This week’s Hammer of Math explores the Gambits deck in the Leviathan Mission Deck from Warhammer 40k.

In addition to vastly simplifying how secondary objectives work with the Secondary Mission deck, 10th Edition also includes a completely new way to score Victory Points: Gambits. These add an interesting element to the game, giving players a kind of low probability “hail mary” play they can attempt in a last-ditch effort to win.

How it Works

At the end of the third battle round, players have the option to go for a high-risk alternative to the Primary Mission that was selected at the beginning of the game. Each player has access to a deck of three Gambits (the decks are identical), of which two are randomly drawn along with a “Proceed as Planned” option. If a player chooses Proceed as Planned, then they continue to score primary VP as normal. On the other hand if a player selects one of the Gambits, then they can no longer score from the Primary Mission. Instead, if they achieve the requirement specified on the card they score 30 VP. Otherwise they score nothing.

Each of the Gambits has a different effect, with different requirements and different ways of improving their odds of success.

Delaying Tactics

This one’s complicated. You start by determining a target number (the Distraction target). That number will be equal to half the number of enemy units that are within Engagement Range of one or more units from your army (rounding up) at the end of your fifth turn. If that number is less than 4, it becomes 4.

At the end of your fifth turn, you roll a D6 for every enemy unit within Engagement Range of one or more units form your army; on a 4+ that unit counts as being “delayed”. The roll has a +1 bonus if the target unit is Battle-shocked and -1 if one or more friendly units within Engagement Range of that unit are Battle-shocked.

If your number of delayed units (4+ rolls) meets or beats your target number, then the enemy unit has been successfully delayed. Note that this means it’s entirely possible to fail this Gambit if your opponent simply doesn’t have enough units for you to tie up in combat. The ideal result here is that your opponent has 8 units for you to tie up in melee, each of which ends up tied up in melee with your army.

Success in this case is driven by a binomial probability function. Assuming the most likely outcome of having four target units and a base 50% chance of success, 4 rolls will only have a 6.3% chance of getting at least 4 successes. 5 rolls is 18.7%, 6 is 34.3%, 7 is 50%, and 8 gives you the highest chance of success at 63.6%.

After that the odds get worse, though it becomes a bit of a rollercoaster, as that “rounding up” clause prevents you from getting the advantage and going above that 63.6% mark. At 9 rolls and a target of 5 success your odds drop to 50%, then rise to 62.3% for 10 rolls with 5 successes.  12 rolls needing 6 successes happens 61.3% of the time. In other words the chance of success stabilizes around 60%, but the most likely outcome due to attrition and battlefield circumstances is a vastly lower probability. You can improve this by getting Battleshock results, and if you truly live the dream and end up in combat with all eight of your opponent’s units at the end of turn 5 and every one of them is battle-shocked (and none of yours are), you’ll have a 91% chance of success.

Rob: This is easily the hardest Gambit to influence, as for starters you can’t control whether your opponent has enough units in their army to make this work. Tying every one of them up is also a massive hurdle and so the reality is that most of the time you’re looking at trying to even get four dice rolls so you can chase that 6% chance.

Emergency Evacuation

This Gambit is similar to Delaying Tactics in gives you a target number to beat – equal to half the number of units from your army on the battlefield at the end of the battle, rounding up. That is the number of units must meet a specific condition and successfully pass a D6 roll to evacuate from the battlefield. As with Delaying Tactics, this one also rounds up, and if your target number is less than 4, it becomes 4.

In this case the goal is to have friendly units be wholly within 6” of the center of the battlefield and pass a D6 roll on a 4+ (with a -1 penalty if the unit is Battle-shocked). The probability of success is identical to Delaying Tactics here, requiring at least 7 units be wholly within 6” of the center of the battlefield to have an even chance of getting four successes. This doesn’t include the challenge of physically fitting that many units wholly within the circle, which could prove particularly difficult for Knight players.

Rob: The only way you’re pulling this off is if you have a bunch of units with 1-3 models and your opponent has somehow managed to give up the middle while outscoring you on primary by flanking you on both sides.

Orbital Strike Coordinates

Unlike the other two Gambits, Orbital Strike Coordinates depends on the success of a single die roll at the end of the mission. When you select this Gambit, your goal is to have at least one unit wholly within 9” of a corner outside your deployment zone and not subject to the Battle-shocked condition. If this condition is met, then at the end of the game you make a single 2D6 roll. You add 1 to the roll for every other corner which has a friendly unit wholly within 9” (units that are Battle-shocked or within Engagement Range of any enemies do not count). If the final result is 12 or more, the Gambit is completed. The probability of this roll is easily calculated. With only a single corner, the probability of success is 2.8%. With two corners the probability is 8.3%, with three the probability is 16.7%, and at four the probability caps out at 27.8%.

Rob: This is going to be the most common Gambit chosen, if for no other reason than it always gives you a flat boxcars chance to succeed.

Some Early Stats

Rob: GW’s US Open in Tacoma gives us our first very large competitive event sample with Gambit data, though we’re still waiting on final stats. Current estimates based on BCP data put the number of games using Gambits at 8%, with 10% of those actually succeeding at achieving the Gambit once it was taken. That seems high, but it may just mean that players were smarter about picking Gambits when they were achievable. This means around 3-4 successful Gambits during the event (including one used against me), but it doesn’t mean an outcome was affected – I believe only a single game was determined by a successful Gambit over the weekend, but I think GW can confirm that one in their next Metawatch.

Wrapping Up

Unsurprisingly the Gambit cards really do represent a high stakes roll of the dice. From a purely mathematical standpoint Orbital Strike Coordinates represents the option with the lowest chance of success, where even the best possible conditions only provide a roughly 1 in 4 probability. To put that into context that’s less than the chance of getting 4 results of a 4+ on 6 D6 rolls. What makes Delaying Tactics and Emergency Evacuation interesting is how the probability of success trends towards 60% once the target number increases. That said, while they offer much higher potential odds for success on paper, they also make it likely you won’t be able to succeed at all. 

Whether or not a Gambit is worth it depends heavily on the condition of your army as well as your opponent’s, as well as your ability to maneuver into position to achieve the desired effect. Good luck.

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