Focusing versus Intransitivity Geometrical Aspects of Co-evolution
Recently, a minimal domain dubbed the numbers game has been proposed to illustrate well-known issues in co-evolutionary dynamics. The domain permits controlled introduction of features like intransitivity, allowing researchers to understand failings of a co-evolutionary algorithm in terms of the domain. In this paper, we show theoretically that a large class of co-evolution problems closely resemble this minimal domain. In particular, all the problems in this class can be embedded into an ordered, n-dimensional Euclidean space, and so can be construed as greater-than games. Thus, conclusions derived using the numbers game are more widely applicable than might be presumed. In light of this observation, we present a simple algorithm aimed at remedying focusing problems and relativism in the numbers game. With it we show empirically that, contrary to expectations, focusing in transitive games can be more troublesome for co-evolutionary algorithms than intransitivity. Practitioners should therefore be just as wary of focusing issues in application domains.
KeywordsPartial Order Linear Extension Decomposition Theorem Minimal Realizer Linear Realizer
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- 1.Watson, R., Pollack, J.: Coevolutionary Dynamics in a Minimal Substrate. In Spector, L., Goodman, E., Wu, A., Langdon, W., Voigt, H.M., Gen, M., Sen, S., Dorigo, M., Pezeshk, S., Garzon, M., Burke, E., eds.: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO-2001, San Francisco, CA, Morgan Kaufmann Publishers (2001)Google Scholar
- 2.de Jong, E., Pollack, J.B.: Ideal Evaluation from Coevolution. Evolutionary Computation (to appear)Google Scholar
- 3.Ficici, S.G., Pollack, J.B.: Pareto Optimality in Coevolutionary Learning. In: European Conference on Artificial Life. (2001) 316–325Google Scholar
- 4.Noble, J., Watson, R.A.: Pareto coevolution: Using performance against coevolved opponents in a game as dimensions for Pareto selection. In Spector, L., Goodman, E., Wu, A., Langdon, W., Voigt, H.M., Gen, M., Sen, S., Dorigo, M., Pezeshk, S., Garzon, M., Burke, E., eds.: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO-2001, San Francisco, CA, Morgan Kaufmann Publishers (2001) 493–500Google Scholar
- 5.Bucci, A., Pollack, J.B.: A Mathematical Framework for the Study of Coevolution. In: FOGA 2002: Foundations of Genetic Algorithms VII, San Francisco, CA, Morgan Kaufmann Publishers (2002)Google Scholar
- 6.Scheinerman, E.R.: Mathematics: A Discrete Introduction. 1st edn. Brooks/Cole, Pacific Grove, CA (2000)Google Scholar
- 8.Mahfoud, S.W.: Crowding and Preselection Revisited. In Männer, R., Manderick, B., eds.: Parallel Problem Solving from Nature 2, Amsterdam, North-Holland (1992) 27–36Google Scholar
- 10.Bucci, A., Pollack, J.B.: Order-Theoretic Analysis of Coevolution Problems: Coevolutionary Statics. In Barry, A.M., ed.: GECCO 2002: Proceedings of the Bird of a Feather Workshops, Genetic and Evolutionary Computation Conference, New York, AAAI (2002) 229–235Google Scholar