Top-to-bottom, left-to-right, the minimax values are 7;6,7,4;9,6,9,7,6,7,4.

Labelling the nodes A through K top-to-bottom, left-to-right, the $(\alpha, \beta)$ windows passed to each node and the values returned are

Node	$(\alpha, \beta)$	Value
A	$(-\infty, \infty)$	7
B	$(-\infty, \infty)$	4
C	$(4, \infty)$	7
D	$(7, \infty)$	6
E	$(-\infty, \infty)$	4
F	$(-\infty, 4)$ [prunes 2nd child]	6
G	$(-\infty, 4)$ [prunes 2nd child]	7
H	$(4, \infty)$	7
I	$(4, 7)$ [prunes 2nd, 3rd children]	9
J	$(7, \infty)$	9
K	$(7, 9)$	6

Labelling the nodes A through K top-to-bottom, left-to-right, the $(\alpha, \beta)$ windows passed to each node (nodes that are examined multiple times are given with each window passed to them and each returned value) and the values returned are

Node	$(\alpha, \beta)$	Value
A	$(-\infty, \infty)$	7
B	$(-\infty, \infty)$	4
C	$(4, 5)$ $(7, \infty)$ [prunes second child]	7 7
D	$(7, 8)$	6
E	$(-\infty, \infty)$	4
F	$(3, 4)$ [prunes 2nd child]	6
G	$(3, 4)$ [prunes 2nd child]	7
H	$(4, 5)$ [prunes 2nd child] $(7, \infty)$	7 7
I	$(4, 5)$ [prunes 2nd, 3rd children]	9 9
J	$(7, 8)$ [prunes 2nd, 3rd children]	9
K	$(7, 8)$	6

Labelling the nodes A through K top-to-bottom, left-to-right, the depths and bounds stored in the transposition table are

Node	Depth	Bound
A	3	=7
B	2	=4
C	2	=7
D	2	=6
E	1	=4
F	1	≥6
G	1	≥7
H	1	=7
I	1	≥9
J	1	=9
K	1	=6

If we roll on 1 on the 10-sided die anything other than a 1 on either of the other dice then the average regret for each subsequent roll will be non-zero ,and so the limit of the average regret is non-zero. That happens with probability $\frac{1}{10} \cdot (1 - \frac{1}{8}\cdot\frac{1}{6}) > 0$, so the probability that the average regret goes to zero in the limit is less than 1.

The tree policy would have to account for the stochastic element. At random event positions, the tree policy could randomly choose the next position according to the probability distribution of that event.

Predicate Average Sampling Technique: for NFL Strategy, use predicates "trailing by more than 21 points" and "leading by more than 21 points" and keep track of the success of various plays in those situations during random playouts. Then bias play selection towards the plays that are better in those situations when in positions matching those situations.

Search Seeding: use a heuristic function for Kalah (perhaps our simple score margin heuristic) to initialize visit count and reward statistics for newly expanded nodes in the hopes of reducing the amount of time determining that bad moves are in fact bad.

Inputs:

• distance to goal line, normalized to $[0.0, 1.0)$
• time left in quarter, normalized to $[0.0, 1.0)$
• current quarter (1.0 = 1st, 0.66 = 2nd, 0.33 = 3rd, 0.0 = 4th)
• current down (1.0 = 1st, 0.66 = 2nd, 0.33 = 3rd, 0.0 = 4th)
• yards to first down divided by 10 (this one is interesting: we should normalize somehow, but dividing by 100 would result in compressing the most frequently seen positions to a small range; normalizing to standard deviations away from the mean seen during actual play may be better)
• score margin divided by 20 (again, the best normalization may be determined by experimentation)
• (and field position as -1 for left, 0, for center, 1 for right if you wish to include that feature of the commercial game)

Outputs:

• one output for each play, interpreted as the probability of selecting that play (so the output layer should use a softmax activation function)

The value network was trained using positions reached in games played by the policy network obtained after reinforement learning against itself. Instead of training on every position reached during every game, they trained using one position sampled from 30 million different games.