Game Theory

Game Theory Parables for Game Designers

The Stag Hunt

Here is a situation from philosopher Jean-Jacques Rousseau's Discourse on Inequality:

“If it was a matter of hunting a deer, everyone well realized that he must remain faithful to his post; but if a hare happened to pass within reach of one of them, we cannot doubt that he would have gone off in pursuit of it without scruple”

The idea is that it takes two people working together to successfully hunt a stag, but one hunter on their own can catch a single hare.  The stag, though, is a far more valuable prize, but it takes cooperation to obtain it.  There’s a risk, then, that the other player won’t cooperate.

Each player has a choice: do their part to hunt Stag, or bail to hunt Hare.  If they hunt Hare, they’ll get a middling reward (say 1 food) guaranteed.  If they hunt Stag, they’ll get a better reward (3 food) but only if the other player also hunts Stag (otherwise they’ll get nothing).

There’s an element of cooperation here, an element of positive player interaction, but also an element of risk.  Hunting Stag only pays off if the other player hunts Stag as well, whereas hunting Hare always pays off.  Unlike the Prisoner's Dilemma, there is an opportunity for rational players to cooperate here.

Cautious players, who may not trust the other to hunt Stag, will end up hunting Hare.  It provides a guaranteed payoff, but not the best possible payoff for those players.  What is needed to get the players to trust each other enough to both hunt Stag?

One way to encourage Stag hunting is to make it a Focal Point.  This doesn’t change the game at all, but it does make one choice more appealing than the other.  It draws players' attention to a solution that seems natural, special, or relevant.  The hope is that players will choose it when they have to choose their strategies independently.  Focal points can, if successful, eliminate the need for trust, for communication, or for binding agreements. 

If your game was called Stag Hunting, or somehow prominently featured the act of hunting Stag (as Lowlands does with sheep, for example), this could create a focal point.  Likewise, prominent art could create a focal point, as could board layout or flavour text. 

You could also find a way to give players the ability to create focal points, even if you do not want them to change the game itself.  Can they signal their intentions?  Pick a theme?  Creating focal points is all about creating beliefs in the minds of players, and you need to think outside of the box to create focal points without changing the game itself. 

Creating a repeated game environment, where players play the Stag Hunt over and over against randomly chosen opponents (whether it be multiple times in the same game, or multiple different games, as in a tournament environment) could potentially make players more likely to hunt Stag.  Unfortunately for Stag hunters, the prospects are bleak.  In replicator dynamics simulations of the Stag Hunt, populations of players do not, indeed can not, transition from Hare hunting to Stag hunting.  Even if some element of chance variation or mutation is added, Stag hunting becomes possible but not stable: the population can just as easily be displaced back to Hare hunting.  Only if the population is excessively skewed towards Stag hunters (on the order of 3:1 Stag hunters to Hare hunters) will the whole population eventually settle in to Stag hunting.  Getting to this ratio via mutation or experimentation is incredibly unlikely.

There is a mechanism to avoid this inevitability, though.  If players in this repeated game are able to adjust with whom they interact with, then Stag hunters learn to interact with Stag hunters, and Hare hunters learn to interact with other Hare hunters.  Depending on the details of the game, and the environment within which the players play, this can lead to a population all hunting Stag.  As Game Designers, think of ways to allow your players to signal their previous plays, whether that be within the same game or across multiple games, and allow for them to choose who they interact with.

Stag Hunts appear in other interesting social interactions, most notably in the adoption of a new convention, or social contract.  Imagine a society that is doing things one way, and is considering switching to doing things another, better way.  Their “status quo” is the safe, risk dominant Hare hunting equilibrium, and adopting the new convention is the Stag hunting equilibrium — it only pays off if everyone does it.  For a modern example of this, think of the dilemma of switching to a new social media platform.  It is only worthwhile if everyone (well, everyone that matters to you) moves over; otherwise your user-base is fragmented and the new platform dies.  Staying put on the current platform is the safe, Hare hunting equilibrium.

Can the players trust every other player to switch to the new convention, to take the risk and hunt Stag?  It all depends on the structure of the interaction.  How are players paired together?  How large is the population?  Is there a mechanism for learning from past behaviour, changing strategies, and selecting with whom they play?  These considerations are why adopting such conventions is so difficult, and this difficulty looks ripe for exploitation by Game Designers.

Sam HillierComment