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Gamearth by Kevin J.Anderson. It is a trilogy and there is a (brief) Wikipedia article describing the series here.

The scratching off a river hexagon is when they players are trying to remove it from a map:

The hexagon of a lone village was inundated and changed without a trace. Melanie thought of the villagers, the people, their homes and fields.
The River moved on until it emptied into the sea below.
Melanie sat frozen. David turned gray.
Scott jumped up and ran into the kitchen, pulling open drawers until he found the silverware. He came back to the map with a butterknife. He chipped away at one section of the new blue color.

As mentioned in a comment the cover is blue and does show a balloon basket:

You mentioned in a comment that you have been Googling game end, and indeed the last book of the trilogy is called Game's End.

Gamearth by Kevin J.Anderson. It is a trilogy and there is a (brief) Wikipedia article describing the series here.

The scratching off a river hexagon is when they players are trying to remove it from a map:

The hexagon of a lone village was inundated and changed without a trace. Melanie thought of the villagers, the people, their homes and fields.
The River moved on until it emptied into the sea below.
Melanie sat frozen. David turned gray.
Scott jumped up and ran into the kitchen, pulling open drawers until he found the silverware. He came back to the map with a butterknife. He chipped away at one section of the new blue color.

As mentioned in a comment the cover is blue and does show a balloon basket:

You mentioned in a comment that you have been Googling game end, and indeed the last book of the trilogy is called Game's End.

Gamearth by Kevin J.Anderson. It is a trilogy and there is a (brief) Wikipedia article describing the series here.

The scratching off a river hexagon is when they players are trying to remove it from a map:

The hexagon of a lone village was inundated and changed without a trace. Melanie thought of the villagers, the people, their homes and fields.
The River moved on until it emptied into the sea below.
Melanie sat frozen. David turned gray.
Scott jumped up and ran into the kitchen, pulling open drawers until he found the silverware. He came back to the map with a butterknife. He chipped away at one section of the new blue color.

As mentioned in a comment the cover is blue and does show a balloon basket:

Gamearth by Kevin J.Anderson. It is a trilogy and there is a (brief) Wikipedia article describing the series here.

The scratching off a river hexagon is when they players are trying to remove it from a map:

The hexagon of a lone village was inundated and changed without a trace. Melanie thought of the villagers, the people, their homes and fields.
The River moved on until it emptied into the sea below.
Melanie sat frozen. David turned gray.
Scott jumped up and ran into the kitchen, pulling open drawers until he found the silverware. He came back to the map with a butterknife. He chipped away at one section of the new blue color.

As mentioned in a comment the cover is blue and does show a balloon basket:

You mentioned in a comment that you have been Googling game end, and indeed the last book of the trilogy is called Game's End.

Gamearth by Kevin J.Anderson. It is a trilogy and there is a (brief) Wikipedia article describing the series here.

The scratching off a river hexagon is when they players are trying to remove it from a map:

The hexagon of a lone village was inundated and changed without a trace. Melanie thought of the villagers, the people, their homes and fields.
The River moved on until it emptied into the sea below.
Melanie sat frozen. David turned gray.
Scott jumped up and ran into the kitchen, pulling open drawers until he found the silverware. He came back to the map with a butterknife. He chipped away at one section of the new blue color.

Gamearth by Kevin J.Anderson. It is a trilogy and there is a (brief) Wikipedia article describing the series here.

The scratching off a river hexagon is when they players are trying to remove it from a map:

The hexagon of a lone village was inundated and changed without a trace. Melanie thought of the villagers, the people, their homes and fields.
The River moved on until it emptied into the sea below.
Melanie sat frozen. David turned gray.
Scott jumped up and ran into the kitchen, pulling open drawers until he found the silverware. He came back to the map with a butterknife. He chipped away at one section of the new blue color.

As mentioned in a comment the cover is blue and does show a balloon basket: