Theory Thursday–Game Theory Storytelling: The Stag Hunt

TL;DR: Game Theory is not really about games, but it’s about interactions so you can mine it for good drama, and if you don’t care about more than that, you can jump right to it.

Game Theory (hereafter GT) is not really about games as such. It’s a mathematical theory of strategic interactions, and it analyzes them “as if” they were games. But sometimes, it borrows terminology from games-as-entertainment—e.g., coining a whole class of simultaneous zero-sum games “rock-paper-scissors games”—and that muddles things a bit.

Designing game mechanics based on GT is possible, but I don’t think it’s worth the hassle—with a few exceptions, Vickrey auctions being one. That’s not today’s topic, though. Today’s about mining game theory for situations rather than mechanics to present players with challenges and tough choices.

As usual, there’s a quick geeky summary that you can skip about what GT does and how to use it in TTRPGs, and some details about the game I’ve chosen (the Stag Hunt). Next comes the actionable part, where I give a recipe to turn that game into a story structure for TTRPGs.

The Geeky Stuff: Not the Prisoner’s Dilemma

The typical Game Theory approach takes a real-world situation, abstracts the details, then checks if it fits an “archetype game” where players have well-defined preferences and options, and where the optimal strategy for each player is known. The closer a real-world situation is to the archetype, the closer the real-world’s “best response” should be to its optimal strategy.

Game design should start the other way around: find an archetype game with compelling features and add fictional details until it looks like good drama—but still with the archetype game solution. The best-known archetype game is the Prisoner’s Dilemma (PD), but it’s also tricky. It already has a compelling story, and folks often read into that story stuff that’s not in the math—and if you put them in the math, that becomes another game. 

Fortunately, another example has a less misleading and more versatile story, which works better for storytelling and RPGs: the Stag Hunt. The simplest version is as below.

Two hunters share the same hunting grounds. They can hunt hare on their own and are skillful enough that they are almost guaranteed to catch one. Alternatively, they can cooperate to hunt a stag that none of them could catch alone. Still, if they agree to pursue the stag, any of them could break the agreement unilaterally and go after a hare at any moment, preventing the other from catching anything big.

That’s more interesting than PD because the Stag Hunt has two solutions (Nash Equilibria) while PD has only one. There’s a risk-dominant solution: “hunt hare,” less risky; and a payoff-dominant solution: “hunt stag,” more meat. If you know your GT, you can read (almost) all that off of Fig. 1 & 2.

(From Wikipedia)

From a philosophy standpoint, that’s a fantastic model for social contracts. And for evolutionary biology, that’s great to understand the selection of a local optimal over a global one. But even without looking too deep at GT, philosophy, or evolutionary biology, there’s potential for compelling storytelling. 

Stag Hunting for Drama

If you skipped the first part, here’s the gist of it: GT has a handful of archetype games. One of them is the Stag Hunt: a bunch of folks agrees to cooperate towards a high prize, but anybody can strike on their own at any time and still get something. The trouble is that a single person doing so takes the high prize off the table for everyone.

A Stag Hunting Recipe

If you know your GT, all you’d need for prep is a 2×2 matrix, some specific numbers, and a bit of imagination. Storytelling-wise, that’s a bit dry, so we need a narrative recipe to cook a Stag Hunt. First, with the three essential narrative ingredients.

  • There’s a master plan that requires coordination, and if everyone does their part, they strike rich at the end; it’s kinda risky, though.
  • There’s a backup plan for everyone that they could default to at any time. The payoff is lower, but it’s almost a sure thing.
  • Finally, there’s a catch: if someone defaults to their backup plan, the master plan can’t succeed.

Then, with the spices. You can go “fractal” with a backup plan for a sub-group—a stag hunt in the stag hunt, anyone? You can downplay or emphasize the risk factor to tempt PCs to defect. And you can switch communication on—although, and technically, that kinda changes the whole game (see The Fine Prints).

Risk is your main spice. Consider the original story: the hunters are incommunicado as soon as they’ve taken their stag-stalking post. No telling the other, “Know what, I don’t like our odds, I’m going after that hare instead.” So, that story would be about keeping one’s word and establishing trust beforehand. That’s especially true if you incorporate risk dominance (see The Fine Prints).

Keep the same story structure, add cell phones, walkie-talkies, or any real-time communication channel. Now, the PCs can communicate, warn others they’re about to backpedal or try to convince anyone tempted by the quick buck to stick to the plan. The focus shifts to real-time arguments and negotiations, but (again) changes the whole game.

As a general rule, you want to start with a Master Plan that doubles as the Catch. The simplest approach is to hand the PCs a turn-key plan with “falling dominos”: each PC has a specialized task (a domino), and if they bail, the whole operation is out. That works great when the PCs can get intel from an employer, a fixer, or a shot-caller (heists, military operations, etc.)

I’ll use a heist for an example, with a nod to the Fate RPG emplates I made a while ago. The method generalizes to any game with niche protection, from D&D to Apocalypse World

A Stag Hunt Adventure

The Master Plan: The team needs access to the main vault of a high-tech bank full of cash. Getting there requires turning off the central surveillance system (Infiltrator), taking out guards (Heavy), tricking the manager into opening the biometric gate (Smooth Operator), emptying the vault and carrying bags of money (everyone), escaping in a van camouflaged as an ambulance (Talent) and losing the cops in the traffic (Reserve).

The “falling dominos” should be obvious:

  1. Reinforcement will show up if the central surveillance system is not down.
  2. Even with the system down, if the guards are not dealt with, they can call for reinforcement.
  3. Without the manager, there’s no access to the vault.
  4. Without the ambulance there’s no vehicle with both sufficient storage room and inconspicuous appearce.
  5. The ambulance is too slow for a high-speed chase without a great driver and a well-planned escape route

Once you’ve laid down your dominos, make something up that could put every character off-track. The key is to make it easier to get, but not as good as, a share of the high prize, to preserve risk- and payoff dominance. Below are a few read-aloud to tempt the PCs in the above heist.

Infiltrator: “You could hack a backdoor in the bank server to siphon a small amount of money into your bank account every week. It would take a while to discover. Unless there’s a fine-comb system swipe. Like, after a big heist.”

Heavy: “There’s a low-security money transport with ETA during the heist. You’ll have to knock out the guard, so that they don’t mess up with your op. You could make a run with their haul. Probably not much, but a clean escape.”

Smooth Operator: “There’s a secondary high-security vault that the manager alone can open, with some prime blackmail material against an old rival. Too bad there’s notnough time to get there first, then make it to the main vault, within the time frame of the op.”

Talent: “You have to drive the ambulance to the bank’s parking lot, because your driver is double-checking the escape route. The bad news is, your little brother is in a pickle right now, but you won’t have time to help him and deliver the van.”

Reserve: “There’s this beauty of a racecar, parked right there, and it’s a one-in-a-lifetime opportunity because you just happen know the fault in *that* alarm system. There’s no time to steal it, drive it to a hiding place, and be back in time to get everyone out, though.”

I’d keep a communication channel between PCs if I ran this story because it’s a split-party game. But a communication black-out could do wonders too, particularly if compelling The Talent or The Reserve as last-minute twists, forcing everybody to execute their fall-back in a hurry.

The Fine Prints

Communication. The original Stag Hunt is a coordination game: agents act simultaneously without real-time communication. Real-time communication trivializes most coordination games, and in the Stag Hunt, changes the game entirely. Although the stag-or-hare dilemma is still there, the focus is on keeping everybody on the course: it becomes a negotiation game. Also, with communication, the existence of defectors is common knowledge, so everybody ends up defecting, and the second-best payoff becomes inaccessible. The heist example does not lean on that aspect or on risk dominance in general (see next fine prints) but if you want to add it, be aware of the effect of communication.

Risk DominanceIn the Stag Hunt, a hare hunter is better off the fewer hunters there are stomping around and frightening the game—hence the numbers in Fig.2. Risk dominance is, in practice, equivalent to defectors having an incentive not to warn others (in theory, it’s a bit more complicated). The Heist example doesn’t have that, so it’s not “really” a Stag Hunt: it’s an “Assurance Game” where b=d (but communication makes the difference irrelevant, and “An Assurance Game Adventure” sounds lame). You can add risk dominance: tweak the backup plans so the odds are better if fewer are running around trying to pick up scraps. Combine with communication for drama, with an incentive to keep non-defectors in the dark.

Wrapping Up: Stag Hunt Drama

The Stag Hunt’s potential for good drama shouldn’t escape anyone who’s run any TTRPG whatsoever. Just the narrative core and the falling domino method can get you going for a while. If you like fiddling with game mechanics, there’s potential for that, too (but that would be a topic for another day).

The Stag Hunt is about building trust and keeping one’s word. Philosophers often think the Prisoner’s Dilemma is about that too, but they’re mistaken: the way they tell the story transforms it into a Stag Hunt, and they forget to feed that back into the payoff matrix. Don’t fall into that trap.

Other games emphasize other aspects and may suggest different narrative structures. Coordination games, in general, are great, especially those where reaching a solution is tricky (e.g. the Battle of the Sexes). Anti-coordination games are cool, too, especially for action-packed adventures (Hawk-Dove).

I may return to that eventually, but that’s all for today, folks.

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