A Note On Agent-Based Imperfect Competition
The model we discuss in this note is a re-examination of the classical Bertrand model of imperfect competition. The main difference is that consumers are allowed to have some strategic behavior when deciding from which one of the two sellers they will buy.Two sellers offering the same homogeneous product compete against each other in a market with a certain number m of potential buyers. At each period t\quad (t=1,2,\ldots ,T) the market opens (and closes) a fixed number of times R that we will refer to as rounds([footnote] Note the difference between periods and rounds. Each period (for instance, a week) contains R rounds (for instance, 7 days).) . The two sellers decide at the beginning of each period a strategy that specifies the course of action to take at each of its rounds contingent on what happened in previous rounds of the same period. Thus, at the beginning of each round, each of the two sellers sets a price ([footnote] The model could also be thought as a model of competition in some other attribute of the product such as quality or location) for his product according to their strategies for that period. Then, the market opens and the two prices become known to everybody. The m buyers walk in and each of them decides from which of the two sellers will buy based on the observation of this two prices. Trade takes place and a new round begins. After R rounds (end of the period) both sellers update their strategies according to some procedure that will be specified.The idea behind this sequence of events is that each seller uses the R rounds to learn about the behavior of its competitor. Thus, each period represents a step in a learning procedure. The model can also be thought as a evolutionary model where each period represents the end of an old generation of sellers and buyers and the birth of a new generation that inherits the characteristics of its parent generation with, hopefully, some improvements. We will refer to the two intuitions indistinctly.We will approach the resolution of the situation sketched above using two related but different tools. The two of them consider a probabilistic learning (or evolutionary) mechanism and in the two of them we will discuss how the consumers' behavior can affect the competition between the sellers.First, we consider the case in which the learning procedure can be described by a deterministic dynamic system that uses expected values. Using strong simplifying assumptions we are able to solve this case and to produce a complete description of how the learning process behaves. We also discuss the problems involved when we try to relax some of the assumptions made.The second approach is an instance of Agent-Based Computational Economics techniques. We use finite automata (encoded as binary strings) to represent the strategies played by the sellers (and also by the buyers) and a decentralized adaptive process based on the models of genetic algorithms to simulate the stochastic process of learning or evolution. With this technique we can relax some of the strong assumptions used in the first approach and still obtain the same basic results. Additionally, as an agent representation issue, we modify the standard operators used in genetic algorithm techniques to make them more suitable to social (or economic) simulations. First, we use a modified mutation operator in which mutations do not take place on a locus-wise basis of the binarily encoded automata but on a state-wise basis of the underlying automata. The reason for this is that locus-wise mutations induce a non uniform distribution over the set of automata that can be the result of the mutation of a given automata. As a consequence of this, when an automaton mutates, some automata are more likely to be the resulting ``mutant'' than others, which could produce an unwanted erratic behavior of the genetic algorithm dynamics. Second, we modify the standard single point crossover operator to furnish it with a social interaction meaning. In this sense, we introduce the partial imitation crossover operator according to which two parent strings (i.e. two parent automata) can only exchange information that makes sense to exchange. This is important due to the special characteristics of the binary representation of finite automata. Indeed, it can be easily showed that the single point crossover operator could produce, from two identical ``parents'', two automata whose behavior is exactly the opposite from the behavior of their ``parents''. This is not only awkward but could also induced an strange behavior in the underlying dynamics.Finally, we like to think that the limitations of the first approach (analytical) provide a good motivation for the second approach (Agent-Based simulations). Indeed, although both approaches address the same problem, we show that the use of Agent-Based computational techniques allows us to relax hypothesis and overcome the limitations of the analytical approach.
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