Choice Probabilities
Assume the distribution of unknown part is distributed extreme value (Gumbel and type I extreme value)
the variance of this distribution is Π^2/6 (can be normalized)
The difference between two extreme value is distributed logistic:
using this assumption is nearly same as assuming the errors are independently normal, which means that the unobserved portion of utility for one alternative is unrelated to another unobserved portion.
Logit choice probabilities:
The probability that choose alternative i is:
The individual cumulative distributions are：
integral the conditional distributions:
when the utility function is specified as linear in parameters, the logit probabilities become:
Properties:
- The probability p is necessarily between 0 and 1
- The choice probabilities for all alternatives sum to one
- The relation of the logit probability to representative utility is sigmoid, or S-shaped:
(The point at which the increase in representative utility has the greatest effect on the probability of its being chosen is when the probability is close to 0.5)
Power and Limitations of Logit
- Logit can represent systematic taste variation (that is, taste variation that relates to observed characteristics of the decision maker) but not random taste variation (differences in tastes that cannot be linked to observed characteristics)
- implies proportional substitution across alternatives.
- logit can not handle situations where unobserved factors are correlated over time.
Substitution Patterns: 对于任意两个方案，其概率比为：
这意味着，任意两个方案的概率比，与其他方案无关，该特点也叫 independence from irrelevant alternatives, or IIA.
反例： red-bus–blue-bus problem
坐车或者蓝巴士的概率均为1/2,，因此两个方案的概率比为1。 但假设引入了红巴士，且两种巴士的选择概率相同，因此，红、蓝巴士与汽车的概率比均为1，在三个方案的背景下，选择汽车的概率为1/3. 但在现实中，引入巴士往往并不改变汽车的概率，因此仍应该为1/2
Proportional Substitution: 改变方案j的特点对其他方案的影响
可推出其方案i的概率pi的变动为：
显然，方案j的变动对其他方案的影响是成比例的（proportionate shifting）。
反例：对电动汽车的补贴影响更有可能影响小型汽车，而不是大型汽车。
Consumer surplus
3.5, 感觉可以应用的时候再看
Derivatives and Elasticities
当可观察的特点z变动时，对应的概率变动为：
而当效用函数为线性时，可以变为
而一个方案的概率相对于其他方案特点变化下的变动比例为：
从实际的家都来看，一种方案概率的增加，势必意味着另一个方案的改率下降。而对应的弹性为：
Counterfactual Simulations
