Which game should you play?

I give you a choice of two games to play.

In Game #1, I will keep a box in front of you that has 10 red and 10 blue balls. You will have to choose a color and then pick a ball randomly from the box. If the color you chose matches with the color of the ball you pick, I will give you $100; otherwise you will get nothing.

In Game #2, I will adversarially construct a box that will have some number of red and some number of blue balls. The rest of the game is the same. If, for example, I have a hunch that you are going to choose red as your color, I will make sure that the box contains no red balls so that you lose the game.

Which of the games should you pick? Clearly, since I am writing a blog post about this, the answer cannot be Game #1. So the answer is Game #2. But why?

Note that if you pick Game #2 and then choose your color uniformly at random from the set {red, blue}, this game becomes exactly like Game #1, no matter what the ratio of red and blue balls I adversarially choose to put in the box. Thus you can always achieve an expected profit of $50 by using this randomized strategy. In addition, if you know something about me, you can use that information to only enhance the performance. This means you should always pick Game #2.

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