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

Nash Bargaining

Julio D´ avila 2009

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

if they make a deal at 150 euros, both of them are better off

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

if they make a deal at 150 euros, both of them are better off but also if the transactions takes place for 175 euros

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

if they make a deal at 150 euros, both of them are better off but also if the transactions takes place for 175 euros

• r at a price of 125 euros...

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

if they make a deal at 150 euros, both of them are better off but also if the transactions takes place for 175 euros

• r at a price of 125 euros...
• r at any p ∈ (100, 200)

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

if they make a deal at 150 euros, both of them are better off but also if the transactions takes place for 175 euros

• r at a price of 125 euros...
• r at any p ∈ (100, 200)

nevertheless they are not indifferent...

Julio D´ avila Nash Bargaining

a bargaining problem

consider the following situation

• a seller asks for at least 100 euros
• a buyer is willing to pay at most 200 euros

if they make a deal at 150 euros, both of them are better off but also if the transactions takes place for 175 euros

• r at a price of 125 euros...
• r at any p ∈ (100, 200)

nevertheless they are not indifferent...

◮ how should/would they share the surplus of 100 euros?

Julio D´ avila Nash Bargaining

agreements as equilibria

1 n individuals

Julio D´ avila Nash Bargaining

agreements as equilibria

1 n individuals 2 A: a set of outcomes

Julio D´ avila Nash Bargaining

agreements as equilibria

1 n individuals 2 A: a set of outcomes 3 for all i = 1, . . . , n ui ∈ RA

++: utility from outcomes

Julio D´ avila Nash Bargaining

agreements as equilibria

1 n individuals 2 A: a set of outcomes 3 for all i = 1, . . . , n ui ∈ RA

++: utility from outcomes ◮ each agent can propose to replace the statu quo outcome a

with a different one a′, running the risk a breakdown of the negotiations with some probability and nobody getting anything (i.e. 0) as a result

Julio D´ avila Nash Bargaining

agreements as equilibria

1 n individuals 2 A: a set of outcomes 3 for all i = 1, . . . , n ui ∈ RA

++: utility from outcomes ◮ each agent can propose to replace the statu quo outcome a

with a different one a′, running the risk a breakdown of the negotiations with some probability and nobody getting anything (i.e. 0) as a result

◮ for the replacement of a by a′ to take place, the proposal

must get the support of at least one other agent

Julio D´ avila Nash Bargaining

agreements as equilibria

1 n individuals 2 A: a set of outcomes 3 for all i = 1, . . . , n ui ∈ RA

++: utility from outcomes ◮ each agent can propose to replace the statu quo outcome a

with a different one a′, running the risk a breakdown of the negotiations with some probability and nobody getting anything (i.e. 0) as a result

◮ for the replacement of a by a′ to take place, the proposal

must get the support of at least one other agent

◮ the negotiation ends when nobody makes new proposals

Julio D´ avila Nash Bargaining

agreements as equilibria

equilibrium

Julio D´ avila Nash Bargaining

agreements as equilibria

equilibrium an a∗ such that, for all a and all p such that ui(a∗) ≤ pui(a) for some i, it holds uj(a) < puj(a∗) for all j = i,

Julio D´ avila Nash Bargaining

agreements as equilibria

equilibrium If a∗ is such that, for all a and all p such that ui(a∗) ≤ pui(a) for some i, it holds uj(a) < puj(a∗) for all j = i,

Julio D´ avila Nash Bargaining

agreements as equilibria

equilibrium If a∗ is such that, for all a and all p such that ui(a∗) ≤ pui(a) for some i, it holds uj(a) < puj(a∗) for all j = i, then a∗ ∈ arg max

a∈A n

• i=1

ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

consider any a ∈ A

Julio D´ avila Nash Bargaining

agreements as equilibria

consider any a ∈ A

◮ either for some i, 0 = ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

consider any a ∈ A

◮ either for some i, 0 = ui(a) ◮ or for all i, 0 < ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for some i, 0 = ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for some i, 0 = ui(a) then

n

• i=1

ui(a) = 0 <

n

• i=1

ui(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a) then

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a) then either, for all i, ui(a) ≤ ui(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a) then either, for all i, ui(a) ≤ ui(a∗)

• r, for some i, ui(a∗) < ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for all i, ui(a) ≤ ui(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for all i, ui(a) ≤ ui(a∗) then for all i, 0 < ui(a) ≤ ui(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for all i, ui(a) ≤ ui(a∗) then for all i, 0 < ui(a) ≤ ui(a∗) and hence

n

• i=1

ui(a) ≤

n

• i=1

ui(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) then

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) then either, for some j = i ui(a∗) ui(a) < uj(a) uj(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) then either, for some j = i ui(a∗) ui(a) < uj(a) uj(a∗)

• r, for all j = i,

uj(a) uj(a∗) ≤ ui(a∗) ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) and for some j = i ui(a∗) ui(a) < uj(a) uj(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) and for some j = i ui(a∗) ui(a) < uj(a) uj(a∗) then there exists p ∈ [0, 1] such that ui(a∗) ui(a) ≤ p ≤ uj(a) uj(a∗)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) and for some j = i ui(a∗) ui(a) < uj(a) uj(a∗) then there exists p ∈ [0, 1] such that ui(a∗) ≤ pui(a) puj(a∗) ≤ uj(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) and for some j = i ui(a∗) ui(a) < uj(a) uj(a∗) then there exists p ∈ [0, 1] such that ui(a∗) ≤ pui(a) not puj(a∗) > uj(a) for all j!!

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) then either, for some j = i ui(a∗) ui(a) < uj(a) uj(a∗)

• r, for all j = i,

uj(a) uj(a∗) ≤ ui(a∗) ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) then for all j = i, 0 < uj(a) uj(a∗) ≤ ui(a∗) ui(a) < 1 and hence

• j=i uj(a)
• j=i uj(a∗) ≤ ui(a∗)

ui(a)

Julio D´ avila Nash Bargaining

agreements as equilibria

if, for all i, 0 < ui(a), and for some i, ui(a∗) < ui(a) then for all j = i, 0 < uj(a) uj(a∗) ≤ ui(a∗) ui(a) < 1 and hence

n

• i=1

ui(a) ≤

n

• i=1

ui(a∗)

Julio D´ avila Nash Bargaining

Nash bargaining problems

(U, u) ∈ 2Rn × Rn 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) ∈ 2Rn × Rn 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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)B is invariant to coordinate-wise affine transformations iff for all (U, u), (U′, u′) ∈ B such that 1 there exist a ∈ Rn

++ and b ∈ Rn such that

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 ∈ (Rn)B is invariant to coordinate-wise affine transformations iff for all (U, u), (U′, u′) ∈ B such that 1 there exist a ∈ Rn

++ and b ∈ Rn such that

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 ∈ (Rn)B is invariant to coordinate-wise affine transformations iff for all (U, u), (U′, u′) ∈ B such that 1 there exist a ∈ Rn

++ and b ∈ Rn such that

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 ∈ (Rn)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 ∈ (Rn)B preserves symmetry iff for all (U, u) such that 1 for all i = j, ui = uj and

Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions

(SYM) Symmetry preservation a bargaining solution s ∈ (Rn)B preserves symmetry iff for all (U, u) such that 1 for all i = j, ui = uj and 2 for all (u1, . . . , un) ∈ U and all bijective ρ ∈ {1, . . . , n}{1,...,n}, (uρ(1), . . . , uρ(n)) ∈ U

Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions

(SYM) Symmetry preservation a bargaining solution s ∈ (Rn)B preserves symmetry iff for all (U, u) such that 1 for all i = j, ui = uj and 2 for all (u1, . . . , un) ∈ U and all bijective ρ ∈ {1, . . . , n}{1,...,n}, (uρ(1), . . . , uρ(n)) ∈ U it holds, for all i = j, si(U, u) = sj(U, u)

Julio D´ avila Nash Bargaining

desirable properties for bargaining solutions

(EFF) Efficiency a bargaining solution s ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)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 ∈ (Rn)B is

Julio D´ avila Nash Bargaining

Nash theorem

If s ∈ (Rn)B is 1 invariant to point-wise affine transformations

Julio D´ avila Nash Bargaining

Nash theorem

If s ∈ (Rn)B is 1 invariant to point-wise affine transformations 2 symmetry-preserving

Julio D´ avila Nash Bargaining

Nash theorem

If s ∈ (Rn)B is 1 invariant to point-wise affine transformations 2 symmetry-preserving 3 efficient

Julio D´ avila Nash Bargaining

Nash theorem

If s ∈ (Rn)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 ∈ (Rn)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, s(U, u) = arg max

u≤u∈U n

• i=1

(ui − ui)

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui)

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui)

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui) then, for all u ∈ U such that u ≤ u,

n

• i=1

(ui − ui) ≤

n

• i=1

(u∗

i − ui)

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui) then, for all diag(a)u + b ∈ U′ such that diag(a)u + b ≤ diag(a)u + b,

n

• i=1

(aiui + bi − aiui − bi) ≤

n

• i=1

(aiu∗

i + bi − aiui − bi)

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui) then, for all u′ ∈ U′ such that u′ ≤ u′,

n

• i=1

(u′

i − u′ i) ≤ n

• i=1

(aiu∗

i + bi − u′ i).

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui) then, diag(a)u∗ + b = arg max

u′≤u∈U′ n

• i=1

(ui − u′

i)

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 u∗ = arg max

u≤u∈U n

• i=1

(ui − ui) then, diag(a)s(U, u) + b = s(U′, u′).

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

let s ∈ (Rn)B satisfy INV, SYM, EFF, and IND and

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

let s ∈ (Rn)B satisfy INV, SYM, EFF, and IND and let (U, u) ∈ B

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

let s ∈ (Rn)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) 0 <

n

• i=1

(ui − ui) ≤

n

• i=1

(u∗

i − ui)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

let s ∈ (Rn)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) 0 <

n

• i=1

(ui − ui) ≤

n

• i=1

(u∗

i − ui)

it follows that 0 < u∗ − u

Julio D´ avila Nash Bargaining

Nash theorem

• nly 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

• nly 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 a = +

• diag(u∗ − u)

−11 b = −

• diag(u∗ − u)

−1u

Julio D´ avila Nash Bargaining

Nash theorem

• nly 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

• nly s satisfies INV, SYM, EFF, and IND:

1 by INV diag(a)s(U, u) + b = s(U′, u′) 2 also u∗ = arg max

u≤u∈U n

• i=1

(ui − ui)

Julio D´ avila Nash Bargaining

Nash theorem

• nly 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

• i=1

(ui − ui) ≤

n

• i=1

(u∗

i − ui)

Julio D´ avila Nash Bargaining

Nash theorem

• nly 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

• i=1

(aiui + bi − aiui − bi) ≤

n

• i=1

(aiu∗

i + bi − aiui − bi)

Julio D´ avila Nash Bargaining

Nash theorem

• nly 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

• i=1

(u′

i − u′ i) ≤ n

• i=1

(aiu∗

i + bi − u′ i)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

1 by INV diag(a)s(U, u) + b = s(U′, u′) 2 also diag(a)u∗ + b = arg max

u′≤u∈U′ n

• i=1

(ui − u′

i)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

1 by INV diag(a)s(U, u) + b = s(U′, u′) 2 also diag(a)u∗ + b = arg max

u′≤u∈U′ n

• i=1

(ui − u′

i)

3 thus s(U, u) = u∗ if, and only if, diag(a)u∗ + b = s(U′, u′)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

1 by INV diag(a)s(U, u) + b = s(U′, u′) 2 also diag(a)u∗ + b = arg max

u′≤u∈U′ n

• i=1

(ui − u′

i)

3 thus s(U, u) = u∗ if, and only if, 1 = s(U′, 0)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect diag(a)u∗ + b

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect diag(a)u∗ + b =

• diag(u∗ − u)

−1u∗ −

• diag(u∗ − u)

−1u

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect diag(a)u∗ + b =

• diag(u∗ − u)

−1u∗ −

• diag(u∗ − u)

−1u =

• diag(u∗ − u)

−1(u∗ − u)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect diag(a)u∗ + b =

• diag(u∗ − u)

−1u∗ −

• diag(u∗ − u)

−1u =

• diag(u∗ − u)

−1(u∗ − u) = 1

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect u′

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect u′ = diag(a)u + b

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect u′ = diag(a)u + b =

• diag(u∗ − u)

−1u −

• diag(u∗ − u)

−1u

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

in effect u′ = diag(a)u + b =

• diag(u∗ − u)

−1u −

• diag(u∗ − u)

−1u = 0

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n given that diag(a)u∗ + b = arg max

u′≤u∈U′ n

• i=1

(ui − u′

i)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n given that 1 = arg max

0≤u∈U′ n

• i=1

ui

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n given that 1 = max

0≤u∈U′ n

• i=1

ui

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 assume for some u ∈ U′

n

• i=1

ui > n

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 assume for some u ∈ U′

n

• i=1

ui > n then there exists α1 + (1 − α)u ∈ U′, for some 0 < α < 1, such that

n

• i=1

(α1 + (1 − α)ui) > 1 = max

0≤u∈U′ n

• i=1

ui !!

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 1 for all u ∈ U′,

n

• i=1

ui ≤ n 2 a nonempty, compact, convex and symmetric B contains U′ and has 1 on its boundary

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 3 by SYM, for all i, j = 1, . . . , n, si(B, 0) = sj(B, 0)

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 3 by SYM, for all i, j = 1, . . . , n, si(B, 0) = sj(B, 0) 4 by EFF,

n

• i=1

si(B, 0) = n

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 3 by SYM, for all i, j = 1, . . . , n, si(B, 0) = sj(B, 0) 4 by EFF,

n

• i=1

si(B, 0) = n and hence s(B, 0) = 1

Julio D´ avila Nash Bargaining

Nash theorem

• nly s satisfies INV, SYM, EFF, and IND:

s(U′, 0) = 1 since 3 by SYM, for all i, j = 1, . . . , n, si(B, 0) = sj(B, 0) 4 by EFF,

n

• i=1

si(B, 0) = n and hence s(B, 0) = 1 5 by IND, U′ ⊂ B and s(B, 0) ∈ U′, s(U′, 0) = s(B, 0)

Julio D´ avila Nash Bargaining