Nash Bargaining Julio D avila 2009 Julio D avila Nash - PowerPoint PPT Presentation

agreements as equilibria if, for all i , 0 < u i ( a ), and for some i , u i ( a ∗ ) < u i ( a ) then for all j � = i , u j ( a ∗ ) ≤ u i ( a ∗ ) 0 < u j ( a ) u i ( a ) < 1 and hence n n � � u i ( a ∗ ) u i ( a ) ≤ i =1 i =1 Julio D´ avila Nash Bargaining

Nash bargaining problems ( U , u ) ∈ 2 R n × R n is a Nash bargaining problem iff U is nonempty, compact and convex, and there exists u ∈ U such that u < u Julio D´ avila Nash Bargaining

Nash bargaining problems ( U , u ) ∈ 2 R n × R n is a Nash bargaining problem iff U is nonempty, compact and convex, and there exists u ∈ U such that u < u let B be the set of all bargaining problems Julio D´ avila Nash Bargaining

bargaining solutions s ∈ ( R n ) B is a solution for Nash bargaining problems iff for all ( U , u ) ∈ B , s ( U , u ) ∈ U Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (INV) Invariance to coordinate-wise affine transformations a bargaining solution s ∈ ( R n ) B is invariant to coordinate-wise affine transformations iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (INV) Invariance to coordinate-wise affine transformations a bargaining solution s ∈ ( R n ) B is invariant to coordinate-wise affine transformations iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that ++ and b ∈ R n such that 1 there exist a ∈ R n diag ( a ) u + b = u ′ Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (INV) Invariance to coordinate-wise affine transformations a bargaining solution s ∈ ( R n ) B is invariant to coordinate-wise affine transformations iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that ++ and b ∈ R n such that 1 there exist a ∈ R n diag ( a ) u + b = u ′ 2 and u ∈ U iff diag ( a ) u + b ∈ U ′ Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (INV) Invariance to coordinate-wise affine transformations a bargaining solution s ∈ ( R n ) B is invariant to coordinate-wise affine transformations iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that ++ and b ∈ R n such that 1 there exist a ∈ R n diag ( a ) u + b = u ′ 2 and u ∈ U iff diag ( a ) u + b ∈ U ′ it holds diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (SYM) Symmetry preservation a bargaining solution s ∈ ( R n ) B preserves symmetry iff for all ( U , u ) such that Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (SYM) Symmetry preservation a bargaining solution s ∈ ( R n ) B preserves symmetry iff for all ( U , u ) such that 1 for all i � = j , u i = u j and Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (SYM) Symmetry preservation a bargaining solution s ∈ ( R n ) B preserves symmetry iff for all ( U , u ) such that 1 for all i � = j , u i = u j and 2 for all ( u 1 , . . . , u n ) ∈ U and all bijective ρ ∈ { 1 , . . . , n } { 1 ,..., n } , ( u ρ (1) , . . . , u ρ ( n ) ) ∈ U Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (SYM) Symmetry preservation a bargaining solution s ∈ ( R n ) B preserves symmetry iff for all ( U , u ) such that 1 for all i � = j , u i = u j and 2 for all ( u 1 , . . . , u n ) ∈ U and all bijective ρ ∈ { 1 , . . . , n } { 1 ,..., n } , ( u ρ (1) , . . . , u ρ ( n ) ) ∈ U it holds, for all i � = j , s i ( U , u ) = s j ( U , u ) Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (EFF) Efficiency a bargaining solution s ∈ ( R n ) B is efficient iff for all ( U , u ) ∈ B and all u ∈ U such that Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (EFF) Efficiency a bargaining solution s ∈ ( R n ) B is efficient iff for all ( U , u ) ∈ B and all u ∈ U such that 1 u ≤ u , and Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (EFF) Efficiency a bargaining solution s ∈ ( R n ) B is efficient iff for all ( U , u ) ∈ B and all u ∈ U such that 1 u ≤ u , and 2 there exists u ′ ∈ U such that u < u ′ , Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (EFF) Efficiency a bargaining solution s ∈ ( R n ) B is efficient iff for all ( U , u ) ∈ B and all u ∈ U such that 1 u ≤ u , and 2 there exists u ′ ∈ U such that u < u ′ , it holds s ( U , u ) � = u Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (IND) Independence of irrelevant alternatives a bargaining solution s ∈ ( R n ) B is independent of irrelevant alternatives iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (IND) Independence of irrelevant alternatives a bargaining solution s ∈ ( R n ) B is independent of irrelevant alternatives iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that 1 u = u ′ , Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (IND) Independence of irrelevant alternatives a bargaining solution s ∈ ( R n ) B is independent of irrelevant alternatives iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that 1 u = u ′ , 2 U ⊂ U ′ , and Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (IND) Independence of irrelevant alternatives a bargaining solution s ∈ ( R n ) B is independent of irrelevant alternatives iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that 1 u = u ′ , 2 U ⊂ U ′ , and 3 s ( U ′ , u ′ ) ∈ U , Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions (IND) Independence of irrelevant alternatives a bargaining solution s ∈ ( R n ) B is independent of irrelevant alternatives iff for all ( U , u ) , ( U ′ , u ′ ) ∈ B such that 1 u = u ′ , 2 U ⊂ U ′ , and 3 s ( U ′ , u ′ ) ∈ U , it holds s ( U , u ) = s ( U ′ , u ′ ) Julio D´ avila Nash Bargaining

Nash theorem If s ∈ ( R n ) B is Julio D´ avila Nash Bargaining

Nash theorem If s ∈ ( R n ) B is 1 invariant to point-wise affine transformations Julio D´ avila Nash Bargaining

Nash theorem If s ∈ ( R n ) B is 1 invariant to point-wise affine transformations 2 symmetry-preserving Julio D´ avila Nash Bargaining

Nash theorem If s ∈ ( R n ) B is 1 invariant to point-wise affine transformations 2 symmetry-preserving 3 efficient Julio D´ avila Nash Bargaining

Nash theorem If s ∈ ( R n ) B is 1 invariant to point-wise affine transformations 2 symmetry-preserving 3 efficient 4 independent of irrelevant strategies Julio D´ avila Nash Bargaining

Nash theorem If s ∈ ( R n ) B is 1 invariant to point-wise affine transformations 2 symmetry-preserving 3 efficient 4 independent of irrelevant strategies then, for all ( U , u ) ∈ B , n � s ( U , u ) = arg max ( u i − u i ) u ≤ u ∈ U i =1 Julio D´ avila Nash Bargaining

Nash theorem s is well defined: Julio D´ avila Nash Bargaining

Nash theorem s is well defined: since U is non-empty, compact and convex, Julio D´ avila Nash Bargaining

Nash theorem s is well defined: since U is non-empty, compact and convex, then there exists a unique u ∗ satisfying n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ 2 let n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ 2 let n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 then, for all u ∈ U such that u ≤ u , n n � � ( u ∗ ( u i − u i ) ≤ i − u i ) i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ 2 let n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 then, for all diag ( a ) u + b ∈ U ′ such that diag ( a ) u + b ≤ diag ( a ) u + b , n n � � ( a i u ∗ ( a i u i + b i − a i u i − b i ) ≤ i + b i − a i u i − b i ) i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ 2 let n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 then, for all u ′ ∈ U ′ such that u ′ ≤ u ′ , n n � � ( u ′ i − u ′ ( a i u ∗ i + b i − u ′ i ) ≤ i ) . i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ 2 let n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 then, n diag ( a ) u ∗ + b = arg � ( u i − u ′ max i ) u ′ ≤ u ∈ U ′ i =1 Julio D´ avila Nash Bargaining

Nash theorem s satisfies INV, SYM, EFF, and IND: INV: 1 let ( U , u ) , ( U ′ , u ′ ) be such that, for some a ≫ 0 and b , diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ 2 let n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 then, diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) . Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: let s ∈ ( R n ) B satisfy INV, SYM, EFF, and IND and Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: let s ∈ ( R n ) B satisfy INV, SYM, EFF, and IND and let ( U , u ) ∈ B Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: let s ∈ ( R n ) B satisfy INV, SYM, EFF, and IND and let ( U , u ) ∈ B since (i) u ≤ u ∗ , (ii) there exists u ∈ U such that u < u , and (iii) n n � � ( u ∗ 0 < ( u i − u i ) ≤ i − u i ) i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: let s ∈ ( R n ) B satisfy INV, SYM, EFF, and IND and let ( U , u ) ∈ B since (i) u ≤ u ∗ , (ii) there exists u ∈ U such that u < u , and (iii) n n � � ( u ∗ 0 < ( u i − u i ) ≤ i − u i ) i =1 i =1 it follows that 0 < u ∗ − u Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: consider instead ( U ′ , u ′ ) such that diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: consider instead ( U ′ , u ′ ) such that diag ( a ) u + b = u ′ and u ∈ U iff diag ( a ) u + b ∈ U ′ where diag ( u ∗ − u ) � − 1 1 � a = + diag ( u ∗ − u ) � − 1 u � b = − Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also n u ∗ = arg max � ( u i − u i ) u ≤ u ∈ U i =1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also, for all u ∈ U such that u ≤ u n n � � ( u ∗ ( u i − u i ) ≤ i − u i ) i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also, for all u ∈ U such that u ≤ u n n � � ( a i u ∗ ( a i u i + b i − a i u i − b i ) ≤ i + b i − a i u i − b i ) i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also, for all u ′ ∈ U ′ such that u ′ ≤ u ′ n n � ( u ′ i − u ′ � ( a i u ∗ i + b i − u ′ i ) ≤ i ) i =1 i =1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also n diag ( a ) u ∗ + b = arg � ( u i − u ′ max i ) u ′ ≤ u ∈ U ′ i =1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also n diag ( a ) u ∗ + b = arg � ( u i − u ′ max i ) u ′ ≤ u ∈ U ′ i =1 3 thus s ( U , u ) = u ∗ if, and only if, diag ( a ) u ∗ + b = s ( U ′ , u ′ ) Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: 1 by INV diag ( a ) s ( U , u ) + b = s ( U ′ , u ′ ) 2 also n diag ( a ) u ∗ + b = arg � ( u i − u ′ max i ) u ′ ≤ u ∈ U ′ i =1 3 thus s ( U , u ) = u ∗ if, and only if, 1 = s ( U ′ , 0) Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect diag ( a ) u ∗ + b Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect diag ( a ) u ∗ + b diag ( u ∗ − u ) � − 1 u ∗ − diag ( u ∗ − u ) � − 1 u � � = Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect diag ( a ) u ∗ + b diag ( u ∗ − u ) � − 1 u ∗ − diag ( u ∗ − u ) � − 1 u � � = diag ( u ∗ − u ) � − 1 ( u ∗ − u ) � = Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect diag ( a ) u ∗ + b diag ( u ∗ − u ) � − 1 u ∗ − diag ( u ∗ − u ) � − 1 u � � = diag ( u ∗ − u ) � − 1 ( u ∗ − u ) � = = 1 Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect u ′ Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect u ′ = diag ( a ) u + b Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect u ′ = diag ( a ) u + b diag ( u ∗ − u ) � − 1 u − diag ( u ∗ − u ) � − 1 u � � = Julio D´ avila Nash Bargaining

Nash theorem only s satisfies INV, SYM, EFF, and IND: in effect u ′ = diag ( a ) u + b diag ( u ∗ − u ) � − 1 u − diag ( u ∗ − u ) � − 1 u � � = = 0 Julio D´ avila Nash Bargaining