AlphaZero in 2017 was able to master chess and other games without human knowledge by playing millions of games against itself (self-play), with a computation budget running in the tens of millions of dollars. It used a variant of the Monte Carlo Tree Search (MCTS) algorithm, known as PUCT. This paper introduces search-contempt, a novel hybrid variant of the MCTS algorithm that fundamentally alters the distribution of positions generated in self-play, preferring more challenging positions. In addition, search-contempt has been shown to give a big boost in strength for engines in Odds Chess (where one side receives an unfavorable position from the start). More significantly, it opens up the possibility of training a self-play based engine, in a much more computationally efficient manner with the number of training games running into hundreds of thousands, costing tens of thousands of dollars (instead of tens of millions of training games costing millions of dollars required by AlphaZero). This means that it may finally be possible to train such a program from zero on a standard consumer GPU even with a very limited compute, cost, or time budget.