19 Nash Equilibrium

A solution concept is a systematic description of the outcomes (i.e., the strategy profiles) that may emerge in a family of games. Solution concepts are the main tools used to analyze strategic situations and are intended to embody different notions of rational behavior.

A best response for player \(i\) to a specific combination of other players’ actions is the action that maximizes player \(i\)’s utility, assuming the other players follow their specified actions.

Consider the following game:

		Column
		\(l\)	\(r\)
Row	\(u\)	2, 1	0, 0
Row	\(d\)	0, 0	1, 2

Row’s best responses:

If Column plays \(l\): Row’s best response is \(u\) (utility 2 vs 0)
If Column plays \(r\): Row’s best response is \(d\) (utility 1 vs 0)

Column’s best responses:

If Row plays \(u\): Column’s best response is \(l\) (utility 1 vs 0)
If Row plays \(d\): Column’s best response is \(r\) (utility 2 vs 0)

A Nash equilibrium is a strategy profile where every player’s action is a best response to the other players’ actions. At a Nash equilibrium, no player can improve their utility by unilaterally changing their strategy.

In the game above, there are two Nash equilibria:

\((u, l)\): Row plays \(u\) (best response to \(l\)) and Column plays \(l\) (best response to \(u\))
\((d, r)\): Row plays \(d\) (best response to \(r\)) and Column plays \(r\) (best response to \(d\))

19.1 Additional Examples

19.1.1 Example 1: Two Nash Equilibria

		Column
		\(l\)	\(r\)
Row	\(u\)	2, 2	2, 1
Row	\(d\)	1, 2	3, 3

There are two Nash equilibria: \((u, l)\) and \((d, r)\). Even though \((d, r)\) gives both players their best possible payoffs, \((u, l)\) is also a Nash equilibrium. This illustrates that players could be stuck at an inferior equilibrium when a mutually beneficial one exists.

19.1.2 Example 2: Unique Nash Equilibrium

		Column
		\(l\)	\(r\)
Row	\(u\)	2, 2	2, 1
Row	\(d\)	1, 3	10, 2

The unique Nash equilibrium is \((u, l)\). Despite the large payoff of 10 at \((d, r)\), Row cannot achieve it because Column would deviate to \(l\).

19.1.3 Example 3: Nash Equilibria with Indifference

		Column
		\(l\)	\(r\)
Row	\(u\)	0, 1	2, 1
Row	\(d\)	2, 0	2, 2

Nash equilibria: \((u, r)\) and \((d, r)\). When Column plays \(r\), Row receives payoff 2 from either \(u\) or \(d\)—Row is indifferent between these strategies. Both \((u, r)\) and \((d, r)\) are Nash equilibria because Row has no incentive to deviate (being indifferent) and Column strictly prefers \(r\) to \(l\) in both cases.