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I was thinking about what is the 'right' mathematical model to the following problem from game theory. May be it is quite trivial for experienced statisticians.

Suppose, we have two players Pl1 and Pl2 (Pl1 scores against Pl2 and Pl2 scores against Pl1 - the winner of the session is the player with the highest score) - the game is not so important but we write down the scores for every player at the end of uniform/equally distant time intervals during the game session, say: $i_1, i_2,..., i_{10}$ - ten time intervals. So, for every game session between the two players we have two series:

$s_{1,1}, s_{1,2}, ..., s_{1,10}$ (say: 3, 0, 5, ....,4) for the first player and $s_1 = s_{1,1} + ... + s_{1,10}$ is the total score of Pl1 for the game session. Analogically,

$s_{2,1}, s_{2,2},..., s_{2,10}$ (say: 1, 3, 2, ....,1) and $s_2 = s_{2,1} + ... + s_{2,10}$ for the second player Pl2.

Suppose, we have historical data for $N$ (sufficiently large number) game sessions between Pl1 and Pl2 - they could be considered as training data set. And I want to build a model that, given a partial results table for one session between Pl1 and Pl2, is able to predict the rest of the scores. That is, for a new session knowing the scores (for example) until the 7-th interval,

$s_{1,1}, s_{1,2},..., s_{1,7}$ and $s_{2,1}, s_{2,2},..., s_{2,7}$ the model has to predict the scores for the last three time intervals, $i_8, i_9$ and $i_{10}$:

$s_{1,8} ,s_{1,9}, s_{1,10}$ and $s_{2,8}, s_{2,9}, s_{2,10}$ - to be generated by the 'trained' mathematical model.

Do you know if I should take into consideration the 'hot hand' effect? Any suggestions what distribution model I can apply and some basic idea how to build it? I was thinking about some Poisson process combined with Markov model. It would be very helpful if someone can provide references to papers, books or forums discussing such kind of problems.

Thanks in advance!

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One standard option for modelling of points being scored in a soccer/basketball game, etc. is to use Inhomogeneous Poisson point process

Another option could be to consider the problem as a classification one. If $N$ is indeed very big, then this approach might work. More precisely, you can consider the given scores $s_{1,i},s_{2,i}$, $i=1,\ldots7$ as a point in $\mathbb{R}_{14}$ and to look at the last 3 components of the closest points received from the given data in a similar way. You can use these 3 components for those points to estimate the 3 components you are interested in, somehow.

Finally, you can indeed think about incorporating the Markov property in a certain way, but I should also spend some time over this, before editing my answer.

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