
'Snowdrift' game tops 'Prisoner's Dilemma' in explaining cooperation (2007) - MichaelAO
https://phys.org/news/2007-10-snowdrift-game-tops-prisoner-dilemma.html
======
jzl
I've seen a couple of oblique references to this in other comments, but I'll
go a little deeper: if you want to truly understand the prisoner's dilemma
scenario and how it relates to human, or more generally, animal behavior,
there is incredible coverage of the topic in Richard Dawkins' The Selfish
Gene. He pioneered the idea of running PD "competitions" in an attempt to
discover the best evolutionary strategy for PD, and how these strategies might
predict large-scale behavior in animal populations. Each competitor (other
scientists and researchers) submitted a strategy for handling the PD in the
form of written code that could face off against other strategies. Each round
of competition consisted of thousands of iterations of the game. Some
strategies were simple and some were incredibly complex. Across two
competitions, the second with 60 competing strategies, the simple and "non-
cynical" strategy called tit-for-tat won handily. (Tit-for-tat is mentioned
briefly in the original article without any explanation.) This is the
strategy:

\- Unless provoked, the agent will always cooperate.

\- If provoked, the agent will retaliate.

\- The agent is quick to forgive.

\- The agent must have a good chance of competing against the opponent more
than once.

The fact that this strategy was a consistent winner in his competitions has
led Dawkins to argue that "nice guys finish first" and could be a partial
explanation for certain forms of so-called "altruistic" animal behavior --
behavior that seemingly gives no benefit to the altruist, except for the fact
that too may benefit when it is their turn to be helped. See also:

[https://en.wikipedia.org/wiki/Nice_Guys_Finish_First](https://en.wikipedia.org/wiki/Nice_Guys_Finish_First)

~~~
kbenson
> \- The agent must have a good chance of competing against the opponent more
> than once.

I was actually thinking about this the other day. I was wondering how much the
population size and matching algorithm in the iterated Prisoner's Dilemma
matter to the strategy outcome. I suspect it's quite a lot, given that if you
never play the same opponent again, strategies that defect often should come
out quite well, but if you always play the same opponent, strategies that
cooperate should do well. If grouping is bought into the system, such that
there are multiple levels of subgroups with the smaller the group both are
members in having higher chances of replaying in the same round, you might see
interesting emergent behavior.

I also wonder if that might allow for special elements (the "sociopath") to
have interesting divergent strategies that work well as long as they don't
achieve more than a small fraction of the total population and keep a
relatively large amount of interactions to strangers.

~~~
theobon
[http://ncase.me/trust/](http://ncase.me/trust/) is a great explainer that
explores that question. How do different strategies compare depending on
number of repeat interactions, population diversity and noise.

~~~
jonahx
I really like this, but had to stop halfway through because my laptop
overheated and went to sleep.

~~~
kbenson
That seems a bit extreme for a browser app. Are you using a docking station,
and have your laptop closed? I find mine sometimes overheats and shuts down
when it's docked and closed and I'm doing something extreme like playing a
game, but propping the lid open about an inch or two helped immensely with
airflow.

~~~
jonahx
No, just a 5 yo macbook pro.

------
PeachPlum
See also

[https://www.sciencedaily.com/releases/2016/09/160915085719.h...](https://www.sciencedaily.com/releases/2016/09/160915085719.htm)

Two people can hunt deer together, but if they are alone, they can only hunt
rabbits. The person belonging to the Envious group will choose to hunt rabbits
because he or she will be at least equal to the other hunter, or maybe even
better; the Optimist will choose to hunt deer because that is the best option
for both hunters; the Pessimist will go for rabbits because that way he or she
is sure to catch something; and the hunter who belongs to the Trusting group
will cooperate and choose to hunt deer, without a second thought.

The largest group, accounting for 30%, being the Envious -- those who don't
actually mind what they achieve, as long as they're better than everyone else;

Next are the Optimists -- who believe that they and their partner will make
the best choice for both of them -- on 20%.

Also on 20% are the Pessimists -- who select the option which they see as the
lesser of two evils -- and the Trusting group -- who are born collaborators
and who will always cooperate and who don't really mind if they win or lose.

There is a fifth, undefined group, representing 10%,

~~~
arthurcolle
I'm at the edge of my seat!

~~~
damnfine
There are two types of people; those that can extrapolate for incomplete data.

------
lisper
Note that the SD is not actually a dilemma. The PD is a dilemma because you
ALWAYS do better for yourself in any given iteration by defecting no matter
what your opponent does. In the SD, if your opponent defects, you still do
better for yourself by cooperating. So of course people will cooperate more
under those circumstances. The only thing left to argue about is what the
payoff matrix for real-world situations looks like.

~~~
quadrangle
You're correct in most of your description, but it's STILL a dilemma. It's a
game of chicken. The dilemma is wanting to wait to see what the other player
decides, so the dilemma is everyone refusing to decide.

[https://en.wikipedia.org/wiki/Snowdrift_game](https://en.wikipedia.org/wiki/Snowdrift_game)
(redirects to chicken game)

~~~
lisper
You're right. In fact, upon reflection, it is the PD which is not really a
dilemma but rather a paradox. The logically correct move in PD (assuming a
purely utilitarian quality metric) is always clear: if the game is non-
iterated (or iterated with a known horizon) then defect. Otherwise play tit-
for-tat. This is true no matter what your opponent does. In the SD the
logically correct move depends on your prior on what your opponent is likely
to do.

~~~
kobeya
That resolution only holds if you assume the other player to be equally
rational in the PD.

~~~
Smaug123
In fact the strategy is simpler if the other player is exactly as rational as
you: simply cooperate. The other player, being exactly as rational as you,
will follow the same reasoning as you do to come to the same conclusion, and
will do exactly the same thing as you. I would always cooperate with a copy of
myself, for instance.

[https://en.wikipedia.org/wiki/Superrationality](https://en.wikipedia.org/wiki/Superrationality)

(Of course, the assumption "my opponent is exactly as rational as I am" never
holds in real life.)

~~~
chriswarbo
There _are_ variants of the prisoner's dilemma where you can check the
'rationality' of your opponent: the prisoners' decisions are chosen by
computer programs, and each program is given the other program's source code
as an argument. This is often referred to as "program equilibrium", and an
optimal strategy is to cooperate iff running the other program with our own
source code as an argument results in cooperation.

This runs into problems of computability though: if we run the opponent's code
to see what it does, and it turns out that _their_ program runs _our_ code to
see what _we_ do, we can get stuck in an infinite loop.

Like most incomputable problems, we can take a conservative approach: try to
prove that it cooperates, using some incomplete computable algorithm; defect
if we can't prove it. One simple algorithm is to check if the source code is
the same as ours; more sophisticated algorithms would perform static analysis
of the code.

~~~
Smaug123
I never really got my head around the Parametric Bounded Löb paper at
[https://arxiv.org/abs/1602.04184](https://arxiv.org/abs/1602.04184) . Clever
people have persuaded me that it was wrong (for a certain reason related to
the existence of nonstandard models of PA) and that it was true.

------
whack
TLDR: In Prisoner's Dilemma, defecting is a dominant strategy, and both
players defecting is the only Nash equilibrium. In Snowdrift, there is no
single dominant strategy, and no Nash Equilibrium exists at all.

By design, the Snowdrift game rewards cooperation more than Prisoner's
Dilemma, so it's no surprise that people cooperate more in Snowdrift. The real
question is which game better models reality.

[http://www.gametheory.net/dictionary/DominantStrategy.html](http://www.gametheory.net/dictionary/DominantStrategy.html)

[https://en.wikipedia.org/wiki/Nash_equilibrium](https://en.wikipedia.org/wiki/Nash_equilibrium)

~~~
throwa_way_
You are wrong. There do exist Nash equilibria in the game [0].

[0][https://en.wikipedia.org/wiki/Chicken_(game)#Best_response_m...](https://en.wikipedia.org/wiki/Chicken_\(game\)#Best_response_mapping_and_Nash_equilibria)

~~~
whack
You're right, my bad.

------
drvinceknight
The Axelrod library is a Python implementation of the Prisoner's dilemma
(tournaments, evolutionary dynamics etc) with over 200 strategies. It can also
be used to study other games
([http://axelrod.readthedocs.io/en/stable/tutorials/advanced/g...](http://axelrod.readthedocs.io/en/stable/tutorials/advanced/games.html))
so it would be straightforward to study the snowdrift game using it :)

Re some of the other comments about Robert Axelrod's tournament and evolution:

\- As part of the project, the Axelrod library developers have written Python
wrappers for the original Fortran code ([https://github.com/Axelrod-
Python/axelrod-fortran](https://github.com/Axelrod-Python/axelrod-fortran))

\- Have used reinforcement learning to train strategies based on machine
learning paradigms
([https://arxiv.org/abs/1707.06307](https://arxiv.org/abs/1707.06307)). EDIT:
here are a couple of blog posts about recent research work
[http://vknight.org/unpeudemath/math/2017/07/28/sophisticated...](http://vknight.org/unpeudemath/math/2017/07/28/sophisticated-
ipd-strategies-beat-simple-ones.html) and
[http://marcharper.codes/2017-07-31/axelrod.html](http://marcharper.codes/2017-07-31/axelrod.html)

Disclaimer: I'm one of the maintainers of the project.

------
quadrangle
[https://snowdrift.coop](https://snowdrift.coop) is an in-progress fundraising
platform for public goods (particularly free/libre/open software and cultural
works) based specifically on addressing the cooperation needed to solve the
snowdrift dilemma…

[https://wiki.snowdrift.coop/about/snowdrift-
dilemma](https://wiki.snowdrift.coop/about/snowdrift-dilemma) has more
specifically and
[https://wiki.snowdrift.coop/about](https://wiki.snowdrift.coop/about) covers
the concepts for the whole project.

------
scj
I just invented a different snowdrift game, modeled on something that happened
to me a couple of years ago.

You live in a townhouse next to another family that shares a driveway which
turns onto a medium traffic density one-way street (there are traffic lights a
few hundred feet away which control flow). Gaps in traffic tend to exist
around when the lights change.

The frequency of snowstorms this year is higher than normal, and the
overworked snowplows have made a snow bank too tall and wide to see around in
your car. The only way onto the street is to turn blindly and hope that either
no one is coming, or that they see you in time to stop.

Your salary is $x/hour, and it would take two hours to manually trim the
snowbank to the point where you can see above it. Thus, it would cost $2x to
remove the snowbank (assuming you could work overtime and earn the money).

Each party makes a choice if they would shovel the bank alone, with help, or
not at all. If they both do it, they both lose $1x in OT wages. If one does it
alone, $2x. But if neither do it, the hazard remains and each party rolls a
6-sided die. Anybody rolling a 1 loses $kx (where k is larger than x), and no
longer have a car to participate with.

That covers one day. It will take 7 days for the city to remove the snow bank
if no one else does. Any parties with a car will repeat this 6 additional
times until the snow bank is removed.

Then rank the dollars lost and award real world prizes for the top positions.
If there are 10 people and the top 5 get the exact same prize, shoveling might
become attractive (through co-operation, as it is extremely likely at least
one of the two parties per trial will be eliminated).

~~~
rhizome
Is there an option to install a fisheye mirror so people in the driveway can
see perpendicularly down the street?

[https://www.reflectionproducts.com/images/View-
car-200-feet-...](https://www.reflectionproducts.com/images/View-car-200-feet-
half.jpg)

~~~
jerf
First, the point of these discussions is explicitly to stay within the lines.
We all know there's a billion elaborations that you could make, but those are
different problems. Worthy and interesting problems, even, but different
problems.

Second, there isn't always a great option for that, no.

~~~
vacri
'Staying within the lines' is why game theory doesn't actually work as a good
predictor of human behaviour.

~~~
rwnspace
Not when applied without wisdom, but it's an excellent predictor of subsets of
human behaviour, and a subset of those generalise well enough to
satisfactorily explain existing phenomena. Simply because one has to be
selective doesn't make it unworkable. It does make for misleading headlines,
though.

------
jamie_ca
I came across an interactive applet last month (on Reddit maybe?) at
[http://ncase.me/trust/](http://ncase.me/trust/)

It explores IPD with a few different strategies, and I found it quite
interesting. I'd be curious to see how the various mechanics of it would work
with ISD.

------
jdonaldson
As mentioned elsewhere in this thread, the payoff for these games never takes
into consideration reputation effects. That is, what's the result of other
participants observing your actions? Are they more likely to trust you or not
trust you when you interact?

I mention this because it's clear that we're changing the way we trust and
interact with each other online. Whether it's virtue signaling or trolling,
we're adopting new strategies for "winning"... whatever that prize may be.

~~~
jzl
See my other comment. In the competitions discussed in The Selfish Gene, PD
strategies are allowed to have memory and the simulation is run thousands of
times per round. However it is pretty consistently the case that long-term
memory is not an outright win for either the competitor or the opponent,
compared to the simple "tit-for-tat" strategy which only remembers the
previous iteration.

Of course the PD only models certain kinds of human/animal interactions.
Knowing that a particular person is a cheater or is vindictive or whatever is
obviously important for evolutionary strategy as a whole. But the question is
more about heterogenous populations. Is there an optimal strategy that wins
more than any other across a wide range of opposing strategies, including if
everyone else is actually using the same strategy?

------
Danihan
The benefits of cooperation can't be modeled so simplistically, since the
concepts of both long-term reputation and long-term retribution need to be
taken into account.

~~~
AznHisoka
Sometimes I wish life was just like that Black Mirror episode with reputation
scores for everybody. Nobody wants to cooperate because there is hardly any
long term reputation to worry about when interacting with most people (ie
cutting off that car in the highway, being rude to that stranger next to you)

~~~
Danihan
There are reputation scores for everyone: net worth and credit score. ;p

~~~
majewsky
A good reputation score would start out at +/\- 0 and not punish you for being
the lone righteous guy in an otherwise immoral neighborhood.

~~~
Danihan
It does, no one really has a net worth one way or another when they are a
child.

Define righteous? Define immoral?

Capitalism rewards providing actual value to people. It doesn't reward shallow
niceties and self-righteousness, those are common.

~~~
majewsky
> Define righteous? Define immoral?

The point here was that credit scores do not only rate your own behavior, but
also those of people around you. If your neighborhood is bad, your credit
score might reflect that instead of your own performance as a debtor. And if
your neighborhood becomes hip and gains a better reputation, your credit score
might improve even though you're still the same person.

~~~
Danihan
Could you send me something reflecting why you think credit scores are based
on the neighborhood in which you reside?

~~~
majewsky
I live in Germany, where the zip code is a strong factor into the credit
score. Source e.g.
[https://www.welt.de/finanzen/verbraucher/article13255357/Wer...](https://www.welt.de/finanzen/verbraucher/article13255357/Wer-
am-falschen-Ort-wohnt-bekommt-keinen-Kredit.html)

~~~
Danihan
Thank you. Yes, I agree, that's pretty unfair.

------
glenstein
I don't want to be gratuitously negative here. So let's start with the fact
that it's great to have a model of cooperation as something out there to be
improved upon, or refuted, or defaulted to in the absence of obviously better
models.

However, if the idea is that a model is good at "explaining" cooperation by
setting up a hypothetical where people choose it a lot, then I guess the
"best" model would make people choose cooperate all the time.

So you can set up the points for cooperation to be infinity and the points for
all non-cooperation to negative infinity. And you can wrap it all up in a
hypothetical story about what the choices mean. Now you have a perfect model
where people choose cooperate 100% of the time. But I don't feel like I'm any
closer to understanding how cooperation evolved in humans.

~~~
TheSpiceIsLife
I despise these contrived experiments and think they should be ridiculed.

Cooperation evolved in humans because:

a. Our young are helpless, and take over a decade to reach sexual maturity.
Those who cooperate to raise the young have more descendants.

b. Cooperation is a sexual fitness signal because of point a

The end.

Edit: fix a word, thanks.

~~~
burkaman
These experiments are about trying to explain what you mean by "cooperate".
Your points feel right, but even if they're true they aren't very helpful in
actually explaining behavior. For example, your explanation seems to directly
apply to adoption, but most people don't choose to adopt despite the enormous
number of orphans in the world. Why not?

~~~
TheSpiceIsLife
> _These experiments are about trying to explain what you mean by
> "cooperate"._

It's not clear to me what you mean by this.

In the example of the experiment in the example _cooperate_ means the snow is
cleared faster.

In my example _cooperate_ means raise the young together.

> _but most people don 't choose to adopt_

Because our biology drives us to want to have our own offspring. It seems
obvious that if our biology drove us to want to raise other people's offspring
we wouldn't be having this conversation. Although this has been an
extraordinarily successful strategy for dogs and cats.

~~~
burkaman
> Because our biology drives us to want to have our own offspring.

But just a few minutes ago you said our biology drives us to "cooperate to
raise the young." Do you see how this topic might deserve some more study?
Basically, my only issue with your comment is "The end."

------
guiambros
You can't talk about explaining Prisoner's Dilemma these days without
mentioning Nicky Case's excellent "Evolution of Trust". My 5yo simply _loves_
the game.

Btw, if you are into data viz or simply enjoy Bret Victor-inspired interactive
programming, Nicky has tons of similar experiments worth checking out [2].

[1] [http://ncase.me/trust/](http://ncase.me/trust/)

[2] [http://ncase.me/](http://ncase.me/)

------
xaa
I get why game theory is an attractive model to explain human behavior. But I
always wonder why they invent the game, and try to explain behavior in terms
of a particular model, rather than using real behavior to fit model
parameters, so you could get (pseudo-) empirical numbers for "payoffs" etc.

Or maybe another way of putting it is that clearly rational and human behavior
varies with the payoff structure, so it would make sense to include that as
another variable.

------
Nomentatus
Uhhh... what made PD important was that it was, precisely, the extreme of non-
cooperation; a situation where cooperation was logically possible, but not
rational. EVERY other game where cooperation is at least possible is better at
promoting cooperation, not just one. If peeps have thought iterated PDs are
the acme of cooperation, that's pretty crazy, too.

------
chrstphrhrt
Building on this stuff is Scott Aaronson's "Eigenmorality":
[https://www.scottaaronson.com/blog/?p=1820](https://www.scottaaronson.com/blog/?p=1820)

------
jstanley
Incredible that only 48% of people got out of their car to shovel the snow...
how can you feel good about yourself, sitting in your car, watching somebody
else clear a path for you, and not helping?

~~~
TheSpiceIsLife
48% of people over 12 iterations of a _game_.

 _"... which involved 96 participants ... Each pair repeated (“iterated”) both
games 12 times, though were initially told the number of repetitions was
randomly determined. The researchers created global competition by revealing
that the players with the four highest pay-offs would receive monetary
awards."_

What people say they would do in a simulation might be different to what they
would. Maybe they didn't bring a shovel. Maybe there's already 96 people at
the snow face and you'd only be getting in the way.

~~~
pavel_lishin
Exactly. In the real world, this is likely iterated dozens of times every
winter, against neighbors you'll see repeatedly.

In a college classroom, where someone tells you you'll get fifty bucks if you
score high on a game? If they tell me I get bonus points for stealing their
shovel, why wouldn't I choose that option? Nothing is at stake.

~~~
logfromblammo
Also, the real world is rarely so symmetric. Most of the time, one party has
diminished ability to retaliate, and so gets defected against much more often.

------
adrianratnapala
People have been doing Prisoner's Dilemma with different payoff amounts since
the dawn of PD, and of course this can result in qualitatively different
strategies.

But sticking to just the rules, rather than the results. Is is the snowdrift
game just a payoff score tweak? Or is there some structural difference that I
missed?

~~~
ykler
In snowdrift, in the case where your opponent is uncooperative, it is better
for you to be cooperative.

------
bluetwo
For me the point of Prisoner's Dilemma is that math alone can't give you the
answer. The answer lies in your ability to trust the other side and to believe
they will trust you.

------
dandare
I always considered the pay-offs in PD to be variables with various positive
or negative values. That would make SD is just a subset of PD with different
pay-offs.

~~~
shkkmo
There is a difference in the relations between the two payoff schemes.

In PD, if you don't cooperate, I'm better off not cooperating

In SD, if you don't cooperate, I'm better off still cooperating

This means that don't cooperate / don't cooperate is not the nash equilibrium
in SD like it is in PD.

In SD the nash equilibriums are one person shoveling.

------
Buge
>while shoveling snow always helps you out

That appears to be an error in the article. If your opponent is shoveling,
then you shoveling hurts you (300 drops to 200).

------
loa_in_
About the game:

One could back out their car and pretend to have dug out the whole thing :)

------
TheSpiceIsLife
These experiments are nonsense.

Firstly, from the article: _“In principle, natural selection predicts
individuals to behave selfishly”_ is a faulty premise. Obviously humans who
cooperate to raise children to sexual maturity will have more descendants.

Secondly, in these simulated games it seems likely to me the participants
would be, at least to some extent, randomly selecting cooperate / defect
because a potential monetary reward for participation isn't the same as
"cooperate or everyone freezes to death in their car tonight", or whatever
real-world consequences might apply where you don't get "12 iterations".

The way I see is: while our worldview is _" everything is a competition"_ that
is how we will interpret what we see. Evolution and economics being the most
strident examples. Anything that doesn't automatically fit the worldview needs
_explaining by science_ , which is code for "publish or perish". If we had a
different worldview I'm sure we would shoehorn everything to fit.

~~~
rizzom5000
>> “In principle, natural selection predicts individuals to behave selfishly”
is a faulty premise.

I'm fairly certain that cooperating to raise your own offspring is still
considered selfish behavior. Something like taking resources from your
offspring to give to someone else would be more in line with the definition of
selfless (e.g. taking food from your own malnourished child to give to another
child).

~~~
nostromo
Dawkins' book The Selfish Gene is the most fitting way of framing this
conundrum.

Spending time and energy raising one's children isn't selfish of the parent as
an individual. But it's selfish from the view of their genes.

~~~
majewsky
Yeah, but if the definition of "selfish" is so broad that it includes
basically all of human behavior, it's not useful anymore.

------
mslate
Snowdrift is just a variation of the Prisoner's Dilemma with a specific
constraint on payoffs. This is not novel research

~~~
quadrangle
The article is from 2007, but the research provides real value even if the
fundamental concept isn't novel. It says that besides the _hypothesis_ that
this adjustment has a particular impact, here's data from real research
testing that hypothesis.

------
freech
This does not at all explain human behaviour: Most people would not shovel all
the snow themselves while their counterpart chills in their car. And many
people cooperate in prisoner's dilemma situations.

What explains human behaviour is that with every action we program ourselves
to follow certain strategies and rules and that others can judge to some
extent which strategies we programmed ourselves to follow. Thus we have an
incentive to follow strategies that lead others to trust us, so that we'll
have more opportunities for cooperation in the future.

~~~
GuB-42
This is tit-for-tat.

Always start by cooperating and then do the same as the other party :
cooperate if he does, retaliate if he doesn't. It is the most effective
strategy in both games.

~~~
freech
No it's not. What needs to be explained is why you cooperate if you know that
you will never play with the same person again. And the answer is that by
doing so you get in the habit of automatically cooperating in similar
situations in the future and other people recognize this habit and trust you.

------
StanislavPetrov
In my opinion the PD has always been worthless because it is based on a series
of false paradigms.

Most importantly, this scenario involves only a binary decision. In reality,
there are no binary decisions. Every single possible decision involved
countless contingencies to consider which are limited only by the awareness
and imagination of the person making the decision. Trying to apply this false
paradigm to reality is nonsense. You might as well try to discern real-world
data about the number of angels that people believe will fit on the end of a
pin.

>Compare this with the Prisoner’s Dilemma. For a quick synopsis, two prisoners
being questioned each have the choice to either defend the other’s innocence
or betray the other’s guilt.

The underlying assumption is that innocence and guilt are the only factors
driving the prisoners decision. What if the prisoner's primary motivation is
the rejection of coercion by his captors? Making assumptions about motivation
completely nullifies any potentially valuable insights about human behavior.

