<Tr> <Th> Choose' 2' </Th> <Td> − 2, 2 </Td> <Td> − 1, 3 </Td> <Td> 2, 2 </Td> <Td> 4, 0 </Td> </Tr> <Tr> <Th> Choose' 3' </Th> <Td> − 2, 2 </Td> <Td> − 1, 3 </Td> <Td> 0, 4 </Td> <Td> 3, 3 </Td> </Tr> <P> This can be illustrated by a two - player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points . In addition, if one player chooses a larger number than the other, then they have to give up two points to the other . </P> <P> This game has a unique pure - strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other strategy can be improved by a player switching their number to one less than that of the other player . In the adjacent table, if the game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square . Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: (0, 0), (1, 1), (2, 2), and (3, 3). </P>

Every finite matrix game has at least one pure strategy nash equilibrium