The mathematical relationship between betting revenue (i.e., handle) and takeout rates on wagers is not publicly known. It is possible, of course, that racetracks have privately conducted experiments and have data-based evidence of how bettors and handle respond to changes in takeout percentages on various types of bets.
One thing is publicly known: the relationship is nonlinear. A takeout rate is like a tax on betting and eventually the tax becomes so onerous that it curbs betting. At 100% takeout, there would be no betting at all. At the other extreme, a takeout rate of 0% would produce plenty of betting handle but no net pari-mutuel revenue.
Note that net pari-mutuel revenue is what remains of handle once winning bets are paid off. The calculation is simply: betting handle x percentage takeout. The goal of a racetrack is to optimize net pari-mutuel revenue as opposed to maximizing handle.
The well-known Laffer Curve (named after economist Dr. Arthur Laffer) conceptualizes the interaction between government revenue from collecting taxes and tax rates (shown below as rendered by Investopedia). In the Laffer-Curve, substitute net pari-mutuel revenue for tax revenue on the vertical axis and takeout rate for tax rate on the horizontal axis and one sees a plausible hypothesis. (At T* revenue is optimized.) However, instead of the smooth asymmetrical curve shown in the Laffer drawing, the curve could be a trapezoid or some other shape.
Without empirically testing various takeout rates over time and for specific kinds of bets (WPS, exactas, trifectas, etc.) one cannot determine the shape of the curve and the takeout rate that optimizes net revenue. Furthermore, the curve might take different shapes for serious bettors more knowledgeable about takeout rates and less aware casual bettors.
Ignorance will continue to put pari-mutuel wagering at a distinct competitive disadvantage.
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Mathematician Jordan Ellenberg wrote about the Laffer Curve as follows in his 2014 book How Not to Be Wrong, The Power of Mathematical Thinking: “His [Laffer’s] drawing made the fundamental and incontrovertible mathematical point that the relationship between taxation and revenue is necessarily nonlinear.” The same can be said about the relationship between takeout rate and net pari-mutuel handle.