Pete Rose signed 1998 WWF Ring ChairIt's going to be a little difficult to help you with an accurate value. The value of an autograph will vary based on the item signed. A
Baseball is the most common and desired item to sign because of it's connection to the game, and it displays the signature well. Even though Pete Rose got involved with the WWF, the Ringside chair has no connection to baseball and might lower the number of collectors interested to Pete Rose/
Wrestling fans.
Value is based on past auctions and sales. Without finding any Pete Rose memorabilia connected with the WWF I will have to estimate the value based the cost of a WWF Ringside Chair, and the cost of having Pete Rose sign it. I found a few auctions for 1998 WWF Wrestlemania Red Attitude, and
Survivor Series ring Chairs that sold at an average price of $150. Dealers get about $200.-$250. but you are paying a dealers premium. (usually about 40 percent over market value)
A Pete Rose autograph is very common as he is often found somewhere in Vegas sitting on a folding chair signing. He gets about $30. or in some cases you buy something in a memorabilia shop and he signs it. You could find different signed items on eBay from baseballs and bats, to Photos and equipment selling in the $20.-$75.+ price range.
In most cases the cost of having a player sign an item that you bought, and have it authenticated on the spot, usually has a lower resale value at that time, than the money you invested in it.
I would estimate the value of a Pete Rose signed WWF Ringdside Chair at about $150.-$200. Value will vary based on the grade of the signature, how well the signature stands out on the chair (color contrast) and how well the chair displays the signature. Signatures that are not properly authenticated could sell at half the market value or less.