user1816: My understanding is that Alpha Zero's algorithm was fixed after the "learning" phase, so would it come up with the same move for every position? Was the variability in the match entirely due to a Stockfish setting, or did Alpha Zero also display an element of randomness in its move selection?

user825: I've noticed AlphaZero played 1.d4 and 1.Nf3 in http://www.chessgames.com/perl/chess.pl?tid=91944&crosstable=1. Possibilities:

Mutlithreading (any software engineer can tell you it's not deterministic)
Monte Carlo Tree Search (the algorithm draws random paths, so it can't be deterministic)

Please note while the model's parameters are "fixed" (your own words), the actual move might be different. We don't have the source code for AlphaZero, but multithreading and MCTS are likely two major casues.

user19030: The LC0 authors studied the AlphaZero paper very carefully, ran their own experiments, and they came to the following conclusion. AlphaZero, despite calling it's algorithm Monte Carlo tree search (All Monte Carlo methods are by definition explicitly random), does not employ any explicit non-determinism after a certain number of moves (I believe this is 15 moves). 
There are two different parts to MCTS that could introduce randomness into move selection. 

During the search phase, AlphaZero assigns weights on which moves to explore based on 2 factors: the win probability as evaulated by the neural network, and also some sub-tree size statistics of the parent and child move sub-trees. The first factor ensures that more promising lines are evaluated first, and the second factor ensures that less promising lines are eventually searched. In a natural implementation of MCTS, it would use weighted random choices between these weighted possibilities. In reality, AlphaZero chooses between these completely deterministicly, by just choosing the max weight every time. It does this because the randomness in this step does not turn out to be helpful, and only distracts the engine from the more important variations, on average.
During the moving phase, it can make a weighted choice between move candidates which already have deep and accurate evaluations calculated by the search phase. The idea is to make slightly suboptimal moves to try to introduce diversity in its play, so it doesn't play the same game every time. This is especially important during training, but also leads to more interesting match results. It turns out that AlphaZero only does this for the first 15 moves or so, after that it always chooses the best move. This determinism turns out to be essential for its extremely accurate endgame play, where tiny differences in evaluation turn out to be the difference between draws and wins. 

This is true during both training and evaluation. 
In other words, once it is out of the opening, its play is as deterministic as a traditional engine. 
However, note that it will have the same sort of multithreading and time control non-determinism as traditional engines. 

1816: <p>My understanding is that Alpha Zero's algorithm was fixed after the "learning" phase, so would it come up with the same move for every position? Was the variability in the match entirely due to a Stockfish setting, or did Alpha Zero also display an element of randomness in its move selection?</p>
 825: <p>I've noticed AlphaZero played 1.d4 and 1.Nf3 in <a href="http://www.chessgames.com/perl/chess.pl?tid=91944&amp;crosstable=1" rel="noreferrer">http://www.chessgames.com/perl/chess.pl?tid=91944&amp;crosstable=1</a>. Possibilities:</p>

<ul>
<li>Mutlithreading (any software engineer can tell you it's not deterministic)</li>
<li>Monte Carlo Tree Search (the algorithm draws random paths, so it can't be deterministic)</li>
</ul>

<p>Please note while the model's parameters are "fixed" (your own words), the actual move might be different. We don't have the source code for AlphaZero, but multithreading and MCTS are likely two major casues.</p>
 19030: <p>The LC0 authors studied the AlphaZero paper very carefully, ran their own experiments, and they came to the following conclusion. AlphaZero, despite calling it's algorithm Monte Carlo tree search (All Monte Carlo methods are by definition explicitly random), does not employ any explicit non-determinism after a certain number of moves (I believe this is 15 moves). </p>

<p>There are two different parts to MCTS that could introduce randomness into move selection. </p>

<ol>
<li><p>During the search phase, AlphaZero assigns weights on which moves to explore based on 2 factors: the win probability as evaulated by the neural network, and also some sub-tree size statistics of the parent and child move sub-trees. The first factor ensures that more promising lines are evaluated first, and the second factor ensures that less promising lines are eventually searched. In a natural implementation of MCTS, it would use weighted random choices between these weighted possibilities. In reality, AlphaZero chooses between these completely deterministicly, by just choosing the max weight every time. It does this because the randomness in this step does not turn out to be helpful, and only distracts the engine from the more important variations, on average.</p></li>
<li><p>During the moving phase, it can make a weighted choice between move candidates which already have deep and accurate evaluations calculated by the search phase. The idea is to make slightly suboptimal moves to try to introduce diversity in its play, so it doesn't play the same game every time. This is especially important during training, but also leads to more interesting match results. It turns out that AlphaZero only does this for the first 15 moves or so, after that it always chooses the best move. This determinism turns out to be essential for its extremely accurate endgame play, where tiny differences in evaluation turn out to be the difference between draws and wins. </p></li>
</ol>

<p>This is true during both training and evaluation. </p>

<p>In other words, once it is out of the opening, its play is as deterministic as a traditional engine. </p>

<p>However, note that it will have the same sort of multithreading and time control non-determinism as traditional engines. </p>
