Matrix Games
Table of Contents
In a matrix payoff game Game in which all actions are chosen simultaneously. It is conventional to describe a matrix payoff game as played by a row player and a column player. The row player chooses a row in a matrix; the column player simultaneously chooses a column.
1. Prisoner's Dilemma
The most famous game of all—which is known as the prisoner’s dilemma Game in which the strategies are to confess or not to confess; the first player to confess avoids jail..
The strategies are to confess or not to confess.
The prisoner’s dilemma is famous partly because it is readily solvable.
No matter what Column does, Row should choose to confess.
No matter what Row does, Column should choose to confess. That is, Column also has a dominant strategy to confess.
The presence of a dominant strategy makes the prisoner’s dilemma particularly easy to solve. Both players should confess. Note that this gets them 10 years each in prison, and thus isn’t a very good outcome from their perspective; but there is nothing they can do about it in the context of the game, because for each the alternative to serving 10 years is to serve 20 years.
2. Iterated Elimination Of Dominating Strategy
Thus, the equilibrium of the game is for Microsoft to enter and Piuny not to enter. This equilibrium is arrived at by the iterated elimination of dominated strategies Eliminating strategies by sequentially removing strategies that are dominated for a player.
The iterated elimination of dominated strategies is a useful concept, and when it applies, the predicted outcome is usually quite reasonable. Certainly it has the property that no player has an incentive to change his or her behavior given the behavior of others. However, there are games where it doesn’t apply, and these games require the machinery of a Nash equilibrium, named for Nobel laureate John Nash (1928–).
A dominant strategy is a strategy that is best for a player no matter what others choose
Iterated elimination of dominated strategies first removes strategies dominated by others, then checks if any new strategies are dominated and removes them, and so on. In many cases, iterated elimination of dominated strategies solves a game.
