không có một tăng ui chức năng liên tục: như vậy mà ui (xi). (Xem Fishburn và Rubinstein (1982, Định lý 2) 0,4) Lưu ý rằng đối với mỗi giá trị của δ chúng ta có thể tìm thấy một ui chức năng phù hợp, giá trị của δ không được xác định bởi các ưu đãi. | 56 Chapter 3. The Strategic Approach Figure The first two periods of a bargaining game in which Player 2 can opt out only when responding to an offer. The branch labelled x0 represents a typical offer of Player 1 out of the continuum available in period 0 similarly the branch labeled x1 is a typical offer of Player 2 in period 1. In period 0 Player 2 can reject the offer and opt out Q reject the offer and continue bargaining N or accept the offer Y . Proposition Consider the bargaining game described above in which Player 2 can opt out only when responding to an offer as in Figure . Assume that the players have time preferences with the same constant discount factor Ỗ 1 and that their payoffs in the event that Player 2 opts out in period t are 0 Ỗtb where b 1. 1. If b Ỗ 1 Ỗ then the game has a unique subgame perfect equilibrium which coincides with the subgame perfect equilibrium of the game in which Player 2 has no outside option. That is Player 1 always proposes the agreement 1 1 Ỗ Ỗ 1 Ỗ and accepts any proposal y in which yi Ỗ 1 Ỗ and Player 2 always proposes the agreement Ỗ 1 Ỗ 1 1 Ỗ accepts any proposal x in which X2 Ỗ 1 Ỗ and never opts out. The outcome is that agreement is reached immediately on 1 1 Ỗ Ỗ 1 Ỗ . 2. If b Ỗ 1 Ỗ then the game has a unique subgame perfect equilibrium in which Player 1 always proposes 1 b b and accepts any Models in Which Players Have Outside Options 57 proposal y in which yi Ỗ 1 b and Player 2 always proposes Ỗ 1 b 1 5 1 b accepts any proposal X in which X2 b and opts out if X2 b. The outcome is that agreement is reached immediately on the division 1 b b . 3. If b 5 1 Ỗ then in every subgame perfect equilibrium the outcome is an immediate agreement on 1 b b . Proof. Throughout this proof we write SPE for subgame perfect equilibrium . First note that if ỗ 1 Ỗ b then the SPE of the bargaining game of alternating offers given in Theorem is an SPE of the game here. Given the equilibrium strategies Player 2 can never
