May 19, 2003

At the start of every match, the winner of the coin flip can decide whether to serve or receive. Does it matter which you choose? I am not referring to a psychological advantage. To see what I mean consider chess. There is a significant advantage to having white in chess. Even if you prefer defense to offense, you should take white. So, what about table tennis? Is there an actual advantage to serving first?

Let us be explicit about our modeling assumptions. Assume that the probability of winning a point only depends on which player is serving, and in particular, is independent of the score. Also, assume that you are more likely to win a point when you serve than when you receive.

If the game goes deuce, then it does not matter who served first since no matter who wins, each player will have served the same number of points.

What if the game does not go deuce? Consider the following modification of the rules: Rather than stopping when one player reaches 11 points, keep playing until 20 points have been played. If you win the game under the modified rules, then you must win at least 11 of the 20 points, and hence would have won the game under the standard rules. Similarly, if you lose under the modified rules, you also would have lost under the standard rules. But, under the modified rules, both players serve 10 points, and so it does not matter which one served first. Thus the answer to our question is that it does not matter who serves first.

What about handicap matches? Traditionally a handicap match is played as one game to 51. In order to analyze this, modify the rules so that we play a total of 100 points (unless we go deuce). Serve changes when the sum of the scores is a multiple of two, just as in non-handicap games. Let A be the player who serves first, and let B be the player who serves second.

Suppose the handicap is one point. Player A serves one point, B serves two points, and the rest of the game continues normally with each player serving two points at a time. Hence A will serve 49 points, and B will serve 50 points. Therefore you should choose to serve second. If the handicap is two points, then player A will serve 50 points, B will serve 48, and you should serve first. If the handicap is three points, then player A will serve 49 points, B will serve 48, and you should serve first. If the handicap is four points, then both players will serve 48 points, and it does not matter who serves first.

In general, divide the handicap by four, and consider the remainder. If the remainder is zero, then it does not matter who serves first. If the remainder is one, then you should serve second. If the remainder is two or three, then you should serve first.

What about choosing an end in a non-handicap match? Suppose one end is better, e.g., the lighting is better. The the change of ends in the last game occurs as soon as one player reaches five points. However, the midway point of a (non-deuce) game is (as we saw above) when ten points have been played. Therefore, the rules have change of ends occurring too soon. Therefore, you should choose the worse side to start the match, so you’ll have the better side for more than half of the last game. Plus, if the game goes deuce, then you will have the better side for all the deuce points.

The same reasoning applies to choosing whether you or your partner should receive serve in the last game of a doubles match. You should start the game with the serving order that favors your opponents so you’ll have the favorable order after the switch at five.

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