Why would I take the annuity? Because of a specific purchase I would make with my winnings.
I would buy an NFL team. Serious, it's what I want to do. The reason I would take the annuity is because it's a higher locked in number. I could use the annuity payments as collateral for funding a loan that will pay for my team. My $1 billion NFL team probably couldn't be purchased with a windfall of $500 million. But if I had a guaranteed stream of income, I could pay for the interest payments with the annual annuity and pay off the loan with profits from my franchise.