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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 /types/ (Rock, Grass, Ice, Psychic, Ground, Bug, Flying, Fire,
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12 Fighting, Dark, Dragon, Poison, Water, Ghost, Normal, Electric, and
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13 Steel) that interact like an extended version of Rock-Paper-Scissors:
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14 for example, the Fire type is strong against the Grass type but weak
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15 against the Water type. In the table below, we've recorded the
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16 relative strengths of each of the types in the Pok\eacute{}mon type
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17 system; the number in each cell indicates how effective an attack of
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18 the type in the row is against a Pok\eacute{}mon of the type in the
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19 column. We call these numbers /susceptibilities/.
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20
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21 In the Pok\eacute{}mon games, only four susceptibility values (two,
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22 one, one-half, and zero) occur. These numbers indicate particularly
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23 high susceptibility, average susceptibility, particularly low
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24 susceptibility, and no susceptibility (immunity).
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25
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26 - The suceptability of Flying types /against/ Ground is 0, because Ground
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27 attacks cannot hurt Flying pok\eacute{}mon at all. The damage that
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28 a Ground type attack normally does is /multiplied/ by 0 when it is
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29 uesd against a Flying type pok\eacute{}mon.
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30
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31 - The susceptability of Fire types against Water attacks
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32 is 2, because Water type attacks are strong against Fire type
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33 Pok\eacute{}mon. The damage that a Water type attack normally does
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34 is doubled when it is used against a Fire type pok\eacute{}mon.
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35
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36 - The susceptability of Water types against Water attacks is
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37 $\frac{1}{2}$, because Water type attacks are strong against Water
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38 type Pok\eacute{}mon. The damage that a Water type attack normally
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39 does is halved when it is used against a Water type
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40 pok\eacute{}mon.
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41
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42 There are two pok\eacute{}mon type systems in use. The first is the
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43 classic system which was used for the very first pok\eacute{}mon
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44 games, Red, Yellow, and Blue. This old system was used from 1998 to
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45 2000 in America, and is known as the /Generation I Type System/. The
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46 modern pok\eacute{}mon type system was introduced in 2000 with the
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47 introduction of pok\eacute{}mon Gold and Silver, and has been in use
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48 ever since. It is called the /Generation II Type System/.
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49
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50 The definitions of the two Type Systems are included below.
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51
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52 * Generation I and II Type System Data
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53
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54 ** Generation II Type System
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55 #+label: pokemon-matchups
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56 #+tblname: pokemon-table-gen-two
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57 | | normal | fire | water | electric | grass | ice | fighting | poison | ground | flying | psychic | bug | rock | ghost | dragon | dark | steel |
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58 |----------+--------+------+-------+----------+-------+-----+----------+--------+--------+--------+---------+-----+------+-------+--------+------+-------|
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59 | normal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 0 | 1 | 1 | .5 |
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60 | fire | 1 | .5 | .5 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 1 | 2 |
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61 | water | 1 | 2 | .5 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | 1 |
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62 | electric | 1 | 1 | 2 | .5 | .5 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | .5 | 1 | 1 |
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63 | grass | 1 | .5 | 2 | 1 | .5 | 1 | 1 | .5 | 2 | .5 | 1 | .5 | 2 | 1 | .5 | 1 | .5 |
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64 | ice | 1 | .5 | .5 | 1 | 2 | .5 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 |
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65 | fighting | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | .5 | .5 | .5 | 2 | 0 | 1 | 2 | 2 |
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66 | poison | 1 | 1 | 1 | 1 | 2 | 1 | 1 | .5 | .5 | 1 | 1 | 1 | .5 | .5 | 1 | 1 | 0 |
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67 | ground | 1 | 2 | 1 | 2 | .5 | 1 | 1 | 2 | 1 | 0 | 1 | .5 | 2 | 1 | 1 | 1 | 2 |
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68 | flying | 1 | 1 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | 1 | 1 | .5 |
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69 | psychic | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | .5 | 1 | 1 | 1 | 1 | 0 | .5 |
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70 | bug | 1 | .5 | 1 | 1 | 2 | 1 | .5 | .5 | 1 | .5 | 2 | 1 | 1 | .5 | 1 | 2 | .5 |
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71 | rock | 1 | 2 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | .5 |
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72 | ghost | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | .5 | .5 |
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73 | dragon | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | .5 |
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74 | dark | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | .5 | .5 |
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75 | steel | 1 | .5 | .5 | .5 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | .5 |
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76
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77 The rows are attack types, while the columns are defense types. To
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78 see the multiplier for a pok\eacute{}mon attack against a certain type, follow
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79 the row for the attack type to the column of the defending type.
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80
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81 ** Generation I Type System
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82 #+label: pokemon-matchups-gen-1
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83 #+tblname: pokemon-table-gen-one
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84 | | normal | fire | water | electric | grass | ice | fighting | poison | ground | flying | psychic | bug | rock | ghost | dragon |
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85 |----------+--------+------+-------+----------+-------+-----+----------+--------+--------+--------+---------+-----+------+-------+--------|
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86 | normal | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | .5 | 0 | 1 |
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87 | fire | 1 | .5 | .5 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | .5 |
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88 | water | 1 | 2 | .5 | 1 | .5 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | .5 |
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89 | electric | 1 | 1 | 2 | .5 | .5 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 1 | 1 | .5 |
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90 | grass | 1 | .5 | 2 | 1 | .5 | 1 | 1 | .5 | 2 | .5 | 1 | .5 | 2 | 1 | .5 |
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91 | ice | 1 | 1 | .5 | 1 | 2 | .5 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 |
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92 | fighting | 2 | 1 | 1 | 1 | 1 | 2 | 1 | .5 | 1 | .5 | .5 | .5 | 2 | 0 | 1 |
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93 | poison | 1 | 1 | 1 | 1 | 2 | 1 | 1 | .5 | .5 | 1 | 1 | 2 | .5 | .5 | 1 |
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94 | ground | 1 | 2 | 1 | 2 | .5 | 1 | 1 | 2 | 1 | 0 | 1 | .5 | 2 | 1 | 1 |
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95 | flying | 1 | 1 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | .5 | 1 | 1 |
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96 | psychic | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | .5 | 1 | 1 | 1 | 1 |
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97 | bug | 1 | .5 | 1 | 1 | 2 | 1 | .5 | 2 | 1 | .5 | 2 | 1 | 1 | 0 | 1 |
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98 | rock | 1 | 2 | 1 | 1 | 1 | 2 | .5 | 1 | .5 | 2 | 1 | 2 | 1 | 1 | 1 |
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99 | ghost | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 1 |
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100 | dragon | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
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101
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102
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103 This is the old table from Generation I. The differences from
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104 Generation II are:
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105 - Dark and Steel types are missing (these were introduced in
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106 Generation II).
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107 - Bug is super-effective against Poison (not-very-effective in
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108 Generation II).
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109 - Poison is super-effective against Bug (normal in Generation II).
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110 - Bug is regularly effective against Ghost (super-effective in
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111 Generation II).
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112 - Ice is normally effective against Fire, (not-very-effective in
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113 Generation II).
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114 - Ghost is completely ineffective against Psychic, even though the
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115 pok\eacute{}mon anime ran [[http://bulbapedia.bulbagarden.net/wiki/EP022][a three-part series]] about how Ghost
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116 pok\eacute{}mon are the best way to defeat Psychic pok\eacute{}mon,
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117 and the Red, Blue, and Yellow games each have a character who
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118 states "The only thing Psychic pok\eacute{}mon fear are Bugs and
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119 Ghosts!" This is considered to be a programning glitch. Ghost is
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120 super-effective against Psychic in Generation II.
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121
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122 * Representing the Data
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123
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124 After creating the Pok\eacute{}mon types namespace, we store the
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125 tables of susceptibilities above in =pokemon-table-gen-one= and
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126 =pokemon-table-gen-two=, each of which is a simple vector of
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127 vectors. Because a vector of vectors can be cumbersome, we do not
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128 access the tables directly; instead, we use the derivative structures
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129 =attack-strengths= and =defense-strengths=, which are functions which
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130 return hash-maps associating each row (respectively column) of the
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131 table with its corresponding Pok\eacute{}mon type.
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132
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133
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134 #+srcname: header
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135 #+begin_src clojure :results silent
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136 (ns pokemon.types
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137 (:use clojure.set)
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138 (:use clojure.contrib.combinatorics)
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139 (:use clojure.contrib.math)
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140 (:use clojure.contrib.def)
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141 (:use rlm.rlm-commands))
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142 #+end_src
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143
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144 #+srcname: data
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145 #+begin_src clojure :results silent
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146 (in-ns 'pokemon.types)
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147 ;; record type strengths as a vector of vectors
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148 ;; the variables pokemon-table-gen-one and pokemon-table-gen-two
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149 ;; are replaced with the tables above when this file is tangled.
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150 (def pokemon-gen-one pokemon-table-gen-one)
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151 (def pokemon-gen-two pokemon-table-gen-two)
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152
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153 (defn type-names [] (vec (doall (map (comp keyword first) pokemon-gen-two))))
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154
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155 (defn attack-strengths []
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156 (zipmap
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157 (type-names)
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158 (map (comp vec rest) pokemon-gen-two)))
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159
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160 (defn defense-strengths []
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161 (zipmap (type-names)
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162 (map
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163 (apply juxt (map (attack-strengths) (type-names)))
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164 (range (count (type-names))))))
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165 #+end_src
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166
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167 The two statements
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168
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169 #+begin_src clojure :exports code
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170 (def pokemon-gen-one pokemon-table-gen-one)
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171 (def pokemon-gen-two pokemon-table-gen-two)
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172 #+end_src
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173
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174 probably look weird. When the actual source file is created, those
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175 variables are replaced with the data from the tables above.
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176
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177 See [[../src/pokemon/types.clj][types.clj]] to look at the final tangled output.
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178
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179
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180 #+begin_src clojure :results output :exports both
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181 (clojure.pprint/pprint pokemon.types/pokemon-gen-two)
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182 #+end_src
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183
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184 #+results:
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185 #+begin_example
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186 (("normal" 1 1 1 1 1 1 1 1 1 1 1 1 0.5 0 1 1 0.5)
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187 ("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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188 ("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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189 ("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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190 ("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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191 ("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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192 ("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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193 ("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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194 ("ground" 1 2 1 2 0.5 1 1 2 1 0 1 0.5 2 1 1 1 2)
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195 ("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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196 ("psychic" 1 1 1 1 1 1 2 2 1 1 0.5 1 1 1 1 0 0.5)
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197 ("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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198 ("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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199 ("ghost" 0 1 1 1 1 1 1 1 1 1 2 1 1 2 1 0.5 0.5)
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200 ("dragon" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0.5)
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201 ("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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202 ("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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203 #+end_example
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204
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205 =pokemon-gen-two= is a simple list-of-lists data structure.
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206
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207 #+begin_src clojure :results output :exports both
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208 (clojure.pprint/pprint (pokemon.types/defense-strengths))
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209 #+end_src
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210
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211 #+results:
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212 #+begin_example
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213 {: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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214 :psychic [1 1 1 1 1 1 0.5 1 1 1 0.5 2 1 2 1 2 1],
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215 :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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216 :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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217 :ice [1 2 1 1 1 0.5 2 1 1 1 1 1 2 1 1 1 2],
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218 :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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219 :ghost [0 1 1 1 1 1 0 0.5 1 1 1 0.5 1 2 1 2 1],
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220 :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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221 :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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222 :normal [1 1 1 1 1 1 2 1 1 1 1 1 1 0 1 1 1],
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223 :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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224 :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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225 :ground [1 1 2 0 2 2 1 0.5 1 1 1 1 0.5 1 1 1 1],
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226 :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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227 :dark [1 1 1 1 1 1 2 1 1 1 0 2 1 0.5 1 0.5 1],
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228 :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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229 :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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230 #+end_example
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231
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232 =defense-strengths= is a more convenient form of =pokemon-gen-two=,
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233 with key/value pair access.
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234
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235 * Interfacing with the Data
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236
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237 In the pok\eacute{}mon games, a pok\eacute{}mon can have up to two
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238 types at the same time. For example, [[http://bulbapedia.bulbagarden.net/wiki/Zapdos][Zapdos]], the fearsome legendary
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239 bird that can control lightning, has both the Electric and Flying
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240 types. A pok\eacute{}mon with more than one type gains the advantages
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241 and disadvanteags of both types. The suceptibilitys of each type are
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242 multiplied together to produce the hybrid type's susceptibilities. For
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243 example, Electric is weak to Ground (susceptibility of 2), but Flying
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244 is immune to Ground (suceptibility of 0). [[http://bulbapedia.bulbagarden.net/wiki/Zapdos][Zapdos']] type,
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245 Electrig/Flying, is immune to Ground because $2 \times 0 = 0$.
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246
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247 #+srcname: types
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248 #+begin_src clojure :results silent
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249 (in-ns 'pokemon.types)
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250
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251 (defn multitypes "All combinations of up to n types" [n]
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252 (vec
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253 (map vec
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254 (reduce concat
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255 (map (partial combinations (type-names))
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256 (range 1 (inc n)))))))
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257
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258 (defn susceptibility
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259 "Hash-map of the susceptibilities of the given type combination
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260 to each type of attack"
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261 [pkmn-types]
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262 (rlm.map-utils/map-vals
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263 clojure.core/rationalize
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264 (apply hash-map
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265 (interleave (type-names)
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266 (apply (partial map *)
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267 (map (defense-strengths) pkmn-types))))))
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268
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269 (defn susceptance
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270 "The cumulative susceptibility of the given type combination"
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271 [types]
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272 (reduce + (map #(expt % 2) (vals (susceptibility types)))))
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273 #+end_src
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274
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275 Now we can work out the suceptability of [[http://bulbapedia.bulbagarden.net/wiki/Zapdos][Zapdos]] automatically.
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276
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277 Electric is weak to Ground.
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278 #+begin_src clojure :exports both
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279 (:ground (pokemon.types/susceptibility [:electric]))
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280 #+end_src
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281
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282 #+results:
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283 : 2
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284
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285 Flying is immune to Ground.
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286 #+begin_src clojure :exports both
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287 (:ground (pokemon.types/susceptibility [:flying]))
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288 #+end_src
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289
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290 #+results:
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291 : 0
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292
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293 Together, they are immune to Ground.
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294 #+begin_src clojure :exports both
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295 (:ground (pokemon.types/susceptibility [:electric :flying]))
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296 #+end_src
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297
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298 #+results:
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299 : 0
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300
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301
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302
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303
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304 * Best-First Search
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305
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306 I'd like to find type combinations that are interesting, but the total
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307 number of combinations gets huge as we begin to consider more
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308 types. For example, the total possible number of type combinations
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309 given just 8 possible types is: 17^{8} = 6,975,757,441 combinations.
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310 Therefore, it's prudent to use search.
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311
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312 These functions are a simple implementation of best-first search in
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313 clojure. The idea to start off with a collection of nodes and some way
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314 of finding the best node, and to always expand the best node at every
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315 step.
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316
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317 #+srcname: search
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318 #+begin_src clojure :results silent
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319 (in-ns 'pokemon.types)
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320
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321 (defn comparatize
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322 "Define a comparator which uses the numerical outputs of fn as its criterion.
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323 Objects are sorted in increasing numerical order. Objects with the same fn-value
|
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324 are further compared by clojure.core/compare."
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325 [fun]
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326 (fn [a b]
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327 (let [val-a (fun a)
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328 val-b (fun b)]
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329 (cond
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330 ;; if the function cannot differentiate the two values
|
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331 ;; then compare the two values using clojure.core/compare
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332 (= val-a val-b) (compare a b)
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333 true
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334 ;; LOWER values of the function are preferred
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335 (compare (- val-a val-b) 0)))))
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336
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337 (defn-memo best-first-step [successors [visited unvisited]]
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338 (cond (empty? unvisited) nil
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339 true
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340 (let [best-node (first unvisited)
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341 visited* (conj visited best-node)
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342 unvisited*
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343 (difference
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344 (union unvisited (set (successors best-node)))
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345 visited*)]
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346 (println best-node)
|
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347 [visited* unvisited*])))
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348
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349 ;; memoize partial from core so that for example
|
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350 ;; (= (partial + 1) (partial + 1))
|
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351 ;; this way, best first search can take advantage of the memoization
|
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352 ;; of best-first step
|
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353 (undef partial)
|
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354 (def partial (memoize clojure.core/partial))
|
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355
|
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356 (defn best-first-search
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357 "Searches through a network of alternatives, pursuing
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358 initially-promising positions first. Comparator defines which
|
|
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359 positions are more promising, successors produces a list of improved
|
|
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360 positions from the given position (if any exist), and initial-nodes is
|
|
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361 a list of starting positions. Returns a lazy sequence of search results
|
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362 [visited-nodes unvisited-nodes], which terminates when
|
|
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363 there are no remaining unvisited positions."
|
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364 [comparator successors initial-nodes]
|
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365 (let [initial-nodes
|
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366 (apply (partial sorted-set-by comparator) initial-nodes)
|
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367 initial-visited-nodes (sorted-set-by comparator)
|
|
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|
368 step (partial best-first-step successors)]
|
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|
369 (take-while
|
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370 (comp not nil?)
|
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371 (iterate step [initial-visited-nodes initial-nodes]))))
|
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372
|
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373 #+end_src
|
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374
|
|
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375
|
|
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|
376 Now that we have a basic best-first-search, it's convenient to write a
|
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|
377 few pok\eacute{}mon-type specific convenience functions.
|
|
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|
378
|
|
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|
379 #+srcname: pokemon-search
|
|
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380 #+begin_src clojure :results silent
|
|
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|
381 (in-ns 'pokemon.types)
|
|
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|
382 (defvar type-compare (comparatize susceptance)
|
|
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|
383 "compare two type combinations wrt their susceptibilities")
|
|
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|
384
|
|
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|
385 (defn type-successors
|
|
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|
386 "Return the set of types that can be made by appending a single type
|
|
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|
387 to the given combination."
|
|
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|
388 [type]
|
|
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|
389 (if (nil? type) '()
|
|
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|
390 (set (map (comp vec sort (partial into type)) (multitypes 1)))))
|
|
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|
391
|
|
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|
392 (defn immortal?
|
|
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|
393 "A type combo is immortal if it is resistant or invulnerable to
|
|
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|
394 every pokemon type. This is because that set of types can just be
|
|
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|
395 repeated to achieve as low a susceptance as desired"
|
|
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396 [type]
|
|
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|
397 (every? (partial > 1) (vals (susceptibility type))))
|
|
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|
398
|
|
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|
399 (defn type-successors*
|
|
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|
400 "Stop expanding a type if it's immortal, or if it is longer than or
|
|
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|
401 equal to limit-size. Also, only return type additions that are
|
|
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|
402 strictly better than the initial type."
|
|
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|
403 [limit-size type]
|
|
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|
404 (if (or (<= limit-size (count type)) (immortal? type)) '()
|
|
rlm@0
|
405 (set (filter #(< 0 (type-compare type %)) (type-successors type)))))
|
|
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|
406
|
|
rlm@0
|
407 (defn pokemon-type-search
|
|
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|
408 "Search among type-combos no greater than length n, limited by limit
|
|
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|
409 steps of best-first-search."
|
|
rlm@0
|
410 ([n] (pokemon-type-search n Integer/MAX_VALUE))
|
|
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|
411 ([n limit]
|
|
rlm@0
|
412 (first (last
|
|
rlm@0
|
413 (take
|
|
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|
414 limit
|
|
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|
415 (best-first-search
|
|
rlm@0
|
416 type-compare
|
|
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|
417 (partial type-successors* n)
|
|
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|
418 (multitypes 1)))))))
|
|
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|
419
|
|
rlm@0
|
420 (defvar immortals
|
|
rlm@0
|
421 (comp (partial filter immortal?) pokemon-type-search)
|
|
rlm@0
|
422 "find all the immortal pokemon types ")
|
|
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|
423
|
|
rlm@0
|
424 #+end_src
|
|
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|
425
|
|
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|
426 Because there are so many type combinations, it's important to narrow
|
|
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|
427 down the results as much as possible. That is why =type-successors*=
|
|
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|
428 only returns types that are actually better than the type it is given.
|
|
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|
429
|
|
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|
430 Best-first search can get caught optimizing a single type forever, so
|
|
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|
431 it's also important to limit the search space to be finite by setting
|
|
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|
432 an upper bound on the length of a type combo.
|
|
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|
433
|
|
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|
434 * Results
|
|
rlm@0
|
435 ** The best dual-type combo
|
|
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|
436
|
|
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|
437 #+begin_src clojure :results cache verbatim :exports both
|
|
rlm@0
|
438 (first (pokemon.types/pokemon-type-search 2))
|
|
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|
439 #+end_src
|
|
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|
440
|
|
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|
441 #+results:
|
|
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|
442 : [:dark :ghost]
|
|
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|
443
|
|
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|
444 Dark and Ghost, which additionally has the property of having no
|
|
rlm@9
|
445 weaknesses to any other type, is the best type combo in terms of
|
|
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|
446 susceptance.
|
|
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|
447
|
|
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|
448 The Dark and Steel types were introduced many years after
|
|
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|
449 pok\eacute{}mon started. In addition to the additional types, the
|
|
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|
450 pok\eacute{}mon games gained a few new rules concerning some of the
|
|
rlm@9
|
451 matchups of the original types. Therefore, it's also interesting to
|
|
rlm@9
|
452 see what type combination was most powerful before those types and new
|
|
rlm@9
|
453 rules were introduced.
|
|
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|
454
|
|
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|
455 The easiest way to do this with my setup is to just rebind the
|
|
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|
456 =pokemon-gen-two= table to the =pokemon-gen-one= table. Since
|
|
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|
457 everything that references this variable is a function and we're not
|
|
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|
458 doing anything too crazy with lazy-sequences and late-binding, this
|
|
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|
459 simple macro will do the job.
|
|
rlm@0
|
460
|
|
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|
461 #+srcname: old-school
|
|
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|
462 #+begin_src clojure :results silent
|
|
rlm@0
|
463 (in-ns 'pokemon.types)
|
|
rlm@0
|
464
|
|
rlm@0
|
465 (defmacro old-school
|
|
rlm@0
|
466 [& forms]
|
|
rlm@0
|
467 `(binding [pokemon-gen-two pokemon-gen-one] ~@forms))
|
|
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|
468 #+end_src
|
|
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|
469
|
|
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|
470 Using the =old-school= macro, it's easy to find answers for the
|
|
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|
471 original 15 pokemon types as well as the expanded pokemon types
|
|
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|
472 introduced later.
|
|
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|
473
|
|
rlm@0
|
474 #+begin_src clojure :results verbatim :exports both :cache yes
|
|
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|
475 (pokemon.types/old-school (first (pokemon.types/pokemon-type-search 2)))
|
|
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|
476 #+end_src
|
|
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|
477
|
|
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|
478 #+results[f43470fdf460ed546e9c57879abc9eda56da129f]:
|
|
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|
479 : [:ghost :psychic]
|
|
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|
480
|
|
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|
481 Ghost and Psychic also manages to have no weaknesses to any of the original
|
|
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|
482 types, using the old Generation I rules.
|
|
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|
483
|
|
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|
484 #+begin_src clojure :results output :exports both
|
|
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|
485 (clojure.pprint/pprint
|
|
rlm@0
|
486 (pokemon.types/old-school
|
|
rlm@0
|
487 (pokemon.types/susceptibility [:ghost :psychic])))
|
|
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|
488 #+end_src
|
|
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|
489
|
|
rlm@0
|
490 #+results:
|
|
rlm@0
|
491 #+begin_example
|
|
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|
492 {:water 1,
|
|
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|
493 :psychic 1/2,
|
|
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|
494 :dragon 1,
|
|
rlm@0
|
495 :fire 1,
|
|
rlm@0
|
496 :ice 1,
|
|
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|
497 :grass 1,
|
|
rlm@0
|
498 :ghost 0,
|
|
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|
499 :poison 1/2,
|
|
rlm@0
|
500 :flying 1,
|
|
rlm@0
|
501 :normal 0,
|
|
rlm@0
|
502 :rock 1,
|
|
rlm@0
|
503 :electric 1,
|
|
rlm@0
|
504 :ground 1,
|
|
rlm@0
|
505 :fighting 0,
|
|
rlm@0
|
506 :bug 0}
|
|
rlm@0
|
507 #+end_example
|
|
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|
508
|
|
rlm@0
|
509 ** An Immortal Type
|
|
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|
510 It's possible to quickly find an immortal type by giving the search
|
|
rlm@0
|
511 a long enough maximum type length. 50 rounds of search with a max
|
|
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|
512 type limit of 10 is enough to find an immortal type.
|
|
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|
513
|
|
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|
514 #+begin_src clojure :results scalar :exports both
|
|
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|
515 (first (pokemon.types/pokemon-type-search 10 50))
|
|
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|
516 #+end_src
|
|
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|
517
|
|
rlm@0
|
518 #+results:
|
|
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|
519 : [:dragon :fire :flying :ghost :grass :ground :steel :steel :water :water]
|
|
rlm@0
|
520
|
|
rlm@0
|
521
|
|
rlm@0
|
522 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
523 (clojure.pprint/pprint
|
|
rlm@0
|
524 (pokemon.types/susceptibility
|
|
rlm@0
|
525 [:dragon :fire :flying :ghost :grass :ground :steel :steel :water :water]))
|
|
rlm@0
|
526 #+end_src
|
|
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|
527
|
|
rlm@0
|
528 #+results:
|
|
rlm@0
|
529 #+begin_example
|
|
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|
530 {:water 1/4,
|
|
rlm@0
|
531 :psychic 1/4,
|
|
rlm@0
|
532 :dragon 1/2,
|
|
rlm@0
|
533 :fire 1/2,
|
|
rlm@0
|
534 :ice 1/2,
|
|
rlm@0
|
535 :grass 1/8,
|
|
rlm@0
|
536 :ghost 1/2,
|
|
rlm@0
|
537 :poison 0,
|
|
rlm@0
|
538 :flying 1/2,
|
|
rlm@0
|
539 :normal 0,
|
|
rlm@0
|
540 :rock 1/2,
|
|
rlm@0
|
541 :electric 0,
|
|
rlm@0
|
542 :ground 0,
|
|
rlm@0
|
543 :fighting 0,
|
|
rlm@0
|
544 :dark 1/2,
|
|
rlm@0
|
545 :steel 1/32,
|
|
rlm@0
|
546 :bug 1/16}
|
|
rlm@0
|
547 #+end_example
|
|
rlm@0
|
548
|
|
rlm@0
|
549 ** Explanations for Common Pok\eacute{}mon Strategies
|
|
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|
550
|
|
rlm@8
|
551 Many people start out a battle with either a Normal pok\eacute{}mon or an
|
|
rlm@8
|
552 Electric pok\eacute{}mon. Here's some justification for that choice.
|
|
rlm@0
|
553
|
|
rlm@0
|
554 #+srcname: weaknesses
|
|
rlm@0
|
555 #+begin_src clojure :results silent
|
|
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|
556 (in-ns 'pokemon.types)
|
|
rlm@0
|
557 (defn critical-weaknesses [type]
|
|
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|
558 (count (filter #(> % 1) (vals (susceptibility type)))))
|
|
rlm@0
|
559 #+end_src
|
|
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|
560
|
|
rlm@0
|
561 #+begin_src clojure :exports both :results output
|
|
rlm@0
|
562 (clojure.pprint/pprint
|
|
rlm@0
|
563 (sort-by pokemon.types/critical-weaknesses (pokemon.types/multitypes 1)))
|
|
rlm@0
|
564 #+end_src
|
|
rlm@0
|
565
|
|
rlm@0
|
566 #+results:
|
|
rlm@0
|
567 #+begin_example
|
|
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|
568 ([:normal]
|
|
rlm@0
|
569 [:electric]
|
|
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|
570 [:water]
|
|
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|
571 [:fighting]
|
|
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|
572 [:poison]
|
|
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|
573 [:ghost]
|
|
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|
574 [:dragon]
|
|
rlm@0
|
575 [:dark]
|
|
rlm@0
|
576 [:fire]
|
|
rlm@0
|
577 [:ground]
|
|
rlm@0
|
578 [:flying]
|
|
rlm@0
|
579 [:psychic]
|
|
rlm@0
|
580 [:bug]
|
|
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|
581 [:steel]
|
|
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|
582 [:ice]
|
|
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|
583 [:grass]
|
|
rlm@0
|
584 [:rock])
|
|
rlm@0
|
585 #+end_example
|
|
rlm@0
|
586
|
|
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|
587 Electric and Normal are among the best types with which to start the
|
|
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|
588 game, since they have the fewest weaknesses among all the types.
|
|
rlm@0
|
589
|
|
rlm@9
|
590 At the beginning of the pok\eacute{}mon games, players are given a
|
|
rlm@9
|
591 choice between the Fire pok\eacute{}mon [[http://bulbapedia.bulbagarden.net/wiki/Charmander][Charmander]], the Water
|
|
rlm@9
|
592 pok\eacute{}mon [[http://bulbapedia.bulbagarden.net/wiki/Squirtle][Squirtle]], or the Grass/Poison pok\eacute{}mon
|
|
rlm@9
|
593 [[http://bulbapedia.bulbagarden.net/wiki/Bulbasaur][Bulbasaur]].
|
|
rlm@0
|
594
|
|
rlm@0
|
595 #+begin_src clojure :exports both :results verbatim
|
|
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|
596 (sort-by pokemon.types/susceptance [[:fire] [:water] [:grass :poison]])
|
|
rlm@0
|
597 #+end_src
|
|
rlm@0
|
598
|
|
rlm@0
|
599 #+results:
|
|
rlm@0
|
600 : ([:water] [:fire] [:grass :poison])
|
|
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|
601
|
|
rlm@9
|
602 As can be seen, the Water pok\eacute{}mon [[http://bulbapedia.bulbagarden.net/wiki/Squirtle][Squirtle]] is the most solid
|
|
rlm@0
|
603 choice starting out, insofar as susceptance is concerned.
|
|
rlm@0
|
604
|
|
rlm@0
|
605 ** The Worst Pok\eacute{}mon Types
|
|
rlm@0
|
606
|
|
rlm@0
|
607 #+srcname: weak-types
|
|
rlm@0
|
608 #+begin_src clojure :results silent
|
|
rlm@0
|
609 (in-ns 'pokemon.types)
|
|
rlm@0
|
610
|
|
rlm@0
|
611 (defn type-compare-weak
|
|
rlm@0
|
612 "compare first by total number of critical-weaknesses,
|
|
rlm@0
|
613 then by overall susceptance, favoring weaker types."
|
|
rlm@0
|
614 [type-1 type-2]
|
|
rlm@0
|
615 (let [measure (memoize (juxt critical-weaknesses susceptance))]
|
|
rlm@0
|
616 (if (= (measure type-2) (measure type-1))
|
|
rlm@0
|
617 (compare type-2 type-1)
|
|
rlm@0
|
618 (compare (measure type-2) (measure type-1)))))
|
|
rlm@0
|
619
|
|
rlm@0
|
620 (defn resistant?
|
|
rlm@0
|
621 "might as well get rid of types that are resistant to any type"
|
|
rlm@0
|
622 [type]
|
|
rlm@0
|
623 (not (every? #(< 0 %) (vals (susceptibility type)))))
|
|
rlm@0
|
624
|
|
rlm@0
|
625 (defn type-successors-weak
|
|
rlm@8
|
626 "Generate ways to weaken the given type combination. Discard type
|
|
rlm@8
|
627 combinations that either strengthen the given type combination or
|
|
rlm@8
|
628 that make it stronger"
|
|
rlm@0
|
629 [limit type]
|
|
rlm@0
|
630 (set (if (<= limit (count type)) '()
|
|
rlm@0
|
631 (filter #(< 0 (type-compare-weak type %))
|
|
rlm@0
|
632 (remove resistant? (type-successors type))))))
|
|
rlm@0
|
633
|
|
rlm@0
|
634 (defn pokemon-type-search-weak
|
|
rlm@0
|
635 "Search among type-combos no greater than length n, limited by limit
|
|
rlm@8
|
636 steps of best-first-search. Find the weakest type combination
|
|
rlm@8
|
637 possible in terms of susceptance."
|
|
rlm@0
|
638 ([n] (pokemon-type-search-weak n Integer/MAX_VALUE))
|
|
rlm@0
|
639 ([n limit]
|
|
rlm@0
|
640 (first (last
|
|
rlm@0
|
641 (take
|
|
rlm@0
|
642 limit
|
|
rlm@0
|
643 (best-first-search
|
|
rlm@0
|
644 type-compare-weak
|
|
rlm@0
|
645 (partial type-successors-weak n)
|
|
rlm@0
|
646 (multitypes 1)))))))
|
|
rlm@0
|
647 #+end_src
|
|
rlm@0
|
648
|
|
rlm@0
|
649
|
|
rlm@0
|
650 #+begin_src clojure :results scalar :exports both
|
|
rlm@0
|
651 (first (pokemon.types/pokemon-type-search-weak 1))
|
|
rlm@0
|
652 #+end_src
|
|
rlm@0
|
653
|
|
rlm@0
|
654 #+results:
|
|
rlm@0
|
655 : [:rock]
|
|
rlm@0
|
656
|
|
rlm@0
|
657 Poor Rock. It's just not that good a type. Maybe this is why Brock
|
|
rlm@0
|
658 (who has rock pok\eacute{}mon) is the first gym leader in the games.
|
|
rlm@0
|
659
|
|
rlm@0
|
660 #+begin_src clojure :results scalar cache :exports both
|
|
rlm@0
|
661 (first (pokemon.types/pokemon-type-search-weak 2))
|
|
rlm@0
|
662 #+end_src
|
|
rlm@0
|
663
|
|
rlm@0
|
664 #+results:
|
|
rlm@0
|
665 : [:grass :ice]
|
|
rlm@0
|
666
|
|
rlm@0
|
667 # ;;bonus convergently immortal type combo
|
|
rlm@0
|
668 # (susceptance (vec (concat (repeat 150 :water) (repeat 50 :poison) (repeat 50 :steel) [:ghost :normal :flying :ground :dark])))
|
|
rlm@0
|
669
|
|
rlm@0
|
670 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
671 (clojure.pprint/pprint
|
|
rlm@0
|
672 (pokemon.types/susceptibility [:grass :ice]))
|
|
rlm@0
|
673 #+end_src
|
|
rlm@0
|
674
|
|
rlm@0
|
675 #+results:
|
|
rlm@0
|
676 #+begin_example
|
|
rlm@0
|
677 {:water 1/2,
|
|
rlm@0
|
678 :psychic 1,
|
|
rlm@0
|
679 :dragon 1,
|
|
rlm@0
|
680 :fire 4,
|
|
rlm@0
|
681 :ice 1,
|
|
rlm@0
|
682 :grass 1/2,
|
|
rlm@0
|
683 :ghost 1,
|
|
rlm@0
|
684 :poison 2,
|
|
rlm@0
|
685 :flying 2,
|
|
rlm@0
|
686 :normal 1,
|
|
rlm@0
|
687 :rock 2,
|
|
rlm@0
|
688 :electric 1/2,
|
|
rlm@0
|
689 :ground 1/2,
|
|
rlm@0
|
690 :fighting 2,
|
|
rlm@0
|
691 :dark 1,
|
|
rlm@0
|
692 :steel 2,
|
|
rlm@0
|
693 :bug 2}
|
|
rlm@0
|
694 #+end_example
|
|
rlm@0
|
695
|
|
rlm@0
|
696 This miserable combination is weak to 6 types and double-weak to
|
|
rlm@0
|
697 Fire. No pok\eacute{}mon in the games actually has this type.
|
|
rlm@0
|
698
|
|
rlm@0
|
699 * Conclusion
|
|
rlm@0
|
700
|
|
rlm@0
|
701 Searching for a type that is weak to everything takes a very long time
|
|
rlm@0
|
702 and fails to reveal any results. That's the problem with a search
|
|
rlm@0
|
703 over this large problem space --- if there's an easy solution, the
|
|
rlm@0
|
704 search will find it quickly, but it can be very hard to determine
|
|
rlm@0
|
705 whether there is actually a solution.
|
|
rlm@0
|
706
|
|
rlm@0
|
707 In the [[./lpsolve.org][next installment]], I'll use =lp_solve= to solve this problem in
|
|
rlm@0
|
708 a different way.
|
|
rlm@0
|
709
|
|
rlm@0
|
710 * COMMENT main program
|
|
rlm@0
|
711 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :exports none
|
|
rlm@0
|
712 <<header>>
|
|
rlm@0
|
713 #+end_src
|
|
rlm@0
|
714
|
|
rlm@0
|
715 ## this is necessary to define pokemon-table inside the source code.
|
|
rlm@0
|
716
|
|
rlm@0
|
717 #+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
|
718 <<data>>
|
|
rlm@0
|
719 #+end_src
|
|
rlm@0
|
720
|
|
rlm@0
|
721 #+begin_src clojure :noweb yes :tangle ../src/pokemon/types.clj :exports none
|
|
rlm@0
|
722 <<types>>
|
|
rlm@0
|
723 <<search>>
|
|
rlm@0
|
724 <<pokemon-search>>
|
|
rlm@0
|
725 <<old-school>>
|
|
rlm@0
|
726 <<weaknesses>>
|
|
rlm@0
|
727 <<weak-types>>
|
|
rlm@0
|
728 #+end_src
|
|
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
|
|
rlm@0
|
741
|
|
rlm@0
|
742
|
|
rlm@0
|
743
|
|
rlm@0
|
744
|
|
rlm@0
|
745
|
|
rlm@0
|
746
|
