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Evolution of Altruistic Utility in Socially Connected Populations Altenberg Workshop in Theoretical Biology Ted Bergstrom September, 2011 Ted Bergstrom Evolution of Altruistic Utility in Socially Connected

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Page 1: Evolution of Altruistic Utility in Socially Connected …econ.ucsb.edu/~tedb/Lectures/ViennaKLI.pdfEvolution of Altruistic Utility in Socially Connected Populations Altenberg Workshop

Evolution of Altruistic Utility inSocially Connected Populations

Altenberg Workshop in Theoretical Biology

Ted Bergstrom

September, 2011

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

Page 2: Evolution of Altruistic Utility in Socially Connected …econ.ucsb.edu/~tedb/Lectures/ViennaKLI.pdfEvolution of Altruistic Utility in Socially Connected Populations Altenberg Workshop

Hamilton’s Conjecture:

“The social behavior of a species evolves in such a way thatin each distinct behavior-evoking situation the individual willseem to value his neighbors fitness against his own according tothe coefficients of relationship appropriate to that situation.

“The Genetical Theory of Social Behavior” TheoreticalBiology, 1964.

This conjecture has come to be the center of a body of evolutionary theoryknown as Inclusive Fitness Theory.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Altruistic Payoff Functions in Two PlayerGames

• If a player who plays strategy x and his opponent plays strategy y , letf (x , y) be the expected number of descendants of the x player andf (y , x) that of the y player.

• Define the x player to have an altruistic utility function with degree ofaltruism a if he acts as if his payoff function is

U(x , y , a) = f (x , y) + af (y , x).

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

Page 4: Evolution of Altruistic Utility in Socially Connected …econ.ucsb.edu/~tedb/Lectures/ViennaKLI.pdfEvolution of Altruistic Utility in Socially Connected Populations Altenberg Workshop

Two Player Version of Hamilton’sConjecture

• The coefficient of relatedness between two individuals is theprobability that if one has a rare mutant gene, so does the other.

• In two player games between relatives, natural selection will favorgenes that dictate altruistic utility functions with degree of altruism rwhere r is the coefficient of relatedness between the two relatives.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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General Question

• When can we expect evolution to result in altruistic utility functions?

• What would these utility functions look like?

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Alternative Theories: Strategies or UtilityFunctions?

• Natural selection acts on hardwired strategies.• Compelling for simple strategies in “toy games”.• Plausible for simple creatures in simple environments: bacteria,

university administrators, etc.

• Natural selection acts on preferences over outcomes and endowsanimals with notions of causality,and an ability to survey options andchoose most preferred.

• In complex and variable environments, there are not enough repetitionsfor natural selection to pick optimal response to each environment.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Two interpretations of Hamilton’sConjecture

• Strategies are hard-wired and distribution of strategies that isobserved would be a Nash equilibrium for players with altruistic payofffunctions that weigh relatives’ fitness at r times their own.

• Animals evolve altruistic utility functions that weigh relatives’ fitnessat r times their own and they try to maximize expected utility.Outcomes observed are Nash equilibria for players with these payofffunctions.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Alger Weibull Theory of Utility Evolution

• Players have altruistic utility functions with some degree of altruism.

• When they meet, they learn each others’ degrees of altruism and playcomplete information Nash equilibrium strategies for a game withthese payoff functions.

• They argue that the equilibrium degree of altruism for games betweenrelatives would be greater than or less than r depending on whetherthe strategies of the two players are strategic complements ofsubstitutes.

• The idea is that if actions are substitutes (complements) andopponent knows you are more altruistic, it is in his interest to act less(more) altruistically.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Alternative Theory of Utility Evolution

• Players are unable to determine the type of their opponent beforethey choose their strategy and must choose their best strategy, giventhe probabilistic information that is determined by their own types.

• Motivation: Even siblings can’t observe each other all the time.Altruism is the motivator for treating your brother well when he isn’tlooking. Treating siblings well when they are looking could beincentivized by reciprocity.

• This approach will get the most attention in this talk.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Inclusive Fitness Theory: A Battleground?

Remarkably (to outsiders), Hamilton’s theory has become the scene of afierce battle among reputable biologists.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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The Attackers

Martin Nowak, Corina Tarnita, Edward Wilson

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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The Attack

A 40 page Supplementary Information paper in Nature

“Hamilton’s rule almost never holds.”

“. . . inclusive fitness theory is not a constructive theory that allows auseful mathematical analysis of evolutionary processes.”

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Defenders of the Faith

”we believe that their arguments are based upon a misunderstanding ofevolutionary theory and a misrepresentation of the empirical literature.”

The Paper With 130 Coauthors. Nature, March 23, 2011

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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A Chorus of 130 Voices

Patrick Abbot, Jun Abe, John Alcock, Samuel Alizon, Joao A. C.Alpedrinha, Malte Andersson, Jean-Baptiste Andre, Minus van Baalen,Francois Balloux, Sigal Balshine, Nick Barton, Leo W. Beukeboom, Jay M.Biernaskie, Trine Bilde, Gerald Borgia, Michael Breed, Sam Brown,Redouan Bshary, Angus Buckling, Nancy T. Burley, Max N.Burton-Chellew, Michael A. Cant, Michel Chapuisat, Eric L. Charnov, TimClutton-Brock, Andrew Cockburn, Blaine J. Cole, Nick Colegrave, LedaCosmides, Iain D. Couzin, Jerry A. Coyne, Scott Creel, Bernard Crespi,Robert L. Curry, Sasha R. X. Dall, Troy Day, Janis L. Dickinson, Lee AlanDugatkin, Claire El Mouden, Stephen T. Emlen, Jay Evans, Regis Ferriere,Jeremy Field, Susanne Foitzik, Kevin Foster, William A. Foster, CharlesW. Fox, Juergen Gadau, Sylvain Gandon, Andy Gardner, Michael G.Gardner, Thomas Getty, Michael A. D. Goodisman, Alan Grafen, RickGrosberg, Christina M. Grozinger, Pierre-Henri Gouyon, Darryl Gwynne,Paul H. Harvey, Ben J. Hatchwell, Jrgen Heinze, Heikki Helantera, Ken R.Helms, Kim Hill, Natalie Jiricny, Rufus A. Johnstone, Alex Kacelnik,

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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And More...E. Toby Kiers, Hanna Kokko, Jan Komdeur, Judith Korb, DanielKronauer, Rolf Kmmerli, Laurent Lehmann, Timothy A. Linksvayer,Sbastien Lion, Bruce Lyon, James A. R. Marshall, Richard McElreath,Yannis Michalakis, Richard E. Michod, Douglas Mock, Thibaud Monnin,Robert Montgomerie, Allen J. Moore, Ulrich G. Mueller, Ronald No, SamirOkasha, Pekka Pamilo, Geoff A. Parker, Jes S. Pedersen, Ido Pen, DavidPfennig, David C. Queller, Daniel J. Rankin, Sarah E. Reece, Hudson K.Reeve, Max Reuter, Gilbert Roberts, Simon K. A. Robson, Denis Roze,Francois Rousset, Olav Rueppell, Joel L. Sachs, Lorenzo Santorelli, PaulSchmid-Hempel, Michael P. Schwarz, Tom Scott-Phillips, JanetShellmann-Sherman, Paul W. Sherman, David M. Shuker, Jeff Smith,Joseph C. Spagna, Beverly Strassmann, Andrew V. Suarez, LiselotteSundstrm, Michael Taborsky, Peter Taylor, Graham Thompson, JohnTooby, Neil D. Tsutsui, Kazuki Tsuji, Stefano Turillazzi, Francisco beda,Edward L. Vargo, Bernard Voelkl, Tom Wenseleers, Stuart A. West, MaryJane West-Eberhard, David F. Westneat, Diane C. Wiernasz, Geoff Wild,Richard Wrangham, Andrew J. Young, David W. Zeh, Jeanne A. ZehAndrew Zink Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Battle Cries of the Defenders

• “Nowak et al are incorrect to state that inclusive fitness requires anumber of stringent assumptions such as pairwise interactions, weakselection, linearity, additivity and special population structures.Hamiltons original formulations did not make all these assumptions,and generalizations have shown that none of them is required.”

• “ Inclusive fitness is as general as the genetical theory of naturalselection itself. It simply partitions natural selection into its direct andindirect components.”

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Clarity as Peacemaker?

• A durable academic quarrel usually requires a confusing variety ofvaguely specified claims.

• Those who don’t enjoy doctrinal battles may prefer to try establishingthe truth or falsehood of more modest but clearly stated propositions.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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When does Inclusive Fitness Theory Work?

• Answer depends on:

• What do we mean by ”Inclusive Fitness Theory”?

• What work do we expect it to do?

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Spelling out Hamilton’s Conjecture

• Natural selection will favor genes that dictate altruistic utilityfunctions with degree of altruism r

• Under appropriate selection dynamics, a population consisting entirelyof x̄ strategists will be an equilibrium if and only if having bothplayers play x̄ is a Nash equilibrium for the game in which playershave the “inclusive fitness” utility functions

U(x , y) = f (x , y) + rf (y , x).

• Essentially applies John Maynard Smith’s notion of EvolutionaryStable State using Hamilton’s inclusive fitness.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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A More Demanding Criterion

• A recent article Matthijs van Veelen suggests the criterion that “thesign of the inclusive fitness must determine the direction of selection.”even in populations that are out of equilibrium. (This view seems tobe endorsed by Nowak et al in their paper.)

• But if a population spends most of its time in the neighborhood ofequilibrium, it may be that natural selection “settles for” preferencesthat maximize reproduction rates in the neighborhood of equilibriumbut not necessarily in disequilibrium situations.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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A Less Demanding Criterion

• The chorus of 130 maintains that “Inclusive fitness is as general asthe genetical theory of natural selection itself. It simply partitionsnatural selection into its direct and indirect components.”

• For them, apparently, inclusive fitness theory is a tautology. Theywould somehow redefine inclusive fitness of a strategy to be thatwhich maximizes its growth rate.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Selection dynamics for Two player gameswith assortative matching

• A population has two types, x strategists and x̄ strategists. withprobability a an individual gets matched

• With probability a an individual gets matched with another of its owntype. With probability 1− a an individual gets matched with arandom selection from the population.

• Define V (x , x̄) = af (x , x) + (1− a)f (x , x̄).

• In the limit as the fraction of x̄ strategists in the populationapproaches 1, the expected payoff of x strategists approaches V (x , x̄)and that of x̄ strategists approaches V (x̄ , x̄).

• Therefore a population consisting entirely of x̄ strategists is stable ifand only if

V (x , x̄) ≤ V (x̄ , x̄)

for all feasible strategies x .

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Revising Hamilton’s conjecture

• For two-player games, Hamilton’s conjecture would be correct if theinclusive fitness function

U(x , y) = f (x , y) + af (y , x)

were replaced by

V (x , y) = af (x , y) + (1− a)f (y , x).

• This was observed by Grafen 1979 and by Hines and Maynard Smith1979, who call the function V , “inclusive fitness”.

• Hines and Maynard Smith discovered that if the set of possiblestrategies consists of mixed strategies over a finite number of possiblepure strategies, then a strategy x̄ is a symmetric Nash equiibrium forthe game with payoff functions V only if it is also a symmetric Nashequilibrium for the game with payoff functions U.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Rescue by Redefinition?

• We could rescue Hamilton’s conjecture, simply by redefining theinclusive fitness function to be V (x , y) rather than U(x , y).

• But it is interesting to ask, when Hamilton’s conjecture is correct inits original form: When does natural selection select for altruisticutility functions?

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Strategic Complements and Substitutes

• Define C (x , y) = f (x , x) + f (y , y)− f (x , y)− f (y , x).

• Strategies x and y are “strategic complements” if C (x , y) > 0 and“strategic substitutes” if C (x , y) < 0.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Inclusive Fitness, Personal Fitness, andComplementarity

• Let DU(x , y) = U(x , y)− U(y , y) and DV (x , y) = V (x , y)− V (y , y)

• Then DV (x , y) = DU(x , y) + aC (x , y) for all x and y .

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Coincident Gradients

• A population consisting entirely of x̄ strategists is stable under ourproposed dynamics if and only if x = x̄ maximizes V (x , x̄).

• At the point, x = x̄ , the derivative of V (x , x̄) with respect to x isV1(x , x̄) = f1(x̄ , x̄) + af2(x̄ , x̄).

• But we also have U1(x , x̄) = f1(x̄ , x̄) + af2(x̄ , x̄)

• Does this prove that Hamilton’s conjecture is right?

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Hamilton’s Rule Doesn’t Always Work

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Example with Complements

Let

f (x , y) = 2xy − x − 1

2x2

• Symmetric Nash equilibrium for inclusive fitness players,

x̄ =1

1 + 2a

• Although x = 1/1 + 2a is a global max for U(x , 1/1 + 2a), it is aglobal minimum for V (x , 1/1 + 2a).

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Hamilton’s Rule Fails again

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Example with Substitutes

Let

f (x , y) = −2xy + x +1

2x2

• Symmetric Nash equilibrium for personal fitness players with payofffunctions V (x , x̄) = aV (x , x) + (1− a)V (x , x̄) is

x̄ =1

1 + 2a

• Although x = 1/(1 + 2a) is a global max for V (x , 1/(1 + 2a), it is aglobal minimum for U(x , 1/1 + 2a).

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Hamilton’s Rule Works (Sometimes)

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Hamilton’s Rule Works (Sometimes)

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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General results for two players

• If strategies are strategic complements, then every symmetric Nashequilibrium for V is a symmetric Nash equilibrium for U.

• If strategies are strategic substitutes, every symmetric Nashequilibrium for U is a symmetric Nash equilibrium for V .

• If the fitness function f (x , y) is a concave function, then thesymmetric Nash equilibria for the payoff functions U and V coincide.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

Page 35: Evolution of Altruistic Utility in Socially Connected …econ.ucsb.edu/~tedb/Lectures/ViennaKLI.pdfEvolution of Altruistic Utility in Socially Connected Populations Altenberg Workshop

A general result for N players

• Players are sorted into groups of n players.

• The payoff to each player i is given by F (xi , g(x∼i )) where g is asymmetric function of the vector x∼i of strategies by other the n − 1players.

• There is an incumbent strategy x̄ and a rare mutant strategy x . If anindividual is a mutant, the probability is p(k) that there are exactly kmutants in the mutant’s group.

• Let α = 1n−1

∑n−1k=1 kp(k).

• If F and g are smooth concave functions, then Hamilton’s conjectureapplies where the inclusive fitness utility function is

U(xi , φ(x)) = F (xi , g(x∼i )) + α∑j 6=i

F (xj , g(x∼j)).

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Back to the Field of Battle

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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Overstated Claims

“for inclusive fitness to work, one has to assume that theeffects of one’s behavior on others are linear, additive, andindependent. . . it must be sufficient to look at pairwiseinteractions independently and such interactions can be addedup”, Nowak et al .

My response: Not so. These are sufficient conditions, but not necessaryconditions for inclusive fitness to work. For example, under the dynamicsthat they posit, inclusive fitness theory “works” when the effects ofbehaviors on fitness are described by any concave function for any numberof players.

“Inclusive fitness only works for the limit of weak selection.”,Nowak et al (This means roughly that the theory only showsconditions for stability against “small” mutations.)

My response: Not so. If the payoff functions are concave, local maximaare also global maxima.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations

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From the Choir of 130

“ Inclusive fitness is as general as the genetical theory ofnatural selection itself. It simply partitions natural selection intoits direct and indirect components.”

My response: One could redefine “inclusive fitness theory” to be atautology, but to do so is to miss the most interesting content. Nowak etal correctly argue that simple versions of inclusive fitness theory do notalways work. To understand the process one needs to look closely at thetransmission mechanism and resulting selection dynamics, as well as thenature of interactions.

Ted Bergstrom Evolution of Altruistic Utility in Socially Connected Populations