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❉❡❝✐s✐♦♥ ❚❤❡♦r② ▼❡❡ts ▲✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥ ✭❇❡②♦♥❞ ❈♦♠♣✉t❛t✐♦♥✮ ❚✳ ❆✉❣✉st✐♥ ●✳ ❙❝❤♦❧❧♠❡②❡r ❈✳ ❏❛♥s❡♥ ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤ ❊❈❙◗❆❘❯ ✷✵✶✼✱ ✶✸t❤ ♦❢ ❏✉❧②✱ ▲✉❣❛♥♦ ✶ ✴ ✶✻

❙❝♦♣❡ ♦❢ t❤❡ t❛❧❦ ◮ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ t❤❡♦r② ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ❛ ♣♦✇❡r❢✉❧ t♦♦❧ ❢♦r ✭✐♠♣r❡❝✐s❡✮ ❞❡❝✐s✐♦♥ t❤❡♦r② r❡❣❛r❞✐♥❣ ❜♦t❤ ◮ ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ♦❢ ♦♣t✐♠❛❧ ❛❝ts ✇✳r✳t✳ ❝♦♠♣❧❡① ❝r✐t❡r✐❛ ✭❝❢✳✱ ❡✳❣✳✱ ❑✐❦✉t✐ ❡t ❛❧✳ ✭✷✵✶✷✮ ♦r ❯t❦✐♥ ❛♥❞ ❆✉❣✉st✐♥ ✭✷✵✵✺✮✮ ◮ ♣r♦✈✐❞✐♥❣ t❤❡♦r❡t✐❝❛❧ ✐♥s✐❣❤ts ♦♥ ♣r♦♣❡rt✐❡s ♦❢ ♦♣t✐♠❛❧ ❛❝ts ✭❝❢✳✱ ❡✳❣✳✱ ❲❡✐❝❤s❡❧❜❡r❣❡r ✭✶✾✾✻✮✮ ◮ ♦✉r ♣❛♣❡r ♣r❡s❡♥ts s♦♠❡ ♥❡✇ r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ ❜♦t❤ r❡❣❛r❞s ✐♥❝❧✉❞✐♥❣ ◮ ❧✐♥❡❛r ♣r♦❣r❛♠s ❢♦r ❍♦❞❣❡s ❛♥❞ ▲❡❤♠❛♥♥ ❛♥❞ ❲❛❧❧❡②✬s ♠❛①✐♠❛❧✐t② ◮ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❧❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦rs ❛♥❞ ●❛♠♠❛✲▼❛①✐♠✐♥ ✷ ✴ ✶✻

❙❡t✉♣ ❛♥❞ ♥♦t❛t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ st❛♥❞❛r❞ ♠♦❞❡❧ ♦❢ ✭✜♥✐t❡✮ ❝❛r❞✐♥❛❧ ❞❡❝✐s✐♦♥ t❤❡♦r②✿ ◮ A = { a ✶ , . . . , a n } ✿ s❡t ♦❢ ❛❝ts ◮ Θ = { θ ✶ , . . . , θ m } ✿ s❡t ♦❢ st❛t❡s ♦❢ t❤❡ ✇♦r❧❞ ◮ u : A × Θ → R ✿ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✱ ✇❤❡r❡ u ij := u ( a i , θ j ) ✐s t❤❡ ✉t✐❧✐t② ♦❢ ❝❤♦♦s✐♥❣ ❛❝t a i ❣✐✈❡♥ θ j ✐s t❤❡ tr✉❡ st❛t❡ ♦❢ t❤❡ ✇♦r❧❞ u ( a i , θ j ) θ ✶ · · · θ m u ( a ✶ , θ ✶ ) · · · u ( a ✶ , θ m ) a ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ · · · ✳ u ( a n , θ ✶ ) · · · u ( a n , θ m ) a n ◮ ❢♦r ❡✈❡r② a ∈ A ✱ ❞❡✜♥❡ u a : Θ → R ❜② u a ( θ ) := u ( a , θ ) ❢♦r ❛❧❧ θ ∈ Θ ◮ ❢♦r ❡✈❡r② θ ∈ Θ ✱ ❞❡✜♥❡ u θ : A → R ❜② u θ ( a ) := u ( a , θ ) ❢♦r ❛❧❧ a ∈ A ✸ ✴ ✶✻

❙❡t✉♣ ❛♥❞ ♥♦t❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♥t❡①t✱ ✇❡ ❛❧s♦ ❛❧❧♦✇ ❢♦r r❛♥❞♦♠✐③❡❞ ❛❝ts✿ ◮ ❝❛❧❧ ❡✈❡r② ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ λ ♦♥ ( A , ✷ A ) ❛ r❛♥❞♦♠✐③❡❞ ❛❝t ❛♥❞ ❞❡♥♦t❡ ❜② G ( A ) t❤❡ s❡t ♦❢ ❛❧❧ r❛♥❞♦♠✐③❡❞ ❛❝ts ◮ ❝❤♦♦s✐♥❣ λ ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❧❡❛✈✐♥❣ t❤❡ ✜♥❛❧ ❞❡❝✐s✐♦♥ t♦ ❛ r❛♥❞♦♠ ❡①♣❡r✐♠❡♥t ✇❤✐❝❤ ②✐❡❧❞s ❛❝t a i ✇✐t❤ ♣r♦❜❛❜✐❧✐t② λ ( { a i } ) � u θ � ◮ ❡✈❛❧✉❛t❡ ❝❤♦♦s✐♥❣ λ ❣✐✈❡♥ θ ❜② G ( u )( λ, θ ) := E λ ◮ ❢♦r λ ∈ G ( A ) ✱ ❞❡✜♥❡ G ( u ) λ : Θ → R ❜② G ( u ) λ ( θ ) := G ( u )( λ, θ ) ◮ ✐❞❡♥t✐❢② a ∈ A ✇✐t❤ δ a ∈ G ( A ) ❛♥❞ ♦❜s❡r✈❡ u ( a , θ ) = G ( u )( δ a , θ ) ✹ ✴ ✶✻

❘❛♥❞♦♠✐③❛t✐♦♥✿ ❆ t♦② ❡①❛♠♣❧❡ ◮ ❈♦♥s✐❞❡r ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t✇♦ ♣❧❛②❡rs✿ P✐♥❦② ✭r♦✇s✮ ❛♥❞ ❇r❛✐♥ ✭❝♦❧✉♠♥s✮ ◮ P✐♥❦② ❝❤♦♦s❡s ♠♦✈❡s P = { p ✶ , p ✷ } ✱ ❇r❛✐♥ r❡❛❝ts ❜② ♠♦✈❡s B = { b ✶ , b ✷ } ◮ P✐♥❦②✬s ✉t✐❧✐t② u p : P × B → R ✐s ❣✐✈❡♥ ❜② t❤❡ ❜❡❧♦✇ t❛❜❧❡ ◮ ❇r❛✐♥✬s ✉t✐❧✐t② u b : B × P → R ✐s ❣✐✈❡♥ ❜② u b ( b , p ) := − u p ( p , b ) u p ( · ) P✐♥❦②✬s r❡✇❛r❞ b ✶ b ✷ p ✶ ✶✵ ✷✵ ✶✵ p ✷ ✸✵ ✺ ✺ � p ✶ � p ✷ ◮ P✐♥❦② t♦ss❡s ❛ ✭❢❛✐r✮ ❝♦✐♥✱ ✐✳❡✳ ❝❤♦♦s❡s r❛♥❞♦♠✐③❡❞ ❛❝t λ ≈ ✳ ✵ . ✺ ✵ . ✺ ◮ ❍❡ r❡❝❡✐✈❡s r❡✇❛r❞ ♦❢ ♠✐♥ b G ( u p )( λ, b ) = ✶✷ . ✺ . ✺ ✴ ✶✻

❚✇♦ ✇❛②s ♦❢ ✐♥❝♦r♣♦r❛t✐♥❣ ✐♠♣❡r❢❡❝t ♣r✐♦r ❦♥♦✇❧❡❞❣❡ ❈♦♥s✐❞❡r❡❞ ❤❡r❡✿ ❉❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ st❛t❡s Θ ✳ ■❢ ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ✐s ♣r❡❝✐s❡❧② ❣✐✈❡♥ ❜② ❛♥ ✭✉♥❞♦✉❜t❡❞✮ ♣r♦❜❛❜✐❧✐t② ♦♥ t❤❡ st❛t❡ s♣❛❝❡✱ ❛❝ts ❛r❡ ♠♦st ❝♦♠♠♦♥❧② r❛♥❦❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡✐r ❡①♣❡❝t❡❞ ✉t✐❧✐t② ✈❛❧✉❡s✳ ❖t❤❡r✇✐s❡ ✭♦❢ ✐♥t❡r❡st ❤❡r❡✮✱ ✇❡ ❞✐st✐♥❣✉✐s❤ t✇♦ ❞✐✛❡r❡♥t ❝❛s❡s✿ ✭✶✮ ❯♥❝❡rt❛✐♥t② ❛❜♦✉t ♣r❡❝✐s❡ ♣r♦❜❛❜✐❧✐t✐❡s✿ ❚❤❡r❡ ✐s ❛ ♣r❡❝✐s❡ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② π ♦♥ (Θ , ✷ Θ ) ❛✈❛✐❧❛❜❧❡✱ ❤♦✇❡✈❡r✱ t❤❡r❡ ✐s s♦♠❡ ❞♦✉❜t ❛❜♦✉t ✐ts ❢✉❧❧ ❛♣♣r♦♣r✐❛t❡♥❡ss✳ ❊①❛♠♣❧❡✿ Pr✐♦r ❛✈❛✐❧❛❜❧❡ ❢♦r ❛♥ ❡①♣❡r✐♠❡♥t❀ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ s❡t✉♣ ✭✷✮ ■♠♣r❡❝✐s❡ ♣r♦❜❛❜✐❧✐t✐❡s✿ ❆ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ π ♦♥ t❤❡ st❛t❡ s♣❛❝❡ Θ ❝❛♥♥♦t ❜❡ ❢✉❧❧② s♣❡❝✐✜❡❞✳ ■♥st❡❛❞ ❛ ❝r❡❞❛❧ s❡t M ♦❢ ♣r✐♦r ♣r♦❜❛❜✐❧✐t✐❡s ✐s ❝♦♠✲ ♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ❊①❛♠♣❧❡✿ ❊✈❡♥t E ✶ ✐s ❛t ❧❡❛st ❛s ❧✐❦❡❧② ❛s E ✷ ✱ ✐✳❡✳ M = { π | π ( E ✶ ) ≥ π ( E ✷ ) } ✻ ✴ ✶✻

✭✶✮ ❯♥❝❡rt❛✐♥t② ❛❜♦✉t ♣r❡❝✐s❡ ♣r✐♦rs✿ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥ ❖♥❡ ❝♦♠♠♦♥ ✇❛② t♦ ❞❡❛❧ ✇✐t❤ s✐t✉❛t✐♦♥ ✭✶✮ ✐s t❤❡ ❞❡❝✐s✐♦♥ ❝r✐t❡r✐♦♥ ♦❢ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥✱ ✇❤✐❝❤ ❧✐♥❡❛r❧② tr❛❞❡s ♦❢ ❜❡t✇❡❡♥ ♠❛①✐♠✐♥ ❛♥❞ ❡①♣❡❝t❡❞ ✉t✐❧✐t②✳ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥ ♦♣t✐♠❛❧✐t② ▲❡t π ❞❡♥♦t❡ s♦♠❡ ♣r✐♦r ♦♥ (Θ , ✷ Θ ) ❛♥❞ ❧❡t α ∈ [ ✵ , ✶ ] ❡①♣r❡ss t❤❡ ❛❣❡♥t✬s tr✉st ✐♥ ✐ts ❛♣♣r♦♣r✐❛t❡♥❡ss✳ ❚❤❡ ❢✉♥❝t✐♦♥ Φ π,α : G ( A ) → R ❞❡✜♥❡❞ ❜② � � Φ π,α ( λ ) = ( ✶ − α ) · ♠✐♥ θ G ( u )( λ, θ ) + α · E π G ( u ) λ � �� � � �� � Expected utility Maximin utility ✐s ❝❛❧❧❡❞ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥✲❝r✐t❡r✐♦♥ ✇✳r✳t✳ ( π, α ) ✳ ❆♥② r❛♥❞♦♠✐③❡❞ ❛❝t λ ∗ ∈ G ( A ) ♠❛①✐♠✐③✐♥❣ t❤❡ ❝r✐t❡r✐♦♥ ✐s t❤❡♥ ❝❛❧❧❡❞ Φ π,α ✲♦♣t✐♠❛❧✳ ◆❛t✉r❛❧ q✉❡st✐♦♥✿ ❍♦✇ t♦ ❞❡t❡r♠✐♥❡✴❝♦♠♣✉t❡ Φ π,α ✲♦♣t✐♠❛❧ ❛❝ts❄ ✼ ✴ ✶✻

❉❡t❡r♠✐♥✐♥❣ ♦♣t✐♠❛❧ ❛❝ts ✉♥❞❡r ✭✶✮ ❖♣t✐♠❛❧ r❛♥❞♦♠✐③❡❞ ❛❝ts ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝r✐t❡r✐♦♥ ♦❢ ❍♦❞❣❡s ❛♥❞ ▲❡❤♠❛♥♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠✿ ❍♦❞❣❡s ❛♥❞ ▲❡❤♠❛♥♥ ❛❝ts ❈♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ n � ( ✶ − α ) · ( w ✶ − w ✷ ) + α · E π ( u a i ) · λ i − → ♠❛① ( w ✶ , w ✷ ,λ ✶ ,...,λ n ) i = ✶ ✇✐t❤ ❝♦♥str❛✐♥ts ( w ✶ , w ✷ , λ ✶ , . . . , λ n ) � ✵ ❛♥❞ • � n i = ✶ λ i = ✶ • w ✶ − w ✷ � � n i = ✶ u ij · λ i ❢♦r ❛❧❧ j = ✶ , . . . , m ❚❤❡♥ ❡✈❡r② ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ( w ∗ ✶ , w ∗ ✷ , λ ∗ ✶ , . . . , λ ∗ n ) ✐♥❞✉❝❡s ❛ Φ π,α ✲♦♣t✐♠❛❧ r❛♥❞♦♠✐✲ ③❡❞ ❛❝t λ ∗ ∈ G ( A ) ❜② s❡tt✐♥❣ λ ∗ ( { a i } ) := λ ∗ i ✳ ✽ ✴ ✶✻