= Nash equilibrium
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The best example to look at first is the <penalty kick left right Nash equilibrium>.
Then, a much more interesting example is choosing a deck of a TCG competition: <Magic: The Gathering meta-based deck choice is a bimatrix game>, which is the exact same, but each player has N choices rather than 2.
The next case that should be analyzed is the <prisoner's dilemma>.
The key idea is that:
* imagine that the game will be played many times between two players
* if one player always chooses one deck, the other player will adapt by choosing the anti-deck
* therefore, the best strategy for both players, is to pick decks randomly, each with a certain probability. This type of probabilistic approach is called a <mixed strategy>
* if any player deviates from the equilibrium probability, then the other player can add more of the anti-deck to the deck that the other player deviated, and gain an edge
Therefore, using equilibrium probabilities is the optimal way to play