As a result of this pity for the first player, a whole bunch of special rules (swaps at the end of the game, extra gifts added to the pile at the start or end of the game) have been suggested. This gets too complicated, involved and has its own set of problems, misunderstandings, hard feelings and surprises.
The first person has the pick of the litter of all the gifts. No other player has this advantage. In all the times we have played Dirty Santa at our annual holiday party we have never had the first player feel cheated. Any player could select a dud gift whether they are the first or last player. This is the reason the gifts should be well thought out. They need to have broad appeal to most people and be something worth stealing.