What game does John Buckley mention in S06E09?

According to this transcript, the game is called Jeu Militaire, also known as the French Military Game thought it has nothing to do with France or the military. It is a strategy game that works like this:

The game is played against the computer with the two players alternating turns. You control the three black pieces while the computer controls the remaining red piece. You always go first.

The pieces move along the lines to the adjacent spots. The red computer piece can move in any direction along those lines, while your black pieces cannot go in a backward (downward) direction.

To move, drag a black piece to valid adjacent location. You win the game by trapping the computer where it cannot move within 20 turns. As an example, if the black pieces were at the lower left, middle, and upper left spots and the red piece was trapped at the middle left, you would win.

The computer wins by reaching the very bottom, where it cannot be caught.

There appears to be a downloadable freeware version called Hare - it requires additional downloads, so I have not tested it.

I found a framed game with Union Generals in each corner called Militaire. It has 6 circles of peg-holes and Ironclad ships battling in the center. Manufactured by Newland, Detroit. N.D.