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badger comments on Even Odds - Less Wrong
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A simpler generalization:
Assume there are n people. Let S_i be person i's score for the event in question from your favorite proper scoring rule. Then let the total payment to person i be
(i.e. the person's score minus the average score of everyone else). If T_i is negative, that's a payment that person has to make.
This scheme is always strategyproof and budget-balanced. If the scoring rule is bounded, then the payment is bounded. If the Bregman divergence associated with the scoring rule is symmetric (like it is with the quadratic scoring rule), then each person expects the same payment (fair by your definition).
Or in the language this comment uses, each player divides the square of their surprise equally among all the other players.