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I. Player A and B is about to play a game G. We don't have any information about the players. I assume there is 50-50% chance who wins.

II. Let's say player A wins. How would it change his 50% for the next game? (We still don't have any other information eg. how did he play, did he have 'luck' etc.)

III. Player A and B has a history of playing game G. A leads 12-8 against B. What are the chances for A to win against B in their next game?

IV. Same situation as in III. but we know the distribution of A's games. (1 = win, 0 = lose) 1-1-1-1-0-1-1-1-0-1-1-1-0-1-0-1-0-0-0-0. (This should represent an improving performance of player B against A) Does this information change III. result?

This are more theoretical questions. I do not know if they are mathematically calculable. What are the keywords if I want to learn more about this kind of statistic / mathematics problems?

EDIT

I tried to formulate my questions as theoretical as possible, but I think it is easier to answer them if I define the game G as a sport, eg 1 vs 1 basketball. So the players have skill, they can train / practice to getting better. It is not like they would play coin flip against each other.

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  • $\begingroup$ This could be regarded as essentially a question about Bayesian statistics (you have some prior belief about their relative skill - some representation consistent with your stated ignorance - which you update as you get data). $\endgroup$ – Glen_b -Reinstate Monica Jul 3 '18 at 3:06
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Your question is a prediction problem involving a series of binary variables. The method used to deal with this will depend on the assumptions you are willing to make about the structure of the sequence of games. Without further structure, there are really limitless possibilities, and so you will need to consider some basic things like whether players learn from their past experiences in the game, and how this affects their probability of winning.

The simplest case would be to assume that the games are exchangeable (so that the order of games does not matter). This is tantamount to assuming that the players have some fixed chance of winning, and this does not vary over time. You are then dealing with a series of IID Bernoulli random values, which can be analysed within the Bayesian framework in a way where you update your inference about the probability of A winning the game after each new observation. There is a useful series of papers written on prediction of exchangeable binary sequences of this kind, which looks at the optimal prediction method and the probability of correct prediction (see O'Neill and Puza 2005, O'Neill 2012, and O'Neill 2015). That would be a good place to start if you want to learn about this class of problems.

If you are not willing to assume exchangeability of the sequence of games (e.g., if you want the players to become better/worse at the game as the sequence of games progresses) then you will need to specify the time-dependence structure for your game. You could use some kind of Markov chain, where the probability that A wins a game depends on the results of a finite set of previous games, or you could use a more general model where the probability of winning a game depends on all previous games. For this kind of exercise, you will probably want to look at some references on Markov chains, especially for sequences of auto-correlated binary outcomes.

Since you have also tagged this question with the game-theory tag, it is worth noting that this field looks at optimal actions in games based on their internal structure. If you would like to analyse the game from this perspective, then you will need to formulate its internal structure and then look at the optimal behaviours using standard appeals to Nash equilibria, subgame perfect equilibria, or evolutionary equilibria. Since you are dealing with a sequence of games this would involve looking at the literature on repeated games.

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  • $\begingroup$ game-theory tag was maybe a mistake. The game in my examples is more like sport. (eg. 1vs 1 basketball) $\endgroup$ – user3568719 Jul 3 '18 at 14:31

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