Secretary_Game_With_Rejection.pdf

In[31]:= A[k_, m_] = Gamma[m + 2] / Gamma[k + 1] / Gamma[m - k + 1];
(*expected value of x*)
xexp[k_, m_, n_] = A[k, m] / 2 *
NIntegrate[x^(m - k) * (1 - x)^k * (x * x^(1 - m + n) + (1 + x) * (1 - x^(1 - m + n))), {x, 0, 1}];
N[xexp[3, 37, 0, 100]]
NIntegrate: The integrand (1 - x)k
x-k+m
x2-m+n
+ (1 + x) 1 - x1+Times[2]+n
 has evaluated to non-numerical
values for all sampling points in the region with boundaries {{0, 1}}.
NIntegrate: The integrand (1 - x)k
x-k+m
x2-m+n
+ (1 + x) 1 - x1+Times[2]+n
 has evaluated to non-numerical
values for all sampling points in the region with boundaries {{0, 1}}.
Out[33]= 0.940033
In[27]:= Integrate[(n - m + 1) * y^(n - m), {y, 0, x}]
Out[27]= x1-m+n
if Re[m - n] < 1
(*now we need to adjust for the rate of being rejected*)
success[n_, m_, x_] = (1 - x^(1 - m + n));
failure[n_, m_, x_] = x^(1 - m + n);
(*r is the acceptance rate*)
(*the question we are asking
is: given that the greatest point in the test set is greater than x,
how many points are greater than x*)
(*we have to untangle success again*)
successold[n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m), {y, x, 1}];
(*now given y, we can set up the space between
x and y and compare it to the space below y itself*)
(*deprecated, see below*)
(*greaterx[n_,m_,x_]=Integrate[(n-m+1)*y^(n-m)*(y-x)/y*(n-m),{y,x,1}]*)
(*for each greater than x point y,
we find the number of points that are between x and y, then add them together*)
rejected[n_, m_, x_, r_] = success[n, m, x] * (1 - r)^(greaterx[n, m, x]);
notrejected[n_, m_, x_, r_] = success[n, m, x] * (1 - (1 - r)^(greaterx[n, m, x]));
xexp[k_, m_, n_, r_] =
A[k, m] / 2 * NIntegrate[x^(m - k) * (1 - x)^k * (x * (failure[n, m, x] + rejected[n, m, x, r]) +
(1 + x) * notrejected[n, m, x, r]), {x, 0, 1}];
Plot[N[success[100, 37, x]], {x, 0, 1}, PlotRange → {-0.1, 1}]
(*makes sense, if the first number is large, it's hard to beat it*)
Plot[N[rejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]
Plot[N[notrejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]

Out[169]=
-
(-m + n) (1 - m + n) m + n (-1 + x) + x - m x - x1-m+n

(-1 + m - n) (m - n)
if condition
NIntegrate: The integrand TagBox[TemplateBox[{RowBox[{SuperscriptBox[RowBox[{(, RowBox[{1, -, x}], )}], k],
, SuperscriptBox[x, RowBox[{RowBox[{-, k}], +, m}]],
, RowBox[{(, RowBox[{RowBox[{RowBox[{(, RowBox[{1, -, SuperscriptBox[RowBox[{Plus,
[, RowBox[{, 2, }], ]}], RowBox[{Times, [, RowBox[{, 6, }], ]}]]}],
)}], , RowBox[{(, RowBox[{1, +, x}], )}], , RowBox[{(, RowBox[{1,
-, SuperscriptBox[x, RowBox[{Plus, [, RowBox[{, 3, }], ]}]]}], )}]}],
+, RowBox[{x, , RowBox[{(, RowBox[{SuperscriptBox[x, RowBox[{1, +, RowBox[{Times,
[, RowBox[{, 2, }], ]}], +, n}]],
+, RowBox[{SuperscriptBox[RowBox[{Plus, [, RowBox[{, 2, }],
]}], RowBox[{Times, [, RowBox[{, 6, }], ]}]],
, RowBox[{(, RowBox[{1, +, RowBox[{Times, [, RowBox[{, 2, }], ]}]}],
)}]}]}], )}]}]}],
)}]}], InterpretationBox[DynamicModuleBox[{Typeset`open = False}, TemplateBox[{
Expression, StyleBox[TagBox[TooltipBox[
"condition", RowBox[{RowBox[{(, RowBox[{RowBox[{(, RowBox[{RowBox[{RowBox[{Re,
[, RowBox[{RowBox[{Power, [, RowBox[{, 2, }], ]}], ,
x}], ]}], ≥, 0}], &&, RowBox[{FractionBox[x, RowBox[{Plus,
[, RowBox[{, 2, }], ]}]], ≠, 0}]}], )}],
||, RowBox[{FractionBox[x, RowBox[{1, +, RowBox[{Times, [, RowBox[{,
2, }], ]}]}]], ∉, TemplateBox[{}, Reals]}],
||, RowBox[{RowBox[{Re, [, FractionBox[x, RowBox[{Plus, [, RowBox[{, 2,
}], ]}]], ]}], <, RowBox[{-, 1}]}]}], )}],
&&, RowBox[{(, RowBox[{RowBox[{x, ∉, TemplateBox[{}, Reals]}],
||, RowBox[{RowBox[{Re, [, x, ]}], >, 1}], ||, RowBox[{0, <, RowBox[{Re, [,
x, ]}], <, 1}]}],
)}]}]], Annotation[#1, Short[((Re[Power[<< 2 >>]*x] >= 0 && x/(Plus[<< 2 >>
]) != 0) || NotElement[x/(1 + Times[<< 2 >>]), Reals] || Re[x/(
Plus[<< 2 >>])] < -1) && (NotElement[x, Reals] || Re[x] > 1 ||
Inequality[0, Less, Re[x], Less, 1]), 7],
Tooltip] &],
IconizedCustomName, StripOnInput -> False], GridBox[{{RowBox[{TagBox["Head: ",
IconizedLabel], , TagBox[And, IconizedItem]}]}, {RowBox[{TagBox["Byte count: ",
IconizedLabel], , TagBox[1904,
IconizedItem]}]}}, GridBoxAlignment -> {Columns -> {{Left}}}, DefaultBaseStyle ->
Column, GridBoxItemSize -> {Columns -> {{Automatic}},
Rows -> {{Automatic}}}], Dynamic[Typeset`open]},
IconizedObject]], ((Re[x*<< 2 >>] >= 0 && x/<< 2 >> != 0) || NotElement[x/(1 + << 2 >>),
Reals] || Re[x/<< 2 >>] < -1) && (NotElement[x, Reals] || Re[x] > 1 || Inequality[0, Less, Re[
x], Less, 1]), SelectWithContents -> True, Selectable -> False]},
ConditionalExpression], Short[#1, 5] &] has evaluated to non-numerical values for all sampling points
in the region with boundaries {{0, 1}}.
2

[, RowBox[{, 2, }], ]}], RowBox[{Times, [, RowBox[{, 6, }], ]}]]}],
[, RowBox[{, 2, }], ]}], +, n}]],
]}], RowBox[{Times, [, RowBox[{, 6, }], ]}]],
)}]}]}], )}]}]}],
[, RowBox[{, 2, }], ]}]], ≠, 0}]}], )}],
}], ]}]], ]}], <, RowBox[{-, 1}]}]}], )}],
x, ]}], <, 1}]}],
Tooltip] &],
3

Out[173]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[174]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[175]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
(*that looks much better!*)
4

In[140]:= Plot[xexp[k, 37, 100, 0.1], {k, 1, 15}]
Out[140]=
2 4 6 8 10 12 14
0.60
0.65
0.70
0.75
(*looks like 0.76 and only top 11 or so*)
In[141]:= Plot3D[xexp[k, m, 100, 0.1], {k, 1, 20}, {m, 1, 40}]
Out[141]=
(*it looks like it's telling me top 1 and fire after 4 trials*)
In[176]:= Quiet[FindMaximum[xexp[k, m, 100, 0.1], {k, 10}, {m, 37}]]
Out[176]=
{0.770144, {k → 11.2483, m → 35.5792}}
Out[177]=
{0.794735, {k → 3.64866, m → 14.252}}
In[179]:= Quiet[FindMaximum[{xexp[k, m, 100, 0.1], 1 ≤ k ≤ 15, 1 ≤ m ≤ 40}, {k, m}]]
Out[179]=
{0.794735, {k → 3.64867, m → 14.252}}
5

(*seems like there could be other local maxima but the global one is at k=
3.65 and m = 14.3*)
Out[180]=
{0.962388, {k → 9.65589, m → 171.697}}
In[184]:= Quiet[FindMaximum[{xexp[k, m, 10 000, 0.1], 10 ≤ k ≤ 50, 1000 ≤ m ≤ 2000}, {k, m}]]
Out[184]=
{0., {k → 19.2022, m → 1085.2}}
(*and then it just broke*)
Out[185]=
{0.871543, {k → 5.4209, m → 31.5675}}
Out[188]=
{0.904275, {k → 6.48495, m → 48.9963}}
(*another thing to consider is that (1-r)^(greaterx[n,m,x]) is log-normal,
not normal*)
In[191]:= greaterx["mean"][n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * (n - m), {y, x, 1}]
greaterx["var"][n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * x / y * (n - m), {y, x, 1}]
Out[191]=
-
(-m + n) (1 - m + n) m + n (-1 + x) + x - m x - x1-m+n

(-1 + m - n) (m - n)
if condition
Out[192]=
-
(-m + n) (1 - m + n) x 1 + m + n (-1 + x) - m x - x-m+n

(m - n) (1 + m - n)
if condition
In[197]:= (*I need the mean of (1-r)^(Gaussian) - it's mu + var/2. aside: we have ln[term]=
log(1-r)*Gaussian, this factor doesn't influence mean or std*)
greaterx[n_, m_, x_] = greaterx["mean"][n, m, x] + greaterx["var"][n, m, x] / 2
Out[197]=
-
(-m + n) (1 - m + n) x 1 + m + n (-1 + x) - m x - x-m+n

2 (m - n) (1 + m - n)
-
(-m + n) (1 - m + n) m + n (-1 + x) + x - m x - x1-m+n

(-1 + m - n) (m - n)
if condition
6

In[198]:= rejected[n_, m_, x_, r_] = success[n, m, x] * (1 - r)^(greaterx[n, m, x]);
xexp[k_, m_, n_, r_] =
Plot[N[success[100, 37, x]], {x, 0, 1}, PlotRange → {-0.1, 1}]
(*makes sense, if the first number is large, it's hard to beat it*)
Plot[N[rejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]
Plot[N[notrejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]
[, RowBox[{, 2, }], ]}], RowBox[{Plus, [, RowBox[{, 2, }], ]}]]}],
[, RowBox[{, 2, }], ]}], +, n}]],
]}], RowBox[{Plus, [, RowBox[{, 2, }], ]}]],
)}]}]}], )}]}]}],
[, RowBox[{, 2, }], ]}]], ≠, 0}]}], )}],
}], ]}]], ]}], <, RowBox[{-, 1}]}]}], )}],
x, ]}], <, 1}]}],
Tooltip] &],
7

[, RowBox[{, 2, }], ]}], +, n}]],
)}]}]}], )}]}]}],
[, RowBox[{, 2, }], ]}]], ≠, 0}]}], )}],
}], ]}]], ]}], <, RowBox[{-, 1}]}]}], )}],
x, ]}], <, 1}]}],
Tooltip] &],
8

Out[201]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[202]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[203]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
9

In[205]:= greaterxOld[n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * (n - m), {y, x, 1}]
rejectedOld[n_, m_, x_, r_] = success[n, m, x] * (1 - r)^(greaterxOld[n, m, x]);
Quiet[Plot[N[rejectedOld[100, 37, x, 0.1] - rejected[100, 37, x, 0.1]],
{x, 0, 1}, PlotRange → {-0.1, 1}]]
Out[205]=
-
(-m + n) (1 - m + n) m + n (-1 + x) + x - m x - x1-m+n

(-1 + m - n) (m - n)
if condition
Out[207]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
(*that's good news, we don't reject as many as we thought*)
In[242]:= (*all code together*)
success[n_, m_, x_] = (1 - x^(1 - m + n));
failure[n_, m_, x_] = x^(1 - m + n);
greaterx["mean"][n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * (n - m), {y, x, 1}];
greaterx["var"][n_, m_, x_] = Integrate[(n - m + 1) * y^(n - m) * (y - x) / y * x / y * (n - m), {y, x, 1}];
greaterx[n_, m_, x_] = greaterx["mean"][n, m, x] + greaterx["var"][n, m, x] / 2;
(*r is the acceptance rate*)
rejected[n_, m_, x_, r_] = success[n, m, x] * (1 - r)^(greaterx[n, m, x]);
(*the rejection rate is log-normally distributed, so long tail warning*)
xexp[k_, m_, n_, r_] =
Quiet[Plot[N[success[100, 37, x]], {x, 0, 1}, PlotRange → {-0.1, 1}]]
Quiet[Plot[N[rejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]]
Quiet[Plot[N[notrejected[100, 37, x, 0.1]], {x, 0, 1}, PlotRange → {-0.1, 1}]]
10

[, RowBox[{, 2, }], ]}], +, n}]],
)}]}]}], )}]}]}],
[, RowBox[{, 2, }], ]}]], ≠, 0}]}], )}],
}], ]}]], ]}], <, RowBox[{-, 1}]}]}], )}],
x, ]}], <, 1}]}],
Tooltip] &],
11

[, RowBox[{, 2, }], ]}], +, n}]],
)}]}]}], )}]}]}],
[, RowBox[{, 2, }], ]}]], ≠, 0}]}], )}],
}], ]}]], ]}], <, RowBox[{-, 1}]}]}], )}],
x, ]}], <, 1}]}],
Tooltip] &],
12

Out[250]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[251]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[252]=
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Out[241]=
{0.828168, {k → 3.5195, m → 16.5013}}
13

(*so the median person has it considerably worse than the mean person -
which one am I*)
(*I think I am the mean person. The reason is that the log-
normal distribution gets integrated over all outcomes of the first trial-
phase so that means it becomes Gaussian again
due to the Central Limit Theorem - am I wrong?*)
14

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