Introduction to Game Theory Evolutionary Game Theory

1. DATA MINING AND MACHINE LEARNING
IN A NUTSHELL
EVOLUTIONARY GAME THEORY
Mohammad-Ali Abbasi
http://www.public.asu.edu/~mabbasi2/
SCHOOL OF COMPUTING, INFORMATICS, AND DECISION SYSTEMS ENGINEERING
ARIZONA STATE UNIVERSITY
Arizona State University
http://dmml.asu.edu/
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2. What is a Game
• Game theory is concerned with situations in
which decision-makers interact with one
another,
• and in which the happiness of each participant
with the outcome depends not just on his or
her own decisions but on the decisions made
by everyone.
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3. What is game theory?
• Formal way to analyse interactions between
agents who behave strategically
• Mathematics of decision making in conflict
situations
• Usual to assume players are “rational”
• Widely applied to the study of economics,
warfare, politics, animal behaviour,
sociology, business, ecology and
evolutionary biology
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4. Basic elements of a Game
• Players
– Everyone who has an effect on your earnings
• Strategies
– Actions available to each player
– Define a plan of action for every contingency
• Payoffs
– Numbers associated with each outcome
– Reflect the interests of the players
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5. Nash Equilibrium
• A Nash equilibrium is a situation in which
none of them have dominant Strategy and
each player makes his or her best response
– (S, T) is Nash equilibrium if S is the best strategy to
T and T is the best strategy to S
• John Nash shared the 1994 Nobel prize in
Economic for developing this idea!
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6. Prisoner’s Dilemma
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7. Evolutionary Game Theory
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9. evolutionary stable strategy
• Taller trees get more light, so taller trees
reproduce more.
• Taller trees have to consume more resources
to be tall.
• System converges to a state where only the
tallest trees are present.
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10. Evolutionary stable strategy
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11. Evolutionary Game Theory
1 x1 x1
1 1/ r r…relative fitness of new mutant
n n…population size
1 1/ r
For a neutral mutant, r=1, the fixation probability is 1/n.
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12. Evolutionary Game Theory
• Is the attempt to invent and study
mathematical equations describing
how population change over time due
to mutation and selection (Learning).
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13. GT vs. EGT
• In GT, one assumes that agents are perfectly
rational.
• In EGT, trial and error process gives strategies
that can be selected for by some force
(evolution - biological, cultural, etc…).
• This lack of rationality is the point of
departure between EGT and GT.
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14. Evolutionary game theory
• population of players
• follow different strategies
• frequencies of strategies change over time,
dependent on success relative to other
strategies
• genetic inheritance (mutation) or learning
(innovation)
– Irrational
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15. Evolutionary Biology
Evolutionary biology is based on the idea that an
organism's genes largely determine its observable
characteristics, and hence its fitness in a given
environment.
Organisms that are more fit will tend to produce
more offspring, causing genes that provide
greater fitness to increase their representation in
the population.
In this way, fitter genes tend to win over time,
because they provide higher rates of
reproduction.
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17. Competition for food
• When beetles of the same size compete, they
get equal shares of the food
• When a large beetle competes with a small
beetle, the large beetle gets the majority of
the food.
• In all cases, large beetles experience less of a
fitness benefit from a given quantity of food,
since some of it is diverted into maintaining
their expensive metabolism
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18. Body Size Game
Beetle 2
Small Large
Small 5, 5 1, 8
Beetle 1
Large 8, 1 3, 3
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19. • Small portion of Large Beetles: x
• Small Beetles: 1-x
• Expected Pay off in a population that small is
majority
– Small beetle
• 5(1-x) + 1.X = 5- 4x
– Large beetle
• 8 (1- x) + 3.x = 8-5x
• Small is not evolutionary stable!
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20. • Expected Pay off in a population that Large is
majority
• Large Beetle: 3 * (1-x) + 8 * x = 3 + 5x
• Small Beetle: 1 * (1-x) + 5 * x = 1 + 4x
• Large is evolutionary stable
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24. General Description
Organism 2
S T
S a, a b, c
Organism 1
T c, b d, d
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25. • X -> T
• (1-x) -> S
• Play S
– Expected Payoff: a (1-x) + bx
• Play T
– Expected Payoff: C ( 1- x) + dx
A ( 1- x) + bx > c ( 1-x ) + dx
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26. • In a two-player, two-strategy, symmetric
game, S is evolutionarily stable precisely when
either
a>c
or
a = c and b > d.
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27. Relationship between evolutionary and Nash
Equilibria
• (S, S) is a Nash equilibrium when S is a best
response to the choice of S by the other player
a >= c
• The condition for S to be evolutionarily stable
a>c
Or
a = c and b > d
• If strategy S is evolutionarily stable, then (S, S) is a
Nash equilibrium
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28. • Other direction
– (S, S) is a Nash equilibrium -> S is not ESS
a = c and b < d
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29. Strict Nash Equilibrium
Hunt Hunter 2
Stag or Hare S H
S 4, 4 0, 3
Hunter 1
H 3, 0 3, 3
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30. Strict Nash Equilibrium
Hunt Hunter 2
Stag or Hare S H
S 4, 4 0, 4
Hunter 1
H 4, 0 3, 3
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31. Strict Nash Equilibrium
choice of strategies is a strict Nash equilibrium if
each player is using the unique best response to
what the other player is doing
for symmetric two-player, two-strategy games, the
condition for (S, S) to be a strict Nash equilibrium
is that a > c
the set of evolutionarily stable strategies S is a
subset of the set of strategies S for which (S, S) is
a Nash equilibrium
if (S, S) is a strict Nash equilibrium, then S is
evolutionarily stable
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32. Nash Equilibrium and Evolutionary Stability
• In a Nash equilibrium
– we consider players choosing mutual best responses
to each other's strategy
– This equilibrium concept places great demands on the
ability of the players to chose optimally and to
coordinate on strategies that are best responses to
each other.
• Evolutionary stability
– no intelligence or coordination on the part of the
players
– strategies are viewed as being hard-wired into the
players, perhaps because their behavior is encoded in
their genes
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33. Evolutionarily Stable Mixed
Strategies
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34. General Description
Player 2
S T
S a, a b, c
Player 1
T c, b d, d
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35. Evolutionarily Stable Mixed Strategies
Organism 1:
Play S with probability p and T with (1-p)
Organism 2:
Play S with probability q and T with (1-q)
V (p, q) = pqa + p(1-q)b + (1-p)qc + (1-p)(1-q)d
For p to be ESMS
(1-x)V(p, p) + xV(p, q) > (1-x) V(q, p) + xV(q, q)
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36. Evolutionarily Stable Mixed Strategies
• In the General Symmetric Game, p is an
evolutionarily stable mixed strategy if there is
a (small) positive number y such that when
any other mixed strategy q invades p at any
level x < y, the fitness of an organism playing p
is strictly greater than the fitness of an
organism playing q.
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37. Mohammad-Ali Abbasi (Ali),
Ali, is a Ph.D student at Data Mining
and Machine Learning Lab, Arizona
State University.
His research interests include Data
Mining, Machine Learning, Social
Computing, and Social Media Behavior
Analysis.
http://www.public.asu.edu/~mabbasi2/
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