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Posted prices versus bargaining in markets_2

Tham khảo tài liệu 'posted prices versus bargaining in markets_2', tài chính - ngân hàng, đầu tư chứng khoán phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 58 Chapter 3. The Strategic Approach in which she makes the first offer and Player 2 obtains the same utility in any subgame in which he makes the first offer. Step 5. If S 1 5 b then M1 m1 1 1 S and M2 m2 1 1 S . Proof. By Step 2 we have 1 M1 Sm2 and by Step 1 we have m2 1 SM1 so that 1 Ml S S2M1 and hence Ml 1 1 S . Hence M1 1 1 S by Step 4. Now by Step 1 we have m2 1 SM1 1 1 S . Hence m2 1 1 S by Step 4. Again using Step 4 we have SM2 S 1 S b and hence by Step 3 we have m1 1 SM2 1 S 1 Sm1 . Thus m1 1 1 S . Hence m1 1 1 S by Step 4. Finally by Step 3 we have M2 1 Sm1 1 1 S so that M2 1 1 S by Step 4. Step 6. If b S 1 S then m1 1 b M1 and m2 1 S 1 b M2. Proof. These inequalities follow from the SPE described in the proposition as in Step 4 . Step 7. If b S 1 S then M1 m1 1 b and M2 m2 1 S 1 b . Proof. By Step 2 we have M1 1 b so that M1 1 b by Step 6. By Step 1 we have m2 1 SM1 1 S 1 b so that m2 1 S 1 b by Step 6. Now we show that SM2 b. If SM2 b then by Step 3 we have M2 1 Sm1 1 S 1 SM2 so that M2 1 1 S . Hence b SM2 S 1 S contradicting our assumption that b S 1 S . Given that SM2 b we have m1 1 b by Step 3 so that m1 1 b by Step 6. Further M2 1 Sm1 1 S 1 b by Step 3 so that M2 1 S 1 b by Step 6. Thus in each case the SPE outcome is unique. The argument that the SPE strategies are unique if b S 1 S is the same as in the proof of Theorem 3.4. If b S 1 S then there is more than one SPE in some SPEs Player 2 opts out when facing an offer that gives him less than b while in others he continues bargaining in this case. 3.12.2 A Model in Which Player 2 Can Opt Out Only After Player 1 Rejects an Offer Here we study another modification of the bargaining game of alternating offers. In contrast to the previous section we assume that Player 2 may opt 3.12 Models in Which Players Have Outside Options 59 Figure 3.6 The first two periods of a bargaining game in which Player 2 can opt out only after Player 1 rejects an offer. The branch labelled x0 represents a typical offer