Evolutionary game theory applies game theory concepts to model evolutionary processes. John Nash expanded on early game theory work. George Mennard Smith and John Prise applied game theory to evolution, founding evolutionary game theory. The hawk-dove game models aggression between two animals. When benefits of aggression exceed costs, the hawk strategy dominates; when costs exceed benefits, no single strategy dominates. Mutation, kin selection, and weak altruism can also be modeled. Evolutionary game theory helps explain the evolution of cooperation through concepts like kin selection and altruism.

1. Evolutionary Game Theory
Manik Tembe

2. Game Theory (द् युत सिधददांत)
• Game Theory was formally invented by John Von Neumann and Oskar
Margaret in 1944.
• Mathematician John Nash gave significant extension of the work
done by these two.
• In 1973, George Mennard Smith and John Prise used the concept of
game in theory of evolution and a new branch “ Evolutionary Game
Theory” was invented.

3. Hawk and Dove Game
Player 2
Player 1 ििदणद चदल Hawk कबुतर चदल Dove
ििदणद चदल Hawk (½(b-c) , ½(b-c) ) ( b, 0 )
. कबुतर चदल Dove ( 0, b ) ( b/2 , b/2 )

4. Hawk and Dove Game with b=10 and c=6
Player 2
Player 1 Hawk Dove
Hawk ( 2, 2 ) ( 10, 0 )
Dove
(0, 10 ) ( 5, 5 )

5. Nash Equilibrium
• It it s

6. Nash Equilibrium
• Nash equilibrium is a concept in game theory which determines the
optimal solution in a non-cooperative game in which each player
lacks any incentive to change his/her initial strategy.
• Under Nash equilibrium a player does not gain anything by deviating
from their initially chosen activity.
• It is the most common way to determine solution in non- cooperative
game.
• It is named after mathematician John Nash.

7. Hawk and Dove game with b less than c.
Let b=6 and c=10
Player 2
Hawk Dove
(-2, -2 ) (6, 0 )
(0, 6) ( 3, 3 )

8. If b is less than c, there is no single
equilibrium.
• If player 2 plays hawk then player 1 will get -2 by playing hawk and 0
by playing dove. So he will select dove.
• If player 2 selects dove then player 1 will get 6 by selecting hawk and
3 by selecting dove. So he will select hawk.
• Strategy of player 1 depends on what player 2 selects.
• There is no unique equilibrium.

9. Use of probability.
• Assume that probability of player 2 selecting hawk is p, then
probability of him selecting dove is 1-p.
• Now if player 1 selects hawk, his expected payoff will be
• E( player 1 selecting hawk)= p(-2)+(1-p)6=6-8p
• Similarly if player 1 selects dove, his expected payoff is
• E(player 1 selecting dove)= p(0)+(1-p)3=3-3p
• Now player 1 will select hawk if 6-8p is greater than 3 -3p. This
means if p is less than 3/5. Otherwise he will select dove.

10. Evolutionary game theory
• In evolutionary game theory players are not rational human beings but they are
animals, insects, birds in short any living organisms.
• In evolutionary game theory the payoff is counted in terms of fitness. If a
particular animal wants to survive in the evolution process then it should be
able to reproduce more of it’s own replicas . Hence payoff is calculated in terms
of reproduction ability.
• In regular game theory, players take rational decision by calculating benefits
but in evolutionary game theory players cannot think or calculate instead the
decision is taken by the program which is stored in their genes. According to
Darwin’s rule only successful genes survive in evolutionary process.
• Thus the model of regular game can also represent the evolutionary game.

11. b=6 and c=10 model in evolutionary game
theory
• Assume that in animal kingdom there are two types of species, one is aggressive
using hawk strategy and the other one prefers dove strategy.
• We can apply our previous calculation which tells us that if the proportion of hawk
type animals is less than 3/5 of total population then they will be more successful.
In the evolution process their population will go on increasing. And when the
population of hawk type animals exceeds 3/5 th of total population, dove type will
be more successful and their population and proportion will go on increasing
bringing down the proportion of hawk.
• In general, there will be approximately 3/5th of population of Hawks and remaing
will be dove type in this mixed group.

12. Evolutionary game
• If we go back to our previous example of b=6and c=10 and apply it to
mixed group of dove type submissive and hawk type aggressive
animals staying together, we can conclude that if the proportion of
hawk type animals is less than 3/5 this of total number of animals in
the group then they will be more successful and their population will
go on increasing. When it exceeds 3/5 th, dove type will be more
successful and their population will start increasing.
• In general, the group will be having approximately 3/5 hawk type and
2/5 dove type animals in the mixed group.

13. Effect of mutation.
• Game theory helps in finding out effect of mutation
Assume that in a group of small beetles due to mutation few large
size beetles are born.
• These large size beetles when they fight with small beetles will get
more share because of their bigger size.
• Large size beetles will need more food as compared to small size
beetles or in other words if they get the same amount of food as that
of small beetles then they will get less fitness.
• When large beetles fight with large beetles they will get more injured.

14. Game of size
Large Small
Large ( 4, 4 ) (10, 1 )
small
(1, 10 ) (6, 6 )

15. Game of size
• Assume that the proportion of big beetles is p so the proportion of
small beetles is 1-p
• Expected benefit of large beetles is = 4p+(1-p)10=10-6p.
• Expected benefit of small beetles is = 6(1-p)+ 1p=6-5p
• P is between o and 1 so 10-6p is always greater than 6-5p.
• This means the larger size beetles will always be more successful and
their number will go on increasing and slowly small size beetles will
get distroyed.

16. Altruism ( परदर्थवृत्ती)
• In environmental biology an organism is said to behave altruistically if
that behavior benefits other organism at the cost of itself.
• Benefit is measured in terms of reproductive fitness or expected
number of offsprings.
• This type of behavior is very common in animal kingdom.
• Vampair bats, vervet monkeys, ants, vasps, bees are the examples.

17. Altruism and Darwin’s theory
• From Darwinian point, the existence of altruism in nature is at first
sight puzzling.
• Natural selection leads us to expect animals to behave in ways to
increase their own chance of survival and not of others.
• Darwin was himself aware of this problem and in his book “ Descent
of Man” (1871) suggested the idea of “group selection” to resolve the
contradiction.

18. Group selection
• If natural selection works at group level and not at an individual level
then group consisting of individuals who are ready to sacrifice for
each other will definitely be stronger than a group consisting of all
selfish individuals.
• This theory received opposition since the beginning.
• G.C.Wiliams (1966) and J. Maynard Smith(1964) proved that this
theory cannot be true since within the group selfish animals will be
more successful since they will get help from altruistic animals.
Population of selfish animals will go on increasing and slowly altruistic
animals will get distroyed.

19. Kin Selection( आप्तेष्दांची सिवड)
• Kin selection theory states that altruistic animals prefer helping their
own relatives rather than strangers.
• Close relatives share same type of genes. So when the recipient
getting help will become stronger and his genes having similar
characteristics of altruism will survive in evolution process.
• Darwin himself has discussed this point in his famous book “On the
origin of species” .(1832).

20. Hamilton’s Rule
• In 1964, W.D.Hamilton popularized the theory of Kin selection.
• He stated that if rxB is greater than C the altruistic genes will be
carried into next generation.
• R= genetic relatedness coefficient.
• B= benefit received by the recipient.
• C= cost paid by the animal performing altruistic action.
• R= ½ between father-mother and children and between siblings.
• R=1/4 between grandparents and grandchildren.
• R=1/8 between cousins.

21. Hamilton Rule
• rB should be greater than C so if r is big enough or in other
words the recipient is closely related then the chances of the
altruistic genes getting selected in next generation are more.
• Most of the animals can recognize their relative by their body
smell.
• Most of the times animals will help other animals who are
staying close to them and the chances of such animals being
their close relatives are more.
• Even if animals are not capable of calculate whether rB is
greater than C or not, unknowingly they help their close
relatives to help.
• Kin selection can be modelled using game theory.

22. Kin selection game
Altruistic Selfish
Altruistic ( 11, 11 )
(
(0, 20)
Selfish ( 20, 0 ) ( 5, 5 )

23. Kin selection
• If animals are selecting with whom to interact at random then selfish
animals will obviously be more successful.
• But if probability that Altruistic interact only with altruistic and selfish
interact only with selfish is say 1 and probability of altruistic and
selfish interacting with each other is 0 then the expected benefit of
selfish animal is 5 where as expected benefit of altruistic animal is 11
so altruistic genes will survive.

24. Weak Altruism
• There is one more concept called “ weak altruism” in theory of
evolution.
• An act by which the animal performing it gets some benefits but
other animals of the same type get more benefits is called weak
altruistic act. Example: an act by which the animal will get say 10
offsprings but others will get 15 offsprings.
• Weak altruistic act will increase absolute fitness but reduce relative
fitness.

25. Weak Altruism game
WA S
WA (30, 30 ) (10, 20 )
S
( 20, 10 ) (0, 0 )

26. Thank you.
• manik_tembe@yahoo.com
• Mob. No. 9920031613