It was a showdown in New Mexico earlier this year, as two billionaires, both from Fort Worth, Texas, battled it out to buy two enormous cattle ranches.

When the dust cleared, oil, gas and real-estate investor Bobby Patton, co-owner of the Los Angeles Dodgers, had won the 174,000-acre York Ranch, spending more than the $10.9 million asking price and outmaneuvering D.R. Horton, one of the country’s largest home-building companies. In a separate deal, D.R. Horton, founded by Donald Horton, outbid Mr. Patton for the nearby 292,779-acre...