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1 #+TITLE: Best-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 #+description: Finding interesting pokemon type combinations through Best-First search in clojure.
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5 #+SETUPFILE: ../../aurellem/org/setup.org
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6 #+INCLUDE: ../../aurellem/org/level-0.org
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7
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8 * The Pok\eacute{}mon Type System
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9
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10 The Pok\eacute{}mon type system consists of seventeen different
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11 \ldquo{}types\rdquo{} (Rock, Grass, Ice, Psychic, Ground, Bug, Flying,
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12 Fire, Fighting, Dark, Dragon, Poison, Water, Ghost, Normal, Electric,
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13 and Steel) that interact like an extended version of
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14 Rock-Paper-Scissors: for example, the Fire type is strong against the
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15 Grass type but weak against the Water type. In the table below, we've
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16 recorded the relative strengths of each of the types in the
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17 Pok\eacute{}mon type system; the number in each cell indicates how
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18 effective an attack of the type in the row is against a
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19 Pok\eacute{}mon of the type in the column. We call these numbers
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20 /susceptibilities/.
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21
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22 In the Pok\eacute{}mon games, only four susceptibility values (two,
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23 one, one-half, and zero) occur. These numbers indicate particularly
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24 high susceptibility, average susceptibility, particularly low
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25 susceptibility, and no susceptibility (immunity).
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26
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27 - The suceptability of Flying types /against/ Ground is 0, because Ground
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28 attacks cannot hurt Flying pok\eacute{}mon at all. The damage that
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29 a Ground type attack normally does is /multiplied/ by 0 when it is
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30 uesd against a Flying type pok\eacute{}mon.
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31
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32 - The susceptability of Fire types against Water attacks
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33 is 2, because Water type attacks are strong against Fire type
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34 Pok\eacute{}mon. The damage that a Water type attack normally does
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35 is doubled when it is used against a Fire type pok\eacute{}mon.
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36
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37 - The susceptability of Water types against Water attacks is
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38 $\frac{1}{2}$, because Water type attacks are strong against Water
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39 type Pok\eacute{}mon. The damage that a Water type attack normally
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40 does is halved when it is used against a Water type
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41 pok\eacute{}mon.
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42
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43 There are two pok\eacute{}mon type systems in use. The first is the
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44 classic system which was used for the very first pok\eacute{}mon
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45 games, Red, Yellow, and Blue. This old system was used from 1998 to
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46 2000 in America, and is known as the /Generation I Type System/. The
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47 modern pok\eacute{}mon type system was introduced in 2000 with the
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48 introduction of pok\eacute{}mon Gold and Silver, and has been in use
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49 ever since. It is called the /Generation II Type System/.
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50
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51 Here are the the definitions of the two type systems.
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52
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53 * Generation I and II Type System Data
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54
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55 ** Generation II Type System
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56 #+label: pokemon-matchups
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57 #+tblname: pokemon-table-gen-two
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58 | | normal | fire | water | electric | grass | ice | fighting | poison | ground | flying | psychic | bug | rock | ghost | dragon | dark | steel |
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59 |----------+--------+------+-------+----------+-------+-----+----------+--------+--------+--------+---------+-----+------+-------+--------+------+-------|
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60 | normal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 0 | 1 | 1 | .5 |
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61 | fire | 1 | .5 | .5 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 1 | 2 |
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62 | water | 1 | 2 | .5 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | 1 |
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63 | electric | 1 | 1 | 2 | .5 | .5 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | .5 | 1 | 1 |
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64 | grass | 1 | .5 | 2 | 1 | .5 | 1 | 1 | .5 | 2 | .5 | 1 | .5 | 2 | 1 | .5 | 1 | .5 |
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65 | ice | 1 | .5 | .5 | 1 | 2 | .5 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 |
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66 | fighting | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | .5 | .5 | .5 | 2 | 0 | 1 | 2 | 2 |
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67 | poison | 1 | 1 | 1 | 1 | 2 | 1 | 1 | .5 | .5 | 1 | 1 | 1 | .5 | .5 | 1 | 1 | 0 |
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68 | ground | 1 | 2 | 1 | 2 | .5 | 1 | 1 | 2 | 1 | 0 | 1 | .5 | 2 | 1 | 1 | 1 | 2 |
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69 | flying | 1 | 1 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | 1 | 1 | .5 |
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70 | psychic | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | .5 | 1 | 1 | 1 | 1 | 0 | .5 |
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71 | bug | 1 | .5 | 1 | 1 | 2 | 1 | .5 | .5 | 1 | .5 | 2 | 1 | 1 | .5 | 1 | 2 | .5 |
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72 | rock | 1 | 2 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | .5 |
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73 | ghost | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | .5 | .5 |
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74 | dragon | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | .5 |
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75 | dark | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | .5 | .5 |
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76 | steel | 1 | .5 | .5 | .5 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | .5 |
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77
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78 The rows are attack types, while the columns are defense types. To
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79 see the multiplier for a pokemon attack against a certain type, follow
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80 the row for the attack type to the column of the defending type.
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81
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82 ** Generation I Type System
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83 #+label: pokemon-matchups-gen-1
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84 #+tblname: pokemon-table-gen-one
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85 | | normal | fire | water | electric | grass | ice | fighting | poison | ground | flying | psychic | bug | rock | ghost | dragon |
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86 |----------+--------+------+-------+----------+-------+-----+----------+--------+--------+--------+---------+-----+------+-------+--------|
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87 | normal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 0 | 1 |
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88 | fire | 1 | .5 | .5 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | .5 |
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89 | water | 1 | 2 | .5 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | .5 |
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90 | electric | 1 | 1 | 2 | .5 | .5 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | .5 |
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91 | grass | 1 | .5 | 2 | 1 | .5 | 1 | 1 | .5 | 2 | .5 | 1 | .5 | 2 | 1 | .5 |
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92 | ice | 1 | 1 | .5 | 1 | 2 | .5 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 |
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93 | fighting | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | .5 | .5 | .5 | 2 | 0 | 1 |
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94 | poison | 1 | 1 | 1 | 1 | 2 | 1 | 1 | .5 | .5 | 1 | 1 | 2 | .5 | .5 | 1 |
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95 | ground | 1 | 2 | 1 | 2 | .5 | 1 | 1 | 2 | 1 | 0 | 1 | .5 | 2 | 1 | 1 |
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96 | flying | 1 | 1 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | 1 |
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97 | psychic | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | .5 | 1 | 1 | 1 | 1 |
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98 | bug | 1 | .5 | 1 | 1 | 2 | 1 | .5 | 2 | 1 | .5 | 2 | 1 | 1 | 0 | 1 |
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99 | rock | 1 | 2 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 |
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100 | ghost | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 1 |
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101 | dragon | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
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102
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103
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104 This is the old table from Generation I. The differences from
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105 Generation II are:
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106 - Dark and Steel types are missing (these were introduced in
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107 Generation II).
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108 - Bug is super-effective against Poison (not-very-effective in
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109 Generation II).
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110 - Poison is super-effective against Bug (normal in Generation II).
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111 - Bug is regularly effective against Ghost (super-effective in
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112 Generation II).
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113 - Ice is normally effective against Fire, (not-very-effective in
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114 Generation II).
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115 - Ghost is completely ineffective against Psychic. This is considered
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116 to be a programning glitch.
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117
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118
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119
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120 * Representing the Data
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121
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122 After creating the Pok\eacute{}mon types namespace, we store the
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123 tables of susceptibilities above in =pokemon-table-gen-one= and
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124 =pokemon-table-gen-two=, each of which is a simple vector of
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125 vectors. Because a vector of vectors can be cumbersome, we do not
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126 access the tables directly; instead, we use the derivative structures
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127 =attack-strengths= and =defense-strengths=, which are functions which
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128 return hash-maps associating each row (respectively column) of the
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129 table with its corresponding Pok\eacute{}mon type.
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130
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131
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132 #+srcname: header
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133 #+begin_src clojure :results silent
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134 (ns pokemon.types
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135 (:use clojure.set)
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136 (:use clojure.contrib.combinatorics)
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137 (:use clojure.contrib.math)
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138 (:use clojure.contrib.def)
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139 (:use rlm.rlm-commands))
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140 #+end_src
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141
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142 #+srcname: data
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143 #+begin_src clojure :results silent
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144 (in-ns 'pokemon.types)
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145 ;; record type strengths as a vector of vectors
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146 ;; the variables pokemon-table-gen-one and pokemon-table-gen-two
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147 ;; are replaced with the tables above when this file is tangled.
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148 (def pokemon-gen-one pokemon-table-gen-one)
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149 (def pokemon-gen-two pokemon-table-gen-two)
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150
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151 (defn type-names [] (vec (doall (map (comp keyword first) pokemon-gen-two))))
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152
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153 (defn attack-strengths []
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154 (zipmap
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155 (type-names)
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156 (map (comp vec rest) pokemon-gen-two)))
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157
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158 (defn defense-strengths []
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159 (zipmap (type-names)
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160 (map
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161 (apply juxt (map (attack-strengths) (type-names)))
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162 (range (count (type-names))))))
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163 #+end_src
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164
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165 The two statements
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166
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167 #+begin_src clojure :exports code
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168 (def pokemon-gen-one pokemon-table-gen-one)
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169 (def pokemon-gen-two pokemon-table-gen-two)
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170 #+end_src
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171
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172 probably look weird. When the actual source file is created, those
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173 variables are replaced with the data from the tables above.
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174
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175 See [[../src/pokemon/types.clj][types.clj]] to look at the final tangled output.
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176
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177
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178 #+begin_src clojure :results output :exports both
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179 (clojure.pprint/pprint pokemon.types/pokemon-gen-two)
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180 #+end_src
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181
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182 #+results:
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183 #+begin_example
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184 (("normal" 1 1 1 1 1 1 1 1 1 1 1 1 0.5 0 1 1 0.5)
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185 ("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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186 ("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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187 ("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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188 ("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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189 ("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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190 ("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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191 ("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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192 ("ground" 1 2 1 2 0.5 1 1 2 1 0 1 0.5 2 1 1 1 2)
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193 ("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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194 ("psychic" 1 1 1 1 1 1 2 2 1 1 0.5 1 1 1 1 0 0.5)
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195 ("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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196 ("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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197 ("ghost" 0 1 1 1 1 1 1 1 1 1 2 1 1 2 1 0.5 0.5)
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198 ("dragon" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0.5)
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199 ("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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200 ("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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201 #+end_example
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202
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203 =pokemon-gen-two= is a simple list-of-lists data structure.
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204
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205 #+begin_src clojure :results output :exports both
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206 (clojure.pprint/pprint (pokemon.types/defense-strengths))
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207 #+end_src
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208
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209 #+results:
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210 #+begin_example
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211 {: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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212 :psychic [1 1 1 1 1 1 0.5 1 1 1 0.5 2 1 2 1 2 1],
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213 :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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214 :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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215 :ice [1 2 1 1 1 0.5 2 1 1 1 1 1 2 1 1 1 2],
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216 :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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217 :ghost [0 1 1 1 1 1 0 0.5 1 1 1 0.5 1 2 1 2 1],
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218 :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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219 :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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220 :normal [1 1 1 1 1 1 2 1 1 1 1 1 1 0 1 1 1],
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221 :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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222 :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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223 :ground [1 1 2 0 2 2 1 0.5 1 1 1 1 0.5 1 1 1 1],
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224 :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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225 :dark [1 1 1 1 1 1 2 1 1 1 0 2 1 0.5 1 0.5 1],
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226 :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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227 :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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228 #+end_example
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229
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230 =defense-strengths= is a more convenient form of =pokemon-gen-two=,
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231 with key/value pair access.
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232
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233 * Interfacing with the Data
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234
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235 In the pok\eacute{}mon games, a pok\eacute{}mon can have up to two
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236 types at the same time. For example, Zapdos, the fearsome legendary
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237 that can control lightning, has both the Electric and Flying types. A
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238 pok\eacute{}mon with more than one type gains the advantages and
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239 disadvanteags of both types. The suceptibilitys of each type are
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240 multiplied together to produce the hybrid type's susceptibilities. For
|
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241 example, Electric is weak to Ground (susceptibility of 2), but Flying
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242 is immune to Ground (suceptibility of 0). Zapdos' type,
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243 Electrig/Flying, is immune to Ground because $2 \times 0 = 0$.
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244
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245 #+srcname: types
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246 #+begin_src clojure :results silent
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247 (in-ns 'pokemon.types)
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248
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249 (defn multitypes "All combinations of up to n types" [n]
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250 (vec
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251 (map vec
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252 (reduce concat
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253 (map (partial combinations (type-names))
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254 (range 1 (inc n)))))))
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255
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256 (defn susceptibility
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257 "Hash-map of the susceptibilities of the given type combination
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258 to each type of attack"
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259 [pkmn-types]
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260 (rlm.map-utils/map-vals
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261 clojure.core/rationalize
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262 (apply hash-map
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263 (interleave (type-names)
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264 (apply (partial map *)
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265 (map (defense-strengths) pkmn-types))))))
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266
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267 (defn susceptance
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268 "The cumulative susceptibility of the given type combination"
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269 [types]
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270 (reduce + (map #(expt % 2) (vals (susceptibility types)))))
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271 #+end_src
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272
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273 Now we can work out the suceptability of Zapdos automatically.
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274
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275 Electric is weak to Ground.
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276 #+begin_src clojure :exports both
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277 (:ground (pokemon.types/susceptibility [:electric]))
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278 #+end_src
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279
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280 #+results:
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281 : 2
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282
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283 Flying is immune to Ground.
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284 #+begin_src clojure :exports both
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285 (:ground (pokemon.types/susceptibility [:flying]))
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286 #+end_src
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287
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288 #+results:
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289 : 0
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290
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291 Together, they are immune to Ground.
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292 #+begin_src clojure :exports both
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293 (:ground (pokemon.types/susceptibility [:electric :flying]))
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294 #+end_src
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295
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296 #+results:
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297 : 0
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298
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299
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300
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301
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302 * Best-First Search
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303
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304 I'd like to find type combinations that are interesting, but the total
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305 number of combinations gets huge as we begin to consider more
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306 types. For example, the total possible number of type combinations
|
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307 given just 8 possible types is: 17^{8} = 6,975,757,441 combinations.
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308 Therefore, it's prudent to use search.
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309
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310 These functions are a simple implementation of best-first search in
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311 clojure. The idea to start off with a collection of nodes and some way
|
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312 of finding the best node, and to always expand the best node at every
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313 step.
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314
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315 #+srcname: search
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316 #+begin_src clojure :results silent
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317 (in-ns 'pokemon.types)
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318
|
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319 (defn comparatize
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320 "Define a comparator which uses the numerical outputs of fn as its criterion.
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321 Objects are sorted in increasing numerical order. Objects with the same fn-value
|
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322 are further compared by clojure.core/compare."
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323 [fun]
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324 (fn [a b]
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325 (let [val-a (fun a)
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326 val-b (fun b)]
|
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327 (cond
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328 ;; if the function cannot differentiate the two values
|
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329 ;; then compare the two values using clojure.core/compare
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330 (= val-a val-b) (compare a b)
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331 true
|
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332 ;; LOWER values of the function are preferred
|
|
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333 (compare (- val-a val-b) 0)))))
|
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334
|
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335 (defn-memo best-first-step [successors [visited unvisited]]
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336 (cond (empty? unvisited) nil
|
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337 true
|
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338 (let [best-node (first unvisited)
|
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339 visited* (conj visited best-node)
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340 unvisited*
|
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341 (difference
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342 (union unvisited (set (successors best-node)))
|
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343 visited*)]
|
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344 (println best-node)
|
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345 [visited* unvisited*])))
|
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346
|
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347 ;; memoize partial from core so that for example
|
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348 ;; (= (partial + 1) (partial + 1))
|
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349 ;; this way, best first search can take advantage of the memoization
|
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350 ;; of best-first step
|
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351 (undef partial)
|
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352 (def partial (memoize clojure.core/partial))
|
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353
|
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354 (defn best-first-search
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355 "Searches through a network of alternatives, pursuing
|
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356 initially-promising positions first. Comparator defines which
|
|
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357 positions are more promising, successors produces a list of improved
|
|
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358 positions from the given position (if any exist), and initial-nodes is
|
|
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359 a list of starting positions. Returns a lazy sequence of search results
|
|
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360 [visited-nodes unvisited-nodes], which terminates when
|
|
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|
361 there are no remaining unvisited positions."
|
|
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|
362 [comparator successors initial-nodes]
|
|
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|
363 (let [initial-nodes
|
|
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|
364 (apply (partial sorted-set-by comparator) initial-nodes)
|
|
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|
365 initial-visited-nodes (sorted-set-by comparator)
|
|
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|
366 step (partial best-first-step successors)]
|
|
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|
367 (take-while
|
|
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|
368 (comp not nil?)
|
|
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|
369 (iterate step [initial-visited-nodes initial-nodes]))))
|
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|
370
|
|
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|
371 #+end_src
|
|
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|
372
|
|
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|
373
|
|
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|
374 Now that we have a basic best-first-search, it's convenient to write a
|
|
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|
375 few pokemon-type specific convenience functions.
|
|
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|
376
|
|
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|
377 #+srcname: pokemon-search
|
|
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|
378 #+begin_src clojure :results silent
|
|
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|
379 (in-ns 'pokemon.types)
|
|
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|
380 (defvar type-compare (comparatize susceptance)
|
|
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|
381 "compare two type combinations wrt their susceptibilities")
|
|
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|
382
|
|
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|
383 (defn type-successors
|
|
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|
384 "Return the set of types that can be made by appending a single type
|
|
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|
385 to the given combination."
|
|
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|
386 [type]
|
|
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|
387 (if (nil? type) '()
|
|
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|
388 (set (map (comp vec sort (partial into type)) (multitypes 1)))))
|
|
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|
389
|
|
rlm@0
|
390 (defn immortal?
|
|
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|
391 "A type combo is immortal if it is resistant or invulnerable to
|
|
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|
392 every pokemon type. This is because that set of types can just be
|
|
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|
393 repeated to achieve as low a susceptance as desired"
|
|
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|
394 [type]
|
|
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|
395 (every? (partial > 1) (vals (susceptibility type))))
|
|
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|
396
|
|
rlm@0
|
397 (defn type-successors*
|
|
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|
398 "Stop expanding a type if it's immortal, or if it is longer than or
|
|
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|
399 equal to limit-size. Also, only return type additions that are
|
|
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|
400 strictly better than the initial type."
|
|
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|
401 [limit-size type]
|
|
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|
402 (if (or (<= limit-size (count type)) (immortal? type)) '()
|
|
rlm@0
|
403 (set (filter #(< 0 (type-compare type %)) (type-successors type)))))
|
|
rlm@0
|
404
|
|
rlm@0
|
405 (defn pokemon-type-search
|
|
rlm@0
|
406 "Search among type-combos no greater than length n, limited by limit
|
|
rlm@0
|
407 steps of best-first-search."
|
|
rlm@0
|
408 ([n] (pokemon-type-search n Integer/MAX_VALUE))
|
|
rlm@0
|
409 ([n limit]
|
|
rlm@0
|
410 (first (last
|
|
rlm@0
|
411 (take
|
|
rlm@0
|
412 limit
|
|
rlm@0
|
413 (best-first-search
|
|
rlm@0
|
414 type-compare
|
|
rlm@0
|
415 (partial type-successors* n)
|
|
rlm@0
|
416 (multitypes 1)))))))
|
|
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|
417
|
|
rlm@0
|
418 (defvar immortals
|
|
rlm@0
|
419 (comp (partial filter immortal?) pokemon-type-search)
|
|
rlm@0
|
420 "find all the immortal pokemon types ")
|
|
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|
421
|
|
rlm@0
|
422 #+end_src
|
|
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|
423
|
|
rlm@0
|
424 Because there are so many type combinations, it's important to narrow
|
|
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|
425 down the results as much as possible. That is why =type-successors*=
|
|
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|
426 only returns types that are actually better than the type it is given.
|
|
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|
427
|
|
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|
428 Best-first search can get caught optimizing a single type forever, so
|
|
rlm@0
|
429 it's also important to limit the search space to be finite by setting
|
|
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|
430 an upper bound on the length of a type combo.
|
|
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|
431
|
|
rlm@0
|
432 * Results
|
|
rlm@0
|
433 ** The best dual-type combo
|
|
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|
434
|
|
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|
435 #+begin_src clojure :results cache verbatim :exports both
|
|
rlm@0
|
436 (first (pokemon.types/pokemon-type-search 2))
|
|
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|
437 #+end_src
|
|
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|
438
|
|
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|
439 #+results:
|
|
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|
440 : [:dark :ghost]
|
|
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|
441
|
|
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|
442 Dark and Ghost, which additionally has the property of having no
|
|
rlm@0
|
443 weaknesses to any other type, is the best type combo in terms of
|
|
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|
444 susceptance.
|
|
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|
445
|
|
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|
446 The Dark and Steel types were introduced many years after
|
|
rlm@0
|
447 pok\eacute{}mon started. In addition to the additional types, the
|
|
rlm@0
|
448 pok\eacute{}mon games gained a few new rules concerning some of the
|
|
rlm@0
|
449 matchups of the original types. Therefore, it's also interesting to see what
|
|
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|
450 type combination was most powerful before those types and new rules were introduced.
|
|
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|
451
|
|
rlm@0
|
452 The easiest way to do this with my setup is to just rebind the
|
|
rlm@0
|
453 =pokemon-gen-two= table to the =pokemon-gen-one= table. Since
|
|
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|
454 everything that references this variable is a function and we're not
|
|
rlm@0
|
455 doing anything too crazy with lazy-sequences and late-binding, this
|
|
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|
456 simple macro will do the job.
|
|
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|
457
|
|
rlm@0
|
458 #+srcname: old-school
|
|
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|
459 #+begin_src clojure :results silent
|
|
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|
460 (in-ns 'pokemon.types)
|
|
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|
461
|
|
rlm@0
|
462 (defmacro old-school
|
|
rlm@0
|
463 [& forms]
|
|
rlm@0
|
464 `(binding [pokemon-gen-two pokemon-gen-one] ~@forms))
|
|
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|
465 #+end_src
|
|
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|
466
|
|
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|
467 Using the =old-school= macro, it's easy to find answers for the
|
|
rlm@0
|
468 original 15 pokemon types as well as the expanded pokemon types
|
|
rlm@0
|
469 introduced later.
|
|
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|
470
|
|
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|
471 #+begin_src clojure :results verbatim :exports both :cache yes
|
|
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|
472 (pokemon.types/old-school (first (pokemon.types/pokemon-type-search 2)))
|
|
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|
473 #+end_src
|
|
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|
474
|
|
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|
475 #+results[f43470fdf460ed546e9c57879abc9eda56da129f]:
|
|
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|
476 : [:ghost :psychic]
|
|
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|
477
|
|
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|
478 Ghost and Psychic also manages to have no weaknesses to any of the original
|
|
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|
479 types.
|
|
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|
480
|
|
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|
481 #+begin_src clojure :results output :exports both
|
|
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|
482 (clojure.pprint/pprint
|
|
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|
483 (pokemon.types/old-school
|
|
rlm@0
|
484 (pokemon.types/susceptibility [:ghost :psychic])))
|
|
rlm@0
|
485 #+end_src
|
|
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|
486
|
|
rlm@0
|
487 #+results:
|
|
rlm@0
|
488 #+begin_example
|
|
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|
489 {:water 1,
|
|
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|
490 :psychic 1/2,
|
|
rlm@0
|
491 :dragon 1,
|
|
rlm@0
|
492 :fire 1,
|
|
rlm@0
|
493 :ice 1,
|
|
rlm@0
|
494 :grass 1,
|
|
rlm@0
|
495 :ghost 0,
|
|
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|
496 :poison 1/2,
|
|
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|
497 :flying 1,
|
|
rlm@0
|
498 :normal 0,
|
|
rlm@0
|
499 :rock 1,
|
|
rlm@0
|
500 :electric 1,
|
|
rlm@0
|
501 :ground 1,
|
|
rlm@0
|
502 :fighting 0,
|
|
rlm@0
|
503 :bug 0}
|
|
rlm@0
|
504 #+end_example
|
|
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|
505
|
|
rlm@0
|
506 ** An Immortal Type
|
|
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|
507 It's possible to quickly find an immortal type by giving the search
|
|
rlm@0
|
508 a long enough maximum type length. 50 rounds of search with a max
|
|
rlm@0
|
509 type limit of 10 is enough to find an immortal type.
|
|
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|
510
|
|
rlm@0
|
511 #+begin_src clojure :results scalar :exports both
|
|
rlm@0
|
512 (first (pokemon.types/pokemon-type-search 10 50))
|
|
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|
513 #+end_src
|
|
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|
514
|
|
rlm@0
|
515 #+results:
|
|
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|
516 : [:dragon :fire :flying :ghost :grass :ground :steel :steel :water :water]
|
|
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|
517
|
|
rlm@0
|
518
|
|
rlm@0
|
519 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
520 (clojure.pprint/pprint
|
|
rlm@0
|
521 (pokemon.types/susceptibility
|
|
rlm@0
|
522 [:dragon :fire :flying :ghost :grass :ground :steel :steel :water :water]))
|
|
rlm@0
|
523 #+end_src
|
|
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|
524
|
|
rlm@0
|
525 #+results:
|
|
rlm@0
|
526 #+begin_example
|
|
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|
527 {:water 1/4,
|
|
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|
528 :psychic 1/4,
|
|
rlm@0
|
529 :dragon 1/2,
|
|
rlm@0
|
530 :fire 1/2,
|
|
rlm@0
|
531 :ice 1/2,
|
|
rlm@0
|
532 :grass 1/8,
|
|
rlm@0
|
533 :ghost 1/2,
|
|
rlm@0
|
534 :poison 0,
|
|
rlm@0
|
535 :flying 1/2,
|
|
rlm@0
|
536 :normal 0,
|
|
rlm@0
|
537 :rock 1/2,
|
|
rlm@0
|
538 :electric 0,
|
|
rlm@0
|
539 :ground 0,
|
|
rlm@0
|
540 :fighting 0,
|
|
rlm@0
|
541 :dark 1/2,
|
|
rlm@0
|
542 :steel 1/32,
|
|
rlm@0
|
543 :bug 1/16}
|
|
rlm@0
|
544 #+end_example
|
|
rlm@0
|
545
|
|
rlm@0
|
546 ** Explanations for Common Pok\eacute{}mon Strategies
|
|
rlm@0
|
547
|
|
rlm@0
|
548 Many people start out a battle with either a normal pok\eacute{}mon or an
|
|
rlm@0
|
549 electric pok\eacute{}mon, and here's some justification for that choice.
|
|
rlm@0
|
550
|
|
rlm@0
|
551 #+srcname: weaknesses
|
|
rlm@0
|
552 #+begin_src clojure :results silent
|
|
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553 (in-ns 'pokemon.types)
|
|
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|
554 (defn critical-weaknesses [type]
|
|
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|
555 (count (filter #(> % 1) (vals (susceptibility type)))))
|
|
rlm@0
|
556 #+end_src
|
|
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|
557
|
|
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|
558 #+begin_src clojure :exports both :results output
|
|
rlm@0
|
559 (clojure.pprint/pprint
|
|
rlm@0
|
560 (sort-by pokemon.types/critical-weaknesses (pokemon.types/multitypes 1)))
|
|
rlm@0
|
561 #+end_src
|
|
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|
562
|
|
rlm@0
|
563 #+results:
|
|
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|
564 #+begin_example
|
|
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|
565 ([:normal]
|
|
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|
566 [:electric]
|
|
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567 [:water]
|
|
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|
568 [:fighting]
|
|
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569 [:poison]
|
|
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|
570 [:ghost]
|
|
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|
571 [:dragon]
|
|
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572 [:dark]
|
|
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|
573 [:fire]
|
|
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|
574 [:ground]
|
|
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|
575 [:flying]
|
|
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|
576 [:psychic]
|
|
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|
577 [:bug]
|
|
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|
578 [:steel]
|
|
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|
579 [:ice]
|
|
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|
580 [:grass]
|
|
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|
581 [:rock])
|
|
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|
582 #+end_example
|
|
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|
583
|
|
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|
584 Electric and Normal are among the best types with which to start the
|
|
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|
585 game, since they have the fewest weaknesses among all the types.
|
|
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|
586
|
|
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587 At the beginning of the pok\eacute{}mon games, players are given a choice
|
|
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|
588 between the Fire pok\eacute{}mon Charmander, the Water pok\eacute{}mon Squirtle, or
|
|
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|
589 the Grass/Poison pok\eacute{}mon Bulbasaur.
|
|
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|
590
|
|
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|
591 #+begin_src clojure :exports both :results verbatim
|
|
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|
592 (sort-by pokemon.types/susceptance [[:fire] [:water] [:grass :poison]])
|
|
rlm@0
|
593 #+end_src
|
|
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|
594
|
|
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|
595 #+results:
|
|
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|
596 : ([:water] [:fire] [:grass :poison])
|
|
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597
|
|
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|
598 As can be seen, the Water pok\eacute{}mon Squirtle is the most solid
|
|
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|
599 choice starting out, insofar as susceptance is concerned.
|
|
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|
600
|
|
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|
601 ** The Worst Pok\eacute{}mon Types
|
|
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|
602
|
|
rlm@0
|
603 #+srcname: weak-types
|
|
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|
604 #+begin_src clojure :results silent
|
|
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|
605 (in-ns 'pokemon.types)
|
|
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|
606
|
|
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|
607 (defn type-compare-weak
|
|
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|
608 "compare first by total number of critical-weaknesses,
|
|
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|
609 then by overall susceptance, favoring weaker types."
|
|
rlm@0
|
610 [type-1 type-2]
|
|
rlm@0
|
611 (let [measure (memoize (juxt critical-weaknesses susceptance))]
|
|
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|
612 (if (= (measure type-2) (measure type-1))
|
|
rlm@0
|
613 (compare type-2 type-1)
|
|
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|
614 (compare (measure type-2) (measure type-1)))))
|
|
rlm@0
|
615
|
|
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|
616 (defn resistant?
|
|
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|
617 "might as well get rid of types that are resistant to any type"
|
|
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|
618 [type]
|
|
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|
619 (not (every? #(< 0 %) (vals (susceptibility type)))))
|
|
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|
620
|
|
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|
621 (defn type-successors-weak
|
|
rlm@0
|
622 [limit type]
|
|
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|
623 (set (if (<= limit (count type)) '()
|
|
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|
624 (filter #(< 0 (type-compare-weak type %))
|
|
rlm@0
|
625 (remove resistant? (type-successors type))))))
|
|
rlm@0
|
626
|
|
rlm@0
|
627 (defn pokemon-type-search-weak
|
|
rlm@0
|
628 "Search among type-combos no greater than length n, limited by limit
|
|
rlm@0
|
629 steps of best-first-search."
|
|
rlm@0
|
630 ([n] (pokemon-type-search-weak n Integer/MAX_VALUE))
|
|
rlm@0
|
631 ([n limit]
|
|
rlm@0
|
632 (first (last
|
|
rlm@0
|
633 (take
|
|
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|
634 limit
|
|
rlm@0
|
635 (best-first-search
|
|
rlm@0
|
636 type-compare-weak
|
|
rlm@0
|
637 (partial type-successors-weak n)
|
|
rlm@0
|
638 (multitypes 1)))))))
|
|
rlm@0
|
639 #+end_src
|
|
rlm@0
|
640
|
|
rlm@0
|
641
|
|
rlm@0
|
642 #+begin_src clojure :results scalar :exports both
|
|
rlm@0
|
643 (first (pokemon.types/pokemon-type-search-weak 1))
|
|
rlm@0
|
644 #+end_src
|
|
rlm@0
|
645
|
|
rlm@0
|
646 #+results:
|
|
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|
647 : [:rock]
|
|
rlm@0
|
648
|
|
rlm@0
|
649 Poor Rock. It's just not that good a type. Maybe this is why Brock
|
|
rlm@0
|
650 (who has rock pok\eacute{}mon) is the first gym leader in the games.
|
|
rlm@0
|
651
|
|
rlm@0
|
652 #+begin_src clojure :results scalar cache :exports both
|
|
rlm@0
|
653 (first (pokemon.types/pokemon-type-search-weak 2))
|
|
rlm@0
|
654 #+end_src
|
|
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|
655
|
|
rlm@0
|
656 #+results:
|
|
rlm@0
|
657 : [:grass :ice]
|
|
rlm@0
|
658
|
|
rlm@0
|
659 # ;;bonus convergently immortal type combo
|
|
rlm@0
|
660 # (susceptance (vec (concat (repeat 150 :water) (repeat 50 :poison) (repeat 50 :steel) [:ghost :normal :flying :ground :dark])))
|
|
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|
661
|
|
rlm@0
|
662 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
663 (clojure.pprint/pprint
|
|
rlm@0
|
664 (pokemon.types/susceptibility [:grass :ice]))
|
|
rlm@0
|
665 #+end_src
|
|
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|
666
|
|
rlm@0
|
667 #+results:
|
|
rlm@0
|
668 #+begin_example
|
|
rlm@0
|
669 {:water 1/2,
|
|
rlm@0
|
670 :psychic 1,
|
|
rlm@0
|
671 :dragon 1,
|
|
rlm@0
|
672 :fire 4,
|
|
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|
673 :ice 1,
|
|
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|
674 :grass 1/2,
|
|
rlm@0
|
675 :ghost 1,
|
|
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|
676 :poison 2,
|
|
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|
677 :flying 2,
|
|
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|
678 :normal 1,
|
|
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|
679 :rock 2,
|
|
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|
680 :electric 1/2,
|
|
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|
681 :ground 1/2,
|
|
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|
682 :fighting 2,
|
|
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|
683 :dark 1,
|
|
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|
684 :steel 2,
|
|
rlm@0
|
685 :bug 2}
|
|
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|
686 #+end_example
|
|
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|
687
|
|
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|
688 This miserable combination is weak to 6 types and double-weak to
|
|
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|
689 Fire. No pok\eacute{}mon in the games actually has this type.
|
|
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|
690
|
|
rlm@0
|
691 * Conclusion
|
|
rlm@0
|
692
|
|
rlm@0
|
693 Searching for a type that is weak to everything takes a very long time
|
|
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|
694 and fails to reveal any results. That's the problem with a search
|
|
rlm@0
|
695 over this large problem space --- if there's an easy solution, the
|
|
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|
696 search will find it quickly, but it can be very hard to determine
|
|
rlm@0
|
697 whether there is actually a solution.
|
|
rlm@0
|
698
|
|
rlm@0
|
699 In the [[./lpsolve.org][next installment]], I'll use =lp_solve= to solve this problem in
|
|
rlm@0
|
700 a different way.
|
|
rlm@0
|
701
|
|
rlm@0
|
702
|
|
rlm@5
|
703
|
|
rlm@0
|
704 * COMMENT main program
|
|
rlm@0
|
705 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :exports none
|
|
rlm@0
|
706 <<header>>
|
|
rlm@0
|
707 #+end_src
|
|
rlm@0
|
708
|
|
rlm@0
|
709 ## this is necessary to define pokemon-table inside the source code.
|
|
rlm@0
|
710
|
|
rlm@0
|
711 #+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
|
|
rlm@0
|
712 <<data>>
|
|
rlm@0
|
713 #+end_src
|
|
rlm@0
|
714
|
|
rlm@0
|
715 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :exports none
|
|
rlm@0
|
716 <<types>>
|
|
rlm@0
|
717 <<search>>
|
|
rlm@0
|
718 <<pokemon-search>>
|
|
rlm@0
|
719 <<old-school>>
|
|
rlm@0
|
720 <<weaknesses>>
|
|
rlm@0
|
721 <<weak-types>>
|
|
rlm@0
|
722 #+end_src
|
|
rlm@0
|
723
|
|
rlm@0
|
724
|
|
rlm@0
|
725
|
|
rlm@0
|
726
|
|
rlm@0
|
727
|
|
rlm@0
|
728
|
|
rlm@0
|
729
|
|
rlm@0
|
730
|
|
rlm@0
|
731
|
|
rlm@0
|
732
|
|
rlm@0
|
733
|
|
rlm@0
|
734
|
|
rlm@0
|
735
|
|
rlm@0
|
736
|
|
rlm@0
|
737
|
|
rlm@0
|
738
|
|
rlm@0
|
739
|
|
rlm@0
|
740
|
