Seeking the min-max point of an unknown function which has several saddle points is one of the most important problems to obtain the optimal mixed strategy in a general game. However, no useful method is available for the problem.
This paper presents a powerful method for obtaining multi-dimensional, multi-saddle payoff functions. The search procedure consists of six main steps: i) to fit local models according to the observed data, ii) to estimate the model's uncertainty due to the lack of data, iii) to estimate the upper limit function and the lower limit function of the unknown function, iv) to estimate a point which is thought to be min-max with a maximum possibillity, v) to ovserve the unknown function at the estimated point, vi) to check if the observed point is the true min-max point.
Simulation studies on unintentionally sellected four test functions of two scalar variables produced fairly good results. The correct min-max points are detected with less than 0.3% error after about 100 observations. The number of the observation is about one-thousandth of that which would be required in applying a uniform lattice search.