Strategies

 

When sufficiently widespread adoption of shopbots by buyers forces sellers to become more competitive, it is likely that sellers will respond with the creation of pricebots that automatically set prices in attempt to maximize profitability. It seems unrealistic, however, to expect that pricebots will simply compute a Nash equilibrium and fix prices accordingly. The real business world is fraught with uncertainties, undermining the validity of traditional game-theoretic analyses: sellers lack perfect knowledge of buyer demands, and they have an incomplete understanding of their competitors' strategies. In order to be deemed profitable, pricebots will need to learn from and adapt to changing market conditions.

In this section, we introduce a series of pricebot strategies, and which we later simulate in order to compare the resulting price and profit dynamics with the game-theoretic equilibrium. In 1838, Cournot showed that the outcome of learning via a simple best-reply dynamic is a pure strategy Nash equilibrium in a quantity-setting model of duopoly [7]. Recently, empirical studies of more sophisticated learning algorithms have revealed that learning tends to converge to pure strategy Nash equilibria in games for which such equilibria exist [17]. As there does not exist a pure strategy Nash equilibrium in the shopbot model, it is of particular interest to study the outcome of adaptive pricing schemes.

We consider several pricing strategies, each of which makes different demands on the required level of informational and computational power of agents:

GT

The game-theoretic strategy is designed to reproduce the mixed strategy Nash equilibrium. It therefore generates a price chosen at random according to the probability density function derived in the previous section, assuming its competitors utilize game-theoretic pricing as well, and making full use of information about the buyer population. GT is a constant function that ignores historical observations.

MY

The myopically optimal, or myoptimal, pricing strategy (see, for example,  [24]) uses information about all the buyer characteristics that factor into the buyer demand function, as well as competitors' prices, but makes no attempt to account for competitors' pricing strategies. Instead, it is based on the assumption of static expectations: even if one seller is contemplating a price change under myoptimal pricing, this seller does not consider that this will elicit a response from its competitors. The myoptimal seller s uses all available information and the assumption of static expectations to perform an exhaustive search for the price that maximizes its expected profit . The computational demands of MY can be reduced greatly if the price quantum -- the smallest amount by which one seller may undercut another -- is sufficiently small (see Appendix A). Under such circumstances, the optimal price is guaranteed to be either the monopolistic price or below some competitor's price, limiting the search for to S possible values. In our simulations, we choose .

DF

The derivative-following strategy is less informationally intensive than either the myoptimal or the game-theoretic pricing strategies. In particular, this strategy can be used in the absence of any knowledge or assumptions about one's competitors or the buyer demand function. A derivative follower simply experiments with incremental increases (or decreases) in price, continuing to move its price in the same direction until the observed profitability level falls, at which point the direction of movement is reversed. The price increment is chosen randomly from a specified probability distribution; in the simulations described here the distribution was uniform between 0.01 and 0.02.

NR

The no regret pricing strategies are probabilistic learning algorithms which specify that players explore the space of actions by playing all actions with some non-zero probability, and exploit successful actions by increasing the probability of employing those actions that generate high profits. In this study, we confine our attention to the no external regret algorithm due to Freund and Schapire [14] and the no internal regret algorithm of Foster and Vohra [12]. As the no regret algorithms are inherently non-deterministic, they are candidates for learning mixed strategy equilibria.