Programming

 Variance reduction of rollouts

 From: Jim Williams Address: jimw@giga-net.com Date: 11 June 1997 Subject: Proposed Algorithm for Roll Outs Forum: rec.games.backgammon Google: 339F21F8.3585@giga-net.com

```The classic algorithm for doing roll outs is to start in
position which is to be analyzed.  From this position,
the game is repeatedly rolled out to completion.  A equity
value is assigned to the completion result.  The equity
of the starting position is approximated by the average value
of the completion equities.  The implicit assumption is that
the best move is always made, but subject to this assumption
the average value of the completion equity will converge
stochastically to the true equity of the starting position.

The proposed algorithm will converge to the same limit, but can
potentially converge much faster.  This algorithm assumes
the existance of an evaluation function which approximates
the equity of a given position.  The important feature of
this algorithm is that the better the evaluation function is,
the faster the convergence will be, but no matter how inaccurate
the evaluation function is, the convergence limit is unaffected.

For any give  position, the "disparity" of that position is computed
as follows.  For each possible dice roll, the best move is determined
and the evaluation function is applied to the resulting position.
The disparity is then equal to the weighted average of the evaluations
of all these resulting positions minus the evalution of the
original position.  The better the evaluation funcation is,
the smaller the disparity will be.

The roll out proceeds normally, but the result of each trial is
equal to the equity value determined by applying the evalution function
to the original position plus the sum of the disparities of all the
positions that occurred in the roll out (including the original
position).

The proof of correctness is inductive.  It is assumed that the
evaluation function correctly evaluates a "game over" position.
The expected value of the roll out result of a position is equal to
the weighted average of the expected values of the roll outs of
each of the next possible positions,
plus the evaluation of the current position, minus the weighted average
of the evaluations of the next postions, plus the disparity of the
current position.  This in turn is equal to just the weighted average
of the expected values of the roll outs of the next positions.
These are correct by the induction hypothesis.
```

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