Afriat and Revealed Preference Theory
Afriat and Revealed Preference Theory
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阿夫里亚特和显性偏好理论
DOI：
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发表时间：
1973
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通讯作者：
W. Diewert
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作者：
W. Diewert
假设我们可以观察到一些决策xi(其中xi是一个非负的N维向量，对于i = 1,2，…， I)，这是某个决策单元所做的决策，我们进一步假设决策xi的每个向量满足线性约束形式pTxi < 1，对于I = 1,2，…，其中是给定的N维向量，有正分量在上述框架下，我们可以提出以下问题：观察到的决策集{xi}是否符合决策者选择决策xi （for i = 1,2，…）的假设？， I)因为它最大化了N个变量的实值函数，，受约束pTx < 1？一个相关的问题是我们如何使用观测到的数据{pi；Xi} I = 1,2，…在假设存在这样一个b的情况下，为了构造决策者真实0的近似值。在下面的第3节中，我们将使用观测数据{pi；Xi}构造线性规划问题[4]的系数。如果该线性规划有正解，则表明观测数据不满足一致性假设。另一方面，如果线性规划的目标函数有一个等于零的解，那么我们可以利用线性规划的解构造一个实值函数0，使得对于i = 1,2，…，其中向量xi是以下约束最大化问题的解：
Suppose that we can observe a number of decisions xi (where xi is a non-negative N dimensional vector for i = 1, 2, ..., I) which some decision-making unit has made and let us further suppose that each vector of decisions xi satisfies a linear constraint of the form pTxi < 1 for i = 1, 2, ..., I where pi is a given N dimensional vector which has positive components.3 Given the above framework, we may ask the following question: is the observed set of decisions {xi} consistent with the hypothesis that the decision-maker chose decision xi (for i = 1, 2, ..., I) because it maximized a real valued function of N variables, ., subject to the constraint pTx < 1? A related question is how may we use the observed data {pi; xi} i = 1, 2, ..., I in order to construct an approximation to the decision-maker's true 0, assuming that such a b exists. In Section 3 below, we will give an answer to the above two questions by using the observed data {pi; xi} to construct the coefficients of a linear programming problem [4]. If this linear programme has a positive solution, then it turns out that the observed data does not satisfy the hypothesis of consistency. If on the other hand, the objective function of the linear programme has a solution equal to zero, then we may use the solution to the linear programme to construct a real valued function 0 such that for i = 1, 2, ..., I the vector xi is a solution to the following constrained maximization problem: