| Game Theory
Game Theory is a brand of Economics (and decision mathematics) that looks at the different strategies a player (a person making a decision) can make. The main pioneer of Game Theory was John von Neumann whose book and work published in 1928 created the sub-field of Game Theory. John Nash also created the theory of a Nash equilibrium, a point were both players have no incentive to adopt another decision.
The different decisions a player can make are given in a pay-off matrix. Usually convention dictates that the player listed in the row has his outcomes stated first. For example:
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B Plays 1 |
B Plays 2 |
|
A Plays 1 |
(2,7) |
(4,9) |
|
A Plays 2 |
(3,1) |
(5,5) |
If A plays 1 and B plays 1 then the outcome is 2 units for player A and 7 units for player B. Pay-off matrices may also be zero-sum, this means that one players gain is another's loss. An example of a zero-sum pay-off matrix:
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B Plays 1 |
B Plays 2 |
| A Plays 1 |
5 |
-7 |
|
A Plays 2 |
-3 |
4 |
For a zero-sum game, the pay-off matrix usually only gives the pay-offs for one player (as for the other player the pay-off will be the value*-1), conventionally this is normally Player A (the player represented by the rows) unless otherwise stated. Below is the same pay-off matrix as above, but the pay-offs are for player B; note, to find this we have simply multiplied each cell by -1.
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B Plays 1
|
B Plays 2
|
A Plays 1
|
-5 |
7 |
A Plays 2
|
3 |
-4 |
Keywords
Nash Equilibrium - A
situation where no player has a reason to change their strategy.
Zero-Sum Game - A zero-sum game is one in which one players gain is the other players loss. The pay-off matrix is usually given in terms of Player A, to find out Player B's pay-off matrix multiply each cell by -1.
Play - The choice or decision made by a player.
Saddle Point - A point where it isn't advantageous to adopt a different strategy, this will only occur if the row maximin = column minimax.
Maximin - The Maximum value of the Minimum outcomes from choosing a play. The maximin is usually found for the rows.
Minimax - The Minimum value of the Maximum outcomes from choosing a play, this would be used to find the best choice for a Player B (column player) in a zero-sum game.
Dominate - A row or column dominates another row or column if every single cell has a better pay-off.
Play Safe Strategy - A strategy where a player plays the choice that results in him losing the least. This can be obtained by calculated the worst outcome in each row/column and then selecting the maximum value of these worst outcomes.
Pay-Off Matrix - A matrix representing the pay-offs to the players for playing a certain decision.
Stable Solution - See Saddle Point
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