Williams, Trevor

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Utah State University DigitalCommons@USU Research on Capitol Hill Browse Undergraduate Research Events 1-29-2015 The Game of Thrones: A Study of Power Networks and How They Change Trevor Williams Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/roch  Part of the Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Williams, Trevor, "The Game of Thrones: A Study of Power Networks and How They Change" (2015). Research on Capitol Hill. Paper 17. https://digitalcommons.usu.edu/roch/17 This Poster is brought to you for free and open access by the Browse Undergraduate Research Events at DigitalCommons@USU. It has been accepted for inclusion in Research on Capitol Hill by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. Figure	  2	  –	  	  An	  example	  of	  a	  king. II.	  Methods	    The	  game	  is	  played	  on	  a	  paired	  comparison	  directed	  graph	  (PCD).	  A	  PCD	  depicts	  the	  complete	  paired	  comparison	  among	  n	  objects	  (countries,	  people,	  companies)	  with	  an	  arrow	  between	  objects	  whose	  head	  points	  to	  the	  “winner”	  (whatever	  context	  engenders	  in	  that	  word)	  of	  the	  comparison.	   	   It	  can	  be	  proved	  that	  it	  is	  unlikely	  there	  will	  be	  an	  overall	  winner	  (an	  object	  with	  an	  arrow	  poinGng	  to	  every	  other	  object)	  so	  we	  use	  a	  more	  general	  noGon	  to	  model	  dominance.	  	  	  If	  A	  “wins”	  when	  paired	  against	  B,	  we	  say	  “A	  beats	  B”.  Suppose	  object	  K	  either	  beats	  every	  other	  object	  or,	  for	  any	  object	  Y	  that	  K	  does	  not	  beat,	  K	  beats	  an	  object	  X	  that	  in	  turn	  beats	  Y.	  	  	  The	  object	  K	  is	  called	  a	  king.	  	  It	  is	  king-­‐ness	  we	  analyze	  via	  the	  game.	  	  We	  say	  an	  object	  dominates	  a	  PCD	  if	  it	  is	  the	  only	  king.  One	  variant	  of	  the	  game	  is	  played	  by	  removing	  kings	  from	  the	  PCD	  and	  examining	  the	  relaGonships	  within	  the	  resulGng	  PCD.	  Kings	  are	  removed	  from	  the	  PCD	  unGl	  the	  players	  are	  leP	  with	  a	  PCD	  containing	  a	  single	  King.	    This	  variant	  models	  something	  like	  assassinaGon	  and	  captures	  the	  idea	  that	  some	  object	  “wants”	  to	  be	  the	  sole	  king.	  	  Other	  versions	  of	  the	  game	  grant	  the	  model	  the	  flexibility	  to	  conform	  to	  many	  different	  real	  world	  applicaGons.	    I.	  IntroducGon	    Who	  has	  power	  may	  be	  easy	  to	  spot,	  but	  who	  will	  have	  power	  is	  more	  subtle.	  	  History	  provides	  plenty	  of	  data	  that	  allows	  the	  ad	  hoc	  analysis	  of	  what	  relaGonships	  in	  a	  power	  structure	  yield	  other	  relaGonships	  aPer	  some	  change	  has	  been	  made.	  	    But	  what	  about	  a	  method	  for	  predicGng	  how	  a	  power	  structure	  will	  change	  if	  certain,	  perhaps	  controllable,	  events	  occur?	  	    By	  abstracGng	  power	  structures	  into	  networks	  (or	  directed	  graphs),	  the	  changes	  	  affected	  by	  assassinaGon,	  imprisonment,	  or	  other	  too-­‐costly-­‐to-­‐gather-­‐data	  methods	  	  can	  also	  be	  abstracted	  and	  analyzed	  as	  though	  the	  network	  were	  a	  game.	  	  Chess	  is	  a	  	  game	  that	  uses	  a	  hierarchical	  ranking	  of	  pieces	  named	  aPer	  historical	  societal	  posiGons,	  but	  the	  ways	  in	  which	  the	  pieces	  interact	  hardly	  models	  the	  circumstances	  aPer	  which	  the	  pieces	  are	  named.	  	    The	  game	  constructed	  and	  considered	  here	  was	  built	  to	  the	  ends	  of	  allowing	  quanGtaGve	  and	  qualitaGve	  relaGonships	  of	  power	  to	  be	  determined	  and	  manipulated	  in	  ways	  that	  model	  real-­‐world	  power	  exchanges,	  such	  as	  assassinaGon,	  in	  the	  safe	  confines	  of	  abstract	  MathemaGcs. The	  Game	  of	  Thrones:	  A	  study	  of	  power networks	  and	  how	  they	  change Trevor	  Williams Utah	  State	  University  Dr.	  David	  Brown Utah	  State	  University III.	  Results	    The	  study	  of	  this	  game	  has	  yielded	  interesGng	  results.	  	  Objects	  which	  “lose”	  to	  only	  a	  single	  object	  may	  dominate	  the	  PCD	  in	  a	  single	  move.	  This	  may	  seem	  overt,	  but	  a	  comprehensive	  invesGgaGon	  reveals	  the	  fact	  that	  there	  may	  be	  as	  many	  as	  3	  such	  objects	  in	  a	  single	  PCD.	  Notwithstanding	  the	  PCD	  may	  sGll	  be	  dominated	  in	  a	  move.	   IV.	  Conclusions  This	  finding	  may	  suggest	  strategic	  consideraGons	  regarding	  power	  structures	  developed	  to	  prevent	  singular	  dominaGon,	  such	  as	  a	  democraGc	  government.	  There	  are	  no	  stable	  PCD's	  that	  have	  3	  or	  4	  objects.	  The	  smallest	  stable	  PCD	  contains	  5	  objects.	  	  SuggesGng	  that	  a	  stable	  democraGc	  government	  should	  be	  built	  on	  5	  branches	  rather	  than	  3. Study	  conducted	  with	  funding	  from	  a	  USU	  Summer	  Undergraduate	  Research	  and	  CreaGve	  Opportunity	  Grant	  and	  the	  USU	  Department	  of	  MathemaGcs	  and	  StaGsGcs. Trevor	  Williams Utah	  State	  University Department	  of	  MathemaGcs	  and	  StaGsGcs johndoe314@gmail.com Figure	  1	  –	  SomeGmes	  less	  acceptable	  ways	  of	  gaining	  power	  are	  used	  such	  as	  assassinaGon,	  coup,	  and	  imprisonment. Figure	  3–	  The	  smallest	  stable	  PCD. Figure	  2	  –	  An	  example	  progression	  of	  the	  game.	  	  Red	  objects	  are	  kings.	  	  There	  is	  a	  dominant	  object	  in	  two	  moves. In	  order	  to	  facilitate	  the	  study	  of	  the	  model	  an	  electronic	  version	  of	  the	  game	  was	  developed.	  This	  electronic	  version	  expedited	  the	  gathering	  of	  data.  Because	  of	  the	  incredible	  flexibility	  of	  the	  model	  there	  is	  much	  more	  work	  that	  can	  be	  done	  and	  many	  more	  uses	  for	  the	  model.	   We	  think	  of	  a	  network	  as	  unstable	  	  if	  it	  can	  be	  represented	  as	  a	  PCD	  that	  can	  be	  dominated	  in	  one	  move.	  	  An	  unstable	  PCD	  represenGng	  a	  real	  power	  structure	  clearly	  facilitates	  compeGGon	  for	  dominance	  via	  the	  single	  object	  whose	  deleGon	  yields	  an	  overall	  “winner”.	  	  We	  regard	  a	  network	  as	  stable	  otherwise.	  	   

The Game of Thrones: A Study of Power Networks and How They Change

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The Game of Thrones: A Study of Power Networks and How They Change

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