Game Theory: Pure and Mixed Strategies

I had to produce a short report on pure and mixed strategies in game theory this week giving a brief explenation and examples of real world occurances;

Strategies form the basis of Game Theory, without strategies there would be no way of performing actions in a game. A strategy is that set of actions that will be performed for whichever situation that could arise in a game. A strategy may be completely random or on the other hand a thoroughly thought out and complex set of plays, it may even be a mixture of both.

The Strategy

A strategy profile is a set of strategies that are pre-determined by a player to be the actions they perform in response to any given situation in a game. This could be illustrated by a simple finite state diagram. Consider a simple game where players roll a dice and need to bet if their dice roll would beat their opponents. Player 1’s strategy could be to only bet that his dice will be a higher value than his opponent, if he lost his last game (very crudely trying to follow the law of averages) this could be represented as follows.

In other words a player’s Strategy Profile is a subset (but not necessarily a proper subset) of the player’s Strategy Set (which is all possible strategies).
For example the game Rock-paper-scissors gives players a strategy set of {Rock, Paper, Scissors}, a players Strategy Profile could however be to only play {Rock} every time.


Pure Strategy

The Strategy Set available to a player is the set of pure strategies available to a player, a players strategy profile will then consist of a proper or in-proper subset of that strategy set.
This will provide the player with a complete definition of how they will re-act to satiations in that game. In the Rock-paper-scissors example if a players Strategy Profile consists of always playing Rock and then Scissors, so {(R, S, R, S, R, S, ...)} they would be playing a pure strategy.

People have a mixed view on the concept of free will, some believe we are in charge of what we do and other believe that we are merely puppets in the game of life (played by fate) and that all our actions have been pre-determined. If you look at life from this perspective, that we do not have any free will, then life could be seen as a game were the players (fate and fate’s cousin) are playing a pure strategy, from the Strategy set that encompasses all possible actions our lives are played out by a ordered set of actions.

A simpler and less open to debate example of a pure strategy would be that of a person who play exactly the same numbers in the national lottery every week. Out of all the possible combinations in the Strategy Set (49*48*47..*43 possible combinations) were each combination is a pure strategy. If that player plays only one combination (and the same one) every week, it could be said that they are playing a pure strategy.

Mixed Strategy

With a Mixed Strategy a player assigns a probability of play to each pure strategy, this allows a player to select a pure strategy at random, for instance with the Rock-paper-scissors example, if the player rolled a six sided dice and on a 1 or 2 played Rock, 3 or 4 played Paper and a 5 or 6 played Scissors, they would be using a mixed strategy. { ((1,2)->R,(3,4)->P, (5,6)->S))}

Lastly I’d like to mention Nash Equilibriums and strategies, a set of strategies is a Nash equilibrium if no player can do any better by unilaterally changing their strategy, in other words if I player new the opponents strategy beforehand but could still not indefinitely improve their own strategy then their strategy would be a Nash equilibrium.

Pure and Mixed strategies can be a Nash equilibrium this however becomes tricky when your take into account the effect that peoples beliefs have on a game, the issue here is that people lack the ability to (by themselves at least) generate random numbers. We require the ability to generate random numbers if we wish to use a mixed strategy due to its use of probabilities.

So in the end, people hinder the predictive power of a Nash Equilibrium as well as the ability to play any Strategy correctly (whether pure or mixed). In essence a pure strategy is just a mixed strategy were the played strategy has a probability of 1 and all other strategies have a probability of 0. There will always be one person who will go against the grain of a well thought out strategy and in their stupidity break our “intelligent” reasoning.