Consider search designs for searching one nonzero 2- or 3-factor interaction under the search linear model. In the noisy case, search probability is given by Shirakura et al. (Ann. Statist. 24(6) (1996) 2560). In this paper some new properties of the searching probability are presented. New properties of the search probability enable us to compare designs, which depend on an unknown parameter ?, for all values of the parameter without needing to check for different values of ?. Two new quantitative interval scale and parameter free criteria based on search probabilities are proposed for design comparison. These criteria are used to compare given designs. The equivalent search designs is defined based on new proposed criteria and present a class of equivalent designs which are orthogonal arrays of strength 2.