As mentioned at Human Compatible by Stuart J. Russell (2019), game theory can be seen as the part of artificial intelligence that deas with scenarios where multiple intelligent agents are involved.
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
When taking a penalty kick in soccer, the kicker must chose left or right.
And before he kicks, the goalkeeper must also decide left or right, because there is no time to see where the ball is going.
Because the kicker is right footed however, he kicker kicks better to one side than the other. So we have four probabilities:
  • goal kick left keeper jumps left
  • goal kick right keeper jumps right
  • goal kick left keeper jumps right. Note that it is possible that this won't be a goal, even though the keeper is nowhere near the ball, as the ball might just miss the goal by a bit.
  • kick right and keeper jumps left. Analogous to above
Related ideas: