectively. In Fig. 1(a), most of the probability density is concentrated just above , where sellers expect low margins but high volume. In contrast, in Fig. 1(b), the probability density is concentrated either just above , where sellers expect low margins but high volume, or just below , where they expect high margins but low volume. In both figures, the lower boundary increases as the fraction of random shoppers increases; in other words, decreases as the fraction of shopbot buyers increases. Moving from S = 2 to S = 5 further decreases , which is consistent with Eq. 9. These relationships have a straightforward interpretation: shopbots catalyze competition among sellers, and moreover, small fractions of shopbot users induce competition among large numbers of sellers.

  
Figure 1: CDFs for .

Recall that the profit earned by each seller is , which is strictly positive so long as . It is as though only buyers of type 1 are contributing to sellers' profits, although the actual distribution of contributions from buyers of type 1 vs. buyers of type i > 1 is not as one-sided as it appears. In reality, buyers of type 1 are charged less than on average, and buyers of type i > 1 are charged more than r on average, although total profits are equivalent to what they would be if the sellers practiced perfect price discrimination. In effect, buyers of type 1 exert negative externalities on buyers of type i > 1, by creating surplus profits for sellers.