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Bargaining and Markets phần 3

Giả định tiếp theo rất đơn giản hoá cấu trúc ưu đãi. Nó đòi hỏi rằng các ưu đãi giữa chỉ phụ thuộc vào Nếu các lệnh tôi đáp ứng A5 A2 qua A4 sau đó có một Ui chức năng tiện ích đại diện cho sở thích I trong hơn có một dạng cụ thể: mỗi delta ∈ (0, 1) | 34 Chapter 3. The Strategic Approach The next assumption greatly simplifies the structure of preferences. It requires that the preference between x t and y s depend only on x y and the difference s t. Thus for example it implies that if x 1 j y 2 then x 4 ị y 5 . A5 Stationarity For any t e T x e X and y e X we have x t i y t 1 if and only if x 0 i y 1 . If the ordering j satisfies A5 in addition to A2 through A4 then there is a utility function Uj representing i s preferences over X X T that has a specific form for every Ỗ e 0 1 there is a continuous increasing function ui 0 1 R such that Ui xi t ỗtui xi . See Fishburn and Rubinstein 1982 Theorem 2 .4 Note that for every value of Ỗ we can find a suitable function ui the value of Ỗ is not determined by the preferences. Note also that the function ui is not necessarily concave. To facilitate the subsequent analysis it is convenient to introduce some additional notation. For any outcome x t it follows from A2 through A4 that either there is a unique y e X such that Player i is indifferent between x t and y 0 in which case A3 implies that if xi 0 and t 1 then yi xi or every outcome y 0 including that in which yi 0 is preferred by i to x t . Define vi 0 1 X T 0 1 for i 1 2 as follows . xt y if y 0 i x t - 31 Vj xj t 0 if y 0 i x t for all y e X. 1 The analysis may be simplified by making the more restrictive assumption that for all x t and for i 1 2 there exists y such that y 0 j x t . This restriction rules out some interesting cases and therefore we do not impose it. However to make a first reading of the text easier we suggest that you adopt this assumption. It follows from 3.1 that if vi xi t 0 then Player i is indifferent between receiving vi xi t in period 0 and xi in period t. We slightly abuse the terminology and refer to vi xi t as the present value of x t for Player i even when vi xi t 0. Note that y 0 hi x t whenever yi Vj xj t 3.2 and y t -i x s whenever vi yi t vi xi s . If the preference ordering ffi .