Q4 — Games: Strategies, Alpha-Beta, Force MinMax

A 3-ply game tree — strategy I,K, best line B,D at 46, α-β pruning, and forcing the MinMax value to 36.

The Question

3-ply game tree. Root = MAX, three MIN children, each with two MAX children, each with two horizon leaves A–L.

#	Question	Official answer
4.1	Strategy for MAX (A: A,B · B: A,B,E,F,I,J · C: I,K · D: A,E,I,K)	C
4.2	Best strategy leaves	`B,D`
4.3	Alpha-Beta pruned	`D,G,H,K,L`
4.4	Force MinMax to 36 (one node, largest value)	`B,36`

The Topic in 60 Seconds

Back up: MAX takes max, MIN takes min.
Strategy: MAX commits to one child per MAX node; MIN branches everywhere.
Force MinMax: change one leaf's eval to make the root value a target.

Deep-dive: Game Strategies, Force MinMax Value, Alpha-Beta Pruning.

Step 0 — Back up

$N_{11} = \max(29, 46) = 46$ ; $N_{12} = \max(75, 96) = 96$
$N_{21} = \max(13, 36) = 36$ ; $N_{22} = \max(99, 28) = 99$
$N_{31} = \max(33, 26) = 33$ ; $N_{32} = \max(37, 18) = 37$
$N_1 = \min(46, 96) = 46$ ; $N_2 = \min(36, 99) = 36$ ; $N_3 = \min(33, 37) = 33$
Root $= \max(46, 36, 33) = \mathbf{46}$ via $N_1$

4.1 — Strategy for MAX → C (I,K)

A strategy for MAX = one leaf per MAX node on a single root-to-horizon commitment: MAX fixes one child at the root, then one leaf inside each surviving MAX grandchild. Option C: (I, K) — Root → $N_3$ ✓; $N_3$ (MIN) keeps both $N_{31}$ and $N_{32}$ ✓; $N_{31}$ picks I ✓; $N_{32}$ picks K ✓. Two leaves, one per MAX node — valid.

Option A = (A, B): both leaves sit under $N_{11}$ ; $N_{12}$ is left without a move. Option B = (A, B, E, F, I, J): six leaves spread over all three subtrees — MAX cannot commit to three root children at once. Option D = (A, E, I, K): A forces the root to $N_1$ while E forces it to $N_2$ — contradiction. → C ✓

4.2 — Best strategy → `B,D`

Root → $N_1$ (46). $N_1$ keeps $N_{11}$ and $N_{12}$ . $N_{11}$ picks B (46 beats 29). $N_{12}$ picks D (96 beats 75). Leaves (B, D), guarantee min(46, 96) = 46 = game value ✓

4.3 — Alpha-Beta prunes → `D,G,H,K,L`

Node	Window	Happening
$N_1$ (MIN)	$(-\infty, \infty)$	$N_{11}$ (MAX): A=29, B=46 → returns 46. $\beta = 46$
$N_{12}$ (MAX)	$(-\infty, 46)$	C = 75 ≥ 46 → β-cutoff — D pruned!
Root	α = 46
$N_2$ (MIN)	$(46, \infty)$	$N_{21}$ (MAX): E=13, F=36 → returns 36 ≤ 46 → α-cutoff — $N_{22}$ (G, H) pruned!
Root	α = 46
$N_3$ (MIN)	$(46, \infty)$	$N_{31}$ (MAX): I=33, J=26 → returns 33 ≤ 46 → α-cutoff — $N_{32}$ (K, L) pruned!

Pruned: D, G, H, K, L ✓

4.4 — Force MinMax to 36 → `B,36`

Current root = 46 via $N_1 = \min(N_{11}, N_{12}) = \min(46, 96)$ . To lower the root to 36:

Lower $N_{11}$ to 36: $N_{11} = \max(A, B)$ . Set B = 36 → $N_{11} = \max(29, 36) = 36$ . Then $N_1 = \min(36, 96) = 36$ → Root = max(36, 36, 33) = 36 ✓
B = 36 is the largest value for B that achieves root = 36 (any higher and $N_{11} > 36$ → root > 36).

→ B, 36 ✓

Watch the whole tree resolve

1 / 5

Leaves A-L carry the static evaluations. Nothing backed up yet.

backing upevaluated / backed uppruned by alpha-betabest line / force point

Tricks & Tips

Answers: 4.1 C · 4.2 B,D · 4.3 D,G,H,K,L · 4.4 B,36