The board game Risk generally favors an aggressive strategy, partly owing to the rules defining battle outcomes based on dice throws. However, small changes in these rules can lead to drastically different results.
Using classic Risk rules, the attacker has the advantage per roll if and only if the attacker throws more dice than the defender. Here's a table showing the probabilities per throw.
+-----+-------+-------+-------+-------+-------+-------+
| A/D | AVG | A1D0 | A0D1 | A2D0 | A1D1 | A0D2 |
+-----+-------+-------+-------+-------+-------+-------+
| 1/1 | 41.67 | 41.67 | 58.33 | | | |
| 1/2 | 25.46 | 25.46 | 74.54 | | | |
| 2/1 | 57.87 | 57.87 | 42.13 | | | |
| 2/2 | 38.97 | | | 22.76 | 32.41 | 44.83 |
| 3/1 | 65.97 | 65.97 | 34.03 | | | |
| 3/2 | 53.95 | | | 37.17 | 33.58 | 29.26 |
+-----+-------+-------+-------+-------+-------+-------+
The way to read the table is to first determine the row by taking the
number of dice the attacker throws and the number of dice the defender
throws. For example, if the attacker throws two dice and the defender
throws one die, then use the row with 2/1 in its A/D column.
In that row, the AVG cell specifies the average number of armies the
attacker will win per 100 army deaths. A bigger number is better for
the attacker. For example, if the AVG value is 57.87, then, over
the course of the attacker and defender losing a combined total of 100
armies, the defender will average losing 57.87 armies and attacker will
average losing the other 42.13 armies. As another example, an AVG
value of 75.00 suggests that the attack will lose about one army for
every three armies the defender loses.
The other columns specify the probabilities of specific outcomes,
expressed as percentages. For example, a value of 60.00 in the A1D0
column would mean that there is a 60% chance per roll that the attacker
will win one army and lose none. Whereas, a 70.00 value in the A1D1
column would mean that there is a 70% chance per roll that the attacker
and defender will each lose one army.
One simple change to Risk is to allow the attacker and/or defender to roll more dice. Here's a table showing probabilities for using as many as four dice per attacker and defender.
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| A/D | AVG | A1D0 | A0D1 | A2D0 | A1D1 | A0D2 | A3D0 | A2D1 | A1D2 | A0D3 | A4D0 | A3D1 | A2D2 | A1D3 | A0D4 |
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| 1/1 | 41.67 | 41.67 | 58.33 | | | | | | | | | | | | |
| 1/2 | 25.46 | 25.46 | 74.54 | | | | | | | | | | | | |
| 1/3 | 17.36 | 17.36 | 82.64 | | | | | | | | | | | | |
| 1/4 | 12.59 | 12.59 | 87.41 | | | | | | | | | | | | |
| 2/1 | 57.87 | 57.87 | 42.13 | | | | | | | | | | | | |
| 2/2 | 38.97 | | | 22.76 | 32.41 | 44.83 | | | | | | | | | |
| 2/3 | 25.33 | | | 12.59 | 25.48 | 61.93 | | | | | | | | | |
| 2/4 | 17.66 | | | 7.59 | 20.13 | 72.27 | | | | | | | | | |
| 3/1 | 65.97 | 65.97 | 34.03 | | | | | | | | | | | | |
| 3/2 | 53.95 | | | 37.17 | 33.58 | 29.26 | | | | | | | | | |
| 3/3 | 36.90 | | | | | | 13.76 | 21.47 | 26.47 | 38.30 | | | | | |
| 3/4 | 25.02 | | | | | | 7.33 | 14.84 | 23.41 | 54.42 | | | | | |
| 4/1 | 70.74 | 70.74 | 29.26 | | | | | | | | | | | | |
| 4/2 | 62.49 | | | 45.91 | 33.16 | 20.93 | | | | | | | | | |
| 4/3 | 50.84 | | | | | | 25.11 | 26.29 | 24.60 | 24.00 | | | | | |
| 4/4 | 35.21 | | | | | | | | | | 8.74 | 15.12 | 18.90 | 22.72 | 34.52 |
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
An interesting case is the 3/3 roll, which significantly disfavors the
attacker compared to the standard 3/2 roll, with the AVG down from
53.95 to 36.90. Of course, the defender would need at least
three armies in their territory to be eligible for this roll, but this
one difference could tip the equilibrium for the winning strategy from
aggressiveness to defensiveness. It's hard to say.
What about replacing Risk's standard six-sided dice with the dice from Betrayal at House on the Hill? In that game, the dice are six-sided but show only the three values 0, 1, and 2, with each number appearing on two faces. Here's the table, again calculated for allowing more dice.
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| A/D | AVG | A1D0 | A0D1 | A2D0 | A1D1 | A0D2 | A3D0 | A2D1 | A1D2 | A0D3 | A4D0 | A3D1 | A2D2 | A1D3 | A0D4 |
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| 1/1 | 33.33 | 33.33 | 66.67 | | | | | | | | | | | | |
| 1/2 | 18.52 | 18.52 | 81.48 | | | | | | | | | | | | |
| 1/3 | 11.11 | 11.11 | 88.89 | | | | | | | | | | | | |
| 1/4 | 7.00 | 7.00 | 93.00 | | | | | | | | | | | | |
| 2/1 | 48.15 | 48.15 | 51.85 | | | | | | | | | | | | |
| 2/2 | 28.40 | | | 13.58 | 29.63 | 56.79 | | | | | | | | | |
| 2/3 | 17.28 | | | 7.00 | 20.58 | 72.43 | | | | | | | | | |
| 2/4 | 10.84 | | | 3.70 | 14.27 | 82.03 | | | | | | | | | |
| 3/1 | 55.56 | 55.56 | 44.44 | | | | | | | | | | | | |
| 3/2 | 41.98 | | | 24.28 | 35.39 | 40.33 | | | | | | | | | |
| 3/3 | 25.10 | | | | | | 5.76 | 16.05 | 25.93 | 52.26 | | | | | |
| 3/4 | 16.26 | | | | | | 2.97 | 9.79 | 20.30 | 66.94 | | | | | |
| 4/1 | 59.67 | 59.67 | 40.33 | | | | | | | | | | | | |
| 4/2 | 50.34 | | | 31.14 | 38.41 | 30.45 | | | | | | | | | |
| 4/3 | 37.24 | | | | | | 12.21 | 23.59 | 27.89 | 36.31 | | | | | |
| 4/4 | 22.79 | | | | | | | | | | 2.48 | 8.78 | 15.73 | 23.41 | 49.60 |
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
These dice tip the odds in the defender's favor for the 3/2 roll, down from 53.95 in classic Risk to 41.98. In other words, the attacker is trading about three armies for the defender's two.
Again, I wonder if this is a difference that would change the winning strategy from offense to defense. Notice, however, that the 3/1 roll is still favorable for the attacker—though only slightly so—which may be important when the attacker is steamrolling through territories defended with only one army, as is common in the end game.
Let's replace the standard six-sided dice with two-sided dice—otherwise known as fair coins. Here's the table, calculated for allowing up to four coins per player.
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| A/D | AVG | A1D0 | A0D1 | A2D0 | A1D1 | A0D2 | A3D0 | A2D1 | A1D2 | A0D3 | A4D0 | A3D1 | A2D2 | A1D3 | A0D4 |
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| 1/1 | 25.00 | 25.00 | 75.00 | | | | | | | | | | | | |
| 1/2 | 12.50 | 12.50 | 87.50 | | | | | | | | | | | | |
| 1/3 | 6.25 | 6.25 | 93.75 | | | | | | | | | | | | |
| 1/4 | 3.12 | 3.12 | 96.88 | | | | | | | | | | | | |
| 2/1 | 37.50 | 37.50 | 62.50 | | | | | | | | | | | | |
| 2/2 | 18.75 | | | 6.25 | 25.00 | 68.75 | | | | | | | | | |
| 2/3 | 10.94 | | | 3.12 | 15.62 | 81.25 | | | | | | | | | |
| 2/4 | 6.25 | | | 1.56 | 9.38 | 89.06 | | | | | | | | | |
| 3/1 | 43.75 | 43.75 | 56.25 | | | | | | | | | | | | |
| 3/2 | 29.69 | | | 12.50 | 34.38 | 53.12 | | | | | | | | | |
| 3/3 | 15.62 | | | | | | 1.56 | 9.38 | 23.44 | 65.62 | | | | | |
| 3/4 | 9.90 | | | | | | 0.78 | 5.47 | 16.41 | 77.34 | | | | | |
| 4/1 | 46.88 | 46.88 | 53.12 | | | | | | | | | | | | |
| 4/2 | 37.50 | | | 17.19 | 40.62 | 42.19 | | | | | | | | | |
| 4/3 | 24.48 | | | | | | 3.91 | 16.41 | 28.91 | 50.78 | | | | | |
| 4/4 | 13.67 | | | | | | | | | | 0.39 | 3.12 | 10.94 | 21.88 | 63.67 |
+-----+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
This further exaggerates favor for the defender, decreasing the
AVG from 53.95 in classic Risk down to 29.69—roughly
equivalent to the attacker trading seven armies for every three from the
defender. Even the 3/1 case has an unfavorable AVG for the attacker,
at 43.75, making even a steamroll maneuver costly.
I struggle to imagine this scenario not leading to stalemate, with players amassing huge armies on their borders and no rational player willing to engage on offense. Hence, I dub this “Zapp Brannigan Risk,” where victory is possible only with overwhelming numbers and a willingness to send wave after wave of your men at the enemy.