It's because the numbers R2 and B2 are fixed. If I have unlimited tokens of either color, then the result might be different. But... imagine you only have blue cards left, by some fluke of chance. By this time, I have a certain number of blue tokens and red tokens left. No matter which order I play them the result will equal the number of blue tokens I have left.
Similarly, if you only have red cards left, no matter what order I play my remaining red tokens, the score will equal the number of red tokens I have left at this point.
The issue is that, I cannot predict in what order you will play your cards, therefore by the time I know that I should have more red tokens ideally, or more blue tokens ideally, it's too late.
Let's do reductio ad absurdum, and let's say there is just 1 token of each color, and 1 card of each color.
If I know you will play red then blue, then I should play red then blue.
But, at the time I play, e.g., red, I don't know whether you will play red or blue. Perhaps you play red, perhaps you play blue. By the time it gets to my second token, well, I know what you played, but I can no longer control the fact that my second token is now blue ;-)
Now, going back to my very first sentence: if I had unlimited tokens, then I can look at what you have left, and play the color you have most of. But since my own tokens are fixed, and I have exactly the same number of tokens that you have cards, then the order I play them makes no difference. Summary is:
- from where I am, I can control the order that I play my remaining tokens, but I cannot control the proportion of the color of my remaining tokens. If I had unlimited tokens of either color, I could look at your cards, and see you have more blues, then I could play more blue tokens, and win. But I don't.
Oh, I see, you mean, imagine during the first third of moves, you play only red, so you have more blue than red, but still have some blue left. Then, I could play red for a bit, until such time as you switch to have more blue than red, and then I play blue for a bit.
The problem is, that this means that you have to play more blue in proportion to your remaining red, followed by more red. If I know you're going to do that, then I can play more blue first, and then more red. But since you're playing randomly, for all I know, you're going to play more red first, and then more blue later.
Basically:
- I can see you have proportionally more of one color now, but I can't control what proportion I have now, it's too late
- I can't guess whether you're now going to play first more red, and then more blue; or first more blue and then more red
- if you play evenly, same proportion of blue/red, for the rest of the match, it matters not whether I play my reds first, or my blue first
- if I know you will play blues first, then reds; or reds first then blue; I could change my sequence accordingly
- but without that prior knowledge, changing my own sequence might increase my score, or it might decrease my score, and I won't know which, until I've already played, by which time it's too late.
