Informally, a Nash Equilibrium is: If all players write down the strategy they are going to play on a sheet of paper and show them to all other players (can in

Informally, a Nash Equilibrium is: If all players write down the strategy they are going to play on a sheet of paper and show them to all other players (can include randomisation) then no player would have a strict incentive to alter their strategy. With rock paper scissors, if me and my opponent both write down 1/3,1/3,1/3 randomisation, then neither of us would strictly benefit from doing something else. Your [1/3,1/3,1/3], [1,0,0] is not a Nash Equilibrium. Whilst the second player does not have a strict incentive to change strategies, the first player can switch to [0,0,1] and win every time.