本文是Game Theory An Introduction (by Steven Tadelis) 的学习笔记。
术语
概率分布函数(probability distribution function)
一个简单投机(lottery)(行动a \in A)在结果 X = { x_1, x_2, \cdots, x_n }上的概率分布记做
p = (p(x_1|a), p(x_2|a), \cdots, p(x_n|a)), \\ where \\ p(x_k|a) \geq 0 \text{: the probability that } x_k \text{ occurs when take action a} \\ \sum_{k=1}^n p(x_k|a) = 1
累积分布函数(cumulative distribution function)
一个简单投机(lottery)行动a \in A,在结果区间X = [\underline{x}, \overline{x}]上的累积分布函数:
F : X \to [0, 1] \\ where \\ f(\hat{x} | a) = Pr{x \leq \hat{x}} \text{: the probability that the outcome is less than or equal to } \hat{x}.
期望收益(expected payoff from the lottery function)
一个简单投机(lottery)行动a \in A,在结果区间X = [x_1, x_2, \cdots, x_n]上的期望收益函数:
E[u(x)|a] = \sum_{k=1}^n p_k u(x_k) \\ where \\ u(x) \text{: the payoff function} \\ p = (p_1, p_2, \cdots, p_n) \text{: probability distribution}
连续案例:期望收益(expected payoff from the lottery function)
一个简单投机(lottery)行动a \in A,在结果区间X = [\underline{x}, \overline{x}]上的期望收益函数:
E[u(x) | a] = \int_{\underline{x}}^{\overline{x}} u(x)f(x)dx \\ where \\ u(x) \text{: the payoff function} \\ f(x|a) \text{: the cumulative distribution function}
经济人2
我们称一个人是理性的,如果这个人选择最大期望收益。
\text{choose } a^* \in A \iff v(a^*) = E[u(x)|a^*] \geq E[u(x)|a^*] = v(a), a \in A