D.8 Subgame equilibrium | Game Theory - Microeconomics

in game theory a sub-game is a subset of any game that includes an initial node which has to be independent from any information set and all its successor nodes it's quite easy to understand how sub games work using the extensive form when describing the game in this game tree there are six separate sub games other than the game itself we have two main sub games which in turn contain two more sub games each when dealing with sequential games other than the equilibrium for the main game we can also look for an equilibrium in each of its sub games let's see an example in this game we have two players player 1 and player 2 player 1 has to decide between going up or down while player 2 has to decide between going left or right we can use the extensive form to represent this game player 1 who plays first has to decide whether going up or down after player 1 decides what to do it's the turn of player two who can either go right or left final outcomes are represented at the end of each final branch for example the outcome that results from the combination of left will have a payoff of 2 for player 1 and a payoff of 5 for player 2 let's say the final payoffs for each outcome are as follows 5 - 0 0 + 3 1 it is important to remember that the first payoff corresponds to player 1 and the second player 2 we can also represent the game using the strategic form rows represent player 1 strategies up and down while columns correspond to player 2 strategies right and left if layer 1 goes up and player 2 goes right final payoffs equal R 5 - using the same procedure we are able to fill out the entire matrix let's look for the Nash equilibrium of this game player one knows that player two will rather go left since his expected payoff would be greater therefore player 1 chooses to go up since this will derive in a higher payoff player 2 knows that player 1 will rather go up since his expected payoff would be greater therefore player 2 chooses to go left since this will derive in a higher payoff therefore we find that up left is a Nash equilibrium since going up is the best strategy player 1 can choose and going left is the best strategy for player 2 considering the other players strategy however since this is a sequential game this is not a perfect equilibrium in order to find the sub game perfect equilibrium we must do a backwards induction starting at the last move of the game then proceed to the second-to-last move and so on in this particular case we know that player 2 will choose left if player 1 goes up and right if player 1 goes down since these are the moves that maximize player two's payoff because there is complete information and therefore each player's payoffs are known player 1 knows these choices in advance and will therefore choose to go down because his final payoff will be greater therefore down-right is the perfect sub game equilibrium sub games are of special importance when analyzing repeated games also known as super games since repetition is easier to analyze using a game tree with multiple branches

Channel: Policonomics

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Keywords: subgames, subgame equilibrium, subgame perfect equilibrium, subgame Nash perfect equilibrium, game theory, microeconomics, economics, Policonomics, dictionary of economics

Id: 8fCEfbx5ECE

Length: 3min 44sec (224 seconds)

Published: Tue May 10 2016