# 3.4 Does It Matter Who Serves First?

## Description

This article is from the Table Tennis (Ping Pong) FAQ, by ttennis@bu.edu with numerous contributions by others.

# 3.4 Does It Matter Who Serves First?

(See p31 of Jan/Feb 91 TTTopics)

At the start of every match, assuming you win the coin flip (or the roll of
the ball), you must decide if you want to serve or to receive. Does it
matter which you choose? Now, I don't mean is there 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. Or consider a game of volleyball. In volleyball your
team only scores points when it is serving. It is intuitively clear that,
given a choice, you should serve first.

So what about table tennis? Is there an actual advantage to serving first?
Before reading further, try to answer this question.

Let's 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. First note that if the game goes
deuce, then it doesn't matter who served first since no matter who wins,
each player will have served the same number of times.

What if the game doesn't go deuce? Consider the following modification of
the rules: Rather than stopping when one player reaches 21, keep playing
until 40 points have been played. If you win the game under the modified
rules, then you must win at least 21 of the 40 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 20 times and so it doesn't
matter which one served first. So the answer to our question is: No, it
doesn't matter who serves first.

How about handicap matches? Traditionally a handicap match is played as one
game to 51. In order to analyze this, modify the rules so we'll play a
total of 100 points (unless we go deuce). Serve changes when the sum of the
scores is a multiple of 5, 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 1 point. Player A serves 4 points and then B serves
5 points, and the rest of the game continues normally with each player
serving 5 points at a time. Hence A will serve a total of 49 points and B
will serve a total of 50. Therefore you should choose to serve second
(unless you are weird and are more likely to win a point when your opponent
serves). Now let's consider a handicap of 5. Then player A will serve 50
points and B will serve 45. Therefore you should serve first. If the
handicap is 10, then both players will serve 45 and it doesn't matter who
serves first.

Let's summarize what you should do for handicap games. Only the last digit
matters (so you want to do the same thing for a handicap of 17 as for a
handicap of 7). If the last digit of the handicap is 0, then it doesn't
matter who serves first. If the last digit of the handicap is 1, 2, 3, or
4, then you want to serve second. If the last digit of the handicap is 5,
6, 7, 8, or 9, then you want to serve first.

We'll leave doubles for a future article or you might try it as a
(difficult) homework problem. It might also be interesting to analyze a 2
out of 3 handicap match where each game is to 21.

Perhaps a few words about psychological advantage is in order. If there is
no real advantage and the players know this, then there shouldn't be any
psychological advantage. However, if you know there is no real advantage,
but your opponent doesn't, then perhaps you can get a psychological
advantage by letting him serve first.

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