 
Summary: 1
The Schema Theorem and Price's Theorem
Lee Altenberg
Institute of Statistics and Decision Sciences
Duke University, Durham, NC, USA 277080251
Internet: altenber@dynamics.org
Abstract
Holland's Schema Theorem is widely taken to be the foundation for explanations of the
power of genetic algorithms (GAs). Yet some dissent has been expressed as to its implica
tions. Here, dissenting arguments are reviewed and elaborated upon, explaining why the
Schema Theorem has no implications for how well a GA is performing. Interpretations of
the Schema Theorem have implicitly assumed that a correlation exists between parent and
offspring fitnesses, and this assumption is made explicit in results based on Price's Covari
ance and Selection Theorem. Schemata do not play a part in the performance theorems
derived for representations and operators in general. However, schemata reemerge when
recombination operators are used. Using Geiringer's recombination distribution represen
tation of recombination operators, a ``missing'' schema theorem is derived which makes
explicit the intuition for when a GA should perform well. Finally, the method of ``adap
tive landscape'' analysis is examined and counterexamples offered to the commonly used
correlation statistic. Instead, an alternative statistic  the transmission function in the
