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I've noticed an interesting pattern when changing the order of magnitude in an expected value problem. I'll use the most recent example I've stumbled across. Let probabilities Π={0.41, 0.49,0.57,0.65,0.72}, negative utilities Φ={29,30,31,32,33}, and positive utilities Ψ={165,215,275,335,415}. In this case and in all cases, all variables are positively correlated. Now, let Φ'=100Φ and let Ψ'=0.01Ψ. Finally, let Ζ=ΦΠ+Ψ(1-Π), Α=Φ'Π+Ψ(1-Π), Β=ΦΠ+Ψ'(1-Π), and Γ=Φ'Π+Ψ'(1-Π), where Ζ, Α, Β, and Γ are all expected value equations. It is surprising to me that Ζ is a parabolic function, where MAX(Ζ)=φ3π3+ψ3(1-π3), but Α, Β, and Γ are linear, where the function is maximized as the with the fifth element of their respective vectors.
Why is it that changing the order of magnitude of the utility vectors changes a quadratic function into a linear function? I'm not a mathematician, so please ask if I'm not making sense with my notation or if I'm not using the correct vocabulary. Thank you Wikipedians! 47.187.81.103 ( talk) 16:15, 11 September 2020 (UTC)
Mathematics desk | ||
---|---|---|
< September 10 | << Aug | September | Oct >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
I've noticed an interesting pattern when changing the order of magnitude in an expected value problem. I'll use the most recent example I've stumbled across. Let probabilities Π={0.41, 0.49,0.57,0.65,0.72}, negative utilities Φ={29,30,31,32,33}, and positive utilities Ψ={165,215,275,335,415}. In this case and in all cases, all variables are positively correlated. Now, let Φ'=100Φ and let Ψ'=0.01Ψ. Finally, let Ζ=ΦΠ+Ψ(1-Π), Α=Φ'Π+Ψ(1-Π), Β=ΦΠ+Ψ'(1-Π), and Γ=Φ'Π+Ψ'(1-Π), where Ζ, Α, Β, and Γ are all expected value equations. It is surprising to me that Ζ is a parabolic function, where MAX(Ζ)=φ3π3+ψ3(1-π3), but Α, Β, and Γ are linear, where the function is maximized as the with the fifth element of their respective vectors.
Why is it that changing the order of magnitude of the utility vectors changes a quadratic function into a linear function? I'm not a mathematician, so please ask if I'm not making sense with my notation or if I'm not using the correct vocabulary. Thank you Wikipedians! 47.187.81.103 ( talk) 16:15, 11 September 2020 (UTC)