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1 #+TITLE: Breadth-first Search for Effective Pokemon Types
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2 #+AUTHOR: Robert McIntyre & Dylan Holmes
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3 #+EMAIL: rlm@mit.edu
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4 #+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
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5 #+STYLE: <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />
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6 #+OPTIONS: H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
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7 #+SETUPFILE: ../templates/level-0.org
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8 #+INCLUDE: ../templates/level-0.org
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9
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10
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11 * The Pok\eacute{}mon Type System
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12
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13 The Pok\eacute{}mon type system consists of seventeen different
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14 \ldquo{}types\rdquo{} (Rock, Grass, Ice, Psychic, Ground, Bug, Flying,
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15 Fire, Fighting, Dark, Dragon, Poison, Water, Ghost, Normal, Electric,
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16 and Steel) that interact like an extended version of
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17 Rock-Paper-Scissors: for example, the Fire type is strong against the
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18 Grass type but weak against the Water type. In the table below, we've
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19 recorded the relative strengths of each of the types in the
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20 Pok\eacute{}mon type system; the number in each cell indicates how
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21 effective an attack of the type in the row is against a
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22 Pok\eacute{}mon of the type in the column. We call these numbers
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23 /susceptibilities/ because we are interested in the column totals,
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24 which quantify the overall vulnerability of each Pok\eacute{}mon type
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25 (as opposed to the row totals, which quantify the overall
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26 effectiveness of each attack type.)
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27
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28 In the Pok\eacute{}mon games, only four susceptibility values (two,
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29 one, one-half, and zero) occur. These numbers indicate particularly
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30 high susceptibility, average susceptibility, particularly low
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31 susceptibility, and no susceptibility
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32 (immunity). Here is the entire Pok\eacute{}mon type chart.
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33
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34
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35
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36 ** TODO add the pokemon chart in a pretty form
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37
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38 * COMMENT Pokemon Table Data
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39
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40 #+caption: The rows are attack types, while the columns are defense types. To see the multiplier for a pokemon attack against a certain type, follow the row for the attack type to the column of the defending type.
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41 #+label: pokemon-matchups
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42 #+tblname: pokemon-table-gen-two
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43 | | normal | fire | water | electric | grass | ice | fighting | poison | ground | flying | psychic | bug | rock | ghost | dragon | dark | steel |
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44 |----------+--------+------+-------+----------+-------+-----+----------+--------+--------+--------+---------+-----+------+-------+--------+------+-------|
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45 | normal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 0 | 1 | 1 | .5 |
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46 | fire | 1 | .5 | .5 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 1 | 2 |
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47 | water | 1 | 2 | .5 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | 1 |
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48 | electric | 1 | 1 | 2 | .5 | .5 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | .5 | 1 | 1 |
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49 | grass | 1 | .5 | 2 | 1 | .5 | 1 | 1 | .5 | 2 | .5 | 1 | .5 | 2 | 1 | .5 | 1 | .5 |
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50 | ice | 1 | .5 | .5 | 1 | 2 | .5 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 |
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51 | fighting | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | .5 | .5 | .5 | 2 | 0 | 1 | 2 | 2 |
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52 | poison | 1 | 1 | 1 | 1 | 2 | 1 | 1 | .5 | .5 | 1 | 1 | 1 | .5 | .5 | 1 | 1 | 0 |
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53 | ground | 1 | 2 | 1 | 2 | .5 | 1 | 1 | 2 | 1 | 0 | 1 | .5 | 2 | 1 | 1 | 1 | 2 |
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54 | flying | 1 | 1 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | 1 | 1 | .5 |
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55 | psychic | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | .5 | 1 | 1 | 1 | 1 | 0 | .5 |
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56 | bug | 1 | .5 | 1 | 1 | 2 | 1 | .5 | .5 | 1 | .5 | 2 | 1 | 1 | .5 | 1 | 2 | .5 |
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57 | rock | 1 | 2 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | .5 |
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58 | ghost | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | .5 | .5 |
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59 | dragon | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | .5 |
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60 | dark | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | .5 | .5 |
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61 | steel | 1 | .5 | .5 | .5 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | .5 |
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62
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63 #+caption: this is the old table from generation 1. The differences are: dark and ghost are missing, Bus is super against Poison, Poison is super against Bug, Bug is regularly effective against Ghost, and Ice is normally effective against Fire. Ghost is not effective against psychic.
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64 #+label: pokemon-matchups-gen-1
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65 #+tblname: pokemon-table-gen-one
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66 | | normal | fire | water | electric | grass | ice | fighting | poison | ground | flying | psychic | bug | rock | ghost | dragon |
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67 |----------+--------+------+-------+----------+-------+-----+----------+--------+--------+--------+---------+-----+------+-------+--------|
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68 | normal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 0 | 1 |
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69 | fire | 1 | .5 | .5 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | .5 |
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70 | water | 1 | 2 | .5 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | .5 |
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71 | electric | 1 | 1 | 2 | .5 | .5 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | .5 |
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72 | grass | 1 | .5 | 2 | 1 | .5 | 1 | 1 | .5 | 2 | .5 | 1 | .5 | 2 | 1 | .5 |
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73 | ice | 1 | 1 | .5 | 1 | 2 | .5 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 |
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74 | fighting | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | .5 | .5 | .5 | 2 | 0 | 1 |
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75 | poison | 1 | 1 | 1 | 1 | 2 | 1 | 1 | .5 | .5 | 1 | 1 | 2 | .5 | .5 | 1 |
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76 | ground | 1 | 2 | 1 | 2 | .5 | 1 | 1 | 2 | 1 | 0 | 1 | .5 | 2 | 1 | 1 |
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77 | flying | 1 | 1 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | 1 |
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78 | psychic | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | .5 | 1 | 1 | 1 | 1 |
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79 | bug | 1 | .5 | 1 | 1 | 2 | 1 | .5 | 2 | 1 | .5 | 2 | 1 | 1 | 0 | 1 |
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80 | rock | 1 | 2 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 |
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81 | ghost | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 1 |
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82 | dragon | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
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83
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84 * Representing the Data
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85
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86 After creating the Pok\eacute{}mon types namespace, we store the table
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87 of susceptibilities in =pokemon-table-gen-one= and
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88 =pokemon-table-gen-two=, each of which is a simple vector of
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89 vectors. Because a vector of vectors can be cumbersome, we do not
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90 access the tables directly; instead, we use the derivative structures
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91 =attack-strengths= and =defense-strengths=, which are functions which
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92 return hash-maps associating each row (respectively column) of the
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93 table with its corresponding Pok\eacute{}mon type.
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94
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95
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96
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97 #+srcname: header
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98 #+begin_src clojure :results silent
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99 (ns pokemon.types
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100 (:use rlm.ns-rlm))
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101 (rlm.ns-rlm/ns-clone rlm.light-base)
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102 (use 'clojure.set)
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103 #+end_src
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104
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105 #+srcname: data(pokemon-table-gen-one=pokemon-table-gen-one, pokemon-table-gen-two=pokemon-table-gen-two)
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106 #+begin_src clojure :results silent
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107 (in-ns 'pokemon.types)
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108 ;; record type strengths as a vector of vectors
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109 (def pokemon-gen-one pokemon-table-gen-one)
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110 (def pokemon-gen-two pokemon-table-gen-two)
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111
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112 (defn type-names [] (vec (doall (map (comp keyword first) pokemon-gen-two))))
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113
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114 (defn attack-strengths []
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115 (zipmap
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116 (type-names)
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117 (map (comp vec rest) pokemon-gen-two)))
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118
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119 (defn defense-strengths []
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120 (zipmap (type-names)
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121 (map
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122 (apply juxt (map (attack-strengths) (type-names)))
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123 (range (count (type-names))))))
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124 #+end_src
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125
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126 #+begin_src clojure :results output :exports both
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127 (clojure.pprint/pprint pokemon.types/pokemon-gen-two)
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128 #+end_src
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129
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130 #+results:
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131 #+begin_example
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132 (("normal" 1 1 1 1 1 1 1 1 1 1 1 1 0.5 0 1 1 0.5)
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133 ("fire" 1 0.5 0.5 1 2 2 1 1 1 1 1 2 0.5 1 0.5 1 2)
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134 ("water" 1 2 0.5 1 0.5 1 1 1 2 1 1 1 2 1 0.5 1 1)
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135 ("electric" 1 1 2 0.5 0.5 1 1 1 0 2 1 1 1 1 0.5 1 1)
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136 ("grass" 1 0.5 2 1 0.5 1 1 0.5 2 0.5 1 0.5 2 1 0.5 1 0.5)
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137 ("ice" 1 0.5 0.5 1 2 0.5 1 1 2 2 1 1 1 1 2 1 0.5)
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138 ("fighting" 2 1 1 1 1 2 1 0.5 1 0.5 0.5 0.5 2 0 1 2 2)
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139 ("poison" 1 1 1 1 2 1 1 0.5 0.5 1 1 1 0.5 0.5 1 1 0)
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140 ("ground" 1 2 1 2 0.5 1 1 2 1 0 1 0.5 2 1 1 1 2)
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141 ("flying" 1 1 1 0.5 2 1 2 1 1 1 1 2 0.5 1 1 1 0.5)
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142 ("psychic" 1 1 1 1 1 1 2 2 1 1 0.5 1 1 1 1 0 0.5)
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143 ("bug" 1 0.5 1 1 2 1 0.5 0.5 1 0.5 2 1 1 0.5 1 2 0.5)
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144 ("rock" 1 2 1 1 1 2 0.5 1 0.5 2 1 2 1 1 1 1 0.5)
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145 ("ghost" 0 1 1 1 1 1 1 1 1 1 2 1 1 2 1 0.5 0.5)
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146 ("dragon" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0.5)
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147 ("dark" 1 1 1 1 1 1 0.5 1 1 1 2 1 1 2 1 0.5 0.5)
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148 ("steel" 1 0.5 0.5 0.5 1 2 1 1 1 1 1 1 2 1 1 1 0.5))
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149 #+end_example
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150
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151 =pokemon-gen-two= is a simple list-of-list data structure.
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152
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153 #+begin_src clojure :results output :exports both
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154 (clojure.pprint/pprint (pokemon.types/defense-strengths))
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155 #+end_src
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156
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157 #+results:
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158 #+begin_example
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159 {:water [1 0.5 0.5 2 2 0.5 1 1 1 1 1 1 1 1 1 1 0.5],
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160 :psychic [1 1 1 1 1 1 0.5 1 1 1 0.5 2 1 2 1 2 1],
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161 :dragon [1 0.5 0.5 0.5 0.5 2 1 1 1 1 1 1 1 1 2 1 1],
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162 :fire [1 0.5 2 1 0.5 0.5 1 1 2 1 1 0.5 2 1 1 1 0.5],
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163 :ice [1 2 1 1 1 0.5 2 1 1 1 1 1 2 1 1 1 2],
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164 :grass [1 2 0.5 0.5 0.5 2 1 2 0.5 2 1 2 1 1 1 1 1],
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165 :ghost [0 1 1 1 1 1 0 0.5 1 1 1 0.5 1 2 1 2 1],
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166 :poison [1 1 1 1 0.5 1 0.5 0.5 2 1 2 0.5 1 1 1 1 1],
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167 :flying [1 1 1 2 0.5 2 0.5 1 0 1 1 0.5 2 1 1 1 1],
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168 :normal [1 1 1 1 1 1 2 1 1 1 1 1 1 0 1 1 1],
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169 :rock [0.5 0.5 2 1 2 1 2 0.5 2 0.5 1 1 1 1 1 1 2],
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170 :electric [1 1 1 0.5 1 1 1 1 2 0.5 1 1 1 1 1 1 0.5],
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171 :ground [1 1 2 0 2 2 1 0.5 1 1 1 1 0.5 1 1 1 1],
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172 :fighting [1 1 1 1 1 1 1 1 1 2 2 0.5 0.5 1 1 0.5 1],
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173 :dark [1 1 1 1 1 1 2 1 1 1 0 2 1 0.5 1 0.5 1],
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174 :steel [0.5 2 1 1 0.5 0.5 2 0 2 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5],
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175 :bug [1 2 1 1 0.5 1 0.5 1 0.5 2 1 1 2 1 1 1 1]}
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176 #+end_example
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177
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178 =defense-strengths= is a more convenient form of =pokemon-gen-two=, with key/value pair access.
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179
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180 * Interfacing with the Data
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181 #+srcname: types
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182 #+begin_src clojure :results silent
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183 (in-ns 'pokemon.types)
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184
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185 (defn multitypes "All combinations of up to n types" [n]
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186 (vec
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187 (map vec
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188 (reduce concat
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189 (map (partial combinations (type-names))
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190 (range 1 (inc n)))))))
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191
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192 (defn susceptibility ;; susceptibility-map
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193 "Hash-map of the susceptibilities of the given type combination
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194 to each type of attack"
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195 [pkmn-types]
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196 (rlm.map-utils/map-vals
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197 clojure.core/rationalize
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198 (apply hash-map
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199 (interleave (type-names)
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200 (apply (partial map *)
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201 (map (defense-strengths) pkmn-types))))))
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202
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203 (defn susceptance ;; susceptibility
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204 "The cumulative susceptibility of the given type combination"
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205 [types]
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206 (reduce + (map sqr (vals (susceptibility types)))))
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207 #+end_src
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208
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209 * Best-First Search
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210
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211 I'd like to find type combinations that are interesting, but the total
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212 number of combinations gets huge as we begin to consider more
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213 types. For example, the total possible number of type combinations
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214 given just 8 possible types is: 17^{8} = 6975757441 combinations.
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215 Therefore, it's prudent to use search.
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216
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217 These functions are a simple implementation of best-first search in
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218 clojure. The idea to start off with a collection of nodes and some way
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219 of finding the best node, and to always expand the best node at every
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220 step.
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221
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222 #+srcname: search
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223 #+begin_src clojure :results silent
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224 (in-ns 'pokemon.types)
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225
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226 (defn comparatize
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227 "Define a comparator which uses the numerical outputs of fn as its criterion.
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228 Objects are sorted in increasing numerical order. Objects with the same fn-value
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229 are further compared by clojure.core/compare."
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230 [fun]
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231 (fn [a b]
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232 (let [val-a (fun a)
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233 val-b (fun b)]
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234 (cond
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235 ;; if the function cannot differentiate the two values
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236 ;; then compare the two values using clojure.core/compare
|
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237 (= val-a val-b) (compare a b)
|
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238 true
|
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239 ;; LOWER values of the function are preferred
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240 (compare (- val-a val-b) 0)))))
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241
|
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242 (defn-memo best-first-step [successors [visited unvisited]]
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243 (cond (empty? unvisited) nil
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244 true
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245 (let [best-node (first unvisited)
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246 visited* (conj visited best-node)
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247 unvisited*
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248 (difference
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249 (union unvisited (set (successors best-node)))
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250 visited*)]
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251 (println best-node)
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252 [visited* unvisited*])))
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253
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254 ;; memoize partial from core so that for example
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255 ;; (= (partial + 1) (partial + 1))
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256 ;; this way, best first search can take advantage of the memoization
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257 ;; of best-first step
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258 (undef partial)
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259 (def partial (memoize clojure.core/partial))
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260
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261 (defn best-first-search
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262 "Searches through a network of alternatives, pursuing
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263 initially-promising positions first. Comparator defines which
|
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264 positions are more promising, successors produces a list of improved
|
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265 positions from the given position (if any exist), and initial-nodes is
|
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266 a list of starting positions. Returns a lazy sequence of search results
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267 [visited-nodes unvisited-nodes], which terminates when
|
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268 there are no remaining unvisited positions."
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269 [comparator successors initial-nodes]
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270 (let [initial-nodes
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271 (apply (partial sorted-set-by comparator) initial-nodes)
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272 initial-visited-nodes (sorted-set-by comparator)
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273 step (partial best-first-step successors)]
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274 (take-while
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275 (comp not nil?)
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276 (iterate step [initial-visited-nodes initial-nodes]))))
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277
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278 #+end_src
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279
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280
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281 Now that we have a basic best-first-search, it's convenient to write a
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282 few pokemon-type specific convenience functions.
|
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283
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284 #+srcname: pokemon-search
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285 #+begin_src clojure :results silent
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286 (in-ns 'pokemon.types)
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287 (defvar type-compare (comparatize susceptance)
|
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288 "compare two type combinations wrt their susceptibilities")
|
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289
|
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290 (defn type-successors
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291 "Return the set of types that can be made by appending a single type
|
|
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292 to the given combination."
|
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293 [type]
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294 (if (nil? type) '()
|
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295 (set (map (comp vec sort (partial into type)) (multitypes 1)))))
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|
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296
|
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|
297 (defn immortal?
|
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298 "A type combo is immortal if it is resistant or invulnerable to
|
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299 every pokemon type. This is because that set of types can just be
|
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|
300 repeated to achieve as low a susceptance as desired"
|
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301 [type]
|
|
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|
302 (every? (partial > 1) (vals (susceptibility type))))
|
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303
|
|
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|
304 (defn type-successors*
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|
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305 "Stop expanding a type if it's immortal, or if it is longer than or
|
|
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306 equal to limit-size. Also, only return type additions that are
|
|
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|
307 strictly better than the initial type."
|
|
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|
308 [limit-size type]
|
|
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309 (if (or (<= limit-size (count type)) (immortal? type)) '()
|
|
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310 (set (filter #(< 0 (type-compare type %)) (type-successors type)))))
|
|
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|
311
|
|
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|
312 (defn pokemon-type-search
|
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313 "Search among type-combos no greater than length n, limited by limit
|
|
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|
314 steps of best-first-search."
|
|
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|
315 ([n] (pokemon-type-search n Integer/MAX_VALUE))
|
|
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|
316 ([n limit]
|
|
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|
317 (first (last
|
|
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|
318 (take
|
|
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|
319 limit
|
|
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|
320 (best-first-search
|
|
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|
321 type-compare
|
|
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|
322 (partial type-successors* n)
|
|
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|
323 (multitypes 1)))))))
|
|
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|
324
|
|
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|
325 (defvar immortals
|
|
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|
326 (comp (partial filter immortal?) pokemon-type-search)
|
|
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|
327 "find all the immortal pokemon types ")
|
|
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|
328
|
|
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|
329 #+end_src
|
|
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|
330
|
|
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|
331 Because there are so many type combinations, it's important to narrow
|
|
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|
332 down the results as much as possible. That is why =type-successors*=
|
|
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|
333 only returns types that are actually better than the type it is given.
|
|
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|
334
|
|
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|
335 Best-first search can get caught optimizing a single type forever, so
|
|
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|
336 it's also important to limit the search space to be finite by setting
|
|
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|
337 an upper bound on the length of a type combo.
|
|
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|
338
|
|
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|
339 * Results
|
|
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|
340 ** The best dual-type combo
|
|
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|
341
|
|
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|
342 #+begin_src clojure :results cache verbatim :exports both
|
|
rlm@0
|
343 (first (pokemon.types/pokemon-type-search 2))
|
|
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|
344 #+end_src
|
|
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|
345
|
|
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|
346 #+results:
|
|
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|
347 : [:dark :ghost]
|
|
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|
348
|
|
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|
349 Dark and Ghost, which additionally has the property of having no
|
|
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|
350 weaknesses to any other type, is the best type combo in terms of
|
|
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|
351 susceptance.
|
|
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|
352
|
|
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|
353 The Dark and Steel types were introduced many years after
|
|
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|
354 pok\eacute{}mon started. In addition to the additional types, the
|
|
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|
355 pok\eacute{}mon games gained a few new rules concerning some of the
|
|
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|
356 matchups of the original types. Therefore, it's also interesting to see what
|
|
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|
357 type combination was most powerful before those types and new rules were introduced.
|
|
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|
358
|
|
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|
359 The easiest way to do this with my setup is to just rebind the
|
|
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|
360 =pokemon-gen-two= table to the =pokemon-gen-one= table. Since
|
|
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|
361 everything that references this variable is a function and we're not
|
|
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|
362 doing anything too crazy with lazy-sequences and late-binding, this
|
|
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|
363 simple macro will do the job.
|
|
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|
364
|
|
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|
365 #+srcname: old-school
|
|
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|
366 #+begin_src clojure :results silent
|
|
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|
367 (in-ns 'pokemon.types)
|
|
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|
368
|
|
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|
369 (defmacro old-school
|
|
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|
370 [& forms]
|
|
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|
371 `(binding [pokemon-gen-two pokemon-gen-one] ~@forms))
|
|
rlm@0
|
372 #+end_src
|
|
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|
373
|
|
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|
374 Using the =old-school= macro, it's easy to find answers for the
|
|
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|
375 original 15 pokemon types as well as the expanded pokemon types
|
|
rlm@0
|
376 introduced later.
|
|
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|
377
|
|
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|
378 #+begin_src clojure :results verbatim :exports both :cache yes
|
|
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|
379 (pokemon.types/old-school (first (pokemon.types/pokemon-type-search 2)))
|
|
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|
380 #+end_src
|
|
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|
381
|
|
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|
382 #+results[f43470fdf460ed546e9c57879abc9eda56da129f]:
|
|
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|
383 : [:ghost :psychic]
|
|
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|
384
|
|
rlm@0
|
385 Ghost and Psychic also manages to have no weaknesses to any of the original
|
|
rlm@0
|
386 types.
|
|
rlm@0
|
387
|
|
rlm@0
|
388 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
389 (clojure.pprint/pprint
|
|
rlm@0
|
390 (pokemon.types/old-school
|
|
rlm@0
|
391 (pokemon.types/susceptibility [:ghost :psychic])))
|
|
rlm@0
|
392 #+end_src
|
|
rlm@0
|
393
|
|
rlm@0
|
394 #+results:
|
|
rlm@0
|
395 #+begin_example
|
|
rlm@0
|
396 {:water 1,
|
|
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|
397 :psychic 1/2,
|
|
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|
398 :dragon 1,
|
|
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|
399 :fire 1,
|
|
rlm@0
|
400 :ice 1,
|
|
rlm@0
|
401 :grass 1,
|
|
rlm@0
|
402 :ghost 0,
|
|
rlm@0
|
403 :poison 1/2,
|
|
rlm@0
|
404 :flying 1,
|
|
rlm@0
|
405 :normal 0,
|
|
rlm@0
|
406 :rock 1,
|
|
rlm@0
|
407 :electric 1,
|
|
rlm@0
|
408 :ground 1,
|
|
rlm@0
|
409 :fighting 0,
|
|
rlm@0
|
410 :bug 0}
|
|
rlm@0
|
411 #+end_example
|
|
rlm@0
|
412
|
|
rlm@0
|
413 ** An Immortal Type
|
|
rlm@0
|
414 It's possible to quickly find an immortal type by giving the search
|
|
rlm@0
|
415 a long enough maximum type length. 50 rounds of search with a max
|
|
rlm@0
|
416 type limit of 10 is enough to find an immortal type.
|
|
rlm@0
|
417
|
|
rlm@0
|
418 #+begin_src clojure :results scalar :exports both
|
|
rlm@0
|
419 (first (pokemon.types/pokemon-type-search 10 50))
|
|
rlm@0
|
420 #+end_src
|
|
rlm@0
|
421
|
|
rlm@0
|
422 #+results:
|
|
rlm@0
|
423 : [:dragon :fire :flying :ghost :grass :ground :steel :steel :water :water]
|
|
rlm@0
|
424
|
|
rlm@0
|
425
|
|
rlm@0
|
426 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
427 (clojure.pprint/pprint
|
|
rlm@0
|
428 (pokemon.types/susceptibility
|
|
rlm@0
|
429 [:dragon :fire :flying :ghost :grass :ground :steel :steel :water :water]))
|
|
rlm@0
|
430 #+end_src
|
|
rlm@0
|
431
|
|
rlm@0
|
432 #+results:
|
|
rlm@0
|
433 #+begin_example
|
|
rlm@0
|
434 {:water 1/4,
|
|
rlm@0
|
435 :psychic 1/4,
|
|
rlm@0
|
436 :dragon 1/2,
|
|
rlm@0
|
437 :fire 1/2,
|
|
rlm@0
|
438 :ice 1/2,
|
|
rlm@0
|
439 :grass 1/8,
|
|
rlm@0
|
440 :ghost 1/2,
|
|
rlm@0
|
441 :poison 0,
|
|
rlm@0
|
442 :flying 1/2,
|
|
rlm@0
|
443 :normal 0,
|
|
rlm@0
|
444 :rock 1/2,
|
|
rlm@0
|
445 :electric 0,
|
|
rlm@0
|
446 :ground 0,
|
|
rlm@0
|
447 :fighting 0,
|
|
rlm@0
|
448 :dark 1/2,
|
|
rlm@0
|
449 :steel 1/32,
|
|
rlm@0
|
450 :bug 1/16}
|
|
rlm@0
|
451 #+end_example
|
|
rlm@0
|
452
|
|
rlm@0
|
453 ** Explanations for Common Pok\eacute{}mon Strategies
|
|
rlm@0
|
454
|
|
rlm@0
|
455 Many people start out a battle with either a normal pok\eacute{}mon or an
|
|
rlm@0
|
456 electric pok\eacute{}mon, and here's some justification for that choice.
|
|
rlm@0
|
457
|
|
rlm@0
|
458 #+srcname: weaknesses
|
|
rlm@0
|
459 #+begin_src clojure :results silent
|
|
rlm@0
|
460 (in-ns 'pokemon.types)
|
|
rlm@0
|
461 (defn critical-weaknesses [type]
|
|
rlm@0
|
462 (count (filter #(> % 1) (vals (susceptibility type)))))
|
|
rlm@0
|
463 #+end_src
|
|
rlm@0
|
464
|
|
rlm@0
|
465 #+begin_src clojure :exports both :results output
|
|
rlm@0
|
466 (clojure.pprint/pprint
|
|
rlm@0
|
467 (sort-by pokemon.types/critical-weaknesses (pokemon.types/multitypes 1)))
|
|
rlm@0
|
468 #+end_src
|
|
rlm@0
|
469
|
|
rlm@0
|
470 #+results:
|
|
rlm@0
|
471 #+begin_example
|
|
rlm@0
|
472 ([:normal]
|
|
rlm@0
|
473 [:electric]
|
|
rlm@0
|
474 [:water]
|
|
rlm@0
|
475 [:fighting]
|
|
rlm@0
|
476 [:poison]
|
|
rlm@0
|
477 [:ghost]
|
|
rlm@0
|
478 [:dragon]
|
|
rlm@0
|
479 [:dark]
|
|
rlm@0
|
480 [:fire]
|
|
rlm@0
|
481 [:ground]
|
|
rlm@0
|
482 [:flying]
|
|
rlm@0
|
483 [:psychic]
|
|
rlm@0
|
484 [:bug]
|
|
rlm@0
|
485 [:steel]
|
|
rlm@0
|
486 [:ice]
|
|
rlm@0
|
487 [:grass]
|
|
rlm@0
|
488 [:rock])
|
|
rlm@0
|
489 #+end_example
|
|
rlm@0
|
490
|
|
rlm@0
|
491 Electric and Normal are among the best types with which to start the
|
|
rlm@0
|
492 game, since they have the fewest weaknesses among all the types.
|
|
rlm@0
|
493
|
|
rlm@0
|
494 At the beginning of the pok\eacute{}mon games, players are given a choice
|
|
rlm@0
|
495 between the Fire pok\eacute{}mon Charmander, the Water pok\eacute{}mon Squirtle, or
|
|
rlm@0
|
496 the Grass/Poison pok\eacute{}mon Bulbasaur.
|
|
rlm@0
|
497
|
|
rlm@0
|
498 #+begin_src clojure :exports both :results verbatim
|
|
rlm@0
|
499 (sort-by pokemon.types/susceptance [[:fire] [:water] [:grass :poison]])
|
|
rlm@0
|
500 #+end_src
|
|
rlm@0
|
501
|
|
rlm@0
|
502 #+results:
|
|
rlm@0
|
503 : ([:water] [:fire] [:grass :poison])
|
|
rlm@0
|
504
|
|
rlm@0
|
505 As can be seen, the Water pok\eacute{}mon Squirtle is the most solid
|
|
rlm@0
|
506 choice starting out, insofar as susceptance is concerned.
|
|
rlm@0
|
507
|
|
rlm@0
|
508 ** The Worst Pok\eacute{}mon Types
|
|
rlm@0
|
509
|
|
rlm@0
|
510 #+srcname: weak-types
|
|
rlm@0
|
511 #+begin_src clojure :results silent
|
|
rlm@0
|
512 (in-ns 'pokemon.types)
|
|
rlm@0
|
513
|
|
rlm@0
|
514 (defn type-compare-weak
|
|
rlm@0
|
515 "compare first by total number of critical-weaknesses,
|
|
rlm@0
|
516 then by overall susceptance, favoring weaker types."
|
|
rlm@0
|
517 [type-1 type-2]
|
|
rlm@0
|
518 (let [measure (memoize (juxt critical-weaknesses susceptance))]
|
|
rlm@0
|
519 (if (= (measure type-2) (measure type-1))
|
|
rlm@0
|
520 (compare type-2 type-1)
|
|
rlm@0
|
521 (compare (measure type-2) (measure type-1)))))
|
|
rlm@0
|
522
|
|
rlm@0
|
523 (defn resistant?
|
|
rlm@0
|
524 "might as well get rid of types that are resistant to any type"
|
|
rlm@0
|
525 [type]
|
|
rlm@0
|
526 (not (every? #(< 0 %) (vals (susceptibility type)))))
|
|
rlm@0
|
527
|
|
rlm@0
|
528 (defn type-successors-weak
|
|
rlm@0
|
529 [limit type]
|
|
rlm@0
|
530 (set (if (<= limit (count type)) '()
|
|
rlm@0
|
531 (filter #(< 0 (type-compare-weak type %))
|
|
rlm@0
|
532 (remove resistant? (type-successors type))))))
|
|
rlm@0
|
533
|
|
rlm@0
|
534 (defn pokemon-type-search-weak
|
|
rlm@0
|
535 "Search among type-combos no greater than length n, limited by limit
|
|
rlm@0
|
536 steps of best-first-search."
|
|
rlm@0
|
537 ([n] (pokemon-type-search-weak n Integer/MAX_VALUE))
|
|
rlm@0
|
538 ([n limit]
|
|
rlm@0
|
539 (first (last
|
|
rlm@0
|
540 (take
|
|
rlm@0
|
541 limit
|
|
rlm@0
|
542 (best-first-search
|
|
rlm@0
|
543 type-compare-weak
|
|
rlm@0
|
544 (partial type-successors-weak n)
|
|
rlm@0
|
545 (multitypes 1)))))))
|
|
rlm@0
|
546 #+end_src
|
|
rlm@0
|
547
|
|
rlm@0
|
548
|
|
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|
549 #+begin_src clojure :results scalar :exports both
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550 (first (pokemon.types/pokemon-type-search-weak 1))
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551 #+end_src
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552
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553 #+results:
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554 : [:rock]
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555
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556 Poor Rock. It's just not that good a type. Maybe this is why Brock
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557 (who has rock pok\eacute{}mon) is the first gym leader in the games.
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558
|
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559 #+begin_src clojure :results scalar cache :exports both
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560 (first (pokemon.types/pokemon-type-search-weak 2))
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561 #+end_src
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562
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563 #+results:
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564 : [:grass :ice]
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565
|
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566 # ;;bonus convergently immortal type combo
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567 # (susceptance (vec (concat (repeat 150 :water) (repeat 50 :poison) (repeat 50 :steel) [:ghost :normal :flying :ground :dark])))
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568
|
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569 #+begin_src clojure :results output :exports both
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570 (clojure.pprint/pprint
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571 (pokemon.types/susceptibility [:grass :ice]))
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572 #+end_src
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573
|
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574 #+results:
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575 #+begin_example
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576 {:water 1/2,
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577 :psychic 1,
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|
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578 :dragon 1,
|
|
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579 :fire 4,
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580 :ice 1,
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|
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581 :grass 1/2,
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582 :ghost 1,
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583 :poison 2,
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584 :flying 2,
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585 :normal 1,
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586 :rock 2,
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587 :electric 1/2,
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588 :ground 1/2,
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589 :fighting 2,
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590 :dark 1,
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591 :steel 2,
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592 :bug 2}
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593 #+end_example
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594
|
|
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595 This miserable combination is weak to 6 types and double-weak to
|
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596 Fire. No pok\eacute{}mon in the games actually has this type.
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597
|
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598 * Conclusion
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599
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600 Searching for a type that is weak to everything takes a very long time
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601 and fails to reveal any results. That's the problem with a search
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602 over this large problem space --- if there's an easy solution, the
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603 search will find it quickly, but it can be very hard to determine
|
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604 whether there is actually a solution.
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|
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|
605
|
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606 In the [[./lpsolve.org][next installment]], I'll use =lp_solve= to solve this problem in
|
|
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|
607 a different way.
|
|
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|
608
|
|
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|
609
|
|
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|
610 * COMMENT main program
|
|
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|
611 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :exports none
|
|
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|
612 <<header>>
|
|
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|
613 #+end_src
|
|
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|
614
|
|
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|
615 ## this is necessary to define pokemon-table inside the source code.
|
|
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|
616
|
|
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|
617 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :var pokemon-table-gen-one=pokemon-table-gen-one :var pokemon-table-gen-two=pokemon-table-gen-two :exports none
|
|
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|
618 <<data>>
|
|
rlm@0
|
619 #+end_src
|
|
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|
620
|
|
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|
621 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :exports none
|
|
rlm@0
|
622 <<types>>
|
|
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|
623 <<search>>
|
|
rlm@0
|
624 <<pokemon-search>>
|
|
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|
625 <<old-school>>
|
|
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|
626 <<weaknesses>>
|
|
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|
627 <<weak-types>>
|
|
rlm@0
|
628 #+end_src
|
|
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|
629
|
|
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|
630
|
|
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|
631
|
|
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|
632
|
|
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|
633
|
|
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|
634
|
|
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|
635
|
|
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|
636
|
|
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|
637
|
|
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|
638
|
|
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|
639
|
|
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|
640
|
|
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|
641
|
|
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|
642
|
|
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|
643
|
|
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|
644
|
|
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|
645
|
|
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|
646
|
