Proposition: Deduplication changes the reference task

If the three distinct functions $A,B,D$ are instead sampled with equal marginal probabilities, while remaining hidden and independent of the points, the population optimum is

Thus removing the duplicated ID and balancing the remaining functions is a protocol change.

Proof. In each same-sign quadrant, the conditional optimum is the average of $1$ and $2/3$, namely $5/6$. In each opposite-sign quadrant it is the average of $2/3$ and $1/3$, namely $1/2$. Average the four quadrant values.