Types of prediction markets and appropriate LMSR\nPublic icon\nTruthcoin_test/LMSR_function_demo\n\nCategorical markets\n\s> If the content to be bet on is binary, that is in the case of binary decision, use the following cost function:\n\s> \n\s> C(q0,q1)=b∗ln(eq0/b+eq1/b)\n\s> \n\s> where\n\s> \n\s> b ... liquidity parameter specified by the author. Higher values ensure liquidity at the start of the market (avoids no one making bet) but the cost for the author is larger when creating the market\n\s> e ... the base of the natural logarithm\n\s> For example, if you want to buy 13 units of outcome 0, the cost is C(q0+13,q1) and if you want to sell 150 units of outcome 1, the cost is C(q0,q1−150).\n\s> \n\s> Even if you want to vote "looking only" at a specific dimension in a multi-dimensional decision, there is an advantage in using log for sufficient accuracy and incentive compatibility.\n\nCategorical markets\n\s>If the content to be bet on is categorical (i.e. more than three choices)\n\s> C(q)=blog\n\s> ∑\n\s> o∈O\n\s> e\n\s> q\n\s> 0\n\s> /b\n\s> C(q)=blog∑o∈Oeq0/b\n\s> \n\s> where,\n\s> O\n\s> is the set of all possible outcomes, and\n\s> o\n\s> is the item to be bought.\n\s> \n\s> The cost of buying q0 outcomes when r shares are purchased is\n\s> C(q+r)−C(q)\n\s> When r is small enough, the derivative of the cost function becomes the actual cost.\n\s> \n\s> As you can see, the cost function depends only on the item being purchased and does not depend on the state of the entire market, making it easy to calculate even in complex markets or with low cost of cognitive effort when purchasing.\n\s> \n\s> There may be numerical computation difficulties because the exponential part before taking the logarithm can become very large.\n\nScalar markets\n\s> Price function\n\s> Let the price of the outcome o's security at a specific time be po (0≤po≤1), which is a function of qo, and the calculation formula is called a price function. In the case of LMSR, the price function is as follows:\n\s> \n\s> po(q)=∂C(q)∂qo=eqo/b∑o′∈Oeq′o/b\n\s> \n\s> LMSR is fully differentiable, making it easy to handle in price calculations, and the sum of qo is 1 (dollar) so it can be treated as a probability distribution and used as "current market beliefs about the probability of the event o occurring".\n\s> When using LMSR, the upper limit of the amount that market makers can lose is blog|O|.
An actual example of its usage is CME Group, one of the world's leading derivative providers.