Lots of things can be represented as games, but you can't have a game without a few necessary bits. In particular, a game is made of:
Some players
Some things the players can do (actions)
A way to show how good (or bad) those actions are
When we look at a game, we want to find equilibrium strategies. These are basically the action(s) for each player that give them both the best outcomes. Put another way, at equilibrium, neither player would want to change their strategy because a change would make them worse off.
To start, here's a game theoretic version of an actual game: Rock, paper, scissors.
P2 | |||||
R | P | S | |||
R | 0,0 | -1,1 | 1,-1 | ||
P1 | P | 1,-1 | 0,0 | -1,1 | |
S | -1,1 | 1,-1 | 0,0 | ||
Rock, paper, scissors has two players, 1 and 2. Each player can play either (R)ock, (P)aper, or (S)cissors. The numbers here are each player's utility values for a pair of actions. So if Player 1 picks Paper, and Player 2 picks Rock, Player 1 gets 1 while Player 2 gets -1. Bigger numbers are better, so both players prefer winning (1) to a tie (0) or a loss (-1). Both players pick their actions simultaneously, and without knowing what the other player is going to do. This is an example of a strategic form game.
In order to find the optimal strategies for each player (which is what we're after in this whole mess), we compare the utility values for one player's actions assuming the other player is playing a fixed strategy. So if Player 2 plays Rock, it's best for Player 1 to play Paper, etc. A pure strategy equilibrium is when when a pair of actions is best for both players. However, there aren't any of those here, so we need a slightly different equilibrium concept. Without going into the math, it turns out that the best strategy for each player is to play each action with a probability of 1/3. This is called a mixed strategy equilibrium.
For anyone who's played rock, paper, scissors, this makes a lot of sense. You have to change what you play, or you'll lose quite often. Note that playing each action 1/3 of the time does not mean you should optimally play each action once and then repeat that pattern. That would be silly, and you would also begin losing as soon as the other person caught on.
There are other types of games where the players take turns acting, or where the players don't know something about the other players, but I'll explain those if they come up.
Questions, comments, have I left something really important out? As always, leave a comment or email and I'll try to be less confusing!
