Risk Strategy Briefing

# Risk Strategy Briefing (Classic Rules) — Probabilities + Practical Strategy

This reference uses **exact** dice probabilities (uniform d6, ties go to defender) and dynamic programming for multi-roll battles.

**Dice rules reminder:** attacker rolls up to `min(3, A−1)` dice; defender up to `min(2, D)` dice, where `A` and `D` are the armies currently in the attacking and defending territories.

## 1) Complete Single-Roll Attack/Defense Probability Tables (Exact)

To make the tables uniform and ‘printable’, every dice matchup reports the probabilities of these casualty outcomes (some are impossible and shown as 0):

- **A−2**: attacker loses 2 armies (only possible when 2 comparisons happen: defender rolls 2 dice)
- **Each−1**: each side loses 1 army (only possible when 2 comparisons happen)
- **D−2**: defender loses 2 armies (only possible when 2 comparisons happen)
- **A−1**: attacker loses 1 army (single-comparison matchups)
- **D−1**: defender loses 1 army (single-comparison matchups)

### 3 attack dice vs 2 defense dice

| Outcome | Exact probability | Percentage |
|---|---:|---:|
| A−2 | 2275/7776 | 29.257% |
| Each−1 | 2611/7776 | 33.578% |
| D−2 | 1445/3888 | 37.166% |
| A−1 | 0/1 | 0.000% |
| D−1 | 0/1 | 0.000% |

**Expected loss per roll:** attacker 2387/2592 = 0.920910, defender 2797/2592 = 1.079090.

### 3 vs 1

| Outcome | Exact probability | Percentage |
|---|---:|---:|
| A−2 | 0/1 | 0.000% |
| Each−1 | 0/1 | 0.000% |
| D−2 | 0/1 | 0.000% |
| A−1 | 49/144 | 34.028% |
| D−1 | 95/144 | 65.972% |

**Expected loss per roll:** attacker 49/144 = 0.340278, defender 95/144 = 0.659722.

### 2 vs 2

| Outcome | Exact probability | Percentage |
|---|---:|---:|
| A−2 | 581/1296 | 44.830% |
| Each−1 | 35/108 | 32.407% |
| D−2 | 295/1296 | 22.762% |
| A−1 | 0/1 | 0.000% |
| D−1 | 0/1 | 0.000% |

**Expected loss per roll:** attacker 791/648 = 1.220679, defender 505/648 = 0.779321.

### 2 vs 1

| Outcome | Exact probability | Percentage |
|---|---:|---:|
| A−2 | 0/1 | 0.000% |
| Each−1 | 0/1 | 0.000% |
| D−2 | 0/1 | 0.000% |
| A−1 | 91/216 | 42.130% |
| D−1 | 125/216 | 57.870% |

**Expected loss per roll:** attacker 91/216 = 0.421296, defender 125/216 = 0.578704.

### 1 vs 2

| Outcome | Exact probability | Percentage |
|---|---:|---:|
| A−2 | 0/1 | 0.000% |
| Each−1 | 0/1 | 0.000% |
| D−2 | 0/1 | 0.000% |
| A−1 | 161/216 | 74.537% |
| D−1 | 55/216 | 25.463% |

**Expected loss per roll:** attacker 161/216 = 0.745370, defender 55/216 = 0.254630.

### 1 vs 1

| Outcome | Exact probability | Percentage |
|---|---:|---:|
| A−2 | 0/1 | 0.000% |
| Each−1 | 0/1 | 0.000% |
| D−2 | 0/1 | 0.000% |
| A−1 | 7/12 | 58.333% |
| D−1 | 5/12 | 41.667% |

**Expected loss per roll:** attacker 7/12 = 0.583333, defender 5/12 = 0.416667.

## 2) Extended Battle Outcome Probabilities (Rolling Until Conquest or Failure)

Assume the attacker continues rolling (always maximum dice) until either the **defender is eliminated** (territory conquered) or the attacker is reduced to **1 army** (cannot continue).

### Common Starting Battles

| Start (A vs D) | P(attacker takes territory) | Exact (fraction) | E[attacker losses] | E[defender losses] |
|---:|---:|---:|---:|---:|
| 5v3 | 64.162% | 402241935565/626913312768 | 2.078 | 2.235 |
| 6v4 | 63.829% | 1166853115687865/1828079220031488 | 2.851 | 3.115 |
| 8v5 | 73.640% | 144709205741850981768695/196510150862874095910912 | 3.759 | 4.314 |
| 10v5 | 87.294% | 10671214934543021258195122645/12224503464877671758426013696 | 3.927 | 4.695 |
| 12v8 | 79.412% | 95091610626472432093687816033287888905/119745374439224305280465125993560932352 | 6.285 | 7.351 |
| 15v10 | 83.457% | 1546942875361685794687479009888379961625917389565/1853579490823177765520981419568340531793952243712 | 8.035 | 9.448 |
| 20v10 | 96.504% | 80741035596617172070857365965808792043038293069474677680624165/83666423453096234037579724162955786641867687038701502046666752 | 8.267 | 9.900 |
| 20v15 | 82.208% | 2021211797369232788803375509191853627237481625002236327917842308797741765/2458668501301446091275194300620606315798531579132791303430002451333251072 | 12.172 | 14.281 |

### Win probability: P(attacker conquers)

| A\D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| 2 | 41.7% | 10.6% | 2.7% | 0.7% | 0.2% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% |
| 3 | 75.4% | 36.3% | 20.6% | 9.1% | 4.9% | 2.1% | 1.1% | 0.5% | 0.3% | 0.1% | 0.0% | 0.0% | 0.0% |
| 4 | 91.6% | 65.6% | 47.0% | 31.5% | 20.6% | 13.4% | 8.4% | 5.4% | 3.3% | 2.1% | 0.8% | 0.2% | 0.0% |
| 5 | 97.2% | 78.5% | 64.2% | 47.7% | 35.9% | 25.3% | 18.1% | 12.3% | 8.6% | 5.7% | 2.6% | 0.7% | 0.1% |
| 6 | 99.0% | 89.0% | 76.9% | 63.8% | 50.6% | 39.7% | 29.7% | 22.4% | 16.2% | 11.8% | 5.9% | 1.9% | 0.3% |
| 7 | 99.7% | 93.4% | 85.7% | 74.5% | 63.8% | 52.1% | 42.3% | 32.9% | 25.8% | 19.3% | 10.7% | 4.1% | 0.7% |
| 8 | 99.9% | 96.7% | 91.0% | 83.4% | 73.6% | 64.0% | 53.6% | 44.6% | 35.7% | 28.7% | 17.3% | 7.3% | 1.5% |
| 9 | 100.0% | 98.0% | 94.7% | 88.8% | 81.8% | 73.0% | 64.3% | 54.7% | 46.4% | 38.0% | 24.7% | 11.9% | 2.8% |
| 10 | 100.0% | 99.0% | 96.7% | 93.0% | 87.3% | 80.8% | 72.6% | 64.6% | 55.8% | 48.0% | 33.4% | 17.3% | 4.8% |
| 12 | 100.0% | 99.7% | 98.8% | 97.2% | 94.3% | 90.5% | 85.2% | 79.4% | 72.3% | 65.4% | 50.7% | 30.9% | 11.1% |
| 15 | 100.0% | 100.0% | 99.8% | 99.3% | 98.5% | 97.0% | 94.9% | 91.8% | 88.2% | 83.5% | 72.4% | 53.8% | 25.9% |
| 20 | 100.0% | 100.0% | 100.0% | 99.9% | 99.9% | 99.7% | 99.3% | 98.7% | 97.8% | 96.5% | 92.5% | 82.2% | 57.7% |

### Expected attacker losses (total, until battle ends)

| A\D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| 2 | 0.58 | 0.89 | 0.97 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| 3 | 0.67 | 1.41 | 1.66 | 1.86 | 1.92 | 1.97 | 1.98 | 1.99 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 |
| 4 | 0.57 | 1.41 | 1.89 | 2.29 | 2.54 | 2.71 | 2.82 | 2.89 | 2.93 | 2.96 | 2.98 | 3.00 | 3.00 |
| 5 | 0.53 | 1.52 | 2.08 | 2.67 | 3.02 | 3.34 | 3.53 | 3.69 | 3.79 | 3.86 | 3.94 | 3.98 | 4.00 |
| 6 | 0.52 | 1.51 | 2.18 | 2.85 | 3.37 | 3.79 | 4.12 | 4.36 | 4.55 | 4.68 | 4.84 | 4.95 | 4.99 |
| 7 | 0.52 | 1.54 | 2.23 | 3.01 | 3.59 | 4.15 | 4.56 | 4.92 | 5.19 | 5.41 | 5.69 | 5.89 | 5.98 |
| 8 | 0.52 | 1.54 | 2.27 | 3.07 | 3.76 | 4.38 | 4.91 | 5.36 | 5.73 | 6.02 | 6.44 | 6.78 | 6.96 |
| 9 | 0.52 | 1.55 | 2.28 | 3.14 | 3.85 | 4.56 | 5.16 | 5.71 | 6.15 | 6.55 | 7.12 | 7.61 | 7.92 |
| 10 | 0.52 | 1.54 | 2.30 | 3.16 | 3.93 | 4.67 | 5.35 | 5.95 | 6.50 | 6.96 | 7.70 | 8.39 | 8.85 |
| 12 | 0.52 | 1.55 | 2.30 | 3.19 | 3.99 | 4.80 | 5.57 | 6.29 | 6.96 | 7.56 | 8.59 | 9.70 | 10.60 |
| 15 | 0.52 | 1.55 | 2.31 | 3.21 | 4.03 | 4.88 | 5.70 | 6.51 | 7.28 | 8.03 | 9.40 | 11.07 | 12.83 |
| 20 | 0.52 | 1.55 | 2.31 | 3.21 | 4.04 | 4.90 | 5.74 | 6.59 | 7.43 | 8.27 | 9.89 | 12.17 | 15.25 |

### Expected defender losses (total, until battle ends)

| A\D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| 2 | 0.42 | 0.36 | 0.35 | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 | 0.34 |
| 3 | 0.75 | 0.91 | 1.07 | 1.10 | 1.13 | 1.14 | 1.15 | 1.15 | 1.15 | 1.15 | 1.15 | 1.15 | 1.15 |
| 4 | 0.92 | 1.44 | 1.83 | 2.07 | 2.23 | 2.33 | 2.39 | 2.43 | 2.45 | 2.47 | 2.48 | 2.49 | 2.49 |
| 5 | 0.97 | 1.65 | 2.24 | 2.63 | 2.94 | 3.14 | 3.29 | 3.38 | 3.45 | 3.50 | 3.55 | 3.58 | 3.58 |
| 6 | 0.99 | 1.83 | 2.54 | 3.11 | 3.56 | 3.90 | 4.15 | 4.35 | 4.48 | 4.58 | 4.69 | 4.76 | 4.79 |
| 7 | 1.00 | 1.90 | 2.72 | 3.40 | 3.99 | 4.46 | 4.84 | 5.12 | 5.35 | 5.51 | 5.72 | 5.87 | 5.94 |
| 8 | 1.00 | 1.95 | 2.83 | 3.63 | 4.31 | 4.91 | 5.39 | 5.80 | 6.11 | 6.37 | 6.71 | 6.97 | 7.10 |
| 9 | 1.00 | 1.97 | 2.90 | 3.76 | 4.54 | 5.23 | 5.83 | 6.33 | 6.76 | 7.10 | 7.59 | 8.00 | 8.24 |
| 10 | 1.00 | 1.98 | 2.94 | 3.85 | 4.70 | 5.47 | 6.16 | 6.77 | 7.28 | 7.73 | 8.39 | 8.98 | 9.36 |
| 12 | 1.00 | 2.00 | 2.98 | 3.94 | 4.87 | 5.76 | 6.58 | 7.35 | 8.04 | 8.66 | 9.67 | 10.69 | 11.49 |
| 15 | 1.00 | 2.00 | 3.00 | 3.99 | 4.97 | 5.93 | 6.87 | 7.77 | 8.64 | 9.45 | 10.90 | 12.62 | 14.30 |
| 20 | 1.00 | 2.00 | 3.00 | 4.00 | 5.00 | 5.99 | 6.98 | 7.97 | 8.94 | 9.90 | 11.75 | 14.28 | 17.56 |

## 3) Strategy With Probabilities

### Optimal attack thresholds (minimum attacker armies for a target win chance)

These are computed for **attacker armies A** vs **defender armies D** where **A includes the 1 army that must stay behind**.


**Target:** P(win) ≥ 50%

| Defender D | Min attacker A | P(win at that A) |
|---:|---:|---:|
| 1 | 3 | 75.4% |
| 2 | 4 | 65.6% |
| 3 | 5 | 64.2% |
| 4 | 6 | 63.8% |
| 5 | 6 | 50.6% |
| 6 | 7 | 52.1% |
| 7 | 8 | 53.6% |
| 8 | 9 | 54.7% |
| 9 | 10 | 55.8% |
| 10 | 11 | 56.8% |
| 11 | 12 | 57.6% |
| 12 | 12 | 50.7% |
| 13 | 13 | 51.8% |
| 14 | 14 | 52.8% |
| 15 | 15 | 53.8% |
| 16 | 16 | 54.7% |
| 17 | 17 | 55.5% |
| 18 | 18 | 56.3% |
| 19 | 18 | 50.4% |
| 20 | 19 | 51.3% |

**Target:** P(win) ≥ 75%

| Defender D | Min attacker A | P(win at that A) |
|---:|---:|---:|
| 1 | 3 | 75.4% |
| 2 | 5 | 78.5% |
| 3 | 6 | 76.9% |
| 4 | 8 | 83.4% |
| 5 | 9 | 81.8% |
| 6 | 10 | 80.8% |
| 7 | 11 | 80.0% |
| 8 | 12 | 79.4% |
| 9 | 13 | 79.0% |
| 10 | 14 | 78.7% |
| 11 | 15 | 78.4% |
| 12 | 16 | 78.3% |
| 13 | 17 | 78.2% |
| 14 | 18 | 78.1% |
| 15 | 19 | 78.1% |
| 16 | 20 | 78.1% |
| 17 | 21 | 78.1% |
| 18 | 22 | 78.1% |
| 19 | 23 | 78.1% |
| 20 | 24 | 78.2% |

**Target:** P(win) ≥ 90%

| Defender D | Min attacker A | P(win at that A) |
|---:|---:|---:|
| 1 | 4 | 91.6% |
| 2 | 7 | 93.4% |
| 3 | 8 | 91.0% |
| 4 | 10 | 93.0% |
| 5 | 11 | 91.6% |
| 6 | 12 | 90.5% |
| 7 | 14 | 92.5% |
| 8 | 15 | 91.8% |
| 9 | 16 | 91.2% |
| 10 | 17 | 90.7% |
| 11 | 18 | 90.2% |
| 12 | 20 | 92.5% |
| 13 | 21 | 92.1% |
| 14 | 22 | 91.8% |
| 15 | 23 | 91.5% |
| 16 | 24 | 91.3% |
| 17 | 25 | 91.0% |
| 18 | 26 | 90.8% |
| 19 | 27 | 90.6% |
| 20 | 28 | 90.5% |

### Rule-of-thumb ratios (backed by the threshold tables)

For mid-sized fights (D ≳ 5), a remarkably stable approximation is:

- **~50% odds:** A ≈ D + 1 (very rough; small D is swingy)
- **~75% odds:** A ≈ D + 4
- **~90% odds:** A ≈ D + 7 to D + 8

Convert to a ‘committed attackers’ view by subtracting 1 from A (because one army must remain behind).


### Continent holding probabilities: border model (approximate but useful)

Holding a continent is primarily about defending **its borders**. A simple probabilistic model:

- Suppose each turn, each border faces one serious break attempt by an attacker stack of size S.
- If you defend that border with T, the break probability is b = pwin(S,T).
- With k borders and N turns, if you assume independence (approx.), then P(hold N turns) ≈ (1−b)^{kN}.


**Defender troops needed (T) so an attacker stack S has low chance to break:**

| Attacker stack S | T for ≤25% break | P(break) | T for ≤10% break | P(break) |
|---:|---:|---:|---:|---:|
| 6 | 8 | 22.4% | 11 | 8.3% |
| 8 | 11 | 22.2% | 14 | 10.0% |
| 10 | 14 | 21.9% | 18 | 8.3% |
| 12 | 17 | 21.0% | 21 | 8.6% |
| 15 | 21 | 22.0% | 26 | 8.0% |
| 20 | 27 | 24.6% | 33 | 8.9% |

**Border counts (Classic map):**

| Continent | Bonus | # territories | Practical borders to defend | Notes |
|---|---:|---:|---:|---|
| Australia | +2 | 4 | **1** (Siam ↔ Indonesia) | Easiest hold; low bonus; great for safety + card farming. |
| South America | +2 | 4 | **2** (Venezuela, Brazil) | Still easy; forces defense split; good springboard to Africa. |
| Africa | +3 | 6 | **3** (North Africa, Egypt, East Africa) | Harder: multiple entry lines; bonus decent; position central. |
| North America | +5 | 9 | **3** (Alaska, Greenland, Central America) | Strong bonus but spread-out; Greenland is a liability. |
| Europe | +5 | 7 | **4** (Iceland, Western Europe, Southern Europe, Ukraine) | High border count; Ukraine is huge liability. |
| Asia | +7 | 12 | **5–6** (Ural, Afghanistan, Middle East, Siam, Kamchatka, sometimes Yakutsk) | Big bonus but extremely porous; usually endgame only. |

### Card timing (classic increasing set values)

Classic escalating turn-in sequence: **4, 6, 8, 10, 12, 15, 20, 25, 30, 35, ...** (then +5 each set).

Key quantitative idea: **a set is worth much more when it converts directly into territory captures**, because every captured territory gives an extra card and snowballs tempo.

Practical rules:

- **Cash-in timing is about ‘power spike per turn’, not raw value.**
- **Avoid dying with 3–4 cards** (you become profitable to eliminate).
- **Engineer turns where a cash-in produces a conquest and a card**, enabling future cash-ins.
- **Forced cash at 5+ cards** means you should usually set up a cash before you hit 5 unless you’re very safe.


### Fortification math: concentrate vs spread

Because attackers roll up to 3 dice while defenders roll up to 2, you generally want to **concentrate** defense on borders that would otherwise be breakable.

- Fortifying +x onto a key border is ‘worth it’ if it meaningfully reduces pwin(S,T) for the likely enemy stack S.
- Spreading x is better only if you expect multiple independent attacks on different borders.


### Early / mid / late game strategy shifts

- **Early:** prioritize position and low-cost expansion; avoid grinding marginal fights.
- **Mid:** attempt a continent only if you can defend its borders at low break probability; otherwise play for cards + mobility.
- **Late:** card sets dominate; focus on kill chains and denying opponent cash-ins.


### Territory connection value analysis

High-leverage territories have high adjacency and act as choke points. Top examples on the classic map:

| Territory | Why it matters |
|---|---|
| **Ukraine** | Europe↔Asia hinge; a single break can collapse Europe. |
| **Middle East** | Connects Africa/Asia/Europe; staging for continent breaks. |
| **North Africa** | Connects South America to Europe/Africa; critical for Africa/South America dynamics. |
| **Siam** | Asia↔Australia choke; dictates whether Australia is ‘farmable’. |
| **Central America** | NA↔SA choke; controls whether SA is safe. |
| **Greenland / Iceland** | NA↔EU linkage; pressure valve in NA/EU conflicts. |

## 4) Key Strategic Principles

### When to attack vs. turtle

Attack when:
- Your key conquest is **≥75%** (or payoff is huge).
- You can end your turn with a defensible shape (borders not breakable).
- You are denying an opponent a continent or cash-in.

Turtle when:
- You’d be taking lots of <60% fights that only trade armies.
- You’re setting up a major cash-in / kill chain turn.


### Continent priority (probability-aware, typical table)

1) Australia, 2) South America, 3) North America, 4) Africa, 5) Europe, 6) Asia.


### Diplomatic considerations

- Visible strength attracts coalitions; sometimes the best ‘probability play’ is to look non-threatening.
- Stable borders reduce the number of serious break attempts you face, compounding your hold probability over turns.
- Players with 3–4 cards are elimination targets; don’t be one of them.


### Endgame

- Large sets enable **kill chains**: cash → eliminate → gain cards → cash again.
- In late game, **breaking** continents and denying sets is usually higher EV than holding small bonuses.