Risk — The Attrition Constant, and Why Big Battles Are Predictable

A Risk battle solved exactly as an absorbing Markov chain, and the three structural facts that collapse it. The headline is a design result, not a math result: because the engagement frontage is capped at 3-vs-2 dice regardless of stack size, Risk implements Lanchester’s linear law — concentration of force buys nothing — and the attrition price is the exact rational 2387/2797 ≈ 0.853 attacker armies per defender killed. Two consequences run in opposite directions: there is no tactical deterrence (matched stacks of 12+ favour the attacker, and the defensive premium needed to hold shrinks as borders grow, so a big border is an offensive asset), but a won battle costs ~85% of the attacking force, so in a symmetric three-player standoff the winner drops from a third of the board’s force to 13.6% while the bystander rises to 86.4%. Risk omits tactical deterrence and recovers stability from the player count — and that stability decays as players are eliminated, since the restraint was always the bystander. Applied to two pieces of common table advice: matching an escalating border is the most expensive way to stay unprotected, and interior garrisons should be 2, never 3 (the 2nd army buys the defender’s second die; the 3rd buys nothing).

Links: BattleValue (the direct contrast — BV = sqrt(Attack × HP) is a Square Law metric; Risk is the linear-law counterexample), D&D Monster Tournament — Exact Markov Chains Instead of Dice (same method, sibling specimen), Yahtzee — 259 Trillion → 405 Million (state-space collapse as the enabling move), Battleship — 30 Billion Boards (structure creates norms, substrate fully enumerated), D&D Spell Damage Model, Randomness as the Termination Mechanism (the sibling stabilizer — the brake here is the cost of winning, not randomness), The Multiplayer Coalition Problem (where the strategic half of this page lives — relative position is everything, and the bystander math below is that thesis with the arithmetic filled in), The Three-Layer Method

Tool: tools/risk-battle-odds.py — two independent engines, --selftest cross-checks them on every run.

Specimen trigger (2026-08-13). A Risk short: a stack of 299 attacks a territory of 300 and takes it, moving in 54 and leaving 1 behind. The table-side claim was that ~50 survivors is roughly par and 54 is only slightly lucky. That claim is correct — the exact expectation is 45.3 given a win, and 54 sits at the 65th percentile.

The exact answer

299-stack (so 298 committed) against 300 defenders, fought to the death, both sides always rolling maximum dice, defender winning ties:

Why the matrix collapses

The brute-force (a, d) grid is mostly waste, and the game says so. While a ≥ 3 and d ≥ 2 the round is always 3-vs-2, and a 3-vs-2 round kills exactly two armies — never one, never three. Three consequences follow:

1. Parity is invariant. a + d falls by exactly 2 each round, so its parity never changes and half the grid is unreachable. Measured on the 299-vs-300 case: of 89,999 cells, exactly 50.7% are ever touched (half, plus the boundary).
2. The bulk is one-dimensional. In that regime the dice odds do not depend on (a, d) at all, so attacker losses are a running sum of IID draws from {0, 1, 2}. Not a 2-D chain — a random walk.
3. The endgame is a thin closed shell. The regime is left only through a ≤ 2 or d ≤ 1, and that region is closed under transitions, so it solves separately in O(A+D) — 595 states against a 90,000-cell grid.

Honest accounting: the reduction is not asymptotically faster. Both engines are O(A·D)/2, because a forward sweep that skips zero-probability cells already avoids the dead half implicitly. What the reduction actually buys is a structurally independent oracle, and fact 2’s closed forms — which brute force cannot give you.

The closed forms (derived, not fitted)

Per 3-vs-2 round the attacker loses 7161/7776 armies and the defender 8391/7776; they sum to exactly 2, because two armies always die. So the attacker’s price per defender killed is the exact rational

c = 7161/8391 = 2387/2797 = 0.8534143725…

Because the bulk increments are IID, Z = a − c·d is a martingale — expected change per round exactly zero — so optional stopping gives E[survivors] = A − c·D directly. The same argument on the variance gives the part everyone misjudges:

sd(survivors) ≈ 1.447 · √D

Spread grows with the square root of the battle while force grows linearly, so big battles are proportionally more predictable. Risk feels swingy because a single 3-vs-2 roll is; a 278-round battle is not. The back-of-envelope players actually use — “attacker wins 7 for every 6 lost” — is 7/6 = 1.1667 against the exact ratio 2797/2387 = 1.1718: 0.435% off, and slightly understating the attacker’s edge.

The design result: Risk is Lanchester-LINEAR

Lanchester’s Square Law says concentrated force is superlinearly better — the basis of BattleValue’s BV = sqrt(Attack × HP). Risk does not obey it, and the reason is a design choice: the engagement frontage is capped at 3 dice vs 2 regardless of stack size. A 300-army stack brings exactly as much force to bear per round as a 3-army stack. Fixed frontage is the classic condition that produces linear-law attrition, and the exact chain confirms it:

E[S]/N converges to the predicted 1 − c = 0.1466, and sd/√N to the predicted 1.447. Doubling both sides multiplies survivors by 1.795 → 1.880 → 1.950, converging on 2 — linear, not square.

The sharpest test is direct. Is one concentrated 200-vs-100 fight better than two separate 100-vs-50 fights?

• 200 v 100 in one battle: E[S] = 114.88
• 100 v 50, twice: E[S] = 115.10 total
• Difference: −0.22 armies. Zero concentration advantage (the sign is an endgame artifact — each smaller battle gets its own favourable tail).

So numerical superiority in Risk is purely additive. This is a real design lever, and it explains a familiar table dynamic: massing a doom-stack buys staying power but no force multiplier, which is exactly what keeps a Risk leader killable and the game terminating — the same function randomness serves in N≥3 games. A Square-Law Risk (dice scaling with stack size) would snowball uncontrollably.

Tactical: there is no defensive deterrence, and big borders make it worse

Because an attacking army is worth 1/c = 1.172 defending armies — the third die more than pays for the defender’s tie-break — parity favours the attacker, and linear attrition means that per-army edge multiplies by the count instead of washing out. The crossover is sharp. If both players hold a stack of N and the attacker commits N−1:

Stacks of 12 are the tipping point — above that, matched borders favour the attacker, and the edge runs away with scale. The break-even frontier converges on the attrition constant itself: the attacker needs only A/D → 0.8534 for even odds. And the defensive premium required to actually hold (P(hold) ≥ 0.9) shrinks as the border grows — 1.90× the attacker’s numbers at 10 armies, 1.41× at 100, 1.31× at 300.

A big border is not a defensive structure. It is an offensive one — a stack large enough to hold is already large enough to attack, and the bigger it gets the truer that becomes. Risk has no tactical deterrent; it was never built in.

Strategic: winning is how you lose (the bystander)

The brake is not on the board, it is in the player count — and the exact chain quantifies precisely how expensive a won battle is. Take a symmetric three-player standoff, each holding a stack of N. P1 attacks P2 with everything; P3 does nothing:

At N = 300: P1 spends 253 armies, P2 spends all 300, P3 spends nothing — and P1 goes from a third of the board’s force to 13.6% by winning, while the bystander rises to 86.4%. Since retention converges on 1 − c = 14.7%, this gets worse at scale, not better. Hence the closed form for the price of admission:

To attack a stack of N without falling behind an untouched bystander, you need (1 + c) = 1.853 × N — measured at 1.833 → 1.853 as N grows.

In Risk, relative strength is the only strength, so a border stack’s value is as an unspent threat. Spending it converts a tactical certainty into a strategic collapse. That is the same brake randomness provides in N≥3 games and the coalition problem describes socially, arriving here through the combat arithmetic instead — which is a design result worth naming: Risk deliberately omits tactical deterrence and recovers stability from the player count.

What is computed here and what is not. The two-player battle is exact. The three-player table is arithmetic layered on top of it under a deliberately bare model — symmetric stacks, no turn order, no cards, no coalition choice, and no reinforcement income. That last one is the real limitation and cuts against the thesis: conquest takes territory, territory pays reinforcements, and continent bonuses can repay 253 armies over enough turns. So this is a snapshot of force, not a flow — it shows what a battle costs, not whether the conquest was worth it. The claim that a third player actually converts the advantage is an assumption imported from the two pages above, not a result of this solver.

Reading the signal: what to do about a big border

Matching it is the worst available option — not merely insufficient, but the most expensive way to remain unprotected:

You spend 300 armies and still lose 95% of the time if they commit — while both of you feed the bystander. So the correct read of an escalating neighbour is don’t follow, and the reason is not stinginess: the escalation cannot be answered tactically at all, because there is no tactical deterrent to buy.

The phase transition — stability decays as players are eliminated. Since an attacker retains S − c·D, an attack only improves their relative standing if S − c·D ≥ B for a bystander at B. The required border therefore depends on the third player, and rises sharply as the table empties:

This is the mechanism under the N≥3 stability thesis: the restraint was never in the combat system, and it evaporates as players die. The endgame knife fight is not a change in mood — it is B → 0 removing the only constraint that was operating.

The signal to read is not your neighbour’s stack size. It is their stack size relative to the largest untouched player. A 300-border while someone else also sits at 300 is a bluff that cannot be profitably cashed; the same border in a three-player endgame with a crippled third is a genuine threat.

⚠ Heuristic, not a rule. The D > (S − B)/c threshold is directionally sound in the mid-range but degenerates at both ends, and the failure is instructive: at B = S it claims a 300-stack cannot profitably take even a 1-army territory, i.e. “defend with 1 army.” That is the no-income assumption showing through — taking a 1-army territory for ~0.85 armies is plainly correct in real Risk, because territory pays reinforcements and advances the card track. Use the direction (a larger third player means a smaller border needed); do not use the number.

Interior garrisons: the folk advice is half right, and the half is “2, never 3”

Common table advice is to hold interior territories with 2 or 3 armies rather than 1. The exact chain says the entire effect is the second army, because the defender rolls min(2, d) dice — the 2nd buys a second die, the 3rd buys nothing:

A 1.65× improvement, so the instinct is sound. But it is fully spent at two. Holding the same 12-army budget:

Three-stacks are marginally worse than two-stacks for the same budget and cover a third fewer territories. The advice should be “2”, never “3”.

What it actually buys is denial of the one-turn cascade. A rolling stack clears ten 1-army territories for 5.2 armies; ten 2-army territories cost it 15.5. That is frequently the difference between losing your whole interior in a single turn and stalling the stack halfway. It is an anti-sweep measure, not general defense.

Why it is still usually wrong. Per army, offense beats even the best defensive arrangement: an attacking army removes 1/c = 1.172 defending armies against 0.853 absorbed — 1.37×, and 2.27× against 1-stacks. ⚠ But the dominant term is outside this model: mobility. Risk permits one fortify per turn, so armies spread two-deep across a dozen interior territories are not merely lower-value, they are stranded — they cannot be reassembled into the concentration a winning attack requires. That mechanic, not the 1.37×, is the real argument against garrisoning, and this solver cannot see it.

The exception is when the offense multiplier has nothing to multiply. The 1.37× premium only pays if the attack changes your standing — and per the bystander section above, a won battle costs ~85% of the committed force and lowers your relative position. A player far enough behind that no available attack improves their position holds armies with an offensive value of effectively zero, leaving 0.853-per-army survival as the only remaining return. Garrisoning at 2 is what is left when attacking has stopped paying — which makes it a symptom of a losing position rather than a route out of one.

Verification ledger

Four routes, agreeing — the point of the exercise, per the three-layer method:

Per-round dice odds are checked against the published Risk tables (3v2 = 2890/2611/2275 over 7776). The MC’s P(win) first landed 3σ high — two further seeds at 0.95σ and 0.47σ showed that was an unlucky draw, not a bug. A single-seed Monte Carlo is not a verification.

Open threads

• Multi-territory campaigns. One battle is solved; a chain of territories (each conquest leaving 1 behind, feeding the next attack) is the real strategic object and is not yet modelled.
• The stop-early decision. This assumes fighting to the death. The genuine decision problem — when to break off — turns this from a chain into an MDP, the same fixed-policy-vs-decision boundary that makes the D&D tournament tractable.
• Does the linear-law finding generalize? Any game with a capped engagement frontage should be linear-law. Candidate check across the vault’s combat games — and a genuine falsifier for BV’s Square-Law framing wherever frontage is capped.

Tags

games, game-theory, game-design, strategy