can anyone recommend resources that explain game theory and how to find the equilibrium? looking from a textbook, i understand the basic concept of two players and a they both have two choices. however i run into trouble when it comes to figuring out if there is an equilibrium, what is the equilibrium if it exists, and if there are more than one equilibrium that the players can choose from…
any suggestions are appreciated
The usual notion of equilibrium in a competitive (i.e., non-cooperating) game between two players is the Nash equilibrium. The equilibrium consists of a strategy for each player such that neither player has any advantage in unilaterally deviating from the strategy. The strategies need not be the same, but in some symmetric games like the prisoners’ dilemma they are. A strategy need not be an action, but could be a random choice among actions. For example in rock, paper, scissors, the Nash equilibrium is to pick each action randomly with probability 1/3
The mathematician John Nash proved that in any game with a finite set of actions there must exist at least one Nash equilibrium pair of strategies (possibly mixed strategies). I believe there is a theorem that says there must be an odd number of equilibria, but I’ve forgotten the exact conditions required for this.
There are various refinements to the Nash equilibrium like subgame perfect and trembling hand. The former says that any “threats” of actions involved in an equilibrium must be credible – that is the threat must be one that once it would be invoked would be a Nash equilibrium just considering future actions. Trembling hand is a stability issue. It must be that if you deviate slightly from the equilibrium action, the best thing to do would be to return to it rather than go to a different equilibrium (much as in the notion of a stable equilibrium in physics).
Repeated games also introduce a whole set of complications.
The wiki article on Nash equilibrium provides a good introduction with several worked out examples. I found the Fudenberg Tirole book Game Theory to be good, but the right introduction depends a great deal on what formality you want.
As confusing as it is, game theory and combinatorial game theory are two pretty different disciplines. There’s really not much in the way of overlap between the two.