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

qq

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

qq

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

qq

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

qq

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 =