You might want to ask them why they need the affidavit of eligibility. If you agreed to provide it when you initially signed up for their drawings and awards they might be able to make some type of claim against you if you do not provide in info in return for their payments. It is likely they want to use the info for marketing purposes only, but I would not totally blow them off without at least calling and speaking to someone.
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Requesting a social security number from you allows them to issue a Form 1099 to you that reports the income. Copies of the 1099 are then sent to the IRS and FTB. I see no harm in providing PCH with your SSN. As for the advertisements, I agree that you can decline. Finally, even if no 1099 is issued to you, you must report the $750 of income when you file your tax returns.
I agree with Mr. Wagner that you can safely blow off PCH, since you will be reporting the $750 of prizes as income on your tax return. The fact PCH failed to obtain your SSN before awarding the prizes to you is their problem, not yours. Also, I view the chance of PCH filing an action against you due to your failure to provide your SSN as being slim to none, with none leading by two lengths.
The answer to this question does not establish an attorney-client relationship. Moreover, this attorney is licensed to practiced law ONLY in the State of California. Answers to questions from users in other jurisdictions or states are meant to provide only general information. Users should contact a local attorney in their jurisdiction or state to address their specific tax issue.