However, here's a story that may help to reduce rivalry between little kids.
When my daughter was little I wanted to be a stay-at-home mom, but needed to earn some money, so I ran a family daycare home. Normally the children I taught were from almost-three to Kindergarten age, but once a neighbor had an emergency and I agreed to care for her infant for a few weeks. None of the children had ever acted jealous of each other before, but they all seemed jealous of the baby and I had no idea why.
Finally an articulate little boy told me it was because I loved the baby more than I loved them. When I asked what gave him that idea he said it was because I was always hugging the baby. I explained that holding the baby wasn't the same as hugging and I had to hold the baby a lot because it couldn't eat or move from one place to another by itself. Immediately all signs of jealousy from any of the children disappeared.
Perhaps if parents explain that to the big brothers and sisters as soon as a new baby joins the family it will help reduce the sibling rivalry.