<P> where the last inequality follows since σ i ∗ (\ displaystyle \ sigma _ (i) ^ (*)) is a non-zero vector . But this is a clear contradiction, so all the gains must indeed be zero . Therefore, σ ∗ (\ displaystyle \ sigma ^ (*)) is a Nash equilibrium for G (\ displaystyle G) as needed . </P> <P> If a player A has a dominant strategy s A (\ displaystyle s_ (A)) then there exists a Nash equilibrium in which A plays s A (\ displaystyle s_ (A)). In the case of two players A and B, there exists a Nash equilibrium in which A plays s A (\ displaystyle s_ (A)) and B plays a best response to s A (\ displaystyle s_ (A)). If s A (\ displaystyle s_ (A)) is a strictly dominant strategy, A plays s A (\ displaystyle s_ (A)) in all Nash equilibria . If both A and B have strictly dominant strategies, there exists a unique Nash equilibrium in which each plays their strictly dominant strategy . </P> <P> In games with mixed - strategy Nash equilibria, the probability of a player choosing any particular strategy can be computed by assigning a variable to each strategy that represents a fixed probability for choosing that strategy . In order for a player to be willing to randomize, their expected payoff for each strategy should be the same . In addition, the sum of the probabilities for each strategy of a particular player should be 1 . This creates a system of equations from which the probabilities of choosing each strategy can be derived . </P> <Table> Matching pennies <Tr> <Th> Player B Player A </Th> <Th> Player B plays H </Th> <Th> Player B plays T </Th> </Tr> <Tr> <Th> Player A plays H </Th> <Td> − 1, + 1 </Td> <Td> + 1, − 1 </Td> </Tr> <Tr> <Th> Player A plays T </Th> <Td> + 1, − 1 </Td> <Td> − 1, + 1 </Td> </Tr> </Table>

The nash equilibrium is a noncooperative dominant strategy equilibrium