The evolution of fuzzy rules as strategies in two-player games.

1. Introduction

How to model players' strategies has been an important issue in the study of repealed games. The problem has been reasonably well addressed for the case in which stralegy choices in the stage game are finite because the history at any point in time consists only of a series of choices from a finite set of stage game strategies. For example, Miller (1996) and Linster (1992, 1994) used finite automata to encode strategies in the repeated prisoner's dilemma. These finite automata, or Moore machines, can capture the idea of bounded rationality, but they can only be used when strategy choices are finite in number.

The purpose of this paper is to illustrate the use of fuzzy strategies in two familiar two-player games with continuous strategy spaces. We chose games for which analytic solutions are available so as to enable comparisons between equilibrium outcomes of our fuzzy models and Coumot-Nash, competitive, and collusive outcomes. These simulations based on fuzzy strategies can be used in models for which analytic solutions do not exist or are computationally very difficult to obtain.

A rule described by a small number of parameters can express only limited rationality within the context of a complex model, but these rules might be "good enough" to describe human behavior in a variety of situations. Consider as an example the task of running an automobile's heating and cooling system prior to the advent of climate control. A heater temperature/air conditioner lever position is continuously and minutely variable. No doubt a complicated optimal nonlinear response function could be specified in which temperature readings are taken and minute adjustments are made to the lever position. Instead, we hypothesize that as few as three simple rules might be enough to adequately control the automobile's temperature. If the car is hot, turn on the air conditioning to maximum cooling. If the car is cold, turn on the heat to maximum heating. If the car temperature is "just right," do nothing. Interpolation over these three rules can specify a proper response to any automobile temperature. We propose tha t much as these three simple rules should eventually arrive at just the right temperature, strategies involving fuzzy rules can be used to find the equilibrium outcome in a variety of two-player games. The equilibrium will not be found with the precision of analytically derived optimal-response functions, but we believe our results illustrate the possibility for boundedly rational agents to achieve a near-equilibrium outcome. Although we consider exclusively two-player games, the techniques we illustrate can easily be extended to games with more players.

This research has both theoretical and empirical components. From a theoretical perspective, this model captures some elements of boundedly rational behavior and adaptive learning at a very elementary level. The strategies are able to learn and adapt as the generations pass. This model has the advantages of enabling us to tractably model continuous strategies in discrete ways and allowing us to evaluate how these strategies evolve over time relative to predetermined benchmarks like Cournot-Nash equilibrium behavior, collusive behavior, and perfecfly competitive behavior. At an empirical level, this paper introduces new methods for performing economic simulations that can be modified for other situations.

Consider the following thought experiment. Suppose two players are about to engage in a repeated version of a game in which the strategic choice comes from a continuous set. The players are not fully rational. Although they play the game repeatedly with the same opponent, they always predict their opponent's future behavior using a simple rule. The predictions may be incorrect, but these naive players never alter the way they make their predictions. Each player is required before the game to submit an initial choice, as well as rules for making a choice in the next period as a function of past play. Initially, the players have minimal information about payoffs, so the rules are randomly chosen. After each generation, the players' payoffs and the rules they used become common knowledge. The players may submit new rules, and the game is played again. What sort of rules should prosper in an environment like this?

The current state of the art in the study of repeated games offers little hope of revealing what outcomes will emerge. The strategy choices available for an individual player in this environment are functions like [q.sub.t]: [0, N] [right arrow] [0, N]. That is, player i uses a rule [q.sub.i] to assign an output level in response to any possible level of output by his or her opponent. In order to model these functions, we employ fuzzy rules as strategies. The fuzzy rules are coded so that they can evolve according to a genetic algorithm.

We assume that a player's strategy can be represented by a set of rules like, "If I think my opponent will choose action x, I will choose action y." We operationalize the idea of selecting strategies based on how well some players are doing with a genetic algorithm. Here, strategic choices are explored in the context of a simple duopoly game and a politically contestable rent game using fuzzy rules to define the strategy and a genetic algorithm to capture the notion of evolving strategies.

2. Literature Review

In considering how to model a game with an infinite strategy space, our first task was to determine how strategies should use information from opponents' past play. Axelrod (1984) published results of a computer tournment in which game theorists were invited to submit strategies to play in an iterated prisoner's dilemma game with the following properties:

1. The interactions were between pairs of players.

2. Each player had two available choices on each move: cooperate or defect. Choices were made simultaneously.

3. The payoffs (Axelrod and Dion 1988) were fixed before play and announced to all players.

4. At each move in the game, each player had access to the history of the game up to that move--in short, there was no noise in the transmission of strategy choices between players.

The winning strategy for this tournament was the tit-for-tat strategy--to cooperate in the first round and mimic one's opponent's most recent behavior in subsequent rounds. Although an iterated prisoner's dilemma game has a finite set of choices (cooperate or defect) as opposed to the infinite set in the games we consider, we sought to emulate the properties of a tit-for-tat strategy in what we call a Cournot adaptive scheme. In a Cournot adaptive scheme, a strategy responds only to the immediate past action of its opponent. To experiment with the effect of additional information on the rate of convergence, we ran an additional set of simulations in which the strategy responds to the unweighted average or mean value of all past actions of its opponent. We call this the fictitious play adaptive scheme.

In their survey of the evolution of cooperation, Axelrod and Dion (1988, p. 1388) address limitations of previous tournaments in the generation of new strategies:

One method of overcoming this limitation is to use a genetic approach to develop new strategies for playing the IPD (iterated prisoner's dilemma). A good method of implementing this on a computer is Holland's (1975) genetic algorithm. A strategy can be represented as a "chromosome" describing what to do in each different context that the strategy can distinguish. The genetic algorithm simulates the evolutionary process by mutating the chromosomes and by allowing sexual reproduction to recombine features from two different strategies. It then subjects the resulting offspring to competition with other programs in the population.

Miller (1996) employed this approach to a repeated prisoner's dilemma simulation.

Huck, Normann, and Oechssler (1999) investigated the role of information in learning in a Cournot oligopoly. They note the general intuition that over time, an oligopoly game will converge to the Cournot-Nash equilibrium. This is the result of players' choosing best responses to their opponents' previous play. However, under alternative learning dynamics, the Cournot-Nash equilibrium need not be the expected result. If learning occurs through the imitation of successful strategies, a Walrasian, or competitive, equilibrium is expected. Of further interest with regard to our work, Huck, Normann, and Oechssler found that increasing the information available to firms about the quantities and profits of opponents increases the competitiveness of the market. They cite the imitation of successful firms' actions as the likely cause.

3. Fuzzy Numbers and Fuzzy Rules

Consider an extremely simple version of the first game we will evaluate. Imagine that two firms play a standard repeated duopoly game where firms can produce levels of output we will call small, medium, and large. While infinitely rational players might have a strategy that has the firm choose to produce f([x.sup.*]) units when the other firm is expected to produce [x.sup.*] units, a firm using fuzzy rules might have rules like, "if I think my opponent will produce about y units, I will produce about z units." The fuzzy numbers will overlap, and multiple rules will usually be used to different degrees. The way we operationalize this idea will become apparent as we discuss the model used in our simulations.

It seems completely plausible that people and firms tend to use fuzzy numbers and sets when faced with decisions. Rather than requiring a continuous reaction function (or at least one defined over very many discreet levels of output), a firm can get by using only a few fuzzy rules that answer questions like, What should the player do if it expects the other to choose a small amount? A medium amount? Or a large amount? Using these fuzzy rules, a firm's entire strategy set has only nine members.

Suppose, for example, that the set of possible outcomes can range from 0 to 4000. It is obvious that dealing with the potential number of such rules would be unwieldy (about [4000.sup.2] = 1.6 X [10.sup.7] combinations of possible rules if we consider only discrete levels of output) when we consider a strategy space that requires an explicit response for every possible predicted level of output. However, when we employ fuzzy numbers and fuzzy rules, (1) we can cut the strategy space to relatively few possibilities. For example, if we consider only fuzzy output levels evenly divisible by 1000 (i.e., 0, 1000, 2000, 3000, and 4000), we can reduce the strategy space to one fuzzy response for each of the five fuzzy output levels. (2) This is now a much more manageable set of 5 (2) = 25 rule combinations or strategic choice possibilities to explore.

For example, suppose a duopolist thinks his opponent will produce 2426 units of output in the next period. How much should he produce? Rather than require a separate rule for every possible level of output, the duopolist can develop a small set of fuzzy rules that will allow him to employ a response function of sorts.

Imagine a duopolist who has certain rules by which he plays the game, two of which are the following: (i) "If I think the other player will produce about 2000 units of output, I will produce about 3000 units in the next stage game," and (ii) "if I think the other player will produce about 3000 units of output, I will produce about 4000 units in the next stage game."

For a duopolist who predicts that his opponent will produce 2426 units of output in the current period, neither of the above rules applies strictly. The response to such a prediction would be neither exactly 3000 nor exactly 4000 units, but somewhere in between. If we think in terms of fuzzy numbers, we can develop a response function with few rules. In order to understand the concept of fuzzy rules, we first need to understand the idea of fuzzy numbers.

Although the number 2426 is not exactly 2000 or 3000, We can say that it is partially in the set of numbers we call 2000 and partially in the set of numbers we call 3000. We can define a membership function for the set of numbers we call i (with i [member of] {0, 1000, 2000, 3000, 4000}) as [[mu].sub.i]: [0,4000] [right arrow] [0, 1]. This function takes any real number between 0 and 4000 and assigns to it a degree of membership in the set of numbers we call i.

For example, the function [[mu].sub.2000](z) will assign a degree of membership in the set of numbers we call 2000 for the number z. If we have a membership function like

[[mu].sub.2000](z) = max {0, 1 - \2000 - z\/1000}, we can describe the "crisp" number 2426 as being in the set of numbers we call 2000 with a membership of 0.574. We can similarly define a membership function for the set of numbers we call 3000 as

[[mu].sub.3000] = max{0, 1 -\3000 - z\/1000 - Z}.

The number 2426 is also in the set of numbers we call 3000, then, with a degree of membership of 0.426. More generally, we can say that z is in the set of numbers we call i with degree of membership

[[mu].sub.i](z) = max{0, 1 - \i - z\/1000},

with i [member of]\0, 1000, 2000, 3000, 4000}.

"Triangular fuzzy numbers" like these can be depicted graphically. If we measure the degree of membership vertically, the membership functions look like those in Figure 1. In other words, any number between 0 and 4000 can be represented by A = ([[mu].sub.0], [[mu].sub.1000], [[mu].sub.2000], [[mu].sub.3000], [[mu].sub.4000]). Specifically, the number 2426 can be described a A = (0, 0, 0.574, 0.426, 0), Specifically, the number 2426 can be described as A = (0, 0, 0.574, 0.426, 0), where the zeros represent zero degrees of membership in the sets of numbers we call 0, 1000, and 4000, while 0.574 and 0.426 represent membership in the sets of numbers we call 2000 and 3000, respectively.

The fuzzy rules we have in mind, then, will take a predicted level of output and respond according to the fuzzy rules. To make this slightly more concrete, suppose the possible levels of production are between 0 and 4000 as described above, and we allow rules like, "if I think the other player will produce about x units of output, I will produce about y units," x, y [member of] {0, 1000, 2000, 3000, 4000}.

We can use a matrix to represent a set of rules. Fuzzy rules that describe what to do with five fuzzy input values and five fuzzy output values can be represented with a 5 X 5 matrix. The fuzzy matrix multiplication operation is similar to classical vector-matrix multiplication except that we replace pairwise multiplication with pairwise minima and we replace sums with maxima. For row vectors A and B representing fuzzy numbers and the 5 X 5 matrix M representing the fuzzy rules, the composition rule we use for A o M = B is

[b.sub.j] = max min {[a.sub.i],[m.sub.ij]}, 1[less than or equal to]i[less than or equal to]5

where a, b, and m represent elements of the vectors/matrix and the subscripts on m represent row and column indices. (3)

Consider this example a little further. Suppose the matrix M below summarizes the rules a player uses. The ith row summarizes what the strategy calls for if the opponent's predicted output is i - 1000 units. If, as in our simulations, [m.sub.ij] [member of] [0, 1], elements of the ith row represent the weights placed on the five discrete responses to interpolate the actual response to the opponent's predicted output of i - 1000 units. Notice that the elements of M need not sum to 1.

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An interpretation of matrix M is rather straightforward. The first row indicates that if I think my opponent will produce about 0 units, I will employ the rules produce about 3000 and produce about 4000 with strengths 0.6 and 0.4, respectively. Similarly, if I think my opponent will produce about 1000 units, I will use the rules produce about 2000, about 3000, and about 4000 with equal strengths.

Employing the matrix above on A = (0, 0, 0.574, 0.426, 0), we get the following fuzzy output vector: A o M = B = (0, 0.426, 0.574, 0, 0). This output vector B must be "defuzzified" to obtain a response. We have experimented with two methods, a simple inverse of the fuzzification process described above, and the fuzzy centroid method. (4) The former process applied to B can be represented as

z = [summation over (i)] i[[mu].sub.i]/[summation over (i)] [[mu].sub.i],

where [[mu].sub.i] is the membership function of the set of numbers we call i, as defined above. In this example, z = 1574.

By contrast, the fuzzy centroid method is based on the mass contained in each membership function and its center of mass, or centroid. Given the symmetric shape of the membership trapezoids in Figure 2, the centers of mass for the membership functions shown are 1000 and 2000. Since the areas of the trapezoids are 670.524 and 818.524, the defuzzified value of vector B is

1000(670.524) + 2000(818.524)/670.524 + 818.524 = 1549.696.

The main difference between the above two defuzzification methods can be seen for values close to 0 or 4000. Consider the fuzzy vector (1, 0, 0, 0, 0). Inverse fuzzification defuzzifies this vector to 0, while the fuzzy centroid method defuzzifies it to 381.966.

We have found results from the duopoly game to be robust with respect to the defuzzification method used. We believe that this is because given the parameters used, all equilibrium output values observed are nowhere near the endpoints of 0 and 4000. Since this is not the case for a contestable rent game, (5) we have chosen to use the inverse fuzzification method for all simulations reported in this paper.

4. The Evolution of Strategies

In each of the games we consider, a strategy consists of all relevant information required to play the game, including 25 real numbers on the interval [0,1] to define the elements of a 5 X 5 fuzzy associative matrix M, as defined above. We employed a genetic algorithm in which all strategies are able to reproduce sexually with probabilities proportional to their profits to replace the 10 poorest performing strategies. Our simulations created the initial set of 30 strategies by randomly generating 30 sets of the required parameters. Each strategy played every other strategy for the same number of repetitions. (6) Average payoffs for each stage game were calculated for each strategy in the population, and a genetic algorithm was employed to generate a new population. The genetic algorithm we employed was as follows: (7)

1. An initial population of 30 strategies is randomly generated.

2. Each strategy plays every other strategy, and average profits are calculated.

3. Each strategy is assigned a probability of mating (with replacement) proportional to its profit. (8)

4. Ten new offspring are formed using a combination of crossover and mutation to replace the 10 least profitable strategies.

5. Repeat steps 2-4.

When two strategies mated, the computer program independently generated two random numbers, a crossover point c and a length l. If the parameters contained within a strategy object are numbered from 1 to 26 (25 parameters to define M and 1 parameter to define the initial choice), the crossover point c defines the point at which genetic material ceases to come from one parent and comes from the other. The length l determines how many parameters are to come from the other parent. For example, consider the following two parents:

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If a crossover point of 5 and a length of 15 were randomly selected, the child would look like this:

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Notice that the second, third, and fourth rows of the M matrix from the first parent have been replaced with those from the second. After crossover, mutation occurred with a probability of 0.01. That is, each parameter in the strategy mutated to a different value with a probability of 0.01. Since this strategy has 26 parameters, the probability of at least one parameter mutating is approximately 0.23.

The three components of this adaptive plan--reproduction based on performance, crossover, and mutation--correspond quite closely to the thought experiment discussed in section 1. The idea that players adaptively learn when submitting their revised strategies is captured in the imitative and innovative elements of the genetic algorithm. The transmission of the top 20 strategies to the subsequent generation unaltered represents the best strategies being copied exactly. Combining different parts of existing strategies (crossover) along with some modifications (mutation) captures the innovative element of the genetic algorithm.

5. Duopoly Game Results

The first game we explored was a very simple linear-demand Cournot duopoly game. The essential elements of this two-firm game are the inverse market demand function that we assume to be P(Q) = max {5.5 - Q, 0} and a linear cost function for each firm, [c.sub.i]([q.sub.i]) [q.sub.i], with i [member of] (1, 21 and Q = [q.sub.1] + [q.sub.2] (where [q.sub.1] and [q.sub.2] are real numbers in the interval [0,4]). The players simultaneously choose quantities in each period that are determined by their respective strategies.

The Cournot-Nash prediction for this model is easily determined. Since the inverse demand for the good is P 5.5 - [q.sub.1] - [q.sub.2] and both firms have cost functions c([q.sub.i]j) = [q.sub.i], firm I's profit is (4.5 - [q.sub.2])[q.sub.i]. The first-order maximization condition for firm 1 requires that we set 4.5 - [2.sub.[q.sup.*.sub.1]] - [q.sub.2] = 0. Solving for [q.sup.*.sub.1], we get

[q.sup.*.sub.1]([q.sub.2]) =2.25 - [q.sub.2]/2. (1)

On the basis of the symmetric nature of the problem, we also have [q.sup.*.sub.2]([q.sub.1]) = 2.25 - [q.sub.1]/2. Solving for both players' best replies simultaneously yields the Cournot-Nash equilibrium level of output, 1.5, for both firms, If the firms were to collude, they would act like a monopolist and maximize (4.5 - Q)Q. Some simple calculus yields a collusive solution involving the production of 1.125 units by each firm. If both firms were to produce 2.25 units, the profit of each firm would be zero, as in a perfectly competitive equilibrium.

In this game, the Cournot and fictitious play adaptive schemes produce somewhat similar results. Figures 3 and 4 are histograms of output with the different adaptive schemes. Simulations for both adaptive schemes began with the same random number seed and a mutation probability of 0.01. The observations are "endpoints" taken at the conclusion of the 1000-generation evolutionary process. This process was repeated 100 times with a different randomly selected initial population each time. We recorded the mean quantity of the 10 most profitable strategies for each of the 100 replications. Thus, 1000 observations (endpoints) for each adaptive scheme are recorded. Table 1 provides summary statistics for the simulations along with the Cournot-Nash prediction.

Our prediction was that the strategies would tend to evolve toward the Cournot-Nash outcome not only on average, but nearly all the time. Visual inspection of the histogram in Figure 3 shows that under the Cournot adaptive scheme, a small number of the most profitable strategies produced an outcome that was lower than the Cournot-Nash predicted outcome of 1.5, with output levels consistent with cooperative or collusive behavior. However, by far the largest probability density was centered in the 1.52 range. Almost none of the most profitable strategies exceeded this output level. The fictitious play adaptive scheme, as seen in Figure 4, generally produced output levels above 1.5. Most output levels were in the range of 1.5 to 1.75 but were not concentrated to the same extent as in the Cournot adaptive scheme. Table 1 reports the mean and standard deviation of output and profit for each adaptive scheme as well as the Cournot-Nash predicted values. Note that given the underlying variation, the output levels for the individual strategies for both adaptive schemes are not statistically different from the Cournot-Nash prediction. However, the Cournot adaptive scheme average output level is <1.5 with a high level of significance. Likewise, the fictitious play adaptive scheme average output level is >1.5 with a high level of statistical significance. Figure 5 illustrates a Cournot adaptive scheme fuzzy response function after 1000 generations chosen at random along with the optimal best-response function given in Equation 1. We have found that fuzzy response functions generally approximate the optimal best-response function in a neighborhood around the Cournot-Nash equilibrium output level but often deviate substantially outside this neighborhood. We believe this has a very simple explanation. Once a group of strategies has converged to a stable equilibrium, there is very little need to formulate a response to output levels far from the equilibrium, as these deviations occur so infrequently.

6. Contestable Rent Game Results

The second game we consider is a contestable rent game from Tullock (1980). In this game, the prize is split among the players in proportion to their contributions, as opposed to a winner take all; thus, the game has a nonlinear best-response function and is substantially different from the linear-demand duopoly game of section 5.

In this game, two firms make contributions toward determining the allocation of a contestable rent. Firm 1 contributes [c.sub.1] and receives [c.sub.1]/([c.sub.1] + [c.sub.2] of rent R. Firm 1's profit function is

[[pi].sub.1] [c.sub.1]/[c.sub.1] + [c.sub.2] R - [c.sub.1].

Differentiating with respect to [c.sub.1] and setting to zero yields firm 1's profit-maximizing output,

[c.sub.1] ([c.sub.2]) = max{[square root of ([c.sub.2]R - [c.sub.2], 0)]}. (2)

Imposing symmetry, the Cournot-Nash equilibrium contribution for each firm is c = R/4. If each firm were to contribute R/2, the rent would be entirely dissipated. If both firms were to make contributions 0 < [c.sub.i] < R/4, this would constitute a collusive outcome. For these simulations, we set R = 2.

To implement this game using fuzzy rules, we employ a structure identical to that used in the duopoly game with the exception of the determination of firm profit. Each firm responds to previous plays by its opponent through interpolation over five fuzzy rules. As before, we performed simulations using the Cournot play adaptive scheme (in which the firm responds only to the immediate past play of its opponent) and the fictitious play adaptive scheme (in which the firm responds to the average of all past plays by its opponent).

We found that due to the more complicated nature of this game, 1000 generations was not enough to obtain convergence. Hence, Figures 6 and 7 represent contribution levels obtained after 5000 generations.

Figure 6 illustrates that a small number of the most profitable strategies using the Cournot play adaptive scheme have made contributions consistent with collusive behavior. However, most of the contribution levels were in the range of 0.51-0.52. The mean contribution level, as seen in Table 2, was below the Cournot-Nash predicted level of 0.5. However, given the variation in these observations, no single contribution level is statistically different from 0.5. Figure 7 indicates that the fictitious play adaptive scheme does not seem to have converged after 5000 generations. Comparison with the optimal response function as shown in Figure 8 indicates that these strategies are not playing on the best response function and have considerable variability. We believe that an explanation lies in the nonlinear nature of the best replies. The series of averages to which a strategy replies does not work nearly as well with these best-reply functions as it did with the linear best replies of the duopoly game. Recall tha t the fictitious play adaptive scheme responds to the mean contribution level of its opponent over all previous plays. To the extent that a strategy begins with a nonoptimal contribution, particularly a large contribution, the average of past contributions may be a very poor predictor of contributions in the current round of the game, and the nonlinearity of the best reply magnifies any errors. This makes convergence very slow and unsteady for the fictitious play adaptive scheme. Figure 8 shows a Cournot adaptive scheme fuzzy response function after 5000 generations chosen at random along with the optimal best-response function (Eqn. 2). As in the duopoly game above, the fuzzy-response function approximates the optimal best-response function in a neighborhood around the Cournot-Nash equilibrium predicted contribution of 0.5 but deviates substantially at both low and high contribution levels. Although the optimal best-response function is nonlinear, it appears that this randomly selected best-response function has a linear response in the neighborhood of 0.5, where the nonlinear best-response function has a very slight curvature.

7. Conclusion

In this paper, we described the use of fuzzy rules in computerized simulations of two-player games with infinite strategy spaces. These strategies evolved according to a genetic algorithm involving crossover and mutation. We applied this technique to two different two-player games--a linear-demand Cournot duopoly game and a contestable rent game. With minor adaptation, insight into a broad class of games can be obtained through simulations with this technique. We are optimistic that with more refinement and application experience, simulations of evolving fuzzy rules will help us understand a variety of economic games in which analytic solutions either do not exist of are computationally difficult to obtain.

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Table 1

Summary of Duopoly Simulation Results

                                                       Cournot-Nash
                   Cournot          Fietitious Play     Prediction

Mean output  1.44636 (0.101014)   1.607784 (0.109383)    1.5 (0)
Mean profit  2.302281 (0.107678)  2.036276 (0.161223)    2.25 (0)

Standard deviations are in parentheses.

Table 2

Summary of Contestable Rent Simulation Results


                         Cournot          Fictitious Play

Mean contribution  0.466897 (0.090305)  0.732057 (0.110353)
Mean profit        0.529666 (0.092052)  0.265485 (0.108316)

                   Cournot-Nash
                    Prediction

Mean contribution    0.5 (0)
Mean profit          0.5 (0)

Standard deviations are in parentheses.

Received February 2001; accepted June 2002.

(1.) This discussion is extremely brief, and we consider only the simplest of models. For a definitive discussion on fuzzy logic and fuzzy mathematics, see Kosko (1992).

(2.) Although we use cardinal terms for the fuzzy sets we define, it is important not to assign cardinal significance to our "fuzzy numbers." These can just as easily he thought of as categories like "very little," "a small amount," and so on. We keep the cardinal designation only for ease of explanation.

(3.) This max-min composition rule for fuzzy vectors corresponds to the fuzzy logic OR-AND operators. If traditional true-false logic were used, the operators would represent standard vector multiplication. See Klir and Foger (1988) for a more detailed explanation and description of this procedure.

(4.) A numher of other alternatives exist. For an exhaustive list of the sorta of "defuzzifying" rules, see Kosko (1992) or Klir and Yuan (1995).

(5.) A collusive equilibrium in this game is where both contestants make equal, minutely small contributions close to zero.

(6.) We experimented with both 30 and 100 sets of strategies but chose 30 to reduce the computational time necessary for the large number of replications we were to perform. We chose to generate parameters of the initial strategies randomly out of a desire to test our algorithm under adverse conditions. Given that the M matrix contains 25 continuously variable real numbers, a systematic uniform initialization scheme is not obvious to us.

(7.) This genetic algorithm is essentially the same as the one used by Miller (1996).

(8.) Strategy i was selected with a probability equal to its profit/total profit. Since negative profits were encountered, payoffs were scaled so that the lowest payoff was equal to zero for these purposes. That is, we added the absolute value of the lowest payoff (if it was negative) to all payoffs to determine mating probabilities.

References

Axelrod, Robert. 1984. The evolution of cooperation. New York: Basic Books.

Axelrod, Robert, and Douglas Dion. 1988. The further evolution of cooperation. Science 242:1385-90.

Holland, J. 1975. Adaptation in natural and artificial systems. Ann Arbor, MI: University of Michigan Press.

Huck, Steffen, Hans-Theo Normann, and Jorg Oechssler. 1999. Learning in Cournot oligopoly--An experiment. The Economic Journal 109:C80-95.

Klir, George, and T. A. Foger. 1988. Fuzzy sets, uncertainty, and information. Englewood Cliffs, NJ: Prentice-Hall.

Klir, George, and Bo Yuan. 1995. Fuzzy sets and fuzzy logic. Englewood Cliffs, NJ: Prentice-Hall.

Kosko, Bart. 1992. Neural networks and fuzzy systems: A dynamical systems approach to machine intelligence. Englewood Cliffs, NJ: Prentice-Hall.

Linster, Bruce. 1992. Evolutionary stability in the infinitely repeated prisoner's dilemma played by two-state Moore machines. Southern Economic Journal 58:880-903.

Linster, Bruce. 1994. Stochastic evolutionary dynamics in the repeated prisoner's dilemma. Economic Inquiry 32:342-57.

Miller, John. 1996. The coevolution of automata in the repeated prisoner's dilemma. Journal of Economic Behavior and Organization 29:87-112.

Tullock. G. 1980. Efficient rent-seeking. In Toward a theory of the rent-seeking society, edited by J. Buchanan, R. Tollison, and G. Tullock. College Station, TX: Texas A&M University Press, pp. 97-112.

James E. West *

Bruce Linster +

* Department of Economics and Geography, United States Air Force Academy, 2354 Fairchild Drive, Suite 6K110, USAF Academy, CO 80840-6299, USA; E-mail [e-mail]: corresponding author.

+ Department of Economics and Geography, United States Air Forte Academy, 2354 Fairchild Drive, Suite 6K110, USAF Academy, CO 80840-6299, USA; E-mail [e-mail]

The authors wish to thank the anonymous referees for their very helpful comments and suggestions.