Let’s think of a really simple scoring system, in which the first player to reach some predefined number of points, N, is the winner. Fencing, for example, uses a first-to-15 format based on the number of touches landed on an opponent.
If players are relatively evenly matched, there’s a nice mathematical formula that tells you how likely the better player is to win the match as the number of points increases. The formula in the image above shows how the probability of a better player can be calculated based on their advantage (a).
Perhaps, unsurprisingly this tells us that the probability of victory increases with their advantage. But it also tells us that if a player has only a slight skill advantage over an opponent, their probability of winning will increase with the number of points played (N). In fact their probability of triumph increases as the square root of the number of points needed for victory. You can see this in the second formula in the image above and illustrated in the graph below.
Keep them hooked
To make sure the winner gets a sufficient advantage, and that the match is long enough to be worth the spectator’s while, we need to make sure a sizeable number of points is required for victory. But we also don’t want players to build up unassailable runaway leads, so matches rarely comprise just one first-to-N points competition. Instead, in sports like table tennis, squash and badminton, they’re broken down into games and the first player to win a fixed number of games wins.
For a competition that is the first to a certain number of games (M) with the first to N points winning a game, we can again work out the probability of a player with the same slight advantage, a, going on to win the whole match, as shown in the third formula in the image above.
