bounding absolute fluctuations of random variable distribution

Question

I have a discrete random variable with values $\underline{y} = (y_1, y_2, \dots , y_d)$ with corresponding probabilities $\underline{x} = (x_1, x_2, \dots , x_d)$ and average $Y = \sum_i x_i y_i$. Some values deviate more than a fixed amount $c$ from the average, i.e. $y_1 - Y > c$ or $y_3 - Y < -c$. I want to build a new distribution (without changing the probabilities $\underline{x}$) such that all $y_i$ are within $\pm c$ of the average $Y$, in the following way; If a value satisfies $|y_i-Y|>c$ I replace it with a value that saturates that bound, i.e. $y_1 \rightarrow Y+c$. However, every time i do this the average changes, and I want it to decrease every time I saturate a bound. Is there a nice way of doing this? For example, a nicer way of finding which variables to bound without simply trying each one and recalculating the average and seeing if all other variables now obey the $c$-bound?