For example, take a binary outcome; A coin toss, heads or tails.
If heads, then those that picked heads receive \$1 and tails receive \$0. To quote the prices for each bet Hanson's LMSR uses logarithms.
Isn't it simple enough to quote the price using a linear model?
For example, in a linear model, the cost for placing a bet on heads is:
x * (q_heads + x) / (q_heads + x + q_tails)
x is the amount of bets to place,
q_heads is the amount of existing bets placed on heads and
q_tails is the amount of existing bets placed on tails.
Computationaly this is far more efficient. What are the disadvatages to using a linear market scoring algorithm vs a logarithmic scoring algo?