Economists at Harvard, Toronto and Yale business schools recently did a study that examined this question. They examined the hiring of low-level service workers at 15 firms. They focused on what happens when an employee is hired based on the algorithm versus what happens when a hiring manager overrules the algorithm and hires based on his or her own judgment.
The results, summarized in Bloomberg Businessweek and available in National Bureau of Economic Research Working Paper No. 21709, are sobering. Job candidates picked by the algorithm stay longer and perform better than those picked by hiring managers. Also, there was a strong correlation between algorithm predictions and actual performance.
One caveat: this study looked at relatively unskilled jobs where performance could be measured objectively. What would happen in more complex jobs such as trial lawyers or university professors? My guess is that we will find out in the not too distant future.