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  1. ❇❛❝❦❣r♦✉♥❞ ❆♥❛❧②s✐s ❉✐s❝✉ss✐♦♥ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t♦❝❦❤♦❧♠✱ ❖❝t♦❜❡r✱ ✷✵✶✹ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✶ ✴ ✶✻

  2. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ ❙❡t✉♣ ✭▼❛②♥❛r❞✲❙♠✐t❤ ❛♥❞ Pr✐❝❡✱ ✶✾✼✸✮ ❆❣❡♥ts ✐♥ ❛ ❧❛r❣❡ ♣♦♣✉❧❛t✐♦♥ ❛r❡ r❛♥❞♦♠❧② ❛♥❞ ❛♥♦♥②♠♦✉s❧② ♠❛t❝❤❡❞ t♦ ♣❧❛② ❛ t✇♦✲♣❧❛②❡r s②♠♠❡tr✐❝ ❣❛♠❡ G = ( A , π ) ✳ A ✲ ✜♥✐t❡ s❡t ♦❢ ❛❝t✐♦♥s✳ ❙tr❛t❡❣② ✭♦r ♠✐①❡❞ ❛❝t✐♦♥✮✿ σ ∈ ∆( A ) ✳ π ✲ ❣❛♠❡ ♣❛②♦✛ ✭r❡♣r❡s❡♥t✐♥❣ ✜t♥❡ss✴s✉❝❝❡ss✮✳ ❘❡str✐❝t✐♦♥ t♦ s②♠♠❡tr✐❝ ❣❛♠❡ ✐s ❲▲❖● ✭s❡❡ ❙❡❧t❡♥✱ ✶✾✽✵✮✳ P❛②♦✛✲♠♦♥♦t♦♥✐❝ ❞②♥❛♠✐❝s ✭❡✳❣✳✱ r❡♣❧✐❝❛t♦r ❞②♥❛♠✐❝s✮✿ ▼♦r❡ s✉❝❝❡ss❢✉❧ str❛t❡❣✐❡s ❜❡❝♦♠❡ ♠♦r❡ ❢r❡q✉❡♥t✳ ❘❛r❡❧② ❢❡✇ ❛❣❡♥ts ✭✏♠✉t❛♥ts✑✮ ❛❞♦♣t ♥❡✇ str❛t❡❣✐❡s✳ ❍♦✇ t♦ ❝❤❛r❛❝t❡r✐③❡ st❛❜❧❡ ♣♦♣✉❧❛t✐♦♥ st❛t❡s❄ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✷ ✴ ✶✻

  3. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ ❙t❛t✐❝ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t✿ σ ∗ ✐s ❛♥ ❊❙❙ ✐❢ ❛♥② s✉✣❝✐❡♥t❧② s♠❛❧❧ ❣r♦✉♣ ♦❢ ♠✉t❛♥ts ✐s str✐❝t❧② ♦✉t♣❡r❢♦r♠❡❞✳ ❚✇♦ ❡q✉✐✈❛❧❡♥t ❢♦r♠✉❧❛t✐♦♥s ✭✜♥✐t❡ ❣❛♠❡s✮✿ ❖r✐❣✐♥❛❧ ❞❡✜♥✐t✐♦♥ ✭▼❙✫P✱ ✶✾✼✸✮✿ ✭✶✮ σ ∗ ✐s ❛ s②♠♠❡tr✐❝ ◆❛s❤ ✶ ❡q✉✐❧✐❜r✐✉♠❀ ✭✷✮ π ( σ , σ ∗ ) = π ( σ ∗ , σ ∗ ) ⇒ π ( σ , σ ) < π ( σ ∗ , σ ) ✳ ❯♥✐❢♦r♠ ✐♥✈❛s✐♦♥ ❜❛rr✐❡r ✭❍♦❢❜❛✉❡r ❛t ❛❧✳✱ ✶✾✼✾✮✿ ∃ ¯ ε > ✵✱ ✷ ∀ σ ′ � = σ ∗ , ✵ < ε < ¯ σ ∗ , σ σ ∗ , ε , σ ′ � σ ′ , σ σ ∗ , ε , σ ′ � � � ε ✿ π > π ) ✳ σ σ ∗ , ε , σ ′ = ( ✶ − ε ) · σ ∗ + ε · σ ′ ✳ ◆❡✉tr❛❧ st❛❜✐❧✐t② ✭◆❙❙✮✿ ✏ ≤ ✑ r❡♣❧❛❝❡s ✏ < ✑✳ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✸ ✴ ✶✻

  4. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ σ ∗ ✐s ✇❡❛❦❧② ✭▲②❛♣✉♥♦✈✮ st❛❜❧❡ ✐❢ ❛❢t❡r ❛ s♠❛❧❧ ✐♥✈❛s✐♦♥✱ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❝♦♠♣♦s✐t✐♦♥ r❡♠❛✐♥s ❝❧♦s❡ t♦ σ ∗ ✳ ■t ✐s str✐❝t❧② ✭❛s②♠♣t♦t✐❝❛❧❧②✮ st❛❜❧❡ ✐❢✱ ✐♥ ❛❞❞✐t✐♦♥✱ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❝♦♥✈❡r❣❡s ❜❛❝❦ t♦ σ ∗ ✳ ❊❙❙ ✐s str✐❝t❧② st❛❜❧❡ ✐♥ s♠♦♦t❤ ♣❛②♦✛✲♠♦♥♦t♦♥✐❝ ❞②♥❛♠✐❝s ✭❚❛②❧♦r ✫ ❏♦♥❦❡rs✱ ✶✾✼✽❀ ❈r❡ss♠❛♥✱ ✶✾✾✼❀ ❙❛♥❞❤♦❧♠✱ ✷✵✶✵✮✳ ◆❙❙ ✐s ✇❡❛❦❧② st❛❜❧❡ ✐♥ t❤❡ r❡♣❧✐❝❛t♦r ❞②♥❛♠✐❝s ✭❚❤♦♠❛s✱ ✶✾✽✺❀ ❇♦♠③❡ ✫ ❲❡✐❜✉❧❧✱ ✶✾✾✺✮✳ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✹ ✴ ✶✻

  5. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ ◆♦♥✲❙✐♠✉❧t❛♥❡♦✉s ●❛♠❡s ❙✐♠✉❧t❛♥❡♦✉s ❣❛♠❡✿ ❊❙❙ ❝❛♣t✉r❡s st❛❜✐❧✐t② ✇❡❧❧✳ ❊❙❙ ❛❧♠♦st ♥❡✈❡r ❡①✐sts ✐♥ ❣❛♠❡s ✇✐t❤ ♠✉❧t✐♣❧❡ st❛❣❡s✿ σ ∗ ❞♦❡s♥✬t r❡❛❝❤ ❛❧❧ ✐♥❢♦r♠❛t✐♦♥ s❡ts ⇒ σ ∗ ✐s♥✬t ❛♥ ❊❙❙✳ ◆❙❙ ✐s ✏t♦♦ ✇❡❛❦✑ ✭❝❛♥ ❜❡ ❛ ✇❡❛❦❧② ❞♦♠✐♥❛t❡❞ str❛t❡❣②✮✳ ❊①❛♠♣❧❡s✳✳✳ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✺ ✴ ✶✻

  6. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ ❊①❛♠♣❧❡s✿ ❊❙❙ ✐s t♦♦ ❙tr♦♥❣ ✫ ◆❙❙ ✐s t♦♦ ❲❡❛❦ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✻ ✴ ✶✻

  7. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ ▲✐♠✐t ❊❙❙ ✭❙❡❧t❡♥✱ ✶✾✽✸✮ P❡rt✉r❜❛t✐♦♥ ζ ✿ ❛ss✐❣♥s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ♥✉♠❜❡r ζ ( a , h ) ❢♦r ❡❛❝❤ ❛❝t✐♦♥ a ❛t ❡❛❝❤ ✐♥❢♦r♠❛t✐♦♥ s❡t h ✳ ▼❛①✐♠❛❧ tr❡♠❜❧❡ ✲ M ( ζ ) = ♠❛① ( a , h ) ζ ( a , h ) . P❡rt✉r❜❡❞ ❣❛♠❡✿ ❡❛❝❤ ♣❧❛②❡r ♠✉st ❝❤♦♦s❡ ❡❛❝❤ ❛❝t✐♦♥ ❛ ❛t ✐♥❢♦r♠❛t✐♦♥ s❡t ❤ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛t ❧❡❛st ζ ( a , h ) ✳ σ ∗ ✐s ❛ ❧✐♠✐t ❊❙❙ ✐❢ ✐t ✐s ❛ ❧✐♠✐t ♦❢ ❊❙❙s ♦❢ ♣❡rt✉r❜❡❞ ❣❛♠❡s ✇❤❡♥ t❤❡ ♠❛①✐♠❛❧ tr❡♠❜❧❡s ❝♦♥✈❡r❣❡ t♦ ✵✳ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✼ ✴ ✶✻

  8. ❇❛❝❦❣r♦✉♥❞ ❊❙❙✱ ◆❙❙ ✫ ❉②♥❛♠✐❝ ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ❉✐s❝✉ss✐♦♥ ▲✐♠✐t ❊❙❙ Pr♦♣❡rt✐❡s ♦❢ ▲✐♠✐t ❊❙❙ ❊❙❙ ⇒ ❧✐♠✐t ❊❙❙ ⇒ s②♠♠❡tr✐❝ s❡q✉❡♥t✐❛❧ ❡q✉✐❧✐❜r✐✉♠✳ ❚r❡♠❜❧❡s ❛❧❧♦✇ ❛❧❧ ✐♥❢♦r♠❛t✐♦♥ s❡ts t♦ ❜❡ r❡❛❝❤❡❞ ⇒ ❧✐♠✐t ❊❙❙ ❡①✐sts ✐♥ ♠❛♥② ❡①t❡♥s✐✈❡✲❢♦r♠ ❣❛♠❡s✳ ■♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ✏❝❛♣t✉r❡s✑ st❛❜✐❧✐t② ✇❡❧❧ ❊✳❣✳✱ ❙❛♠✉❡❧s♦♥ ✫ ❩❤❛♥❣ ✭✶✾✾✷✮❀ ❑✐♠ ✭✶✾✾✸✱ ✶✾✾✹✮❀ ❇♦❧t♦♥ ✭✶✾✾✼✮❀ ▲❡✐♠❛r ✭✶✾✾✼✮❀ ❍❡❧❧❡r ✭✷✵✶✹✱ ❚❊✮✳ ❱❛♥ ❉❛♠♠❡ ✭✶✾✽✼✮ ✏♣r♦✈❡❞✑✿ ❧✐♠✐t ❊❙❙ ⇒ ◆❙❙✳ ❍♦✇❡✈❡r✱ ✐t t✉r♥s ♦✉t t❤❛t t❤❡ ♣r♦♦❢ ❝♦♥t❛✐♥s ❛ ♠✐st❛❦❡✳ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✽ ✴ ✶✻

  9. ❇❛❝❦❣r♦✉♥❞ ❈♦✉♥t❡r ❊①❛♠♣❧❡ ❆♥❛❧②s✐s ❯♥✐❢♦r♠ ▲✐♠✐t ❊❙❙ ❉✐s❝✉ss✐♦♥ ❨✉✈❛❧ ❍❡❧❧❡r ✭❖①❢♦r❞✮ ❙t❛❜✐❧✐t② ❛♥❞ ❚r❡♠❜❧❡s ✐♥ ❊①t❡♥s✐✈❡✲❢♦r♠ ●❛♠❡s ✾ ✴ ✶✻

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