An essential feature of this definition is that each player's payoff depends on the list of all the other players' actions. In particular, a player's payoff does not depend only on her own action.
When using the theory of strategic games to study oligopoly, we will specify the components as follows:
However, the notion of a strategic game can be---and has been---used to study a very wide variety of situations, from tariff wars between countries to electoral competition to the design of legal regimes to sibling rivalry to the mating habits of hermaphroditic fish. In particular, the definition of a strategic game does not put any restrictions on the nature of the players' actions. For example, an action can be a single variable (like an output, or price), or can be a list of variables (like an (output, price) pair), or can be a complicated contingency plan (if X happens, choose x, while if Y happens, choose y, ...).
A list of actions, one for each player in the game, is called an action profile (or, sometimes, a strategy profile or strategy combination).
We can compactly represent a strategic game with two players in a table, like the following one.
Player 2 | |||||||
L | R | ||||||
Player 1 | T |
| |||||
B |
This table represents a strategic game in which player 1's actions are T and B and player 2's actions are L and R. The first number in each box is player 1's payoff to the pair of actions that define the box, while the second number in each box is player 2's payoff to the pair of actions that define the box. Thus, for example, if player 1 chooses the action B and player 2 chooses the action L then player 1's payoff is 3 and player 2's payoff is 0.
A Nash equilibrium of a strategic game is an action profile (list of actions, one for each player) with the property that no player can increase her payoff by choosing a different action, given the other players' actions.Note that nothing in the definition suggests that a strategic game necessarily has a Nash equilibrium, or that if it does, it has a single Nash equilibrium. A strategic game may have no Nash equilibrium, may have a single Nash equilibrium, or may have many Nash equilibria.
Player 2 | |||||||
L | R | ||||||
Player 1 | T |
| |||||
B |
There are four action profiles ((T,L), (T,R), (B,L), and (B,R)); we can examine each in turn to check whether it is a Nash equilibrium.
Notice that in this equilibrium both players are worse off than they are in the action profile (T,L). Thus they would like to achieve (T,L); but their individual incentives point them to (B,R).
This game is called the Prisoner's dilemma; it has been used to model a very wide variety of situations. The story that gives the game its name is the following. Two suspects in a major crime are in separate cells. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). If they are both quiet, each will be convicted of the minor offense and spend one year in prison. If one and only one of them finks, she will be freed and used as a witness against the other, who will spend four years in prison. If they both fink, each will spend three years in prison.
Assign each player the payoff of 0 for a four-year jail term, the payoff of 1 for a three-year term, the payoff of 2 for a one-year term, and the payoff of 3 for freedom, and associate T and B for player 1 with the actions Quiet and Fink, and L and R for player 2 with the actions Quiet and Fink. Then the game above represents this situation.
We conclude from our analysis of the Nash equilibrium of this game that the outcome will be that both players Fink and wind up in jail for three years.
Examples and exercises on Nash equilibrium in games in which each player has finitely many actions
Call the action of player 1 that maximizes her payoff, given that player 2's action is a_{2}, player 1's best response to a_{2}. Similarly, call the action of player 2 that maximizes her payoff, given that player 1's action is a_{1}, player 2's best response to a_{1}. (I am assuming that each player has a single best response.)
Given this definition of best responses, a pair (a_{1}, a_{2}) of actions is a Nash equilibrium if and only if
If we denote player 1's best response to a_{2} by b_{1}(a_{2}) and player 2's best response to a_{1} by b_{2}(a_{1}) then we can write the condition for a Nash equilibrium more compactly:
the pair (a_{1}, a_{2}) of actions is a Nash equilibrium if and only if a_{1} = b_{1}(a_{2}) and a_{2} = b_{2}(a_{1}).
The method of finding the players' best response functions and then solving the two simultaneous equations is most useful when considering a game in which each player has infinitely many actions, but it can be applied also to a game in which each player has finitely many actions. Consider, for example, the Prisoner's dilemma:
Player 2 | |||||||
L | R | ||||||
Player 1 | T |
| |||||
B |
Player 1's best response to L is B, and her best response to R is also B. Similarly, player 2's best response to T is R and her best response to B is R. Thus we have
b_{1}(L) = B and b_{1}(R) = BWe see that the only pair of actions (a_{1}, a_{2}) with the property that a_{1} = b_{1}(a_{2}) and a_{2} = b_{2}(a_{1}) is (B,R): the Nash equilibrium that we found previously.
and b_{2}(T) = R and b_{2}(B) = R.
Examples and exercises on finding Nash equilibria using best response functions