1 . 2 Application : Lower Bounds on Las Vegas Randomized Algorithms

In this section, we will first prove that any 2-player zero-sum finite game has a Nash equilibrium of mixed strategies (a special case of Nash’s theorem), and a Nash equilibrium can be found through solving linear programs. We will further show how to prove lower bounds on randomized complexity of Las Vegas algorithms using this result. A 2-player game is a zero-sum game if the sum of the payoffs of the 2 players (row player and column player) is zero for any choices of strategies. We consider finite game here, i.e., the number of players and the strategy set of each player are both finite. Let A = {A}m×n be the payoff matrix of row player. From the definition of zero-sum game, we have −A is the payoff matrix of column player. Let X = {row vector x ∈ [0, 1]m| ∑ xi = 1} and Y = {column vector y ∈ [0, 1]n| ∑ yi = 1} be the mixed strategy sets of the row player and the column player, respectively.