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Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
On Sustainable Equilibria
Hari Govindan, Rida Laraki & Lucas Pahl
Game Theory World SeminarAugust, 10, 2020
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
The Beginning: Essays in Honor of Selten 65th Birthday
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Myerson Essay (1995)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Desirable Criteria for Refinement
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Sustainable Equilibria in the Battle of the Sexes
Myerson considers, and then dismisses existing refinements thatyield the same prediction in the Battle-of-Sexes game:
Persistent Equilibria (Kalai-Samet): fail invariance
ESS (Maynard-Smith): fail existence
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Sustainable Equilibria in the Battle of the Sexes
Myerson considers, and then dismisses existing refinements thatyield the same prediction in the Battle-of-Sexes game:
Persistent Equilibria (Kalai-Samet): fail invariance
ESS (Maynard-Smith): fail existence
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Sustainable Equilibria in the Battle of the Sexes
Myerson considers, and then dismisses existing refinements thatyield the same prediction in the Battle-of-Sexes game:
Persistent Equilibria (Kalai-Samet): fail invariance
ESS (Maynard-Smith): fail existence
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Sustainable Equilibria in the Battle of the Sexes
Myerson considers, and then dismisses existing refinements thatyield the same prediction in the Battle-of-Sexes game:
Persistent Equilibria (Kalai-Samet): fail invariance
ESS (Maynard-Smith): fail existence
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Intuition from Fixed Point Theory
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustration: odd number of crosses, sum of indices = +1
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Link with Fixed Point and Equilibria
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Myerson’s Conjecture
Myerson was seemingly unaware of the existence at thatmoment of an index theory for Nash equilibria:
Gül, Pearce & Stacchetti (1993) Ritzberger (1994)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Myerson’s Conjecture
Myerson was seemingly unaware of the existence at thatmoment of an index theory for Nash equilibria:
Gül, Pearce & Stacchetti (1993) Ritzberger (1994)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of equilibria for economists: THE BOOK
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Myerson’s Conclusion
Hofbauer (2000) formulated a precise definition ofsustainability for generic games and some conjectures.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Myerson’s Conclusion
Hofbauer (2000) formulated a precise definition ofsustainability for generic games and some conjectures.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Formulation of Myerson’s Criteria
I (A1) Strict Nash equilibria are sustainable.
I (A2) Battle of sexes: only strict equilibria are sustainable.
I (A3) If a game has a unique equilibrium, it is sustainable.
I (A4) Every generic game has a sustainable equilibrium.
I (A5) Sustainable equilibria are invariant under addition orremoval of inferior replies.
A5 is a form of independence of irrelevant alternative.
It is a very strong axiom as, combined with A3, they imply A1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Definition of Sustainability
I Hofbauer defines an equivalence relation among pairs(G, σ) where G is a game and σ is an equilibrium of G.
I (G, σ) ∼ (Ĝ, σ̂) if σ = σ̂ (up to a relabelling) and therestriction of G and Ĝ to the best replies to σ and σ̂,resp., are the same game (up to a relabelling).
I By A5 (IIA) and A3 (uniqueness => sustainability):If an equilibrium σ of a game G is unique in anequivalent pair (Ĝ, σ̂), it must be sustainable.
Hofbauber cleverly combined A5 & A3 with minimality:
I An equilibrium of a game G is sustainable iff it isthe unique equilibrium in an equivalent pair.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Definition of Sustainability
I Hofbauer defines an equivalence relation among pairs(G, σ) where G is a game and σ is an equilibrium of G.
I (G, σ) ∼ (Ĝ, σ̂) if σ = σ̂ (up to a relabelling) and therestriction of G and Ĝ to the best replies to σ and σ̂,resp., are the same game (up to a relabelling).
I By A5 (IIA) and A3 (uniqueness => sustainability):If an equilibrium σ of a game G is unique in anequivalent pair (Ĝ, σ̂), it must be sustainable.
Hofbauber cleverly combined A5 & A3 with minimality:
I An equilibrium of a game G is sustainable iff it isthe unique equilibrium in an equivalent pair.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Definition of Sustainability
I Hofbauer defines an equivalence relation among pairs(G, σ) where G is a game and σ is an equilibrium of G.
I (G, σ) ∼ (Ĝ, σ̂) if σ = σ̂ (up to a relabelling) and therestriction of G and Ĝ to the best replies to σ and σ̂,resp., are the same game (up to a relabelling).
I By A5 (IIA) and A3 (uniqueness => sustainability):If an equilibrium σ of a game G is unique in anequivalent pair (Ĝ, σ̂), it must be sustainable.
Hofbauber cleverly combined A5 & A3 with minimality:
I An equilibrium of a game G is sustainable iff it isthe unique equilibrium in an equivalent pair.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Definition of Sustainability
I Hofbauer defines an equivalence relation among pairs(G, σ) where G is a game and σ is an equilibrium of G.
I (G, σ) ∼ (Ĝ, σ̂) if σ = σ̂ (up to a relabelling) and therestriction of G and Ĝ to the best replies to σ and σ̂,resp., are the same game (up to a relabelling).
I By A5 (IIA) and A3 (uniqueness => sustainability):If an equilibrium σ of a game G is unique in anequivalent pair (Ĝ, σ̂), it must be sustainable.
Hofbauber cleverly combined A5 & A3 with minimality:
I An equilibrium of a game G is sustainable iff it isthe unique equilibrium in an equivalent pair.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Definition of Sustainability
I Hofbauer defines an equivalence relation among pairs(G, σ) where G is a game and σ is an equilibrium of G.
I (G, σ) ∼ (Ĝ, σ̂) if σ = σ̂ (up to a relabelling) and therestriction of G and Ĝ to the best replies to σ and σ̂,resp., are the same game (up to a relabelling).
I By A5 (IIA) and A3 (uniqueness => sustainability):If an equilibrium σ of a game G is unique in anequivalent pair (Ĝ, σ̂), it must be sustainable.
Hofbauber cleverly combined A5 & A3 with minimality:
I An equilibrium of a game G is sustainable iff it isthe unique equilibrium in an equivalent pair.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer Definition of Sustainability
I Hofbauer defines an equivalence relation among pairs(G, σ) where G is a game and σ is an equilibrium of G.
I (G, σ) ∼ (Ĝ, σ̂) if σ = σ̂ (up to a relabelling) and therestriction of G and Ĝ to the best replies to σ and σ̂,resp., are the same game (up to a relabelling).
I By A5 (IIA) and A3 (uniqueness => sustainability):If an equilibrium σ of a game G is unique in anequivalent pair (Ĝ, σ̂), it must be sustainable.
Hofbauber cleverly combined A5 & A3 with minimality:
I An equilibrium of a game G is sustainable iff it isthe unique equilibrium in an equivalent pair.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Battle of the Sexes3 Nash equilibria: 2 strict σ = (t, l) and θ = (b, r), and 1 mixed.
G =l r
t (3, 2) (0, 0)b (0, 0) (2, 3)
By adding two strategies, σ is the unique equilibrium of Ĝ:
Ĝ =
l r yt (3, 2) (0, 0) (0, 1)b (0, 0) (2, 3) (−2, 4)x (1, 0) (4,−2) (−1,−1)
I Hence, the strict equilibrium σ is sustainable in G.
I The mixed equilibrium is not sustainable (prove it?).
I This is in line with Myerson requirements A2.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Battle of the Sexes3 Nash equilibria: 2 strict σ = (t, l) and θ = (b, r), and 1 mixed.
G =l r
t (3, 2) (0, 0)b (0, 0) (2, 3)
By adding two strategies, σ is the unique equilibrium of Ĝ:
Ĝ =
l r yt (3, 2) (0, 0) (0, 1)b (0, 0) (2, 3) (−2, 4)x (1, 0) (4,−2) (−1,−1)
I Hence, the strict equilibrium σ is sustainable in G.
I The mixed equilibrium is not sustainable (prove it?).
I This is in line with Myerson requirements A2.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Battle of the Sexes3 Nash equilibria: 2 strict σ = (t, l) and θ = (b, r), and 1 mixed.
G =l r
t (3, 2) (0, 0)b (0, 0) (2, 3)
By adding two strategies, σ is the unique equilibrium of Ĝ:
Ĝ =
l r yt (3, 2) (0, 0) (0, 1)b (0, 0) (2, 3) (−2, 4)x (1, 0) (4,−2) (−1,−1)
I Hence, the strict equilibrium σ is sustainable in G.
I The mixed equilibrium is not sustainable (prove it?).
I This is in line with Myerson requirements A2.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Battle of the Sexes3 Nash equilibria: 2 strict σ = (t, l) and θ = (b, r), and 1 mixed.
G =l r
t (3, 2) (0, 0)b (0, 0) (2, 3)
By adding two strategies, σ is the unique equilibrium of Ĝ:
Ĝ =
l r yt (3, 2) (0, 0) (0, 1)b (0, 0) (2, 3) (−2, 4)x (1, 0) (4,−2) (−1,−1)
I Hence, the strict equilibrium σ is sustainable in G.
I The mixed equilibrium is not sustainable (prove it?).
I This is in line with Myerson requirements A2.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Battle of the Sexes3 Nash equilibria: 2 strict σ = (t, l) and θ = (b, r), and 1 mixed.
G =l r
t (3, 2) (0, 0)b (0, 0) (2, 3)
By adding two strategies, σ is the unique equilibrium of Ĝ:
Ĝ =
l r yt (3, 2) (0, 0) (0, 1)b (0, 0) (2, 3) (−2, 4)x (1, 0) (4,−2) (−1,−1)
I Hence, the strict equilibrium σ is sustainable in G.
I The mixed equilibrium is not sustainable (prove it?).
I This is in line with Myerson requirements A2.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
A Sustainable Mixed EquilibriumG has 7 equilibria, all symmetric: 3 strict, 3 mixed (playersrandomise between 2 strategies), and 1 completely mixed.
G1 =
l m rt (10, 10) (0, 0) (0, 0)m (0, 0) (10, 10) (0, 0)b (0, 0) (0, 0) (10, 10)
The 3 strict equilibria and the completely mixed aresustainables, as Ĝ shows (von Schemde & von Stengel).
Ĝ1 =
l m r x y zt (10, 10) (0, 0) (0, 0) (0, 11) (10, 5) (0,−10)m (0, 0) (10, 10) (0, 0) (0,−10) (0, 11) (10, 5)b (0, 0) (0, 0) (10, 10) (10, 5) (0,−10) (0, 11)
3 remaining mixed equilibria are not sustainable (prove it?)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
A Sustainable Mixed EquilibriumG has 7 equilibria, all symmetric: 3 strict, 3 mixed (playersrandomise between 2 strategies), and 1 completely mixed.
G1 =
l m rt (10, 10) (0, 0) (0, 0)m (0, 0) (10, 10) (0, 0)b (0, 0) (0, 0) (10, 10)
The 3 strict equilibria and the completely mixed aresustainables, as Ĝ shows (von Schemde & von Stengel).
Ĝ1 =
l m r x y zt (10, 10) (0, 0) (0, 0) (0, 11) (10, 5) (0,−10)m (0, 0) (10, 10) (0, 0) (0,−10) (0, 11) (10, 5)b (0, 0) (0, 0) (10, 10) (10, 5) (0,−10) (0, 11)
3 remaining mixed equilibria are not sustainable (prove it?)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
A Sustainable Mixed EquilibriumG has 7 equilibria, all symmetric: 3 strict, 3 mixed (playersrandomise between 2 strategies), and 1 completely mixed.
G1 =
l m rt (10, 10) (0, 0) (0, 0)m (0, 0) (10, 10) (0, 0)b (0, 0) (0, 0) (10, 10)
The 3 strict equilibria and the completely mixed aresustainables, as Ĝ shows (von Schemde & von Stengel).
Ĝ1 =
l m r x y zt (10, 10) (0, 0) (0, 0) (0, 11) (10, 5) (0,−10)m (0, 0) (10, 10) (0, 0) (0,−10) (0, 11) (10, 5)b (0, 0) (0, 0) (10, 10) (10, 5) (0,−10) (0, 11)
3 remaining mixed equilibria are not sustainable (prove it?)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
A Sustainable Mixed EquilibriumG has 7 equilibria, all symmetric: 3 strict, 3 mixed (playersrandomise between 2 strategies), and 1 completely mixed.
G1 =
l m rt (10, 10) (0, 0) (0, 0)m (0, 0) (10, 10) (0, 0)b (0, 0) (0, 0) (10, 10)
The 3 strict equilibria and the completely mixed aresustainables, as Ĝ shows (von Schemde & von Stengel).
Ĝ1 =
l m r x y zt (10, 10) (0, 0) (0, 0) (0, 11) (10, 5) (0,−10)m (0, 0) (10, 10) (0, 0) (0,−10) (0, 11) (10, 5)b (0, 0) (0, 0) (10, 10) (10, 5) (0,−10) (0, 11)
3 remaining mixed equilibria are not sustainable (prove it?)Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Notations
I Set of players is N = { 1, . . . , N }.
I ∀n, a finite set Sn of pure strategies.
I The set of pure strategy profiles S ≡∏
n∈N Sn.
I Σn is player n’s set of mixed strategies.
I The set of mixed strategy profiles Σ ≡∏
n∈N Σn.
I For each n, let S−n =∏
m6=n Sm and Σ−n =∏
m 6=n Σm.
I G : S → RN is the payoff vector function.
I Γ ≡ RN×S : is the space of games as payoffs vary.
I E = { (G, σ) ∈ Γ× Σ | σ is a Nash equilibrium ofG }is the Nash equilibrium correspondence.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Index of Equilibria
I Let G be a game and U be a neighbourhood of Σ in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Hofbauer-Myerson conjectureHofbauer-Myerson conjecture: A regular equilibrium issustainable if and only if it has index +1.
I von Schemde & von Stengel (2008) proved the conjecturefor 2-player games using polytopal geometry.
I We prove the Hofbauer-Myerson conjecture for all N -playergames using algebraic topology.
I Corollary 1: since the sum of the indices of equilibria is+1, any regular game has a sustainable equilibrium.
I Corollary 2: Since the set of regular games is open anddense, almost every game has a sustainable equilibrium.
This implies Myerson requirements A1, A2, A4 and A5.
As our proof extends to isolated equilibria, we obtain A3(because a unique equilibrium is isolated and has index +1).
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Proof: the Easy Direction
Let σ be a regular equilibrium of game G.
I Let (G, σ) ∼ (Ĝ, σ) and σ = unique equilibrium of Ĝ.
I Let G∗ be obtained from G by deleting inferior replies to σ.
I It follows from a property of the index that:the index of σ in G = the index of σ in G∗.
I G∗ is also obtained from Ĝ by deleting inferior replies.
I Therefore, the index of σ in G∗ = index of σ in Ĝ.
I As σ is the unique equilibrium of Ĝ:the index of σ in Ĝ = +1.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 1: Index=DegreeThe projection map Π projects a pair (G, σ) in the equilibriumcorrespondence E to the game G ∈ G.
Definition: The degree of an equilibrium σ of a game G= the local orientation of projection map Π at (G, σ).
Theorem Govindan & Wilson (2005): Degree = Index
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 1: Index=DegreeThe projection map Π projects a pair (G, σ) in the equilibriumcorrespondence E to the game G ∈ G.
Definition: The degree of an equilibrium σ of a game G= the local orientation of projection map Π at (G, σ).
Theorem Govindan & Wilson (2005): Degree = Index
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 1: Index=DegreeThe projection map Π projects a pair (G, σ) in the equilibriumcorrespondence E to the game G ∈ G.
Definition: The degree of an equilibrium σ of a game G= the local orientation of projection map Π at (G, σ).
Theorem Govindan & Wilson (2005): Degree = Index
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 1: Index=DegreeThe projection map Π projects a pair (G, σ) in the equilibriumcorrespondence E to the game G ∈ G.
Definition: The degree of an equilibrium σ of a game G= the local orientation of projection map Π at (G, σ).
Theorem Govindan & Wilson (2005): Degree = IndexGovindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 2: Use Hopf Extension TheoremLet G be a regular game and σ a +1 equilibrium.
I Let f be the better reply map f defined in Nash’s PhDwhose fixed points are the equilibria of G.
I Since the sum of degrees over all equilibria is +1, the sumof degrees over all equilibria other than σ is zero.
I This implies that we can alter f outside a neighbourhoodU of σ so that the new map f0 has only one fixed point: σ.
I The possibility of such a construction follows from a deepresult in differential topology: Hopf Extension Theorem.
Hopf Theorem Let W be a compact, connected, oriented k+ 1dimensional manifold with boundary, and let f : ∂W → Sk be asmooth map. Then f extends to a globally defined mapF : W → Sk, with F = f , if and only if the degree of f is zero.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 2: Use Hopf Extension TheoremLet G be a regular game and σ a +1 equilibrium.
I Let f be the better reply map f defined in Nash’s PhDwhose fixed points are the equilibria of G.
I Since the sum of degrees over all equilibria is +1, the sumof degrees over all equilibria other than σ is zero.
I This implies that we can alter f outside a neighbourhoodU of σ so that the new map f0 has only one fixed point: σ.
I The possibility of such a construction follows from a deepresult in differential topology: Hopf Extension Theorem.
Hopf Theorem Let W be a compact, connected, oriented k+ 1dimensional manifold with boundary, and let f : ∂W → Sk be asmooth map. Then f extends to a globally defined mapF : W → Sk, with F = f , if and only if the degree of f is zero.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 2: Use Hopf Extension TheoremLet G be a regular game and σ a +1 equilibrium.
I Let f be the better reply map f defined in Nash’s PhDwhose fixed points are the equilibria of G.
I Since the sum of degrees over all equilibria is +1, the sumof degrees over all equilibria other than σ is zero.
I This implies that we can alter f outside a neighbourhoodU of σ so that the new map f0 has only one fixed point: σ.
I The possibility of such a construction follows from a deepresult in differential topology: Hopf Extension Theorem.
Hopf Theorem Let W be a compact, connected, oriented k+ 1dimensional manifold with boundary, and let f : ∂W → Sk be asmooth map. Then f extends to a globally defined mapF : W → Sk, with F = f , if and only if the degree of f is zero.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 2: Use Hopf Extension TheoremLet G be a regular game and σ a +1 equilibrium.
I Let f be the better reply map f defined in Nash’s PhDwhose fixed points are the equilibria of G.
I Since the sum of degrees over all equilibria is +1, the sumof degrees over all equilibria other than σ is zero.
I This implies that we can alter f outside a neighbourhoodU of σ so that the new map f0 has only one fixed point: σ.
I The possibility of such a construction follows from a deepresult in differential topology: Hopf Extension Theorem.
Hopf Theorem Let W be a compact, connected, oriented k+ 1dimensional manifold with boundary, and let f : ∂W → Sk be asmooth map. Then f extends to a globally defined mapF : W → Sk, with F = f , if and only if the degree of f is zero.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 2: Use Hopf Extension TheoremLet G be a regular game and σ a +1 equilibrium.
I Let f be the better reply map f defined in Nash’s PhDwhose fixed points are the equilibria of G.
I Since the sum of degrees over all equilibria is +1, the sumof degrees over all equilibria other than σ is zero.
I This implies that we can alter f outside a neighbourhoodU of σ so that the new map f0 has only one fixed point: σ.
I The possibility of such a construction follows from a deepresult in differential topology: Hopf Extension Theorem.
Hopf Theorem Let W be a compact, connected, oriented k+ 1dimensional manifold with boundary, and let f : ∂W → Sk be asmooth map. Then f extends to a globally defined mapF : W → Sk, with F = f , if and only if the degree of f is zero.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 2: Use Hopf Extension TheoremLet G be a regular game and σ a +1 equilibrium.
I Let f be the better reply map f defined in Nash’s PhDwhose fixed points are the equilibria of G.
I Since the sum of degrees over all equilibria is +1, the sumof degrees over all equilibria other than σ is zero.
I This implies that we can alter f outside a neighbourhoodU of σ so that the new map f0 has only one fixed point: σ.
I The possibility of such a construction follows from a deepresult in differential topology: Hopf Extension Theorem.
Hopf Theorem Let W be a compact, connected, oriented k+ 1dimensional manifold with boundary, and let f : ∂W → Sk be asmooth map. Then f extends to a globally defined mapF : W → Sk, with F = f , if and only if the degree of f is zero.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustrative Example
L RL (1, 1) (0, 0)R (0, 0) (1, 1)
I For simplicity, focus on symmetric strategies.
I A symmetric profile is represented by a number x ∈ [0, 1].
I Two strict equilibria x = 1 and x = 0 (index=degree= +1);one mixed x = 1/2 (index=degree −1).
I The Nash map f of this game is:
f(x) ≡
{x
1−2x2+x if x ∈ [0, 1/2]x−2x2+3x−11−2x2+3x−1 if x ∈ (1/2, 1]
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustrative Example
L RL (1, 1) (0, 0)R (0, 0) (1, 1)
I For simplicity, focus on symmetric strategies.
I A symmetric profile is represented by a number x ∈ [0, 1].
I Two strict equilibria x = 1 and x = 0 (index=degree= +1);one mixed x = 1/2 (index=degree −1).
I The Nash map f of this game is:
f(x) ≡
{x
1−2x2+x if x ∈ [0, 1/2]x−2x2+3x−11−2x2+3x−1 if x ∈ (1/2, 1]
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustrative Example
L RL (1, 1) (0, 0)R (0, 0) (1, 1)
I For simplicity, focus on symmetric strategies.
I A symmetric profile is represented by a number x ∈ [0, 1].
I Two strict equilibria x = 1 and x = 0 (index=degree= +1);one mixed x = 1/2 (index=degree −1).
I The Nash map f of this game is:
f(x) ≡
{x
1−2x2+x if x ∈ [0, 1/2]x−2x2+3x−11−2x2+3x−1 if x ∈ (1/2, 1]
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustrative Example
L RL (1, 1) (0, 0)R (0, 0) (1, 1)
I For simplicity, focus on symmetric strategies.
I A symmetric profile is represented by a number x ∈ [0, 1].
I Two strict equilibria x = 1 and x = 0 (index=degree= +1);one mixed x = 1/2 (index=degree −1).
I The Nash map f of this game is:
f(x) ≡
{x
1−2x2+x if x ∈ [0, 1/2]x−2x2+3x−11−2x2+3x−1 if x ∈ (1/2, 1]
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustrative Example
L RL (1, 1) (0, 0)R (0, 0) (1, 1)
I For simplicity, focus on symmetric strategies.
I A symmetric profile is represented by a number x ∈ [0, 1].
I Two strict equilibria x = 1 and x = 0 (index=degree= +1);one mixed x = 1/2 (index=degree −1).
I The Nash map f of this game is:
f(x) ≡
{x
1−2x2+x if x ∈ [0, 1/2]x−2x2+3x−11−2x2+3x−1 if x ∈ (1/2, 1]
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustrative Example
L RL (1, 1) (0, 0)R (0, 0) (1, 1)
I For simplicity, focus on symmetric strategies.
I A symmetric profile is represented by a number x ∈ [0, 1].
I Two strict equilibria x = 1 and x = 0 (index=degree= +1);one mixed x = 1/2 (index=degree −1).
I The Nash map f of this game is:
f(x) ≡
{x
1−2x2+x if x ∈ [0, 1/2]x−2x2+3x−11−2x2+3x−1 if x ∈ (1/2, 1]
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustration of Hopf Extension Theorem
Figure: Graphs of f (black) and f0 (green)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Illustration of Hopf Extension Theorem
Figure: Graphs of f (black) and f0 (green)
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 3: add payoff-irrelevant strategies
We have a map f0 with no fixed points other than σ.
I The map f0 is meant to be a “better-reply” function.
I But f0n is defined over Σ and not just Σ−n.
I Construct an equivalent game G̃ that incorporates this.
I Strategy set of player n is S̃n ≡ Sn × Sn+1, where the2nd-coordinate is payoff-irrelevant.
I Construct a map f̃0n : Σ̃−n → Σ̃n which uses player n− 1’s2nd-coordinate choice to determine nth component in f0.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 3: add payoff-irrelevant strategies
We have a map f0 with no fixed points other than σ.
I The map f0 is meant to be a “better-reply” function.
I But f0n is defined over Σ and not just Σ−n.
I Construct an equivalent game G̃ that incorporates this.
I Strategy set of player n is S̃n ≡ Sn × Sn+1, where the2nd-coordinate is payoff-irrelevant.
I Construct a map f̃0n : Σ̃−n → Σ̃n which uses player n− 1’s2nd-coordinate choice to determine nth component in f0.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 3: add payoff-irrelevant strategies
We have a map f0 with no fixed points other than σ.
I The map f0 is meant to be a “better-reply” function.
I But f0n is defined over Σ and not just Σ−n.
I Construct an equivalent game G̃ that incorporates this.
I Strategy set of player n is S̃n ≡ Sn × Sn+1, where the2nd-coordinate is payoff-irrelevant.
I Construct a map f̃0n : Σ̃−n → Σ̃n which uses player n− 1’s2nd-coordinate choice to determine nth component in f0.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 3: add payoff-irrelevant strategies
We have a map f0 with no fixed points other than σ.
I The map f0 is meant to be a “better-reply” function.
I But f0n is defined over Σ and not just Σ−n.
I Construct an equivalent game G̃ that incorporates this.
I Strategy set of player n is S̃n ≡ Sn × Sn+1, where the2nd-coordinate is payoff-irrelevant.
I Construct a map f̃0n : Σ̃−n → Σ̃n which uses player n− 1’s2nd-coordinate choice to determine nth component in f0.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 3: add payoff-irrelevant strategies
We have a map f0 with no fixed points other than σ.
I The map f0 is meant to be a “better-reply” function.
I But f0n is defined over Σ and not just Σ−n.
I Construct an equivalent game G̃ that incorporates this.
I Strategy set of player n is S̃n ≡ Sn × Sn+1, where the2nd-coordinate is payoff-irrelevant.
I Construct a map f̃0n : Σ̃−n → Σ̃n which uses player n− 1’s2nd-coordinate choice to determine nth component in f0.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Difficult Direction 3: add payoff-irrelevant strategies
We have a map f0 with no fixed points other than σ.
I The map f0 is meant to be a “better-reply” function.
I But f0n is defined over Σ and not just Σ−n.
I Construct an equivalent game G̃ that incorporates this.
I Strategy set of player n is S̃n ≡ Sn × Sn+1, where the2nd-coordinate is payoff-irrelevant.
I Construct a map f̃0n : Σ̃−n → Σ̃n which uses player n− 1’s2nd-coordinate choice to determine nth component in f0.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
The proof needs a Delaunay triangulation of Σ̃n
Figure: Horizontal axis represents the duplicated strategy set in theequivalent game. Vertical axis represents the original strategy set.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
It is NOT a Delaunay triangulation
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
It is NOT a Delaunay triangulation
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Delaunay triangulation for points in general position
I Let C = co{x0, x1, . . . , xk } in
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
VERY Difficult Direction 5: Add Bonus
I Consider a sufficiently fine Delaunay triangulation of Σ̃n.
I Add vertices of the triangulation as pure strategies to G.
I Give a bonus to the new strategies using f̃0 (followingseveral tricks, as in Govindan & Wilson 2005)
I The ρ in Delaunay triangulation will be part of the bonus.
I Conclude using that f0 has only one fixed point σ.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
VERY Difficult Direction 5: Add Bonus
I Consider a sufficiently fine Delaunay triangulation of Σ̃n.
I Add vertices of the triangulation as pure strategies to G.
I Give a bonus to the new strategies using f̃0 (followingseveral tricks, as in Govindan & Wilson 2005)
I The ρ in Delaunay triangulation will be part of the bonus.
I Conclude using that f0 has only one fixed point σ.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
VERY Difficult Direction 5: Add Bonus
I Consider a sufficiently fine Delaunay triangulation of Σ̃n.
I Add vertices of the triangulation as pure strategies to G.
I Give a bonus to the new strategies using f̃0 (followingseveral tricks, as in Govindan & Wilson 2005)
I The ρ in Delaunay triangulation will be part of the bonus.
I Conclude using that f0 has only one fixed point σ.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
VERY Difficult Direction 5: Add Bonus
I Consider a sufficiently fine Delaunay triangulation of Σ̃n.
I Add vertices of the triangulation as pure strategies to G.
I Give a bonus to the new strategies using f̃0 (followingseveral tricks, as in Govindan & Wilson 2005)
I The ρ in Delaunay triangulation will be part of the bonus.
I Conclude using that f0 has only one fixed point σ.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
VERY Difficult Direction 5: Add Bonus
I Consider a sufficiently fine Delaunay triangulation of Σ̃n.
I Add vertices of the triangulation as pure strategies to G.
I Give a bonus to the new strategies using f̃0 (followingseveral tricks, as in Govindan & Wilson 2005)
I The ρ in Delaunay triangulation will be part of the bonus.
I Conclude using that f0 has only one fixed point σ.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Importance of not adding/removing best replies in IIA
Consider the following three-player game G, where player 3 hasa unique action (is a dummy player).
G =l r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
Three Nash equilibria
I Two strict, (t, l) and (b, r) (index +1)
I One completely mixed σ, with index −1.
I Adding strategies leads to larger game where σ is theunique equilibrium.
I This cannot be done for a two player game!
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Importance of not adding/removing best replies in IIA
Consider the following three-player game G, where player 3 hasa unique action (is a dummy player).
G =l r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
Three Nash equilibria
I Two strict, (t, l) and (b, r) (index +1)
I One completely mixed σ, with index −1.
I Adding strategies leads to larger game where σ is theunique equilibrium.
I This cannot be done for a two player game!
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Importance of not adding/removing best replies in IIA
Consider the following three-player game G, where player 3 hasa unique action (is a dummy player).
G =l r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
Three Nash equilibria
I Two strict, (t, l) and (b, r) (index +1)
I One completely mixed σ, with index −1.
I Adding strategies leads to larger game where σ is theunique equilibrium.
I This cannot be done for a two player game!
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Importance of not adding/removing best replies in IIA
Consider the following three-player game G, where player 3 hasa unique action (is a dummy player).
G =l r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
Three Nash equilibria
I Two strict, (t, l) and (b, r) (index +1)
I One completely mixed σ, with index −1.
I Adding strategies leads to larger game where σ is theunique equilibrium.
I This cannot be done for a two player game!
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Importance of not adding/removing best replies in IIA
Consider the following three-player game G, where player 3 hasa unique action (is a dummy player).
G =l r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
Three Nash equilibria
I Two strict, (t, l) and (b, r) (index +1)
I One completely mixed σ, with index −1.
I Adding strategies leads to larger game where σ is theunique equilibrium.
I This cannot be done for a two player game!
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
−1 equilibrium can become unique by new strategies!
L R
Tl r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
(3, 3, 0)
B (3, 0, 1) (0, 3, 1)
L R
Tl r
t (−3, 0, 4) (1, 4, 0)b (1, 4, 0) (1, 4, 0)
(1, 0, 1)
B (3, 0, 0) (0, 3, 0)
L R
Tl r
t (1, 4, 0) (1, 4, 0)b (1, 4, 0) (−3, 0, 4)
(3, 0, 1)
B (3, 0, 0) (0, 3, 0)
Something wrong?
NO: (G, σ) is not equivalent to (Ĝ, σ) because some ofstrategies we added are best replies to the equilibrium!
Open problem: Can any equilibrium be made unique byadding (non necessarily inferior replies) strategies and players?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
−1 equilibrium can become unique by new strategies!
L R
Tl r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
(3, 3, 0)
B (3, 0, 1) (0, 3, 1)
L R
Tl r
t (−3, 0, 4) (1, 4, 0)b (1, 4, 0) (1, 4, 0)
(1, 0, 1)
B (3, 0, 0) (0, 3, 0)
L R
Tl r
t (1, 4, 0) (1, 4, 0)b (1, 4, 0) (−3, 0, 4)
(3, 0, 1)
B (3, 0, 0) (0, 3, 0)
Something wrong?
NO: (G, σ) is not equivalent to (Ĝ, σ) because some ofstrategies we added are best replies to the equilibrium!
Open problem: Can any equilibrium be made unique byadding (non necessarily inferior replies) strategies and players?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
−1 equilibrium can become unique by new strategies!
L R
Tl r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
(3, 3, 0)
B (3, 0, 1) (0, 3, 1)
L R
Tl r
t (−3, 0, 4) (1, 4, 0)b (1, 4, 0) (1, 4, 0)
(1, 0, 1)
B (3, 0, 0) (0, 3, 0)
L R
Tl r
t (1, 4, 0) (1, 4, 0)b (1, 4, 0) (−3, 0, 4)
(3, 0, 1)
B (3, 0, 0) (0, 3, 0)
Something wrong?
NO: (G, σ) is not equivalent to (Ĝ, σ) because some ofstrategies we added are best replies to the equilibrium!
Open problem: Can any equilibrium be made unique byadding (non necessarily inferior replies) strategies and players?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
−1 equilibrium can become unique by new strategies!
L R
Tl r
t (6, 6, 1) (0, 0, 1)b (0, 0, 1) (6, 6, 1)
(3, 3, 0)
B (3, 0, 1) (0, 3, 1)
L R
Tl r
t (−3, 0, 4) (1, 4, 0)b (1, 4, 0) (1, 4, 0)
(1, 0, 1)
B (3, 0, 0) (0, 3, 0)
L R
Tl r
t (1, 4, 0) (1, 4, 0)b (1, 4, 0) (−3, 0, 4)
(3, 0, 1)
B (3, 0, 0) (0, 3, 0)
Something wrong?
NO: (G, σ) is not equivalent to (Ĝ, σ) because some ofstrategies we added are best replies to the equilibrium!
Open problem: Can any equilibrium be made unique byadding (non necessarily inferior replies) strategies and players?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Extension to non generic games: a first attempt
I Fact 1: A non isolated equilibrium σ of a game G cannotbe made unique in a larger game Ĝ obtained from G byadding inferior replies (our result is sharp).
Proof: By contradiction, let σn → σ be a sequence ofequilibria of G. Since σ is the unique equilibrium of Ĝ,there is a (up to a subsequence) a fixed player i and a fixedprofitable best reply deviation θi to σn in Ĝ. By continuity,θi is a best reply to σ in Ĝ: a contradiction.
I Fact 2: A strict subset U of an equilibrium component Cin G can never be made the unique component in a largergame Ĝ obtained from G by adding inferior replies.
Proof: By contradiction, U must be closed and since C isconnected, there is σ ∈ U , and σn ∈ C but not in U suchthat σn → σ, then we use same argument as above.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Extension to non generic games: a first attempt
I Fact 1: A non isolated equilibrium σ of a game G cannotbe made unique in a larger game Ĝ obtained from G byadding inferior replies (our result is sharp).
Proof: By contradiction, let σn → σ be a sequence ofequilibria of G. Since σ is the unique equilibrium of Ĝ,there is a (up to a subsequence) a fixed player i and a fixedprofitable best reply deviation θi to σn in Ĝ. By continuity,θi is a best reply to σ in Ĝ: a contradiction.
I Fact 2: A strict subset U of an equilibrium component Cin G can never be made the unique component in a largergame Ĝ obtained from G by adding inferior replies.
Proof: By contradiction, U must be closed and since C isconnected, there is σ ∈ U , and σn ∈ C but not in U suchthat σn → σ, then we use same argument as above.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Extension to non generic games: a first attempt
I Fact 1: A non isolated equilibrium σ of a game G cannotbe made unique in a larger game Ĝ obtained from G byadding inferior replies (our result is sharp).
Proof: By contradiction, let σn → σ be a sequence ofequilibria of G. Since σ is the unique equilibrium of Ĝ,there is a (up to a subsequence) a fixed player i and a fixedprofitable best reply deviation θi to σn in Ĝ. By continuity,θi is a best reply to σ in Ĝ: a contradiction.
I Fact 2: A strict subset U of an equilibrium component Cin G can never be made the unique component in a largergame Ĝ obtained from G by adding inferior replies.
Proof: By contradiction, U must be closed and since C isconnected, there is σ ∈ U , and σn ∈ C but not in U suchthat σn → σ, then we use same argument as above.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Extension to non generic games: a first attempt
I Fact 1: A non isolated equilibrium σ of a game G cannotbe made unique in a larger game Ĝ obtained from G byadding inferior replies (our result is sharp).
Proof: By contradiction, let σn → σ be a sequence ofequilibria of G. Since σ is the unique equilibrium of Ĝ,there is a (up to a subsequence) a fixed player i and a fixedprofitable best reply deviation θi to σn in Ĝ. By continuity,θi is a best reply to σ in Ĝ: a contradiction.
I Fact 2: A strict subset U of an equilibrium component Cin G can never be made the unique component in a largergame Ĝ obtained from G by adding inferior replies.
Proof: By contradiction, U must be closed and since C isconnected, there is σ ∈ U , and σn ∈ C but not in U suchthat σn → σ, then we use same argument as above.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
First attempt does not extend to all games!
I Corollary: Only an (entire) equilibrium component canbe made unique by adding inferior replies.
I BIG PROBLEM: There are games where no equilibriumcomponent can be made unique by adding inferior replies.
Proof: In Hauk & Hurkens example, no equilibriumcomponent has index +1.
I An open problem:
Can a +1 index equilibrium component be made unique byadding inferior replies?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
First attempt does not extend to all games!
I Corollary: Only an (entire) equilibrium component canbe made unique by adding inferior replies.
I BIG PROBLEM: There are games where no equilibriumcomponent can be made unique by adding inferior replies.
Proof: In Hauk & Hurkens example, no equilibriumcomponent has index +1.
I An open problem:
Can a +1 index equilibrium component be made unique byadding inferior replies?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
First attempt does not extend to all games!
I Corollary: Only an (entire) equilibrium component canbe made unique by adding inferior replies.
I BIG PROBLEM: There are games where no equilibriumcomponent can be made unique by adding inferior replies.
Proof: In Hauk & Hurkens example, no equilibriumcomponent has index +1.
I An open problem:
Can a +1 index equilibrium component be made unique byadding inferior replies?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
First attempt does not extend to all games!
I Corollary: Only an (entire) equilibrium component canbe made unique by adding inferior replies.
I BIG PROBLEM: There are games where no equilibriumcomponent can be made unique by adding inferior replies.
Proof: In Hauk & Hurkens example, no equilibriumcomponent has index +1.
I An open problem:
Can a +1 index equilibrium component be made unique byadding inferior replies?
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.
I A2: IIA: If a set is sustainable for a game, it remainssustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:
its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Second attempt: an axiomatic approachWe want to construct a correspondence Φ that selects for eachfinite game G, a set of subsets of Nash equilibria of G (calledsustainable sets for G) satisfying the following axioms:
I A1: Existence: Every game has a sustainable set.I A2: IIA: If a set is sustainable for a game, it remains
sustainable after adding or removing inferior replies.
I A3: Invariance: Equivalent games (obtained by additivepositive transformation of payoffs, or addition/deletion ofduplicate of mixte strategies) have equivalent solutions.
I A4: Robustness: If a set is sustainable for G, then anynearby game has a nearby sustainable set.
I A5: Minimality: If Ψ satisfies A1 to A4 then Φ ⊂ Ψ.
Theorem Φ satisfies A1 to A5 iff it associates to each game:its set of Nash equilibrium components with positive index.
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Robustness in Myerson’s Essai
Govindan, Laraki, & Pahl
Myerson Conjecture Hofbauer Formulation Hofbauer-Myerson Conjecture Proof Extending Sustainability
Govindan, Laraki, & Pahl
Myerson ConjectureHofbauer FormulationHofbauer-Myerson ConjectureProofExtending Sustainability