A half-point ko is the common term for a ko in which nothing is at stake beyond the fate of the stone inside the ko. Despite the name, the miai value of connecting or capturing in a half-point ko is only 1/3 point. As such, a half-point ko is also known as a 1/3-point ko. It is rare to have a play smaller than this but larger than a Japanese dame.
The name half-point ko is there because it is the translation for both the Japanese term 半劫 (hankō) and the Chinese term 半劫 (bànjié), both of which literally mean "half ko." But if we do not know the winner of the ko fight, who gets the point at stake is a 50-50 proposition, and thus the term half-point ko makes sense. Charles Matthews proposed minimal ko as an alternative name.
A ko whose miai value is 1/2 point exists. See real half-point ko.
One way to calculate the miai value of this ko is as follows. The difference between winning and losing the ko is 1 point. There are 3 moves between the winning positions, 1 move for each player to win the ko and 1 move to take the ko. So each move is worth 1/3 point, on average.
Another way of calculating the miai value of a minimal ko is through three kos.
Black plays first. takes a ko. connects since it's no use to fight the ko. Since and are miai, there is no use to try to win both kos, so each player connects their own ko. As Black capture one stone, the value of this position is 1 point.
White plays first. connects a ko, and captures. Then and each connects their own ko. Again, the value of this position is 1 point.
The value of this position is 1 point. Dividing the value over the three kos, the miai value of each ko is 1/3 point.
Often go games end with a fight over a half-point ko. These fights are not shown in professional game records, as a rule. Instead the winner of the ko is indicated, or can be determined from the final score.