Understanding the Stag Hunt Game: How Deer Hunting Explains Why People are Socially Late
Every Tuesday is a Game Theory article at Mind Your Decisions
It’s customary for people to arrive “socially late” to parties. In my social circles, this means arriving an hour after the stated invitation time.
But customary does not mean correct. The practice of being socially late creates inefficiency only a government bureaucrat would be proud of. Arriving late means that people spend less time with each other, and punctual people get punished.
Solving this problem is a snap: have everyone show up on time. Everyone would benefit and it’s virtually costless to implement. But change is hard to come by. I’ve discussed this with many people, and no one has a compelling answer.
But I think I’ve come up with one. The reason is that arriving socially late, though sub-optimal, is actually a safer decision than showing up on time. The reasoning comes out of a famous game about hunting for stag (adult deer).
The stag-hunt or assurance game
Image credit: public domain via wikipedia
The stag hunt game is based on a discussion from the philosopher Jean-Jacques Rousseau. It’s a game about social cooperation.
Imagine you and a companion are out hunting. Each of you has the choice of pursuing a rabbit or a stag.
The stag is the bigger prize and tastier, and can be caught for sure if you both choose to pursue it. The rabbit is smaller but still satisfying, and can be captured for sure regardless of what the other person does.
Before you start hunting, you can discuss strategy with your companion, but the game is about individual survival. You can’t be sure your companion will follow through on any agreement.
What choice do you make?
In the next two subsections, I write out the formal version of the game. If that doesn’t interest you, you can skip to the section labeled What’s going to happen?
The diagram
To get the analysis fixed, I’ll write out payoffs that are consistent with the game description.
Let’s assume capturing a rabbit gives a payoff of 3, capturing the stag gives a payoff of 5 to each person, and capturing nothing is a payoff of 0.
By nature of the game, if a player pursues the rabbit, he’s guaranteed a payoff of 3.
On the other hand, if a player pursues the stag, the payoff depends on the other person’s choice. If the other person also chooses stag, then the stag is captured and each gets a payoff of 5. If the other person chooses rabbit instead, then the player captures nothing and gets a payoff of 0.
Here is the matrix for the game:
The solution
The game can be solved by looking for the best responses. For each choice the other person might make, consider what’s best for you. A Nash equilibrium occurs when both players are picking best responses.
What are the best responses? There are two choices to consider.
First, consider if the other person picked stag. In that case, it makes sense to pick stag (5) over rabbit (3).
Second, consider if the other person picked rabbit. Now, it is more sensible to pick rabbit (3) rather than stag (0).
The best responses for each player are:
–Rabbit is a best response to rabbit
–Stag is a best response to stag
This leads us to two Nash equilibriums in pure strategies (no mixing): both picking stag and both picking rabbit.
What’s going to happen?
The above analysis means there are two reasonable outcomes. It is possible both players go for rabbit, or both players go for stag.
This is a comforting solution as it demonstrates selfish incentives can produce social cooperation. Because the stag is a large prize, it’s possible both players will cooperate and achieve it. In fact, this outcome is the best–each player can be made better than the rabbit outcome. Hence, the stag outcome is said to be Pareto optimal.
But is there something wrong with this outcome? On closer inspection, you might realize the stag equilibrium is risky.
If you pick stag, and the other person does not match you, you end up with nothing. If you were a real life hunter a few hundred years ago, you might feel embarrassed. You would have to go home to your family and explain that you had a chance to bring home rabbit and feed everyone, but you instead were going for the big prize and failed. And the reason everyone is starving, you would suggest, is that your partner was stupid. I imagine such answers were the source of many domestic arguments.
The rabbit equilibrium is less risky, and in this particular story, it has no risk. By choosing rabbit, you are guaranteed a tasty meal and a payoff of 3, regardless of what he other person does.
This is why the rabbit equilibrium is called risk dominant. Although it has lower payoffs to each party than stag, picking rabbit might make sense because it is the “safe” option.
The party arrival time game
Should you show up on time or socially late to a party?
If you show up socially late, then you are likely to arrive when other guests will be there for sure. This means you are guaranteed a payoff.
On the other hand, if you show up on time, your payoff depends on what others do. If others show up on time, then you all get to spend lots of time with each other. If others show up late, however, then you might get annoyed and feel like the party is a wash.
Under these considerations, the payoffs to this game are exactly the same as the stag-hunt game, with the strategies having different labels:
So although everyone would prefer to show up on time and have the big payoff, it is a risky equilibrium. Hence, everyone ends up being socially late.
The exact rules and payoffs are up for debate. But the idea is the party arrival game is like the stag-hunt game, and showing up socially late is a safe option.
Some workarounds
As I’ve written about before, if you aren’t winning, then try to change the game.
Not all parties start wastefully, and the reason is people have figured out ways to change the incentives. The solutions are all about creating commitment and incentives for on-time arrivals.
On that note, here are some common ways to induce punctuality:
–Throw a surprise party
No one wants to miss the birthday person’s reaction at the start. And people that do lose out.
–Get a reputation for being short on food/drinks
I don’t like this one, but some people do it on purpose. Scott Adams toys with the idea in one of his Dilbert newsletters.
–Go to a club that charges cover after a certain time
This isn’t a perfect solution, as lots of people are willing to shell out money. I don’t quite understand it, but the people who show up late have to pay, so it’s their choice.
–Buy tickets for a fixed movie time or event in advance
When I’m in a big group, the benefit to committing to a specific time outweighs the small convenience fee, if any. Often, it’s possible to buy tickets at the theater a day or so in advance and avoid the charge.
How do you manage tardy guests? How do you make punctuality the safe and better option?
10 Responses to “Understanding the Stag Hunt Game: How Deer Hunting Explains Why People are Socially Late”
Oddly enough, I just *count* on people to be late, recently I cooked dinner for some people and was very surprised when one of them showed up on time. The other guests were about 10 minutes late, as expected.
Since I still cooking / cleaning when the first person showed up, she either had to help me or wait around till I was done.
While it might seem silly to penalize people for showing up on time, it probably means they won’t be punctual next time, meaning I can safely throw a party starting at 7pm, but really expect it to start around 7:30pm.
For those who wish to show up really late, I like to throw in the added variable that the party can change location and people can be hard to get a hold off in a loud bar.
By RohoMech on Jun 3, 2008
It’s kind of like the prisoner’s dilemma, but if both hunters decide to shoot a rabbit, everybody wins but to a lesser degree.
On a vaguely related note, eating rabbit is not very nutritious and can result in rabbit starvation, which gives this whole idea a sinister edge.
By Kyle Johnson on Jun 4, 2008
Kyle Are stags *more* nutritions? I feel like either way you’d be missing out on vital things.
By RohoMech on Jun 4, 2008
Kyle Johnson: You’re right this is like the Prisoner’s dilemma, except that the likely outcomes are both cooperative. It’s just that risk prevents getting the best outcome.
Didn’t realize rabbits aren’t nutritious. I’ve heard it’s a common food in some European countries.
By Presh Talwalkar on Jun 4, 2008
Presh,
Each time I read your game theory post I get increasingly intrigued by the field. Is there a not-so-technical introduction to this fascinating field suitable for non-economists?
Thank you,
Pankaj
By Pankaj on Jun 5, 2008
Pankaj: This is a question that is hard to answer because many game theory books are technical, and that’s how I learned it.
But I have been reviewing the field more and recently come across two books. I have read them and would recommend them as non-technical books.
1. A Beautiful Math
This book covers how game theory is being applied all across science. It is full of interesting stories.
Beautiful Math on Amazon
2. Game Theory at Work
This book has plenty of economic and business applications. It reads like Machiavelli’s “The Prince”–short and to the point.
Game Theory at Work on Amazon
There is also a book I’ve had on my list for months called “Thinking Strategically.” Apparently MBAs read this book. The excerpt on Amazon makes me excited to read it, and I’ll have more info when I do read it.
(The authors are famous economists, too)
Thinking Strategically at Amazon
By Presh Talwalkar on Jun 5, 2008
Reading this, and I cant help thinking about how the “super delegates” were “late” to endorse Obama. If everyone was “early” then the party saves some cash, saves some mudslinging etc… (score of 5) if everyone was late, then its kind of ok (score of 3) because they can cast a safe vote for Obama, and if you were an “early” super delegate, and no one else was early, you lose points because you didnt “give clinton a chance” and she might of won, and the super delegate would look bad…
No wonder the whole thing lasted so long
Late was the safe choice.
I’m sure the reality is much more complicated, but your simple story makes real sense to me. Thanks!
By Ibrahim on Jun 6, 2008
Ibrahim: You raise an excellent political analogy. I like it because politicians are self-centered, calculating machines–which is why they are very good at demonstrating game theory outcomes. Nice one.
By Presh Talwalkar on Jun 6, 2008
@Pankaj
There are three or four books which you ought to read.
1. Presh’s suggestion of “Think Strategically” by Dixit and Nalebuff is an excellent introduction.
2. For more business applications, read “Co-opetition” by Adam Brandenburger and Barry Nalebuff.
After that there a number of good skills to acquire by reading.
1. Strategy of Conflict or Micromotives and Macrobehavior by Thomas Schelling.
2. Introduction to Game Theory, a good text by Dixit.
3. And for something harder mathematically, Ken Binmore’s Fun and Games.
I am not a big fan of Miller’s Game Theory at Work.
The Stag and Hare game makes more sense when thought of as n-person game - which is implicit in Presh’s post.
Schelling has an excellent discussion of generalizations of these type of dilemma games in Micro and Macro.
Suffice to say that the interesting property he discovers is the existence of a group of people large enough to coordinate together and get the benefits from not being in the state of nature, but that this large group often will attract free-riders that the group might just have to leave with in order to stay out of the state of nature. (Pretty theoretical political science post, I apologize in advance.)
By Michael Webster on Jun 11, 2008
Michael Webster: Thanks for the additional book recommendations.
Schelling does an excellent job of putting these games into a social context, as you point out.
By Presh Talwalkar on Jun 11, 2008