The cost of things when you get points

Amazon.com gives you 1% points on purchases. Yodobashi.com gives you 10%. Rakuten gives you some random number based on a variety of factors because Rakuten is intentionally a rube goldberg machine. I try not to buy from Amazon, so I often end up comparing Yodobashi or Rakuten to them but it's easy to get the comparison wrong. The reason is that when you pay using points, you don't get points for that.

Let's say I spend 1000¥ on Yodobashi. I get 100 points. So you might thing you got a 10% discount but that's not correct. I can now buy a 100Y item for free but I get no more points. So my 1000¥ got me 1100Y worth of sticker-price goods. That's actually a 9.090909...% discount (1-1/1.1). Similarly on Amazon, I'm not getting a 1% discount, I'm getting a 0.9900990099...% discount.

I started looking at this because I was looking at an item for 1980¥ on yodobashi and the same item was 1800Y¥ on Amazon. Naively if you apply 10% and 1% discounts, the item seems to be be excly 1782¥ on both sites! Reality is that the true cost is 1800¥ (1980¥/1.1)on Yodobashi and 1782.17821782...¥ (1800¥/1.01) on Amazon.

I'll still buy it on Yodobashi because Amazon treat their workers horribly.