[gaming_mouse]

You really need more info than this. That is, the answer to your question depends on the player's variance, and this in turn depends on the how many firsts, seconds, or thirds he gets.

For example, a player who always tries for first, and takes risks when he might instead be able to safely assure a third place (but thereby lessen his chance of a first place) will have much bigger swings.

You really need to assume that the player comes in 3rd x% of the time, 2nd y%, and 3rd z%.