Why does 甲斐 use these particular kanji?

Question

Inspired by a recent question about the proper way to write 生きがい, I wanted to ask a long-standing question.

According to Kotobank, 甲斐 has the following definitions (my poor translations follow):

かい

1. An old Japanese state, lying within modern Yamanashi.
2. A city in western Yamanashi.

がい

1. When attached to the ren'youkei, expresses the effect, value or worth of the action.
2. When attached to a noun, expresses that something commands a quality appropriate to that thing.
3. When attached to a phrase that expresses a negation, expresses the extent to which it is fortunate that it is not so.
4. When attached to a phrase that expresses a wish or desire, expresses the desire to do it as much as one pleases.

I do see the semantic relationship between the four senses of がい — all relate to a sense of worthiness or appropriateness — but why were these kanji, which relate to a place name of all things, applied to this concept?

Answer 1

In the first place, -がい is the rendaku form of an independent noun かい, which is defined under デジタル大辞泉 section of the Kotobank page you cited (I don't understand why 大辞林 第三版 section is missing this definition for かい).

This かい (from archaic かひ) is, according to 小学館's 日本語源大辞典 which I own, a nominalization of archaic verb かふ "to substitute, compensate" (corresponds to modern [代]{か}える). This usage is found as early as in [万葉集]{まんようしゅう}, the oldest poem anthology from the 8th century.

On the other hand, the geographical かい is, the same dictionary says, supposedly derived from archaic かひ "a canyon, gorge". This word is found in 万葉集 too.

So, etymologically speaking, these two words have nothing to do with each other at all. This is a plain ateji case, though I don't know which definition first used the kanji 甲斐.