Transcript Lecture 17

5. Alternative Approaches
1. Introduction
2. Individual Decision Making
3. Basic Topics in Game Theory
4. The Theories and the Real World
5. Alternative Approaches
5.1 Psychology
5.2 Evolution
5.3 Artificial Intelligence
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5.1 Psychology
Remember the main assumptions in Game Theory
Rationality
Players aim to maximize their payoffs
They are selfish, they only care about their payoffs
Players are perfect calculators
Common Knowledge
Each player knows the environment and the rules of the game
Each player knows that each player knows the environment
and the rules
Each player knows that each player knows that each player knows
the environment and the rules
Etc. Etc. Etc.
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5.1 Psychology
Many are criticized since experimental evidence is in contradiction
Rationality
Players aim to maximize their payoffs
They are selfish, they only care about their payoffs
Players are perfect calculators
Common Knowledge
Each player knows the environment and the rules of the game
Each player knows that each player knows the environment
and the rules
Each player knows that each player knows that each player knows
the environment and the rules
Etc. Etc. Etc.
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5.1 Psychology
In general, experiments show that
People do not have (or do not use) all the information
People have limited memory
People have limited computational abilities
Therefore, people's observed behavior often does not match
the predictions of Rational Theories
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5.1 Psychology
In general, experiments show that
People do not have (or do not use) all the information
People have limited memory
People have limited computational abilities
Therefore, people's observed behavior often does not match
the predictions of Rational Theories
This does not mean that people is not rational !
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5.1 Psychology
In general, experiments show that
People do not have (or do not use) all the information
People have limited memory
People have limited computational abilities
Therefore, people's observed behavior often does not match
the predictions of Rational Theories
This does not mean that people is not rational !
People is “boundedly” rational
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Bounded Rationality
The term is thought to have been coined by Herbert Simon.
In Models of My Life, Herbert Simon points out that most people
are only partly rational, and are in fact emotional/irrational in
the remaining part of their actions. In another work, he states
"boundedly rational agents experience limits in formulating and
solving complex problems and in processing (receiving, storing,
retrieving, transmitting) information"
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Bounded Rationality
The term is thought to have been coined by Herbert Simon.
In Models of My Life, Herbert Simon points out that most people
are only partly rational, and are in fact emotional/irrational in
the remaining part of their actions. In another work, he states
"boundedly rational agents experience limits in formulating and
solving complex problems and in processing (receiving, storing,
retrieving, transmitting) information"
Experiments suggest that individual's behavior depends on
previous experiences, a process of “learning”, some intuition,
innovation, experimentation, . . .
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Alternative Approaches
Among others, the following incorporate some of the
criticisms to rationality
Evolution: Nature takes “good” decisions without
any rationality at all, without any computation
Artificial Intelligence: Allows for “full control” of the
degree of rationality,
computational power,
memory restrictions, etc
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1. Introduction
2. Individual Decision Making
3. Basic Topics in Game Theory
4. The Theories and the Real World
5. Alternative Approaches
5.1 Psychology
5.2 Evolution
5.3 Artificial Intelligence
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5.2 Evolutionary Game Theory
Darwinian evolution is based on three fundamental
principles:
reproduction,
mutation and
selection
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Species tend to perpetuate by reproducing themselves, but only those that better fit
in a changing environment will survive. Inheritance causes reproduction. Mutation is
the source of change and, in turn, the way of adapting to the changes. Natural selection
is the paradigm, the way nature uses to decide which species will survive and which
will not. Such a simple explanation of such a complex phenomenon is the main virtue
of that theory. It is not surprising, one might think, that other scientific disciplines
involving some sort of dynamics or evolution (like economics or history among many
others) try to approach their unsolved problems in a similar way. Nevertheless, the
fact is that Darwin's is not the only way to study evolution, not even the first
evolutionary theory to appear. As Hallpike (1985) points out, the theories of social
evolution do not come after Darwin's. Indeed, economists like Malthus, Marx or
Smith, to name only a few, all developed theories of social or economical evolution
independently of any contribution from biology. As a matter of fact, it was when
Darwin happened to read “for amusement” Malthus's Essay on Population that he
got the idea of natural selection (Barlow,1958).
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5.2 Evolutionary Game Theory
Was initiated by John Maynard-Smith and George Price in
1973 and continued by Maynard-Smith in 1981
Originally, it was an application of Game Theory to
Evolutionary Theory
Very soon, it became an application of Evolutionary
Theory to Game Theory
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5.2 Evolutionary Game Theory
Evolutionary Game Theory considers a population
of “organisms” that interact with each other.
The “genotype” of each organism is identified with
an strategy in Game Theory. When (two) organisms
interact, they use the strategies that correspond to
their genotypes and collect some “payoff”.
Organisms that “do well” (get high payoffs) pass the
test of “natural selection” and reproduce themselves.
Organisms that “do poorly” (get low payoffs) will be
invaded and displaced by new “mutant” organisms that
do better
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5.2 Evolutionary Game Theory
The core ingredient of Evolutionary Game Theory is
the so called Evolutionary Stable Strategy (ESS)
“A strategy such that if all members of the population adopt
it, then no mutant can invade the population under the
influence of selection”
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5.2 Evolutionary Game Theory
The core ingredient of Evolutionary Game Theory is
the so called Evolutionary Stable Strategy (ESS)
A genotype s is an Evolutionary Stable Strategy if:
Organism genotype
Population genotype
(1) U(s,s) ≥ U(s',s) for any other genotype s'
(2) If U(s,s) = U(s',s) for some other s', then
U(s,s') > U(s',s')
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A strategy s is an Evolutionary Stable Strategy if:
(1) U(s,s) ≥ U(s',s) for any other strategy s'
(2) If U(s,s) = U(s',s) for some other s', then
U(s,s') > U(s',s')
Condition (1) says that the strategy of the population (s)
is a Best Reply to itself (Equilibrium)
Condition (2) says first that if another organism (a
mutant s' ) can do as well as the population then it might
invade the population ! The second part of this condition says
that it that occurs then being the “original organism” s (not
the mutant s' ) has a higher payoff (it's better for natural
selection) (Stability)
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Example
Think of a population of lions that have the following
instructions embedded in their genotypes (strategy)
In the presence of another lion, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
In the presence of a zebra, chase it and share the
meal with the other lions and cabs
This appears to be a “reasonable” strategy for lions.
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Example
In the same population of lions consider the
following “mutant” with this information
In the presence of another lion, chase it and share
the meal with the other lions and cabs
In the presence of a zebra, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
Most likely, this sort of “mutation” will not be
successful at all !
It will not satisfy condition (1)
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Example
In the same population of lions consider a
different “mutant” with this information
In the presence of another lion, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
In the presence of a zebra, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
Most likely, this will satisfy condition (1)
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Example
In the same population of lions consider a
different “mutant” with this information
In the presence of another lion, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
In the presence of a zebra, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
But will NOT satisfy condition (2). If all the lions
were as this mutant, the whole population will
die of starvation !! In such case, it's better to behave
as the “original” lions
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Example: What does this have to do with
business or economics ?
Think of a population of firms that have the following
instructions embedded in their strategies (genotypes)
In the presence of a partner, act friendly unless
it is behaves aggressively. In such case you should
act aggressively as well
In the presence of a competitor, be aggressive and
share the meal with the other partners
This appears to be a “reasonable” strategy for firms.
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Example: What does this have to do with
business or economics ?
Darwinian evolution is based on three fundamental
principles:
reproduction,
mutation and
selection
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Example: What does this have to do with
business or economics ?
Darwinian Social evolution is based on three fundamental
principles:
reproduction, imitation
mutation and innovation (R&D)
selection competition
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Example: The Hawk-Dove Game
This is version of a Game of Chicken that is very useful in
evolutionary biology
The name "Hawk-Dove" refers to a situation in which two animals
compete for a shared resource and the contestants can choose either
conciliation or conflict.
V is the value of the contested resource, and C is the cost of an
escalated fight. It is (almost always) assumed that the value of the
resource is less than the cost of a fight is, i.e., C > V > 0
If the two animals behave in the same way, the split the resource.
Otherwise, the animal playing Hawk gets the whole resource
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The environment of the game
Players:
Animal 1 and Animal 2
Strategies: Dove (show you intention) or Hawk (attack)
Payoffs:
(see the table)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Conflict (anti-coordination)
Symmetric
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Animal 2
Dove
Dove
V/2 , V/2
Hawk
0,V
Animal 1
Hawk
V,0
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(V-C)/2 , (V-C)/2
The are 2
Nash Equilibria
in pure strategies
Dove
Animal 2
Dove
V/2 , V/2
Hawk
0,V
Animal 1
Hawk
V,0
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(V-C)/2 , (V-C)/2
In real like, some animals behave as doves while others
Are Hawks (mixed strategy !)
This example set the basis for a extremely fruitful
application of Game Theory to Evolutionary Biology
(John Maynard-Smith)
And vice versa, Evolutionary Theory can be applied to
Game Theory !
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Computation of the Evolutionary Stable Strategy
Suppose that V=2 and C=6
Animal 2
Dove
Dove
V/2 , V/2
Hawk
0,V
Animal 1
Hawk
V,0
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(V-C)/2 , (V-C)/2
Computation of the Evolutionary Stable Strategy
Suppose that V=2 and C=6
Animal 2
Dove
Dove
1,1
Hawk
0,2
Animal 1
Hawk
2,0
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove
Dove
1,1
Hawk
0,2
Animal 1
Hawk
2,0
-2 , -2
If all the population behaves like a Dove, it's better to be a Hawk
If all the population behaves like a Hawk, it's better to be a Dove
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Computation of the Evolutionary Stable Strategy
Animal 2
Dove
Dove
Hawk
1,1
0,2
Animal 1
Hawk
2,0
No strategy (genotype) satisfies condition (1) !! ??
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
Suppose that 2/3 of the population are Doves and 1/3
are Hawks
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
If I behave like a Dove, my expected payoff will be:
E(Dove)= (2/3)·( 1 ) + (1/3)· ( 0 ) = 2/3
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
If I behave like a Hawk, my expected payoff will be:
E(Hawk)= (2/3)·( 2 ) + (1/3)· ( -2 ) = 4/3 – 2/3 = 2/3
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
-2 , -2
Thus, if 2/3 of the population are Doves and 1/3 are Hawks, being
a Dove or a Hawk pays the same !!!
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Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
This IS the ESS:
2/3 of the population are Doves and 1/3 are Hawks
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
Notice !!!
Any other “genotype” will satisfy condition (1) !!!
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-2 , -2
Computation of the Evolutionary Stable Strategy
Animal 2
Dove (2/3)
Dove
Hawk (1/3)
1,1
0,2
Animal 1
Hawk
2,0
But if there are more that 2/3 of Doves, then it is better
to be a Hawk (will not satisfy condition (2))
And if there are more that 1/3 of Hawks then it is better
to be a Dove (will not satisfy condition (2))
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-2 , -2
Evolutionary Stable Dynamics
How evolution takes place can be studied dynamically
as well.
We might, for instance, say that the proportion of
organisms that play a given strategy evolves in the
following way:
If the payoff of such strategy is above the average
payoff of the population, then the proportion of
organisms using that strategy will increase
If the payoff of such strategy is below the average
payoff of the population, then the proportion of
organisms using that strategy will decrease
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Evolutionary Stable Dynamics
If the payoff of such strategy is above the average
payoff of the population, then the proportion of
organisms using that strategy will increase
If the payoff of such strategy is below the average
payoff of the population, then the proportion of
organisms using that strategy will decrease
Payoff of s
pt+1(s) = pt(s)·
Average Payoff
This is known as the “Replicator Dynamics”
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Evolutionary Stable Dynamics
If the payoff of such strategy is above the average
payoff of the population, then the proportion of
organisms using that strategy will increase
If the payoff of such strategy is below the average
payoff of the population, then the proportion of
organisms using that strategy will decrease
Payoff of s
pt+1(s) = pt(s)·
Average Payoff
Roughly speaking the solution of the “Replicator Dynamics” coincides
with the Evolutionary Stable Strategy
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Example: The Repeated Prisoners' Dilemma
Consider a population of organisms that play the Prisoners'
Dilemma repeatedly
For simplicity, suppose that there are only three types of
“genotypes”
(C) Always Cooperate
(D) Always Defect
(T) Tit-for-Tat
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Behavior of the “Replicator Dynamics”
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Summary
The main criticism form Psychologists and Experimentalist
to the “rational theories” is that individuals are only
“boundedly rational”
The ideas of Biological Evolution can be successfully
adapted to Social Evolution
Evolutionary Game Theory was initiated as an application
of Game Theory to Biological Evolution but soon became
an application of Biological Evolution to Game Theory
Evolutionary Game Theory is useful to single out those
equilibria that can be achieved without rationality
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