Editing Nash equilibrium (section)

=== Coordination game ===
{{Main|Coordination game}}

{| class="wikitable floatright" style="text-align:center; font-size:95%; margin-left:1em; margin-bottom:1em"
|+ A coordination game showing payoffs for player 1 (row) and player 2 (column) 
!scope="col" rowspan="2"| Player 1<br/> strategy
!scope="col" colspan="4"| Player 2 strategy
|-
!scope="col"| A
!scope="col"| B
|-
!scope="row"| A
|{{diagonal split header| 4 | 4 |transparent}}
|{{diagonal split header| 1 | 3 |transparent}}
|-
!scope="row"| B
|{{diagonal split header| 3 | 1 |transparent}}
|{{diagonal split header| 2 | 2 |transparent}}
|}

The ''coordination game'' is a classic two-player, two-[[strategy (game theory)|strategy]] game, as shown in the example [[payoff matrix]] to the right. There are two pure-strategy equilibria, (A,A) with payoff 4 for each player and (B,B) with payoff 2 for each. The combination (B,B) is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will fall from 2 to 1.

{| class="wikitable floatright" style="text-align:center; font-size:95%; margin-left:1em; margin-bottom:1ex"
|+ The stag hunt
!scope="col" rowspan="2"| Player 1<br/> strategy
!scope="col" colspan="4"| Player 2 strategy
|-
!scope="col"| Hunt stag
!scope="col"| Hunt rabbit
|-
!scope="row"| Hunt stag
|{{diagonal split header|2| 2|transparent}}
|{{diagonal split header|0| 1|transparent}}
|-
!scope="row"| Hunt rabbit
|{{diagonal split header|1| 0|transparent}}
|{{diagonal split header|1| 1|transparent}}
|}

A famous example of a coordination game is the [[stag hunt]]. Two players may choose to hunt a stag or a rabbit, the stag providing more meat (4 utility units, 2 for each player) than the rabbit (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit hunter will succeed, for a payoff of 1. The game has two equilibria, (stag, stag) and (rabbit, rabbit), because a player's optimal strategy depends on their expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they think the other will hunt the rabbit, they too will hunt the rabbit. This game is used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation.

Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:
{| class="wikitable floatright" style="text-align:center; font-size:95%; margin-left:1em;"
|+ The driving game
!scope="col" rowspan="2"| Player 1 strategy
!scope="col" colspan="4"| Player 2 strategy
|-
!scope="col"| Drive on the left
!scope="col"| Drive on the right
|-
!scope="row"| Drive on the left
|{{diagonal split header|10| 10|transparent}}
|{{diagonal split header|0| 0|transparent}}
|-
!scope="row"| Drive on the right
|{{diagonal split header|0| 0|transparent}}
|{{diagonal split header|10| 10|transparent}}
|}

In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit [[mixed strategy|mixed strategies]] (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player are (50%, 50%).

{{Clear left}}