Sub-game Perfect Nash Equilibrium Assignment Help

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Sub-game Perfect  Nash  Equilibrium  of  an  Infinitely  Repeated  Game (Selton  1965): 

A  Nash  equilibrium  is  sub-game  perfect  if  the  players' strategies  constitute  a Nash  equilibrium  in  every  sub-game  of  the  infinitely repeated  game.  Sub-game  perfect  Nash  equilibrium  is  refinement  of the concept  of Nash  equilibrium, which means  for  a strategy profile  to be  sub- game  perfect,  it must  be  a Nash equilibrium first  and  then  it must  satisfy  an additional test, that  is,  it must be a Nash equilibrium  in every sub-game of the game.  

Our  objective  of describing all  the  definition  above was  to show that the trigger strategy that we have already defined  in the game of infinitely repeated prisoners'  dilemma  is sub-game perfect. Therefore, we need  to show that the trigger strategy constitute a Nash equilibrium  on  every sub-game  of  that infinitely repeated game. Recall that every sub-game  of an  infinitely repeated game is  identical  to  the game  as a whole.  In  the  trigger  strategy,  Nash equilibrium  of the  infinitely  repeated prisoners'  dilemma sub-games could be grouped into two classes:

i)  sub-games  in which all the outcomes of the earlier  stages have been  (RI, R2)

ii)  sub-games  in which the outcome of at least  one  stage differs  from  (Rl, R2).

If  the players  adopt the trigger strategy for the  game as a whole,  then  (i) the players'  strategies in a sub-game in the first case are again  the trigger strategy, which  we  have already shown  to  be  the Nash equilibrium  of  the  game  as a whole.
The players' strategies  in  a sub-game  in  the  second case are simply  to repeat the stage game equilibrium (L1, L2)  forever, which  is also Nash equilibrium of the game as a whole. Therefore,  the trigger strategy Nash equilibrium  of the infinitely repeated prisoners'  dilemma is sub-game-perfect.

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