Which four-part partition games are isomorphic to Silverman's game?

A four-part partition game on a rectangle is a two-person zero-sum game where the pure strategy sets are intervals and the resulting rectangle is partitioned by three curves into four regions, on each of which the payoff is constant. Such a game is essentially a generalization of Silverman's game, which has been studied extensively by the author and others. It has been shown previously that if two of the partitioning curves of suitable type are given, and the four payoffs are suitably related, there is a unique choice of the third curve for which the game is isomorphic to Silverman's game. In the present note this third curve is expressed explicitly in terms of the first two, provided that one of the two given curves is the middle one of the three.