The log market scoring rule (LMSR) depends on there being an order to the stated probabilities, so the payoffs would be different for the order NS, SD, AK than for the order AK, SD, NS.

Given a particular order, the payoff for the i-th probability submitted is log(p_i^k) - log(p_{i-1}^k) if event k occurs. For example, if the order is NS, SD, AK and the system does work, AK's payoff is log(.35) - log(.75). If the system doesn't work, AK's payoff is log(.65) - log(.25).

I haven't seen this written about anywhere, but if you just have probabilities submitted simultaneously and you don't want to fix an order, one way to score them would be log(p_i^k) - \frac{1}{n} \sum_{j \ne i} log(p_j^k) (the log of the probability person i gives to event k minus the average of the probabilities everyone else gave, including the house, assuming there are n participants plus the house). This is just averaging over the payoffs of every possibly ordering of submission. So, for these probabilities, AK's score if the system worked would be log(.35) - (log(.75) + log(.5) + log(.5))/3.