Stick versus carrot under a fixed budget: the interior optimum
Source:R/ledger_scan2.R
morie_budget_optimum.RdMaximises \(g(\theta) + d(C - \theta)\) over \(\theta \in [0, C]\) on a
grid; with concave g and d the objective is concave
(Research.P8.Budget.concave_sum) and the maximiser is interior as
soon as some interior point beats both corners (interior_optimum).
References
D'Orsogna MR, Perc M (2015). Statistical physics of crime: a review. Physics of Life Reviews 12.
Examples
morie_budget_optimum(g = sqrt, d = function(s) 2 * log1p(s), C = 10)
#> $theta
#> [1] 3.5
#>
#> $value
#> [1] 5.900635
#>
#> $interior
#> [1] TRUE
#>
#> $corner_values
#> stick carrot
#> 4.795791 3.162278
#>
#> $grid
#> theta objective
#> 1 0.00 4.795791
#> 2 0.05 5.010286
#> 3 0.10 5.093753
#> 4 0.15 5.155628
#> 5 0.20 5.206306
#> 6 0.25 5.249812
#> 7 0.30 5.288210
#> 8 0.35 5.322728
#> 9 0.40 5.354164
#> 10 0.45 5.383072
#> 11 0.50 5.409857
#> 12 0.55 5.434824
#> 13 0.60 5.458208
#> 14 0.65 5.480199
#> 15 0.70 5.500948
#> 16 0.75 5.520581
#> 17 0.80 5.539203
#> 18 0.85 5.556902
#> 19 0.90 5.573754
#> 20 0.95 5.589825
#> 21 1.00 5.605170
#> 22 1.05 5.619840
#> 23 1.10 5.633878
#> 24 1.15 5.647323
#> 25 1.20 5.660210
#> 26 1.25 5.672569
#> 27 1.30 5.684427
#> 28 1.35 5.695811
#> 29 1.40 5.706742
#> 30 1.45 5.717242
#> 31 1.50 5.727328
#> 32 1.55 5.737019
#> 33 1.60 5.746330
#> 34 1.65 5.755276
#> 35 1.70 5.763869
#> 36 1.75 5.772123
#> 37 1.80 5.780048
#> 38 1.85 5.787655
#> 39 1.90 5.794954
#> 40 1.95 5.801954
#> 41 2.00 5.808663
#> 42 2.05 5.815089
#> 43 2.10 5.821240
#> 44 2.15 5.827123
#> 45 2.20 5.832743
#> 46 2.25 5.838107
#> 47 2.30 5.843221
#> 48 2.35 5.848090
#> 49 2.40 5.852718
#> 50 2.45 5.857110
#> 51 2.50 5.861271
#> 52 2.55 5.865205
#> 53 2.60 5.868915
#> 54 2.65 5.872405
#> 55 2.70 5.875679
#> 56 2.75 5.878739
#> 57 2.80 5.881588
#> 58 2.85 5.884230
#> 59 2.90 5.886667
#> 60 2.95 5.888901
#> 61 3.00 5.890934
#> 62 3.05 5.892769
#> 63 3.10 5.894407
#> 64 3.15 5.895851
#> 65 3.20 5.897102
#> 66 3.25 5.898161
#> 67 3.30 5.899031
#> 68 3.35 5.899712
#> 69 3.40 5.900205
#> 70 3.45 5.900513
#> 71 3.50 5.900635
#> 72 3.55 5.900572
#> 73 3.60 5.900327
#> 74 3.65 5.899898
#> 75 3.70 5.899287
#> 76 3.75 5.898495
#> 77 3.80 5.897521
#> 78 3.85 5.896366
#> 79 3.90 5.895031
#> 80 3.95 5.893516
#> 81 4.00 5.891820
#> 82 4.05 5.889944
#> 83 4.10 5.887888
#> 84 4.15 5.885652
#> 85 4.20 5.883235
#> 86 4.25 5.880638
#> 87 4.30 5.877859
#> 88 4.35 5.874899
#> 89 4.40 5.871757
#> 90 4.45 5.868432
#> 91 4.50 5.864925
#> 92 4.55 5.861233
#> 93 4.60 5.857357
#> 94 4.65 5.853295
#> 95 4.70 5.849048
#> 96 4.75 5.844612
#> 97 4.80 5.839989
#> 98 4.85 5.835176
#> 99 4.90 5.830172
#> 100 4.95 5.824976
#> 101 5.00 5.819587
#> 102 5.05 5.814003
#> 103 5.10 5.808223
#> 104 5.15 5.802244
#> 105 5.20 5.796067
#> 106 5.25 5.789688
#> 107 5.30 5.783105
#> 108 5.35 5.776318
#> 109 5.40 5.769323
#> 110 5.45 5.762119
#> 111 5.50 5.754704
#> 112 5.55 5.747075
#> 113 5.60 5.739230
#> 114 5.65 5.731166
#> 115 5.70 5.722881
#> 116 5.75 5.714372
#> 117 5.80 5.705636
#> 118 5.85 5.696671
#> 119 5.90 5.687473
#> 120 5.95 5.678039
#> 121 6.00 5.668366
#> 122 6.05 5.658450
#> 123 6.10 5.648288
#> 124 6.15 5.637877
#> 125 6.20 5.627212
#> 126 6.25 5.616289
#> 127 6.30 5.605105
#> 128 6.35 5.593655
#> 129 6.40 5.581935
#> 130 6.45 5.569939
#> 131 6.50 5.557665
#> 132 6.55 5.545105
#> 133 6.60 5.532256
#> 134 6.65 5.519111
#> 135 6.70 5.505666
#> 136 6.75 5.491914
#> 137 6.80 5.477850
#> 138 6.85 5.463467
#> 139 6.90 5.448759
#> 140 6.95 5.433719
#> 141 7.00 5.418340
#> 142 7.05 5.402615
#> 143 7.10 5.386536
#> 144 7.15 5.370095
#> 145 7.20 5.353284
#> 146 7.25 5.336094
#> 147 7.30 5.318517
#> 148 7.35 5.300543
#> 149 7.40 5.282162
#> 150 7.45 5.263364
#> 151 7.50 5.244139
#> 152 7.55 5.224475
#> 153 7.60 5.204361
#> 154 7.65 5.183784
#> 155 7.70 5.162732
#> 156 7.75 5.141192
#> 157 7.80 5.119150
#> 158 7.85 5.096590
#> 159 7.90 5.073498
#> 160 7.95 5.049858
#> 161 8.00 5.025652
#> 162 8.05 5.000863
#> 163 8.10 4.975471
#> 164 8.15 4.949458
#> 165 8.20 4.922803
#> 166 8.25 4.895483
#> 167 8.30 4.867476
#> 168 8.35 4.838756
#> 169 8.40 4.809298
#> 170 8.45 4.779075
#> 171 8.50 4.748057
#> 172 8.55 4.716214
#> 173 8.60 4.683513
#> 174 8.65 4.649919
#> 175 8.70 4.615394
#> 176 8.75 4.579900
#> 177 8.80 4.543394
#> 178 8.85 4.505831
#> 179 8.90 4.467161
#> 180 8.95 4.427335
#> 181 9.00 4.386294
#> 182 9.05 4.343981
#> 183 9.10 4.300328
#> 184 9.15 4.255268
#> 185 9.20 4.208724
#> 186 9.25 4.160613
#> 187 9.30 4.110847
#> 188 9.35 4.059328
#> 189 9.40 4.005949
#> 190 9.45 3.950595
#> 191 9.50 3.893137
#> 192 9.55 3.833435
#> 193 9.60 3.771331
#> 194 9.65 3.706654
#> 195 9.70 3.639211
#> 196 9.75 3.568786
#> 197 9.80 3.495138
#> 198 9.85 3.417995
#> 199 9.90 3.337047
#> 200 9.95 3.251942
#> 201 10.00 3.162278
#>
#> attr(,"theorem")
#> [1] "Research.P8.Budget.interior_optimum"