Nash equilibrium example on Ducks



The example shows that Nash’s equilibrium dictates life (it’s genomes program) in free competition to behave good, and in solidarity with the other ducks – which also maximize their utility. However, if bread was constantly tossed all over the pond – the prime mover in the production of love and intelligence, too-little-energy, would disappear and evil phenotypes could be given fitness over good (according to Head Biotech’s theory). The need to maximize their utility decreases as the energy context in the pound increases. Hence, Nash equilibrium will lose its potency in the selection process along with the phenotypes (love and intelligence) which are demanded to reach the Nash equilibrium. The example is taken from Tom Siegfried’s book: A Beautiful Math (page 73, 74 and 75).

 In the winter of 1979, Cambridge University biologist David
Harper decided it would be fun to feed the ducks.
A flock of 33 mallards inhabited the university’s botanical garden,
hanging out at a particular pond where they foraged for food.
Daily foraging is important for ducks, as they must maintain a
minimum weight for low-stress flying. Unlike landlubber animals
that can gorge themselves in the fall and live off their fat in the
winter, ducks have to be prepared for takeoff at any time. They
therefore ought to be good at finding food fast, in order to maintain
an eat-as-you-go lifestyle.
Harper wanted to find out just how clever the ducks could be at maximizing their food intake. So he cut up some white bread
into precisely weighed pieces and enlisted some friends to toss the
pieces onto the pond.
The ducks, naturally, were delighted with this experiment, so
they all rapidly paddled into position. But then Harper’s helpers
began tossing the bread onto two separated patches of the pond.
At one spot, the bread tosser dispensed one piece of bread every
five seconds. The second was slower, tossing out the bread balls
just once every 10 seconds.
Now, the burning scientific question was, if you’re a duck,
what do you do? Do you swim to the spot in front of the fast
tosser or the slow tosser? It’s not an easy question. When I ask
people what they would do, I inevitably get a mix of answers (and
some keep changing their mind as they think about it longer).
Perhaps (if you were a duck) your first thought would be to go
for the guy throwing the bread the fastest. But all the other ducks
might have the same idea. You’d get more bread for yourself if you
switched to the other guy, right? But you’re probably not the only
duck who would realize that. So the choice of the optimum strategy
isn’t immediately obvious, even for people. To get the answer
you have to calculate a Nash equilibrium.
After all, foraging for food is a lot like a game. In this case, the
chunks of bread are the payoff. You want to get as much as you
can. So do all the other ducks. As these were university ducks, they
were no doubt aware that there is a Nash equilibrium point,
an arrangement that gets every duck the most food possible when
all the other ducks are also pursuing a maximum food-getting
Knowing (or observing) the rate of tosses, you can calculate
the equilibrium point using Nash’s math. In this case the calculation
is pretty simple: The ducks all get their best possible deal if
one-third of them stand in front of the slow tosser and the other
two-thirds stand in front of the fast tosser.
And guess what? It took the ducks about a minute to figure
that out. They split into two groups almost precisely the size that
game theory predicted. Ducks know how to play game theory!
When the experimenters complicated things—by throwing
bread chunks of different sizes—the ducks needed to consider
both the rate of tossing and the amount of bread per toss. Even
then, the ducks eventually sorted themselves into the group sizes
that Nash equilibrium required, although it took a little longer.1
Now you have to admit, that’s a little strange. Game theory
was designed to describe how “rational” humans would maximize
their utility. And now it turns out you don’t need to be rational, or
even human. The duck experiment shows, I think, that there’s
more to game theory than meets the eye. Game theory is not just a
clever way to figure out how to play poker. Game theory captures
something about how the world works.

One thought on “Nash equilibrium example on Ducks

  1. Pingback: Has the Nash Equilibrium lead to any significant economic discoveries? – Best Solution Space

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