Suppose two players $p_0, p_1$ are playing this simple game: a game is a sequence of $m_1, \dots, m_{2n + 1}$ matches and the winner of the game is the player who has won more matches. Each match consists in a simultaneous choice from both players of a value in the set $\lbrace0, 1\rbrace$: given the players choices $c_0$ and $c_1$ the player $p_i$ wins the match if $i = c_0 \oplus c_1$ (where $\oplus$ is the xor operator).
What I would like to achieve is to predict (with some degree of uncertainty) the next choice of a player: suppose you are the player $p_0$ and the game is at match $i$. What is the probability that my adversary (player $p_1$) chooses $0$ if I choose $1$ and vice versa given the history $m_0, \dots, m_{i - 1}$?
I'm more interested in which topics should I study instead of a complete answer.
I assume that the player is not playing randomly and that your goal is exactly to model the opponent's "psychology" when playing the game. We will also ignore the possibility that the opponent will change his/her strategy as a response to you modeling him/her.
In this case, you could simply use logistic regression to predict the probability of the opponent choosing "1" at the next game as a function of your and their most recent N choices (a sort of binary outcome ARX model).
Apart from this simple algorithm, if you want to take a more rigorous approach, you could model your problem as a Markov Decision Process. An excellent resource is the book by Sutton and Barto.