I've been thinking about a probability problem that has been bugging me; I would appreciate any insight.

Problem: Let's say that me and my 9 colleagues each come up with an independent marketing strategy for a product; and let's assume that we are all equally skillful at coming up with a good strategy. What is the probability that I come up with the best strategy? I have 2 "solutions".

1) Comparing all strategies all at once: Since there are 10 possible strategies, I have a 1/10 chance of having the best strategy, or 10%.

2) Comparing strategies one at a time: I have a 50% chance of having a better strategy than colleague A. I have a 25% (50% x 50%) chance of having a better strategy than colleagues A and B. I have a 12.5% (50% x 50% x 50%) chance of having a better strategy than colleagues A, B, and C, so on until I finally conclude that I have a ~2% (0.5^9) chance of having a better strategy than my 9 colleagues A-I.

I'm very sure that solution 1 is the correct solution, but I am having trouble explaining the logical fallacy of solution 2. Could you guys and gals please help provide some statistical explanations? Thank you in advance.