Alternating offers bargaining with risk of breakdown Julio D avila - PowerPoint PPT Presentation

Alternating offers bargaining with risk of breakdown Julio D´ avila 2009 Julio D´ avila Alternating offers bargaining with risk of breakdown

alternating offers with risk of breakdown assume agents a , b are infinitely patient Julio D´ avila Alternating offers bargaining with risk of breakdown

alternating offers with risk of breakdown assume agents a , b are infinitely patient negociations break down with probability 1 − π any time Julio D´ avila Alternating offers bargaining with risk of breakdown

alternating offers with risk of breakdown assume agents a , b are infinitely patient negociations break down with probability 1 − π any time outcomes are lotteries [( x a , x b ) , t ], i.e. ”( x a , x b ) at date t ” with probability π t , otherwise b Julio D´ avila Alternating offers bargaining with risk of breakdown

alternating offers with risk of breakdown assume agents a , b are infinitely patient negociations break down with probability 1 − π any time outcomes are lotteries [( x a , x b ) , t ], i.e. ”( x a , x b ) at date t ” with probability π t , otherwise b that is to say [( x a , x b ) , t ] = { ( x a , x b ) , b ; π t , 1 − π t } Julio D´ avila Alternating offers bargaining with risk of breakdown

preferences over lotteries agents have rational preferences � i over lotteries over S 1 ∪ { b } such that Julio D´ avila Alternating offers bargaining with risk of breakdown

preferences over lotteries agents have rational preferences � i over lotteries over S 1 ∪ { b } such that 1 breakdown is the worst outcome: i.e. for all [ x , p ] b � i [ x , p ] Julio D´ avila Alternating offers bargaining with risk of breakdown

preferences over lotteries agents have rational preferences � i over lotteries over S 1 ∪ { b } such that 1 breakdown is the worst outcome: i.e. for all [ x , p ] b � i [ x , p ] 2 VNM representation u i (with u ′ i > 0 , u ′′ i < 0): i.e. [ x , t ] � i [ x ′ , t ′ ] iff π t u i ( x i ) + (1 − π t ) u i ( b ) ≤ π t ′ u i ( x ′ i ) + (1 − π t ′ ) u i ( b ) Julio D´ avila Alternating offers bargaining with risk of breakdown

preferences over lotteries agents have rational preferences � i over lotteries over S 1 ∪ { b } such that 1 breakdown is the worst outcome: i.e. for all [ x , p ] b � i [ x , p ] 2 VNM representation u i (with u ′ i > 0 , u ′′ i < 0): i.e. [ x , t ] � i [ x ′ , t ′ ] iff π t u i ( x i ) + (1 − π t ) u i ( b ) ≤ π t ′ u i ( x ′ i ) + (1 − π t ′ ) u i ( b ) 3 normalization u i ( b ) = 0 = u i (0) Julio D´ avila Alternating offers bargaining with risk of breakdown

preferences over lotteries agents have rational preferences � i over lotteries over S 1 ∪ { b } such that 1 breakdown is the worst outcome: i.e. for all [ x , p ] b � i [ x , p ] 2 VNM representation u i (with u ′ i > 0 , u ′′ i < 0): i.e. [ x , t ] � i [ x ′ , t ′ ] iff π t u i ( x i ) +(1 − π t ) u i ( b ) ≤ π t ′ u i ( x ′ i )+(1 − π t ′ ) u i ( b ) 3 normalization u i ( b ) = 0 = u i (0) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk let x i < x ′ i 1 � i increasing in x i : for all x i < x ′ i and t [ x , t ] ≺ i [ x ′ , t ] since π t u i ( x i ) < π t u i ( x ′ i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk let x i < x ′ i 1 � i increasing in x i : for all x i < x ′ i and t [ x , t ] ≺ i [ x ′ , t ] since π t u i ( x i ) < π t u i ( x ′ i ) 2 � i decreasing in t : for all x i > 0 and t < t ′ [ x , t ] ≻ i [ x , t ′ ] since π t u i ( x i ) > π t ′ u i ( x i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk let x i < x ′ i 1 � i increasing in x i : for all x i < x ′ i and t [ x , t ] ≺ i [ x ′ , t ] since π t u i ( x i ) < π t u i ( x ′ i ) 2 � i decreasing in t : for all x i > 0 and t < t ′ [ x , t ] ≻ i [ x , t ′ ] since π t u i ( x i ) > π t ′ u i ( x i ) 3 indifference to t if x i = 0: i.e. for all x i = 0 and t ≤ t ′ , [ x , t ] ∼ i [ x , t ′ ] since π t u i (0) = π t ′ u i (0) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk let x i < x ′ i 1 � i increasing in x i : for all x i < x ′ i and t [ x , t ] ≺ i [ x ′ , t ] since π t u i ( x i ) < π t u i ( x ′ i ) 2 � i decreasing in t : for all x i > 0 and t < t ′ [ x , t ] ≻ i [ x , t ′ ] since π t u i ( x i ) > π t ′ u i ( x i ) 3 indifference to t if x i = 0: i.e. for all x i = 0 and t ≤ t ′ , [ x , t ] ∼ i [ x , t ′ ] since π t u i (0) = π t ′ u i (0) 4 stationarity: if [ x , t ] � i [ x ′ , t ′ ] then, for all k , [ x , t + k ] � i [ x ′ , t ′ + k ] since π t + k u i ( x i ) ≤ π t ′ + k u i ( x i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk let x i < x ′ i 5 continuity: i.e. for all x n → x , x ′ n → y such that, for all n , [ x n , t ] � i [ x ′ n , t ′ ], [ x , t ] � i [ x ′ , t ′ ] i ) ≤ π t ′ u i ( x ′ n since π t u i ( x n i ) and u i is continuous Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk let x i < x ′ i 5 continuity: i.e. for all x n → x , x ′ n → y such that, for all n , [ x n , t ] � i [ x ′ n , t ′ ], [ x , t ] � i [ x ′ , t ′ ] i ) ≤ π t ′ u i ( x ′ n since π t u i ( x n i ) and u i is continuous 6 increasing loss from delays: i.e. for all x i < x ′ i , x i − φ i i − φ i i ( x , 1) < x ′ i ( x ′ , 1) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] u i ( φ i i [ x , 1]) = π u i ( x i ) u i ( φ i i [ x ′ , 1]) = π u i ( x ′ i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] − u i ( φ i i [ x , 1]) = − π u i ( x i ) − u i ( φ i i [ x ′ , 1]) = − π u i ( x ′ i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] u i ( x i ) − u i ( φ i i [ x , 1]) = u i ( x i ) − π u i ( x i ) i ) − u i ( φ i u i ( x ′ i [ x ′ , 1]) = u i ( x ′ i ) − π u i ( x ′ i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] u i ( x i ) − u i ( φ i i ( x , 1)) = (1 − π ) u i ( x i ) i ) − u i ( φ i u i ( x ′ i ( x ′ , 1)) = (1 − π ) u i ( x ′ i ) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] u i ( x i ) − u i ( φ i i ( x , 1)) = (1 − π ) u i ( x i ) i ) − u i ( φ i u i ( x ′ i ( x ′ , 1)) = (1 − π ) u i ( x ′ i ) moreover φ i i ( x , 1) < x i and φ i i ( x ′ , 1) < x ′ i φ i i ( x , 1) < φ i i ( x ′ , 1) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] u i ( x i ) − u i ( φ i i ( x , 1)) = (1 − π ) u i ( x i ) i ) − u i ( φ i u i ( x ′ i ( x ′ , 1)) = (1 − π ) u i ( x ′ i ) moreover φ i i ( x , 1) < x i and φ i i ( x ′ , 1) < x ′ i φ i i ( x , 1) < φ i i ( x ′ , 1) and since u ′′ < 0 u i ( x i ) − u i ( φ i i ) − u i ( φ i i ( x , 1)) > u i ( x ′ i ( x ′ , 1)) x i − φ i i − φ i i ( x , 1) x ′ i ( x ′ , 1) Julio D´ avila Alternating offers bargaining with risk of breakdown

equivalence to alternating offers without risk in effect, if x i < x ′ i since [ φ i ( x , 1) , 0] ∼ i [ x , 1] and [ φ i ( x ′ , 1) , 0] ∼ i [ x ′ , 1] u i ( x i ) − u i ( φ i i ( x , 1)) = (1 − π ) u i ( x i ) i ) − u i ( φ i u i ( x ′ i ( x ′ , 1)) = (1 − π ) u i ( x ′ i ) moreover φ i i ( x , 1) < x i and φ i i ( x ′ , 1) < x ′ i φ i i ( x , 1) < φ i i ( x ′ , 1) and since u ′′ < 0 (1 − π ) u i ( x i ) i ( x , 1) > (1 − π ) u i ( x ′ i ) x i − φ i i − φ i x ′ i ( x ′ , 1) Julio D´ avila Alternating offers bargaining with risk of breakdown