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A simple model of group selection that
cannot be analyzed with inclusive fitness
0.0 0.5 1.00.0
0.5
1.0
Hamilton’s rule(biology)
Folk theorem(economics)
Jerry CoyneRichard Dawkins (c)
Andy GardnerAlan Grafen
Laurent KellerLaurent Lehmann
Steven PinkerDavid Queller
Francois RoussetStuart WestGeoff Wild
Leticia AvilesRob Boyd
Samuel BowlesLee DugatkinHerbert Gintis
Charles GoodnightJon Haidt
Pete RichersonArne Traulsen
DS Wilson (c) EO Wilson
Pro-group selection team
Anti-group selection team
1) Has group selection shaped (human, cooperative) behaviour?
Two different issues
2) Is group selection equivalent with inclusive fitness?
'Group selection', even in the rare cases where it is not actually
wrong, is a cumbersome, time-wasting, distracting impediment to
what would otherwise be a clear and straightforward
understanding of what is going on in natural selection.
Richard Dawkins (2012)
Inclusive fitness theory, summarised in Hamilton’s rule, is a
dominant explanation for the evolution of social behaviour. A
parallel thread of evolutionary theory holds that selection
between groups is also a candidate explanation for social
evolution. The mathematical equivalence of these two
approaches has long been known.
Marshall (2011)
No group selection model has ever been constructed where
the same result cannot be found with kin selection theory
West, Griffin & Gardner (2007)
Inclusive fitness models and group selection models are
extremely similar to each other. Their only fundamental
difference is in how they choose to decompose fitness. Other
differences are trivial matters of the form of presentation.
Queller (1992)
Mathematical gene-selectionist (inclusive fitness) models can
be translated into multilevel selection models and vice versa.
One can travel back and forth between these theories with the
point of entry chosen according to the problem being
addressed.
Hölldobler and Wilson (2009)
The Price formulation convinced Hamilton that kin
selection was group selection.
Wade et al. (2010)
Hamilton (1975)
1970 1980 1990 2000 2010
Hamilton’s rule 1964
Williams1966
Karlin & Matessi1983, 1984
Price equation1970, 1972
Queller1992
Inclusive fitness / group selection
Hamilton 1975
“Unto Others” 1998
“Nail in the coffin of group selection”
2009
NTW2010
On the use of the Price equation, 2005
GS ≠ IF 2009
Hamilton’s missing link, 2007
Traulsen& Nowak
2006
Shishi Luo Burt Simon
a b
Individual reproduction Group reproduction
intensity 1
intensity 1intensity 1
a
0 1 2 3
0 1 2 3
b
20 1 3
0 1 2 3
a b
Individual reproduction Group reproduction
The PDE that describes the dynamics
loss in individual reproduction
gain in collective reproduction
“wave” movement to the left
increase (uniform) death rate
increase reproduction rate -groups
large “in the middle”
large if groups heterogeneous
Change in frequency of cooperators at
Change in frequency of cooperators at
Of course, it is now generally understood that the correct
definition of relatedness is that which makes inclusive fitness
theory work.
Marshall / Grafen
A rule is not a rule if it changes from case to case.
Van Veelen, 2012
Go Procrustes, go!
Change in frequency of cooperators at
Standard: 2 players, random matching
Replicator dynamics
14
12
14
Generalization: n players, assortative matching
1 1( )4 8
1 1( )4 8
1 3( )4 8
1 3( )4 8
2 players
Replicator dynamics
0f 1f 2f
3 players
0f 3f1f 2f
It’s the equal gains from switching, st…!
1
1
1
+
_
Queller (1985) but then Price-less
altruism selected altruism selected against
bistability coexistence
+
_
+
_
+
_
+
_
Hamilton
0 1f 2 1f
1 1f
2 1f0 1f
1 1f
Queller
+
_
0
0 0
12
1 12 2
K t t
K t t K t t
ep te e
where
K r b - c
Replicator dynamics
1) the population structure implies a constant r, and
2) the game satisfies generalized equal gains from switching,
⇓
Right hand side
A list of numbers
Left hand side =
If you want to win a game, you should score [at least] one goal more that your opponent
Johan Cruijff
The frequency has gone up because the frequency has gone up
the Price Equation
Is there such a thing as Price’s theorem?
Theorem 1 (biology): If the left hand side in the Price equation is computed as suggested in Price (1970) and the right hand side as well,
then they are equal.
Theorem 1 (football): If team A scores more goals than team B, then team A wins.
Theorem 2 (football): If team A and B have equally able players, and interactions occur according to Assumption 1, … , Assumption N, and
team A plays 4-3-3 and B plays 4-4-2, then team A is more likely to win than team B.
Theorem 2 (biology): If the fitness of an individual depends on its own and the other individual’s behaviour according to Assumption 1, … , Assumption N, than the behaviour that emerges is more likely to be
behaviour A than it is to be behaviour B.
How to quit the Price equation
www.evolutionandgames.com/price
Price (as simple as it gets)
Nq
Nz
Nqz
Nq
Nqz
QQQ i ii ii iii ii ii12
Price (as simple as it gets)
Nq
Nz
Nqz
Q i ii ii ii
Price (as simple as it gets)
qzCovQ ,
Price (as simple as it gets)
Nq
Nz
Nqz
Q i ii ii ii
Correct would be
qz,cov SampleQ
if the numbers are data
… or …
qzCovQE ,
If zi and qi random variables for all i.
Parent generation
Ind. 1
Ind. 2
Offspring generation
Model: draw twice, both times
P (red) = p P (white) = 1 - p
1
2
10
Parent generation Offspring generation
Ind. 1
Ind. 2
Model: draw twice, both times
P (red) = p P (white) = 1 - p
1
2
20
zz
Q Cov z,q
12
Q 1
2 2 2 2i i i ii i i
z q z q
1
2
10
Parent generation Offspring generation
Ind. 1
Ind. 2
Model: draw twice, both times
P (red) = p P (white) = 1 - p
1
2
02
zz
Q Cov z,q
12
Q 1
2 2 2 2i i i ii i i
z q z q
1
2
10
Parent generation Offspring generation
Ind. 1
Ind. 2
Model: draw twice, both times
P (red) = p P (white) = 1 - p
1
2
11
zz
Q Cov z,q
0Q 02 2 2
i i i ii i iz q z q
1
2
10
Parent generation Offspring generation
Ind. 1
Ind. 2
Model: draw twice, both times
P (red) = p P (white) = 1 - p
1
2
11
zz
Q Cov z,q
0Q 02 2 2
i i i ii i iz q z q
Parent generation Possible offspring generations
Ind. 1
Ind. 2
Model: draw twice, both times
P (red) = p P (white) = 1 - p
Q " Cov z,q "
I II III IV
p2
p(1-p)p(1-p)
(1-p)2
1,2
Cov X Y p
Randomly draw a parent (hypothetically)
Properties of the model
X its genotypeY its number of offspring
Parent generation Possible offspring generations
Ind. 1
Ind. 2
Model: draw twice, both times
P (red) = p P (white) = 1 - p
Q " Cov z,q "
I II III IV
p2
p(1-p)p(1-p)
(1-p)2
1) Estimate p
What would a statistician do?
2) Test if p > 0
i i ii
i ii i
z q' qNQ " Cov z,q "z z
Price 2.0
i i ii
i ii i
z q' qNQ " Cov z,q "z z
Price 2.0
Meiosis term
i i ii
i ii i
z q' qNQ " Cov z,q "z z
Price 2.0
i i ii
i ii i
z q' qNQ " Cov z,q "z z
Price 2.0
8 7 1
Parent generation Possible offspring generations
Ind. 1
Ind. 4
I II CCLVI
Price 2.0
Ind. 2
Ind. 3
..... .....
1
2
3
4
10 50 50
qq .q .q
1
2
3
4
2132
zzzz
1
2
3
4
101 30
q 'q 'q ' /q '
i i ii
i ii i
z q' qNQ " Cov z,q "z z
Price 2.0
1 108 8
0i i ii
ii
z q' qE
z
What would a modeler do?
Show that the assumption of fair meiosis implies that
i i ii
ii
z q' qz
What would a statistician do?
Test the hypothesis of fair meiosis, using the realization of
i i ii
i ii i
z q' qNQ " Cov z,q "z z
Projection of intuition onto the Price equation
Parent generation Possible offspring generations
Ind. 1
Ind. n
I II
1,Cov X Yn
Price 3.01 00 1
Ind. 2...
Ind. 1
Ind. n
Ind. 2...
1,Cov X Yn
Model: 1) match randomly
2) play
3) draw each individual with probabilities proportional to payoffs
Dynamical sufficiency
1,Cov X Yn
1,Cov X Yn
Right hand side
A list of numbers
Left hand side =