Answers.Body: <p>I'd argue the process you're talking about can be represented by <code>Markov Chain</code>. It solves the continuous problem raised by @Brian.</p>
  
  <p>I don't know where to find a proper state diagram for your question.The following diagram is for a simple board game. Imagine we model the <strong>transition probability</strong> from <code>win</code> to <code>lose</code> (or <code>win</code> to <code>draw</code>) as the blunder probability, this is similar to what you are asking.</p>
  
  <p><a href="https://i.sstatic.net/QVbay.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QVbay.png" alt="enter image description here"></a></p>
  
  <p>We can represent "local minimum" described in your question as an <code>absorbing state</code>.</p>
  
  <p>More generally, if you want to assume some latent factors for making a blunder (e.g. bad opening, personal issues), you could generalize to <code>hidden markov model</code>. It's possible to use the EM algorithm to fit a model from engine PV evaluations.</p>
  
  <p>Given a fitted model, you can query the probability of a blunder given a chess game.</p>
  
  <blockquote>
    <p>Have there been any studies on chess using such model?</p>
  </blockquote>
  
  <p>No studies that I am aware of.</p>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2017-03-21T14:06:14.170Z
Answers.LastActivityDate: 2017-03-22T05:20:43.717Z
Answers.LastEditDate: 2017-03-22T05:20:43.717Z
Answers.UserDisplayName: SmallChess
Body: <p>Not sure whether this is more of a physics or a chess question.</p>
  
  <p>Looking at the evaluation graph (computer evaluation vs. move number) of a game of chess, I realized that there are some similarities to the physics model of escape from a local potential minimum. Let me explain:</p>
  
  <p>While none of the players make a mistake the evaluation will stay at values around zero corresponding to a draw. This I would associate with the motion of a particle in a local minimum, whose coordinate (=evaluation) might fluctuate a bit around the minimum value.</p>
  
  <p>If a player makes a mistake, the evaluation will increase to let's say +2, corresponding to a won position. Normally (i.e. with a very high probability) white should win this game. Until mate, the evaluation will further increase to +3, +4, +5....</p>
  
  <p>This process is similar to the escape from a local minimum. A blunder can "activate" the particle so that it can leave the local minimum and it will subsequently accelerate to larger coordinates (evaluation increases).</p>
  
  <p><a href="https://i.sstatic.net/4ROZp.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4ROZp.png" alt="simple model for chess"></a></p>
  
  <p>Have there been any studies on chess using such model?</p>
  
  <p>Does it make sense at all?</p>
  
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T11:40:10.537Z
Comments.Text: "Does it make sense at all?" - No! You have shown a continuous function. Chess is discrete not continuous (each move representing a potential step change) and evaluation can jump from +3 to -3 in one move without passing through zero. This question has no relevance to chess.
Comments.UserDisplayName: Brian Towers
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T11:53:35.143Z
Comments.Text: There has been no studies in Chess till now and the concept here is quite abstract . Einstein used to play Chess and he was cleanly associated with a lot of Physics . I hope he could have answered it better .
Comments.UserDisplayName: Seth Projnabrata
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T11:58:39.263Z
Comments.Text: @BrianTowers If you are doing numerics on some (continuous) physics problem you are also dealing with discrete values. I don't see any difference here. Note that the vertical axis, i.e. the potential would be something like "resistance to loss", i.e. describing the (presumed) fact that with perfect play a game ends in draw and that you have some margin (large choice of moves) which all keep the game in draw. Perhaps will ask this question at physics stackexchange.
Comments.UserDisplayName: user1583209
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T11:59:18.767Z
Comments.Text: @ProjnabrataSeth: Einstein did not have computer evaluation though...
Comments.UserDisplayName: user1583209
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T11:59:54.737Z
Comments.Text: Chess evaluation can probably be described as a variation on a random walk, but there's no such thing as a local minimum. Given a +0.5 evaluation, it isn't harder for Black to make a bad move than for White.
Comments.UserDisplayName: Glorfindel
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T12:06:12.247Z
Comments.Text: @Glorfindel: Values are somewhat arbitrary and you could argue that the maxima should be at slightly larger values. Anyway, I would claim that if white has a small advantage (e.g. +0.5, i.e. still draw) a small error by black might turn the game into white's favor (white win), while an equally small error by white might  turn it into +-0 or into a small black advantage of e.g. -0.5 (still draw).
Comments.UserDisplayName: user1583209
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T12:16:07.060Z
Comments.Text: @user1583209 Einstein could have easily found out the relation irrespective of a Computer . He did not focussed much on the game and he had his insights in Physics . A person with such a brilliant imaginative mind could have easily derived this .
Comments.UserDisplayName: Seth Projnabrata
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T13:53:10.933Z
Comments.Text: @Glorfindel It can't be a random walk. That's because the states are highly dependent. Correct model is markov chain.
Comments.UserDisplayName: SmallChess
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T14:21:09.897Z
Comments.Text: @StudentT well, in a random walk, the states (i.e. how far from the x-axis you are) are also dependent. A Markov chain has the problem that it doesn't deal well with the evaluation function being continuous. OTOH, the evaluation function itself is both more and less than the outcome of the game. That's probably subject for another discussion.
Comments.UserDisplayName: Glorfindel
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T14:26:08.647Z
Comments.Text: @Glorfindel Chess evaluation is not truly continuous. The floating value you see is a product of division internally by an engine. Chess scores are always almost an integer. I'd argue it's possible to bin the scale. The logic is like that, chess PV is just heuristic, so 1.56 and 1.67 should not be too different.
Comments.UserDisplayName: SmallChess
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T14:34:11.650Z
Comments.Text: @Glorfindel In random walk, the increments are independent. In chess, we tend to have a trend.
Comments.UserDisplayName: SmallChess
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T14:35:12.703Z
Comments.Text: That's why I said 'variation'. But the mathematical theory is too long ago for me.
Comments.UserDisplayName: Glorfindel
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-03-21T14:39:37.503Z
Comments.Text: Another problem. The expectation of the walk is zero. I am not sure its correct for chess.
Comments.UserDisplayName: SmallChess
Tags: <theory><computer-chess>
Title: Chess as random process
ThreadText: # Chess as random process
  
  (Asked by: username_0 on 2017-03-21)
  
  Not sure whether this is more of a physics or a chess question.
  
  Looking at the evaluation graph (computer evaluation vs. move number) of a game of chess, I realized that there are some similarities to the physics model of escape from a local potential minimum. Let me explain:
  
  While none of the players make a mistake the evaluation will stay at values around zero corresponding to a draw. This I would associate with the motion of a particle in a local minimum, whose coordinate (=evaluation) might fluctuate a bit around the minimum value.
  
  If a player makes a mistake, the evaluation will increase to let's say +2, corresponding to a won position. Normally (i.e. with a very high probability) white should win this game. Until mate, the evaluation will further increase to +3, +4, +5....
  
  This process is similar to the escape from a local minimum. A blunder can "activate" the particle so that it can leave the local minimum and it will subsequently accelerate to larger coordinates (evaluation increases).
  
  [![simple model for chess](https://i.sstatic.net/4ROZp.png)](https://i.sstatic.net/4ROZp.png)
  
  Have there been any studies on chess using such model?
  
  Does it make sense at all?
  
  **Tags:** theory, computer-chess
  
  > **Comments:**
  > **username_1**: "Does it make sense at all?" - No! You have shown a continuous function. Chess is discrete not continuous (each move representing a potential step change) and evaluation can jump from +3 to -3 in one move without passing through zero. This question has no relevance to chess.  
  > **username_2**: There has been no studies in Chess till now and the concept here is quite abstract . Einstein used to play Chess and he was cleanly associated with a lot of Physics . I hope he could have answered it better .  
  > **username_0**: @username_1 If you are doing numerics on some (continuous) physics problem you are also dealing with discrete values. I don't see any difference here. Note that the vertical axis, i.e. the potential would be something like "resistance to loss", i.e. describing the (presumed) fact that with perfect play a game ends in draw and that you have some margin (large choice of moves) which all keep the game in draw. Perhaps will ask this question at physics stackexchange.  
  > **username_0**: @ProjnabrataSeth: Einstein did not have computer evaluation though...  
  > **username_3**: Chess evaluation can probably be described as a variation on a random walk, but there's no such thing as a local minimum. Given a +0.5 evaluation, it isn't harder for Black to make a bad move than for White.  
  > **username_0**: @username_3: Values are somewhat arbitrary and you could argue that the maxima should be at slightly larger values. Anyway, I would claim that if white has a small advantage (e.g. +0.5, i.e. still draw) a small error by black might turn the game into white's favor (white win), while an equally small error by white might  turn it into +-0 or into a small black advantage of e.g. -0.5 (still draw).  
  > **username_2**: @username_0 Einstein could have easily found out the relation irrespective of a Computer . He did not focussed much on the game and he had his insights in Physics . A person with such a brilliant imaginative mind could have easily derived this .  
  > **username_4**: @username_3 It can't be a random walk. That's because the states are highly dependent. Correct model is markov chain.  
  > **username_3**: @StudentT well, in a random walk, the states (i.e. how far from the x-axis you are) are also dependent. A Markov chain has the problem that it doesn't deal well with the evaluation function being continuous. OTOH, the evaluation function itself is both more and less than the outcome of the game. That's probably subject for another discussion.  
  > **username_4**: @username_3 Chess evaluation is not truly continuous. The floating value you see is a product of division internally by an engine. Chess scores are always almost an integer. I'd argue it's possible to bin the scale. The logic is like that, chess PV is just heuristic, so 1.56 and 1.67 should not be too different.  
  > **username_4**: @username_3 In random walk, the increments are independent. In chess, we tend to have a trend.  
  > **username_3**: That's why I said 'variation'. But the mathematical theory is too long ago for me.  
  > **username_4**: Another problem. The expectation of the walk is zero. I am not sure its correct for chess.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_4 on 2017-03-21. Score: 7 | Accepted answer
  
  I'd argue the process you're talking about can be represented by `Markov Chain`. It solves the continuous problem raised by @username_1.
  
  I don't know where to find a proper state diagram for your question.The following diagram is for a simple board game. Imagine we model the **transition probability** from `win` to `lose` (or `win` to `draw`) as the blunder probability, this is similar to what you are asking.
  
  [![enter image description here](https://i.sstatic.net/QVbay.png)](https://i.sstatic.net/QVbay.png)
  
  We can represent "local minimum" described in your question as an `absorbing state`.
  
  More generally, if you want to assume some latent factors for making a blunder (e.g. bad opening, personal issues), you could generalize to `hidden markov model`. It's possible to use the EM algorithm to fit a model from engine PV evaluations.
  
  Given a fitted model, you can query the probability of a blunder given a chess game.
  
  > Have there been any studies on chess using such model?
  
  No studies that I am aware of.
  
  
  ---
  
  