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<section id="sactgroup" revision="$Revision: 1841 $ $Name$"
  vendor="1.19" xmlns:xi="http://www.w3.org/2003/XInclude">

  <title id="stactgroup">Group Actions</title></title>

  <para>It's often interesting to see the characters cooperate
    together for a particularly difficult task, or to put at the head
    of a troup of kobolds and see how they can make them <emphasis
    role="ironic">efficient</emphasis>.  The problem meet most often
    here is that the rules that they know can only be used to rule for
    one character at once, asking for rolling dice for each character
    and each action, which is quite long.  Arpeggii uses some
    shortcuts to manage those kinds of ruling, with only one roll, and
    obtains similar results as if each character have roll for
    themself.  You can them <link linkend="scombinaisons">compute the
    total power of a group</link>, <link linkend="sjetgroupe">count
    the number of success</link> in a group with only one roll, and
    even <link linkend="sjetdist">determine the degrees of
    success</link> of each individual character of the group.</para>

  <section id="scombinaisons">
    <title id="stcombinaisons">Values Combination</title>

    <para>Some actions can be sometime impossible for one individual
      and required the cooperation of the group to be succeed.
      However, attach a young puppy to a group of 10 horses will not
      help a lot to the task.  To find the strength of the group, you
      can <emphasis>combine</emphasis> the characters' Attributes.
      The simple method is to convert all of them in Measures, add
      them and convert it back in Value.  This way can became long and
      tedious however when they are many characters.</para>

    <para>Another way is possible but required a bit of practice from
      the GM to be efficient, and can be long if the group is too much
      heterogenous.  However, as soon as the group is composed of
      individual with identical attributes, the calculation became
      very quick and make it very clear when someone will not add
      something interesting to the total strenght of the group.</para>

    <procedure id="pcombinaison">
      <title id="ptcombinaison">Calculation of Values Combination</title>
      <step id="pscombind">
        <para>Take each individual with the same Value and form a
          group.  The Value of those group will be equal to the common
          Value of each individual, plus the Value corresponding to
          the number of individuals in the group (e.g. +5 is you have
          3 individuals, or +0 if you only have one).</para>
      </step>
      <step id="pscombgrpident">
        <para>Take all groups with the same Value and combine them
          again but this time by adding the Value of the number of
          groups instead of the number of individuals.  For example,
          if you have 3 groups with +2 each, whatever the number of
          individuals in those groups, the new group will have a Value
          of +2 + 5 (the Value of 3), which is +7.</para>
      </step>
      <step id="pscombgrpdiff">
        <para>Take the two groups with the least Value.</para>
        <substeps>
          <step id="pscomb1a3">
            <para>If the difference between both is 3 points or less,
              combine them in a new group with give them the Value of
              the strongest one +2.</para>
          </step>
          <step id="pscomb4a6">
            <para>If the difference is between 4 and 6 included,
              combine them and give the Value of the strongest +1.</para>
          </step>
          <step>
            <para>If the difference is greater than 6 points, discard
              the weakest group and keep the stronger.  The weakest
              group will be ignored in the total but if a new
              individual is add at later time, you'll be able to check
              if he can be add to discard groups first to form a
              strongest group.</para>
          </step>
        </substeps>
      </step>
      <step id="scombfinal">
        <para>Redo the last two steps until you have only one group
          left, except for the discard groups.  The combined Value of
          all individuals will be the Value of this final
          group.</para>
      </step>
    </procedure>
    
    <para>It should be note that's not always possible (or realistic
      if you prefer) to combine Value this way, and the Game Master
      can always impose some penalities and even restrictions to the
      numbers of characters which can participate and for their
      know-how, including asking for a leadership roll to the chief of
      the group.  It's the GM responsability to create the right
      suspension of disbelief and keep the players interest.</para>

    <example id="xcombinaison">
      <title id="xtcombinaison">Group Value Combination</title>

      <para>A group is composed of 6 characters with a score in Body
        of -10, -5, -5, +3, +4 and +5.  Since they are already
        ordered, we can began to group them:</para>

      <orderedlist>
        <listitem>
          <para>First, we group the identical Values: We have two
            individuals with -5.  We can so group them by adding to
            them the Value of 2, which is +3.  The new group value
            will be -5 + 3 = -2.</para>
        </listitem>
        <listitem>
          <para>The two weakest groups can are now -10 and -2.  The
            difference is 8 points, so the -10 group (a single
          individual) can be ignored.</para>
        </listitem>
        <listitem>
          <para>We now compare the -2 and +3 group.  The difference is
            5 points and so we combine both groups and give them the
            value of the stronger (+3) augment of one point, for a
            total of +4.</para>
        </listitem>
        <listitem>
          <para>In the remaining groups, we now have a new identical
            pair, which is two +4 groups.  Again, we can combine them
            in a new group by adding +3 to the common Value, for a
            total of +7.  The group is now composed of 3 subgroups,
            one ignored (-10), a +5 and a +7 group.</para>
        </listitem>
        <listitem>
          <para>We compare again the two weakest, +5 and +7 (the -10
            is still ignored).  The difference is only 2 points.  The
            groups can so be combined in a new group with the highest
            Value (+7) augment of 2 points, for a final group with a
            <emphasis>combined</emphasis> Body Attribute of +9.
        </listitem>
      </orderedlist>

      <para>Remark that if the group at -10 was compared with the -5
        group first, you will got a group with -4, which combined with
        the other group at -5, will also give you a group with -2.
        So, it doesn't make any difference at the end and since the
        combination of identical groups is the only one which can be
        done on multiple groups at the same time, she is favored most
        of the time.</para>
    </example>
  </section>
      
  <section id="sjetgroupe">
    <title id="stjetgroupe">Rolling dice for a group</title>
      
    <para>In a game, the players are often confronted to group of NPC
      sometime numerous and often hostile.  In a combat or a pursuit,
      it will be tedious for the GM to roll dice individually for each
      members of the group.  When the group is sufficiently <anchor
      id="refjgrpcond"/>homogeneous (no more then 5 points of
      difference in their Competency Level for this particular action)
      et that the roll is a <link linkend="sactsimple">simple
      action</link>, a quick alternative can be used instead.  First,
      the Game Master make a normal roll for the whole group.  Then,
      she takes the Value of the total number of members in this
      group.  To this value, she subtracts either the degrees of
      success if this one is positive or the degrees of failure
      elsewhere, and then subtracts 3 more points to the result.  The
      result will always be at most the Value of the group minus 3.
      This Value can then be converted in a Measure and round up to
      the lower whole number.  If the roll was positive, this Measure
      represent the number of individuals that
      <emphasis>failed</emphasis> their roll.  Elsewhere, it represent
      the number of individual that succeed their roll.</para>

    <example id="xreussitegroupe">
      <title>Simple Group Roll</title>

      <para>A group of 25 guards try to catch the characters in a
        mountain road.  The characters decide to destroy a little
        bridge to slow them down.  The guards decide to jump over the
        small gap.  The difficulty is -5 under Ag+Pw, where the guards
        have a total of +3 each.  The Game Master roll the dice and
        get +4, which is a +2 degrees of success.  The group of 25
        guards correspond to a Value of +14.  If you subtract the
        degrees of success and 3, you get +9.  The <xref
        linkend="tmesures"/> indicate that it's equivalent to a
        Measure of 8.  So, 8 guards have fail their roll into a
        dramatic end.  If the roll have give a failure at -3, the
        final value will then be (+14 - 3 - 3 =) +8, and so only 6
        guards will have <emphasis>succeed</emphasis> to jump over the
        gap.  It will have take a degrees of success of +12 for every
        guards to pass the gap.</para>
    </example>
    
    <para>When you roll the dice for a group, it is good advice to
      only make <link
      linkend="sjetdifference"><emphasis>closed</emphasis> difference
      roll (±d10)</link>.  The great range of open rolls create must
      of the time incredible catastrophe or miraculous prowess,
      instead of normal result.</para>
  </section>

  <section id="sjetdist">
    <title id="stjetdist">Group Degrees of Success</title>
    
    <para>The precedent method is very exact if you apply the
      theoretical probability curve on which is based the Harmonies,
      but which are only an approximation of the real dice dispersion.
      A similar method permit not only to be nearer than the real
      value, but also to calculate the level of success of each
      individual or, to be exact, to know the number of individual for
      each level of success.  It is, however, much more longer and
      complex.  It will be to the Game Master to choose which of both
      it will used, but must stay consistent for their players.</para>

    <para>The method need the <link linkend="refjgrpcond">same
      conditions</link> then the preceding method (homogeneous group)
      and at least hundreds of individual if we want to detail each
      level of success (the precision will be lesser elsewhere).  The
      Game Master begin again with a roll for the whole group.  She
      then convert the number of individuals in Value and subtracts
      the number of points indicate in <xref linkend="tjetdist"/>, in
      function of the difference between the degrees of success of the
      roll and the one we are interested in.  The first row of the
      table represent the difference between the degrees of success of
      the roll and the one we want to know, the second row show the
      value to subtract to obtain the Value of the number of
      individuals having exactly this number of degrees of success,
      and the last one give the value to subtract to obtain the number
      of individual that have <emphasis>at least</emphasis> this
      difference in number of degrees of success.  The result can then
      be converted into a Measure and you got the number of
      individuals having this number of degrees of success plus or
      minus the difference.  For columns with fraction (like -2.5),
      you must take the average of the two adjacent values.  For
      example, at -2.5, you must take the average of the Measure of -2
      and -3.</para>

    <table frame="all" id="tjetdist">
      <title id="ttjetdist">Groups Degrees of Success</title>
      
      <tgroup cols="11" align="center">
        <colspec colnum="1" colwidth="1.5in" align="right"/>
        <colspec colnum="2" colwidth="0.5in"/>
        <colspec colnum="3" colwidth="0.5in"/>
        <colspec colnum="4" colwidth="0.5in"/>
        <colspec colnum="5" colwidth="0.5in"/>
        <colspec colnum="6" colwidth="0.5in"/>
        <colspec colnum="7" colwidth="0.5in"/>
        <colspec colnum="8" colwidth="0.5in"/>
        <colspec colnum="9" colwidth="0.5in"/>
        <colspec colnum="10" colwidth="0.5in"/>
        <colspec colnum="11" colwidth="0.5in"/>
        <tbody>
          <row>
            <entry>difference</entry>
            <entry>0</entry>
            <entry>1</entry>
            <entry>2</entry>
            <entry>3</entry>
            <entry>4</entry>
            <entry>5</entry>
            <entry>6</entry>
            <entry>7</entry>
            <entry>8</entry>
            <entry>9</entry>
          </row>
          <row>
            <entry>exactly this difference</entry>
            <entry>-10</entry>
            <entry>-10.5</entry>
            <entry>-11</entry>
            <entry>-11.5</entry>
            <entry>-12</entry>
            <entry>-13</entry>
            <entry>-14</entry>
            <entry>-15</entry>
            <entry>-17</entry>
            <entry>-20</entry>
          </row>
          <row>
            <entry>this difference or more</entry>
            <entry>-2.5</entry>
            <entry>-3.5</entry>
            <entry>-4.5</entry>
            <entry>-5.5</entry>
            <entry>-7</entry>
            <entry>-8</entry>
            <entry>-10</entry>
            <entry>-12</entry>
            <entry>-15</entry>
            <entry>-20</entry>
          </row>
        </tbody>
      </tgroup>
    </table>

    <example id="xjetdist">
      <title>Group Degrees of Success</title>
      
      <para>We have a group of 300 soldiers (Value of +25) who get a
        +2 success margin.  With +25 - 10 = +15, so 30 soldiers that
        got exactly this success margin (+2).  For the number of
        soldiers who got +1 or +3 (1 point of difference), we must
        take the Measure of -10 (which we already have: 30) and -11
        (which is the same as the column 2, which we still don't
        know).  So, for a difference of 2 points, we have +25 - 11 =
        +14, or 25 individuals with 0 degrees of success, and another
        25 with +4 degrees of success.  We can now take the sum of 30
        and 25, which is 55, that we can split in two: there will be
        28 soldiers with +1 degree of success, and 27 with +3 degrees
        of success.</para>
    </example>
    
    <para>The last row ask for more explanations.  If you're looking
      for the number of individuals with +3 or more degrees of
      success, and that you roll a +2, it's easy.  You just have to
      convert for a difference of +1 or more.  But if you were looking
      for those with 0 or more, you're in trouble!  You can take only
      those with 2 points of difference or more, because you will only
      get those with +4 or more!  What's you really looking for is
      those with +0 (2 points of difference exactly), +1 (1 points of
      difference exactly) and +2 and more (0 or more points of
      difference).  But a shorter way will be to looking for the
      opposite.  Find all of them that get -1 or less (3 points of
      difference and more) and subtract them from the total of
      individual in the group to find those who get +0 and
      more.</para>

    <example id="xjetdist2">
      <title>Group Degrees of Success 2</title>
      
      <para>With the same group as before, we are looking for the
        number of those that got +0 and more degrees of success.  This
        represent a difference of 2 points which include the result of
        the roll.  We can then find those who got less than zero
        degrees of success (a difference of 3 points or more) and
        subtract it to the Measure of the group.  The column 3 got us
        with -5.5, which mean that we must take the average of -5 and
        -6.  +25 - 6 = +19, or 80 individuals.  +25 - 5 = +20, or 100
        individuals.  The average is (100+80)/2 = 90 soldiers who has
        less than zero degree of success.  On the initial group of 300
        soldiers, that's mean that (300-90) = 210 soldiers have got +0
        or more.  Using the method view in the preceding section, we
        would get 100 individuals that have failed (+25 -2 -3 = +20),
        so 200 individual that have succeed.</para>

    </example>

    <note userlevel="adv">
      <title>Choosing between the two methods</title>
      
      <para>The maximum error is 6% or less on the total of
        individuals for the first method, and less than 1% for the
        second method.  You have to choose between a more precise but
        longer roll, or a quickest but less exact one.  At the same
        time, You can choose to only take the lowest Value instead of
        calculating the average when you got an .5 factor to subtract.
        Again, this is just a compromise on speed and
        exactitude.</para>

    </note>

  </section>
</section>
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