Earlier this week, I offered a rough outline of the economics of fixing tennis matches, calculating the expected prize money that players forgo at various levels when they lose on purpose. The vast gulf between prize money, especially at lower-level events, and fixing fees suggests that gamblers must pay high premiums to convince players to do something ethically repugnant and fraught with risk.

So much for match-level fixes. What about single service games? In Ben Rothenberg’s recent report, a shadowy insider offers the following data points:

Buying a service break at a Futures event cost $300 to $500, he said. A set was $1,000 to $2,000, and a match was $2,000 to $3,000.

In other words, a service break is valued at between 10% and 25% the cost of an entire match. The article doesn’t mention service-break prices at higher levels, so we’ll have to use the Futures numbers as our reference point.

Selling a service break might be a way to have your cake and eat it too, taking some cash from gamblers while retaining the chance to advance in the draw and earn ranking points. But it won’t always work out that way.

I ran some simulations to see how much a service break *should* cost, based on the simplifying assumption that prices correspond to chances of winning and, by extension, forgone prize money. It turns out that the range of 10% to 25% is exactly right.

Let’s start with the simplest scenario: Two equal men with middle-of-the-road serves, which win them 63% of service points. In an honest match, these two would each have a 50% chance of winning. If one of them guarantees a break in his second service game, he is effectively lowering his chances of winning the match to 38.5%. dropping his expected prize money for the tournament by 23%.

If our players have weaker serves, for instance each winning 55% of service points, the fixer’s chances of winning the match fall to about 42%, only a 16% haircut. With stronger serves, using the extreme case of 70% of points going the way of the server, the fixer’s chances drop to 34%, a loss of 32% in his expected prize money.

This last scenario–two equal players with big serves–is the one that confers the most value on a single service break. We can use that 32% sacrifice as an upper bound for the worth of a single fixed break.

Fixed contests have more value to gamblers when the better player is guaranteed to lose, and in those cases, a service break doesn’t have as much impact on the outcome of the match. If the fixer is considerably better than his opponent, he was probably going to break serve a few times more than his opponent would, so losing a single game is less likely to determine the outcome of the match.

Let’s take a few examples:

- If one player wins 64% of service points and other wins 62%, the favorite has a 60% chance of winning. If he fixes one service break, his chances of winning fall to just below 48%, about a 20% drop in expected prize money.
- When one player wins 65% of service points against an opponent winning 61%, his chances in an honest match are 69.3%. Giving up one fixed service break, his odds fall to 57.4%, a sacrifice of roughly 17%.
- A 67% server facing a 60% server has an 80.8% chance of winning. With one fixed service break, that drops to 70.7%, a loss of 12.5%.
- A huge favorite winning 68% of service points against his opponent’s 58% has an 89.5% chance of advancing to the next round. Guarantee a break in one of his service games, and his odds drop to 82%, a loss of 8.4%.

With the exception of very lopsided matches (for which there might not be as many betting markets), we have our lower bound, not far below 10%.

The average Futures first-rounder, if we can generalize from such a mixed bag of matches, is somewhere in the middle of those examples–not an even contest, but without a heavy favorite. So the typical value of a fixed service break is between about 12% and 20% of the value of the match, right in the middle of the range of estimates given by Rothenberg’s source.

Even in this hidden, illegal marketplace, the numbers we’ve seen so far suggest that both gamblers and players act reasonably rationally. Amid a sea of bad news, that’s a good sign for tennis’s governing bodies: It promises that players will respond in a predictable manner to changing incentives. Unfortunately, it remains to be seen whether the incentives will change.