Evolutionary Dynamics

Gametheory

Economic and strategic conflict Nash eq.

Prisoner's dilemma.

Two strategies: Cooperate, Defect.

Payoff matrix elements: remitted years of prison.

Evolutionary game theory: Maynard Smith 1973

System of Nindividuals (agents,DNA), defined by
- a set of strategies 1...k
- aco-evolutionary fitness ("payoff") matrix k × k
Agents within a population interact according to the payoff matrix
Agents reproduce at a rate increasing with the payoff (... and die ...) Mutation (& Crossover): can be included

Evolutionary game theory in infinite populations

Consider densities of strategies x :=i/N thus 0 ≤ x ≤ 1 ,and
Replicator equations x = x(1−x)(πA(x)−πB(x))
Stable fixed points: Evolutionarily stable strategies (ESS)

Moran evolution dynamics in 2×2 games

Arbitrary payoff matrix:
Frequency-dependent Moran process:

Every agent interacts with a representative sample of the population:

With probability ,a copy of an A agent replaces a randomly chosen individual.

The Moran Process

N individuals
Choose an individual at random proportional to its fitness
Create identical offspringRemove one random individual

Moran, The Statistical Processes of Evolutionary Theory (

962).

The

frequency-dependent

MoranProcess

Nindividuals

interacting by payoff matrix:

(

abcd

)

Choose an individual at random

proportional to its payoff

(

)

〈

〉

(

−

1)+

(

−

)

−

Create identical offspringRemove one random individual

Moran, The Statistical Processes of Evolutionary Theory (

962).

M.A. Nowak, A. Sasaki, C. Taylor, and D. Fudenberg, Nature 42

8, 646 (2004),

Prisoner's Dilemma

Prisoner's Dilemma: TFT

Player B tive to others. A bacterium might easily C Cooperation

I

R= Reward for

mutual cooperation 1

D Defection 7 I Fig. 1. The Prisoner's Dilem-

I ma game. The payoff to player

Sucker's payoff A is with 1 numerical values. The game is

have production of its own bacteriocin dependent on the perceived presence of like hostile products in its environment, but it could not aim the toxin produced toward an offending initiator. From ex- isting evidence, so far from an individual level, discrimination seems to be by spe- cies rather even than variety. For exam- ple, a Rhizobium strain may occur in nodules which it causes on the roots of many species of leguminous plants, but it may fix nitrogen for the benefit of the plant in only a few of these species (20). Thus, in many legumes the Khizohium seems to be a pure parasite. In the light of theory to follow, it would be interest- ing to know whether these parasitized legumes are perhaps less beneficial to free living Khizohirnm in the surrounding soil than are those in which the full symbiosis is established. But the main point of concern here is that such dis- crimination by a Rhizobium seems not to be known even at the level of varieties within a species.

As one moves up the evolutionary ladder in neural complexity, game-play-ing behavior becomes richer. The intelligence of primates, including humans, allows a number of relevant improvements: a more complex memory, more complex processing of information to determine the next action as a function of the interaction so far, a better estimate of the probability of future interaction with the same individual, and a better ability to distinguish between different individuals. The discrimination of others may be among the most important of abilities because it allows one to handle interactions with many individuals with- out having to treat them all the same, thus making possible the rewarding of cooperation from one individual and the punishing of defection from another.

The model of the iterated Prisoner's Dilemma is much less restricted than it may at first appear. Not only can it apply to interactions between two bacteria or interactions between two primates, but it can also apply to the interactions between a colony of bacteria and, say, a primate serving as a host. There is no assumption of commensurability of payoffs between the two sides. Provided that the payoffs to each side satisfy the inequalities that define the Prisoner's Dilemma (Fig. l), the results of the analysis will be applicable.

The model does assume that the choices are made simultaneously and with discrete time intervals. For most analytic purposes, this is equivalent to a continuous interaction over time, with SCIENCE, VOL. 21 1

play"*

Cooperation

D Defection

C--------------i---------------+ ' defined by T > R > P > S ! T=5 I P=l 1 and R > (S + T)/2. ' Temptation to Punishment for

i defect

1 mutual defection I

the interactions between pairs of individuals are random and not repeated, then any population with a mixture of heritable strategies evolves to a state where all individuals are defectors. Moreover, no single differing mutant strategy can do better than others when the population is using this strategy. In these respects the strategy of defection is stable. This concept of stability is essential to the discussion of what follows and it is useful to state it more formally. A strate- gy is evolutionarily stable if a population of individuals using that strategy cannot be invaded by a rare mutant adopting a different strategy (11). In the case of the Prisoner's Dilemma played only once, no strategy can invade the strategy of pure defection. This is because no other strategy can do better with the defecting individuals than the P achieved by the defecting players who interact with each other. So in the single-shot Prisoner's Dilemma, to defect always is an evolutionarily stable strategy. In many biological settings, the same two individuals may meet more than once. If an individual can recognize a previous interactant and remember some aspects of the prior outcomes, then the strategic situation becomes an iterated Prisoner's Dilemma with a much richer set of possibilities. A strategy would take the form of a decision rule which deter- mined the probability of cooperation or defection as a function of the history of the interaction so far. But if there is a known number of interactions between a pair of individuals, to defect always is still evolutionarily stable and is still the only strategy which is. The reason is that defection on the last interaction would be optimal for both sides, and conse- quently so would defection on the next- to-last interaction, and so on back to the first interaction. Our model is based on the more realis- tic assumption that the number of inter- actions is not fixed in advance. Instead, there is some probability, w, that after the current interaction the same two

individuals will meet again. Factors that affect the magnitude of this probability of meeting again include the average life- span, relative mobility, and health of the individuals. For any value of w, the strategy of unconditional defection (ALL D) is evolutionarily stable; if ev- eryone is using this strategy, no mutant strategy can invade the population. But other strategies may be evolutionarily stable as well. In fact, when w is suffi- ciently great, there is no single best strategy regardless of the behavior of the others in the population (19). Just be- cause there is no single best strategy, it does not follow that analysis is hopeless. On the contrary, we demonstrate not only the stability of a given strategy, but also its robustness and initial viability. Before turning to the development of the theory, let us consider the range of biological reality that is encompassed by the game theoretic approach. To start with, an organism does not need a brain to employ a strategy. Bacteria, for exam- ple, have a basic capacity to play games in that (i) bacteria are highly responsive to selected aspects of their environment, especially their chemical environment; (ii) this implies that they can respond differentially to what other organisms around them are doing; (iii) these condi- tional strategies of behavior can certain- ly be inherited; and (iv) the behavior of a bacterium can affect the fitness of other organisms around it, just as the behavior of other organisms can affect the fitness of a bacterium. While the strategies can easily include differential responsiveness to recent changes in the environment or to cumu- lative averages over time, in other ways their range of responsiveness is limited. Bacteria cannot "remember" or "inter- pret" a complex past sequence of changes, and they probably cannot dis- tinguish alternative origins of adverse or beneficial changes. Some bacteria, for example, produce their own antibiotics, bacteriocins; those are harmless to bac- teria of the producing strain, but destruc-

Iterated Games: Iterated Prisoner's Dilemma

Iterated Prisoner's Dilemma: TFT

Tit-for-Tat (TFT) - Anatol Rapoport's strategy in Robert Axelrod's

computer tournament

In the first round, TFT cooperates

In each subsequent round, TFT imitates the opponent's strategy.

CCCCCCC... CCCCCCC...

However, accidental mistakes detriment cooperation

CCCDCDC... CCDCDCD...

Many modified strategies have been proposed! e.g. generous TFT, tit-for-2-tat, ...

Axelrod's tournaments (1981): Let the strategies evolve in a computer tournament!

Axelrod's tournaments

Iterated Prisoner's Dilemma: TFT

TFT won in an evolutionary contest against several other strategies! Tournament Results

TABLE 2 The Contestants: Round One

Rank 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15

Name Anatol Rapoport Nicholas Tideman & Paula Chieruzzi Rudy Nydegger Bernard Grofman Martin Shubik William Stein & Amnon Rapoport James W. Friedman Morton Davis James Graaskamp Leslie Downing Scott Feld Johann Joss Gordon Tullock Name withheld RANDOM

Discipline (if faculty) Psychology Economics

Psychology Political Sci. Economics Mathematics Psychology Economics Mathematics

Psychology Sociology Mathematics Economics

Length of Program

4 41

23 8 16 50

13 6 63 33 6 5 18 77 5

Score

The contestants in the second round are listed in table 4 along with some information about their programs. Each pair of rules was matched in five games of varying lengths, averaging 151 moves each. There were sixty-two entries plus RANDOM, so the tournament score matrix for the second round is a huge 63 by 63 matrix. It is so big that table 5 has to give it in compressed form (see table 5). The average score of each rule with each other rule is shown as a single digit according to the following code:

1 : less than 100 points 2 : 100-199. 9 points (15 1 points is total mutual defection) 3 : 200-299. 9 points 4 : 300-399. 9 points 5 : 400-452. 9 points 6 : exactly 453 points (total mutual cooperation)

193

Axelrod's tournaments

Prisoner's Dilemma: TFT

The tion latest may, data in fact, for be (^1978) deteriorating. suggests that.. the. we situa- may

be For losing examples the war see on House air pollution." Subcommittee on the Environment mentcrl Protection and the Agerzcy's Atmosphere, Resecrrch The Erzviron- Progrcrmwith lfecrlth Prirnury und Environmenrul F,'rnphusis on Surveillance the Community Syatem(CHESS): Printing Office, An lnveatigative Washington, Report D.C., (Government 1976). espe-
cially Without chapters trying^4 to to be 6. entirely rigorous, we will use type an of NSF research definition: which "Basic is directed research toward is that in- crease where of the knowledge primary aim in of science. the investigator It is research is a fuller under knowledge study, rather or understanding than a practical of the application subject thereof." then director This of was NSF, given in by Symposium A. T. Waterman, on RuaicReaeurch, ation for the D. Advancement Wolfle, Ed. of (American Science, Washing- Associ-
ton, For example, D.C., 1959). the p. EPA 20. administrator, D. Costle,

in Proxmire, a letter dated chairman 12 June of (^1978) the to HUD-lndependent Senator William Agencies atrons Committee, Subcommittee said concerning of the Senate enviionmen- Appropri- tal dollar research, decisions "I've over had the to make last year too many without billion the critical made five information years ago, this would sort have of provided." investment,

ment, U.S. Congress, A Review Office if the of Technology U.S. F,'nvironmrntul Assess- Protection Outlook FY Agency I976 through Environmentcrl 1980 (Government Resecrrch
National Printing Office, Academy Washington, of Sciences, D.C., Commission 1976). on Natural U.S. Environmentul Resources, Ar~ul),ticcrl Protection Studies Agency, for vol. the 3, Research tul Protection and Development Agency (National in the Environmen- Academy of
National Sciences, Advisory Washington, Committee D.C., 1977). on Oceans and Atmosphere, Congress, fifth A Report annual lo the report President (Government crnd the
Printing ORD Progrcrm Ofice, Washington, Guide (EPA-60019-79-038, D.C., 1976). Envi- ronmental D.C.. 1979). Protection Agency, Washington,
Kesecrrch vironmental ~rrtlook, Protection 1980 (EPA-60019-80-006, Agency, Washington, En- D.C., environmental 1980). This research presents plan the in response agency's to 5-year stat- utory new report requirement. issued annually. The plan is updated and a
Many EPA faces examples are reported of the environmental in Environmentcrl problems Oul-

look Protection (^1980) Agency, (EPA 60018-80-003, Washington, Environmental D.C., 1980).

At the beginning of the 94th Congress (Janu-

ary nology 1975) received the Committee jurisdiction on Science over and envlron-Tech- mental in the research" rules of the as House a result of of Representatives. several changes The Atmosphere Subcommittee was formed on the Environment to handle this and juris- the diction Congressman and for Brown 4 years as (two chairman, Congresses), had respon- with sibility sult of reorganization for ORD. In within January the 1979, Committee as a re-on Science moved to and the Technology, chair of the Congressman Subcommittee Brown on Science, committee Research, on the Environment and Technology. and Atmosphere The Sub- was sources renamed and Environment Subcommittee and given on Natural some addi- Re-

tional Eni~ironmrnterl jurisdiction. Protection Agency Reseurch undDevelopment House Subcommittee laarres: 1978, on the hearings Environment before and the

the ber Atmosphere, 1978 (Government 19 July Printing and (^13) Office, and (^14) Washing- Septem-

ton, In making D.C., funding 1979). decisions, the agency uses a zero-base programs are budgeting approved (ZBB) by a~consensus process in which of the administrator tors. In the ZBB and the process, six assistant the ORD administra- has only about grams one are vulnerable of six votes, to and a great thus deal research of influence pro- from such a the substantial program role offices. in defining Because the program they play of research administrator ultimately and all conducted assistant by administrators ORD, the were that office. asked to testify on what they expect from
Speciul und Houaton, Urban Air hearings Pollution before Problema: the House Denver Sub- committee sphere, 19 on and the 21 November Environment 1977 and (Government the Atmo-
Printing Long-Term Office, Environmental Washington, Research D.C., 1978). in the En- i~ironmenterl the House Subcommittee Protection Agency, on the hearings Environment before

and Printing the Atmosphere, Office, Washington, 30 June (^1977) D.C., (Government 1978). See the supplemental testimony statement, of R. L. Sansom, p. 52. especially his

H. 10.-~ Kissinger, The Reporter, 5 March 1959, p.
system For example, of research the agency grants has putatively instituted aimed a new at bringing gram. Despite new work this aim of high the published quality into solicitation its pro- for unambiguously grant proposals state that does funding not explicitly decisions and will be lowing based appears: on scientific "Scientific quality. merit Instead and relevance the fol- of factors proposals in the will evaluation be significant procedures and balanced since all projects must be in concert with the Agency's

The Evolution of Cooperation Robert Axelrod and William D. Hamilton

struggle The theory for life of evolution and the survival is based on of the the fittest. Yet cooperation is common be- tween even between members members of the same of different species and spe- cies. Before about 1960, accounts of the evolutionary cooperative phenomena process largely as not dismissed requiring special attention. This position followed from signed a most misreading adaptation of theory to selection that as-at

the cies. level As of a result populations of such or whole misreading, spe- cooperation was always considered adaptive. tionary process, Recent however, reviews of have the shown evolu- no sound basis for a pervasive group- benefit species view or a of population, selection; the at the processes level of of a selection are weak. The original individ- ualistic more valid emphasis (1, 2). of Darwin's theory is 0036-807518110327-1390$01 .SO10 Copyright O 1981 AAAS

budget seems that appropriations." work on highly In other relevant words, matters it might EPA and be funded the Academic even if Community of poor quality. (EPA-60018-[See 80-010, cinnati, Environmental Ohio, 19801, p. Protection 2.1 Agency, Cin-

National sory Board, Academy Repor1 of of Sciences, the Ad Hoc Materials Committ~e Advi- on Principles (National Academy i?fReseurch-Engineerinx of Sciences, Washington, Interuc'tion
D.C., W. 0. 1966), Baker, p. in 16. House Committee on Science and tivity, Technology, und the Natiorzul Serninur Economy, on Reaeurch, 18 Jrrne Produc- I (Government D.C., 1980). Printing Office, Washington,
Testimony ,for the Office of J. of N. Resecrrch Pitts, in (^1980) und Aulhorizcltion Developnzenl, Eni~ironmrntul fore the House Protection Subcommittee Ayency, on hearings Science and be- Technology, ment Printing (^13) Office, and (^15) Washington, February (^1979) D.C., (Govern- 1979).
H. Gorrls, W. Bode, a report in nusic to the Resecrrch House Committee und Nrrtioncrl on Science Sciences, and Washington, Astronautics D.C., (National 1965), Academy p. 74. of
At research present not the only program through offices the ZBB guide process EPA's but also mittees. through These the mechanism committees of translate 13 research program com- ofice ments" needs which into guide "research all EPA research strategy (10). docu-
This Law provision 95-155, the is contained FY 1978 authorization in section 6 of Public act for ORD. see Conference For explanation Report of to congressional Accompcrny intent H.R. 5101, ernment 95th Printing Congress, Office, Report Washington, No. 95-722 (Gov- D.C.,
This 1977).. ,.provision is contained in section 11 of Public report cited Law in 95-155. (10), and For also explanation Report lo see Acc'om- the puny 157 (Government H.R. 5101, 95th Printing Congress, Office, Report Washington, No. 95-
D.C.. This was 1977). cbntained in section 4(a) of H.R. 7099,

the bill. House The provision version of was the deleted FY (^1981) from authorization the final version agency of strenuously the bill at (if least informally) in part because opposed the it and Senate-passed succeeded version in having of the it removed bill. For from cxplana- the tion 7099, of 96th intent, Congress, see Report Report to Accompan)~ No. 96-959 (Gov- H.K. ernment 1980). Printing Office, Washington, D.C.,

J. (Harvard Bronowski, Univ. The Press, Common Cambridge, Senae Mass., of Science 19781, 28. p. We 143. thank A. V. Applegate for substantial assist- ance in the preparation of this paper.

To account for the manifest existence of ior, cooperation such as altruism and related and group restraint behav- in competition, evolutionary theory has re- cently These acquired extensions two are, kinds broadly, of extension. genetical kinship ory (4, theory 5). Most (3)of and the reciprocation recent activity, the- both ments in of field theory, work has and been in further on the develop- side of kinship. Formal approaches have varied, but a gene's-eye kinship theory view has of increasingly natural selection taken (6). mortal A gene, bearer in to effect, interests looks of beyond the poten- its tially in other immortal related set individuals. of its replicas If existing interac- tants are sufficiently closely related, al- research Dr. Axelrod scientist is a professor at the Institute of political ol' Public science Policy and Studies, Dr. Hamilton University is a professor of Michigan, of evolutionary Ann Arbor biology 48109. in Biological the Museum Sciences, of Zoology University and of Michigan. the Division of 1390 SCIENCE, VOL.^21 1,^27 MARCH^1981

Axelrod's tournaments

Dynamics of Axelrod's tournaments

Computer Tournaments

FIGURE 2 Simulated Ecological Success of the Decision Rules

rules in the population will become an even larger propor- tion of the environment of the other rules in the next gen- eration. At first, a rule that is successful with all sorts of rules will proliferate, but later as the unsuccessful rules dis- appear, success requires good performance with other suc- cessful rules. This simulation provides an ecological perspective be- cause there are no new rules of behavior introduced. It differs from an evolutionary perspective, which would al- low mutations to introduce new strategies into the envi- ronment. In the ecological perspective there is a changing distribution of given types of rules. The less successful rules become less common and the more successful rules prolif- erate. The statistical distribution of types of individuals changes in each generation, and this changes the environ-

TIT FOR TAT .14 0

.12 0

.08 0

.06 0

.04 0

.02 0

1 3 2 6 7, 9

10 4 11

18 5 Others 13 14,12,1 5 8 0 200 400 600 800 1000 GENERATIONS

PROPORTIO

F POPULATIO

Axelrod's tournaments

Evolutionary Optimisation?

In Axelrod's tournaments, pepole could submit strategies (code)

But we could also let the strategies evolve.

E.g.: considerNsteps memory for opponent's (and, eventually, focal player's) moves

2 Npossible input patterns! 22

N possible strategies... e.g.N= 4: 16 input patterns, 2^16 = 64kpossible strategies

Spatial Games

Spatial game dynamics: Meanies spreading among TFT

FIGURE 4 Meanies Spreading in a Population of TIT FOR TAT

INITIASITUATIOLN

Conclusions

GENERATION 1 GENERATION 7

GENERATION 14

strategy can resfst invasion even with the help of territori- ality. In such a case, the dynamics of the invasion process can sometimes be extremely intricate and quite fascinating to look at. Figure 4 shows an example of such an intricate pattern. It represents the situation of a single player who always defects invading a territorial population of indi- viduals using TIT FOR TAT. In this case, the shadow of 162

Spatial Games

Spatial game dynamics: Meanies spreading among TFT

The Social Structure of Cooperation

LEGEND: X= ALL D BLANK = TIT FOR TAT

the future has been made quite weak, as reflected in low value of the discount parameter, w = 1/3. The four payoff parameters have been selected to provide an illustration of the intricacies that are possible. In this case T = 56 , R = 29 , P = 6 , and S = 0.^4 With these values, figure 4 shows what happens after one, seven, fourteen, and nineteen gen- erations. The meanies colonize the original TIT FOR TAT population, forming a fascinating pattern of long bor- ders and bypassed islands of cooperators. Another way of looking at the effects of territoriality is to investigate what happens when the players are using a wide variety of more or less sophisticated strategies. A con-

163

GENERATION 19

10 / 13

Spatial Games

Games on di↵erent levels(Putnan 1988, Internat. Organization 42, 427

Diplomacy and domestic politics: the logic of two-level games Robert D. Putnam

Introduction: the entanglements of domestic and international politics

Domestic politics and international relations are often somehow entangled, but our theories have not yet sorted out the puzzling tangle. It is fruitless to debate whether domestic politics really determine international relations, or the reverse. The answer to that question is clearly "Both, sometimes." The more interesting questions are "When?" and "How?" This article offers a theoretical approach to this issue, but I begin with a story that illustrates the puzzle. One illuminating example of how diplomacy and domestic politics can become entangled culminated at the Bonn summit conference of 1978.' In the mid-1970s, a coordinated program of global reflation, led by the "lo- comotive" economies of the United States, Germany, and Japan, had been proposed to foster Western recovery from the first oil shock.^2 This proposal

An earlier version of this article was delivered at the 1986 annual meeting of the American Political Science Association. For criticisms and suggestions, I am indebted to Robert Axelrod, Nicholas Bayne, Henry Brady, James A. Caporaso, Barbara Crane, Ernst B. Haas, Stephan Haggard, C. Randall Henning, Peter B. Kenen, Robert 0. Keohane, Stephen D. Krasner, Jacek Kugler, Lisa Martin, John Odell, Robert Powell, Kenneth A. Shepsle, Steven Stedman, Peter Yu, members of research seminars at the Universities of Iowa, Michigan, and Harvard, and two anonymous reviewers. I am grateful to the Rockefeller Foundation for enabling me to complete this research.

The following account is drawn from Robert D. Putnam and C. Randall Henning, "The Bonn Summit of 1978 : How Does International Economic Policy Coordination Actually Work?" Brookings Discussion Papers in International Economics, no. 53 (Washington, D.C.: Brookings Institution, October 1986), and Robert D. Putnam and Nicholas Bayne, Hanging Together: Cooperation and Conflict in the Seven-Power Summits, rev. ed. (Cambridge, Mass.: Harvard University Press, 1987), pp. 62-94.
Among interdependent economies, most economists believe, policies can often be more effective if they are internationally coordinated. For relevant citations, see Putnam and Bayne, Hanging Together, p. 24.

")!" $" &&#%""$

&&#%))) $"$ "$

!( $%&*"$ !

%'&&"& $"$ &$ %"'%(&

&&#%))) $"$ " $&$ %

11 / 13

Cyclic Games

Rock-Paper-Scissors (RPS)

other names: yanken, yan-ken-pon,ro-cham-beau, sansukumi-ken (jp. san=three-way, sukumi=deadlock, ken=fist-game)

https://en.wikipedia.org/wiki/Rock_paper_scissors

Dates back to the Han dynasty (2400y ago)

RPS is a zero-sum game (but can be modified)

Biology: Lizard mating strategies, Bacteria (Colicin warfares)

AllD, AllC and TFT can also form a cyclic game!

12 / 13

Cyclic games

(

,0)

(

,1)

(

,-1)

(

,-1)

(

,0)

(

,1)

(

,1)

(

,-1)

(

,0)

Autocatalytic cycles: Lotka-Volterra dynamics

−→

̇a

(

−

)

̇b

(

−

)

̇c

(

−

)

 

−

 

Cyclic coevolution: Side-blotched Lizards

(Uta stansburiana)

Cyclical games: Lizards "playing" a rock-scissors-paper g

ame

Orange-throated

males establish large territories holding several females

Can be invaded by

yellow-striped males ("sneakers")

,notcontributingtodefense

Yellow-striped

populations can be invaded by

blue-striped

males,

which defend a small territory that can hold one female and de

fend it against sneakers.

Once sneakers are rare, i.e.

blue-striped

have taken over,

it is advantageous to defend a

large territory holding several females.

This allows for cyclic invasion

→

Zamudio & Sinervo, PNAS 97, 14427 (2000), Sinervo & Lively, N

ature 380, 240 (1996).

Evolutionary game theory: Biological evidence

back

Cyclic coevolution: E.coli "play" RPS in vitro

Kerr, Riley, Feldman, Bohannan,Nature 428, 412 (2004)

Cyclic coevolution: E.coli "play" RPS in vitro ... and in mic

e!

−

Kerr, Riley, Feldman, Bohannan,Nature 428, 412 (2004)

Kirkup and Riley,Nature 428, 412 (2004)

Cyclic games

"Chemical warfare between microbes promotes biodiversity

(Cz

́ar

́an, Hoekstra, Pagie, PNAS 99, 786 (2001))

RSP replicator dynamics -

same, but small interaction cost -

spatial system

Evolutionary cycles of cooperation, defection, and Tit-fo

r-Tat

(Imhof, Fudenberg, Nowak, PNAS 102, 10797 (2005))

Stability of evolutionary cycles: Possible mechanisms? What determines the (in)stability of the fixed point (=coexi

stence)?

Payoff

(fitness) values

(for non-zero-sum games)

Spatial

structure (stabilizes coexistence)

Finiteness

of population (usually destabilizes coexistence)

Dynamics of the

(microscopic)

interaction process

(and resulting replicator equations)

What happens in reality?

E.coli (mixed system): Fixates to border.Lizards: damped oscillations

→

stable fixed point.

Social strategies: Many strategies do coexist.Mating (& parental care) behaviour: Fixates to border (typi

cally).

Biodiversity in cyclic coevolution: Role of non zero-sum games

For E.coli, it is costly when two different strategies meet (s>1 or det<0). Aspatialsystemisrequiredforcoexistence.
For the Lizard system, det > 0 ⇒ gross benefit when two different strategies meet is positive.

Is "space" necessary?

Take - Home: TFT and Cyclic Games

Iterated games: one attempt to explain cooperation
Axelrod's computer tournaments
Framework for agent-based modeling of socio-economic dynamics
Cyclic games: coexistence of strategies
More strategies allow for mor complex behaviour

Evolutionary Dynamics

Gametheory​

Evolutionary game theory: Maynard Smith 1973​

Moran evolution dynamics in 2×2 games​

The Moran Process​

The​

frequency-dependent​

MoranProcess​

Nindividuals​

Choose an individual at random​

Create identical offspringRemove one random individual​

Prisoner's Dilemma: TFT

I

I ma game. The payoff to player

i defect​

Iterated Prisoner's Dilemma: TFT

Iterated Prisoner's Dilemma: TFT

Prisoner's Dilemma: TFT

Dynamics of Axelrod's tournaments

Evolutionary Optimisation?

Spatial game dynamics: Meanies spreading among TFT

Spatial game dynamics: Meanies spreading among TFT

Games on di↵erent levels(Putnan 1988, Internat. Organization 42, 427

Cyclic Games

Cyclic games​

Autocatalytic cycles: Lotka-Volterra dynamics​

Cyclic coevolution: Side-blotched Lizards​

Evolutionary game theory: Biological evidence​

Cyclic coevolution: E.coli "play" RPS in vitro​

Cyclic coevolution: E.coli "play" RPS in vitro ... and in mic​

e!​

Cyclic games​

Evolutionary cycles of cooperation, defection, and Tit-fo​

r-Tat​

Stability of evolutionary cycles: Possible mechanisms? What determines the (in)stability of the fixed point (=coexi​

Biodiversity in cyclic coevolution: Role of non zero-sum games​

Take - Home: TFT and Cyclic Games​

Gametheory

Evolutionary game theory: Maynard Smith 1973

Moran evolution dynamics in 2×2 games

The Moran Process

The

frequency-dependent

MoranProcess

Nindividuals

Choose an individual at random

Create identical offspringRemove one random individual

i defect

Cyclic games

Autocatalytic cycles: Lotka-Volterra dynamics

Cyclic coevolution: Side-blotched Lizards

Evolutionary game theory: Biological evidence

Cyclic coevolution: E.coli "play" RPS in vitro

Cyclic coevolution: E.coli "play" RPS in vitro ... and in mic

e!

Cyclic games

Evolutionary cycles of cooperation, defection, and Tit-fo

r-Tat

Stability of evolutionary cycles: Possible mechanisms? What determines the (in)stability of the fixed point (=coexi

Biodiversity in cyclic coevolution: Role of non zero-sum games

Take - Home: TFT and Cyclic Games