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EFFICIENCY & EQUILIBRIUM
Lecture 2 : Games in Economics and Evolution
Wayne Lawton
Department of Mathematics
National University of Singapore
[email protected]
http://www.math.nus.edu.sg/~matwml
SPS2171 Presentation 28/01/2008
Industrialization and Economic Theory
The Industrial Revolution was a period in the late
18th and early 19th centuries when major changes in
agriculture, manufacturing, and transportation had a
profound effect on socioeconomic and cultural
conditions in Britain and subsequently spread
throughout Europe and North America and eventually
the world.
Classical – Value and Distribution
1776 Adam Smith - The Wealth of Nations
Neoclassical – Equilibrium
1871 William Stanley Jevon - Theory of Political Economy
1871 Carl Menger - Principles of Economics
1874 Leon Walrus - Elements of Pure Economics
http://www.rjc.edu.sg/subjects/economics/Economists/adam%20smith.htm
Quantifying Demand
increasing utility
constant utility curves
constant cost line
Consumers : Maximize Utility within Cost Constraint
Quantifying Supply
constant cost line
constant output curves
200 units
of output
Producers : Maximize Output within Cost Constraint
Supply and Demand
Utility/Production determine Demand/Supply
Intersections determine Equilibrium Prices
Game Theory History
The field of game theory came into being with the 1944 classic
Theory of Games and Economic Behavior by John von Neumann and
Oskar Morgenstern. A major center for the development of game theory
was RAND Corporation where it helped to define nuclear strategies.
Game theory has played, and continues to play, a large role in the
social sciences, and is now also used in many diverse academic fields.
Beginning in the 1970s, game theory has been applied to animal
behaviour, including evolutionary theory. Many games, especially the
prisoner’s dilemna, are used to illustrate ideas in political science and
ethics. Game theory has recently drawn attention from computer
scientists because of its use in artificial intelligence and cybernetics.
Extensive Form
In the game pictured here, there are two players. Player 1 moves first and
chooses either F or U. Player 2 sees Player 1's move and then chooses
A or R. Suppose that Player 1 chooses U and then Player 2 chooses A,
then Player 1 gets 8 and Player 2 gets 2.
Normal Form
Player 2
chooses Left
Player 1 4, 3
chooses Up
Player 1 0, 0
chooses Down
Player 2
chooses Right
–1, –1
3, 4
Suppose that Player 1 plays Up and that Player 2 plays Left. Then
Player 1 gets a payoff of 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player
acts simultaneously or, at least, without knowing the actions of the other.
If players have some information about the choices of other players, the
game is usually presented in extensive form.
Zero Sum Game
Player 2
chooses Left
Player 1 -1, 1
chooses Up
Player 1 0, 0
chooses Down
Player 2
chooses Right
3, –3
-2, 2
In zero-sum games the total benefit to all players in the game, for every
combination of strategies, always adds to zero (more informally, a player
benefits only at the equal expense of others). Poker exemplifies a zerosum game (ignoring the possibility of the house's cut), because one wins
exactly the amount one's opponents lose. Other zero sum games include
matching pennies and most classical board games including Go and
chess.
Mixed Strategy
Player 2
chooses Left
Player 1 -1, 1
chooses Up
Player 1 0, 0
chooses Down
Player 2
chooses Right
3, –3
-2, 2
This example shows that player 1 benefits by keeping player 2 ignorant
about what row she / he chooses through a mixed strategy that consists
of choosing rows Up and Down randomly with probabilities U and D
chosen to maximize the expected payoff to player 1.
Question : What values should player 1 choose for U and D ?
Expected Payoffs
U = probability
that Player 1
chooses Up
D = probability
that Player 1
chooses Down
Payoff to player 1
Payoff to player 2
Player 2
chooses Left
Player 2
chooses Right
-1, 1
3, –3
0, 0
-2, 2
P1L(U)= -U
P2L(U)= U
P1R(U)= 5U-2
P2R(U)= 2-5U
The expected payoffs depend on the column that player 2 chooses.
Note that in our derivations we use the fact that U + D = 1.
Question : what column will player 2 choose (she / he knows U) ?
Expected Payoffs P2L and P2R to Player 2
P2L(U) = U
P2R(U) = 2-5U

U  13
Answer : player 2 chooses Left if
U
U  13 , and Right if U  13 ,
P 2(U )  max{U ,2  5U }
Player 1 minimizes P 2(U ) by choosing U  13 so P1( 13 )   13
to obtain a payoff
Expected Payoffs P2L and P2R to Player 2
P2L(U) = U
P2R(U) = 2-5U

U  13
Answer : player 2 chooses Left if
U
U  13 , and Right if U  13 ,
P 2(U )  max{U ,2  5U }
Player 1 minimizes P 2(U ) by choosing U  13 so P1( 13 )   13
to obtain a payoff
Minimax Theorem
An identical argument shows that player 2 should apply a mixed strategy
by choosing the left column with probability L = 5/6 and choosing the
right column with probability R = 1/6. Then the expected payoff to
player 2 equals P2 = 1/3 and the expected payoff to player 1 equals
P1 = -1/3. These are the same values obtained previously !
This amazing result is not an coincidence but rather a consequence of :
The fundamental theorem of game theory which states that every finite,
zero-sum, two-person game has optimal mixed strategies. It was proved
by John von Neumann in 1928.
Prisoner’s Dilemma
The Prisoner's Dilemma was originally framed by Merrill Flood and
Melvin Dresher working at RAND in 1950. Albert W. Tucker formalized
the game with prison sentence payoffs and gave it the "Prisoner's
Dilemma" name (Poundstone, 1992).
Prisoner B Stays Silent
Prisoner A Stays Silent
Prisoner A Betrays
Prisoner B Betrays
Each serves six months
Prisoner A serves ten years
Prisoner B goes free
Prisoner A goes free
Prisoner B serves ten years
Each serves five years
Betraying is a dominant strategy. The other prisoner reasons similarly,
and therefore also chooses to betray. Yet by both defecting they get a
lower payoff than they would get by staying silent. So rational, selfinterested play results in each prisoner being worse off than if they had
stayed silent.
Nash Equilibrium
In game theory, the Nash equilibrium (named after John Forbes Nash,
who proposed it) is a solution concept of a game involving two or more
players, in which no player has anything to gain by changing only his or
her own strategy unilaterally. If each player has chosen a strategy and no
player can benefit by changing his or her strategy while the other players
keep theirs unchanged, then the current set of strategy choices and the
corresponding payoffs constitute a Nash equilibrium.
Proof of Existence. As above, let σ − i be a mixed strategy profile of all players
except for player i. We can define a best response correspondence for player i, bi. bi is a
relation from the set of all probability distributions over opponent player profiles to a
set of player i's strategies, such that each element of bi(σ − i) is a best response to σ − i.
Define ...One can use the Kakutani fixed point theorem to prove…QED
When Nash made this point to John von Neumann in 1949, von
Neumann famously dismissed it with the words, "That's trivial, you
know. That's just a fixed point theorem."
A Beautiful Mind
John Forbes Nash, Jr., (born June 13,
1928), is an American mathematician
who works in game theory, differential
geometry, and partial differential
equations, serving as a Senior Research
Mathematician at Princeton University.
He shared the 1994 Nobel Prize in
Economics with two other game
theorists, Reinhard Selten and John
Harsanyi. He is the subject of the
Hollywood movie, A Beautiful Mind,
about a mathematical genius and his
struggles with schizophrenia.
Evolution
Zhuangzi was an influential Chinese philosopher
who lived around the 4th century BCE during the
Warring States Period, corresponding to the Hundred
Schools of Thought philosophical summit of Chinese
thought. In Chapter 18 of the Taoist book (named
after him) Zhuangzi also mentions life forms have an
innate ability or power to transform and adapt to their surroundings.
While his ideas don't give any solid proof or mechanism of change such
as Alfred Wallace and Charles Darwin, his idea about the transformation
of life from simple to more complex forms is along the same line of
thought. Zhuangzi further mentioned that humans are also subject to this
process as humans are a part of nature. Zhuangzi also mentions life
forms have an innate ability or power to transform and adapt to their
surroundings. While his ideas don't give any solid proof or mechanism
of change such as Alfred Wallace and Charles Darwin, his idea about the
transformation of life from simple to more complex forms is along the
same line of thought. Zhuangzi further mentioned that humans are also
subject to this process as humans are a part of nature.
Evolution
Charles Robert Darwin (12 February 1809 –
19 April 1882) was an English naturalist. After
becoming eminent among scientists for his field
work and inquiries into geology, he proposed and
provided scientific evidence that all species of
life have evolved over time from one or a few
common ancestors through the process of natural
selection. The fact that evolution occurs became
accepted by the scientific community and the
general public in his lifetime, while his theory of
natural selection came to be widely seen as the
primary explanation of the process of evolution
in the 1930s, and now forms the basis of modern
evolutionary theory. In modified form, Darwin’s
scientific discovery remains the foundation of
biology, as it provides a unifying logical
explanation for the diversity of life.
Evolution
Alfred Russel Wallace OM, FRS (8 January
1823 – 7 November 1913) was a British
naturalist, explorer, geographer, anthropologist
and biologist. He did extensive fieldwork first in
the Amazon River basin, and then in the Malay
Archipelago, where he identified the Wallace line
dividing the fauna of Australia from that of Asia.
He is best known for independently proposing a
theory of natural selection which prompted
Charles Darwin
to publish his
own more
developed
and researched
theory sooner
than intended.
Evolution
The modern evolutionary synthesis refers to a set of ideas from several
biological specialities that were brought together to form a unified theory
of evolution accepted by the great majority of working biologists. This
synthesis was produced over a period of about a decade (1936–1947) and
was closely connected with the development from 1918 to 1932 of the
discipline of population genetics, which integrated the theory of natural
selection with Mendelian genetics.
Julian Huxley invented the term, when he summarized the ideas in his
book, Evolution: The Modern Synthesis in 1942. Though the 'Modern
Synthesis' is the basis of current evolutionary thinking, it refers to a
historical event that took place in the 1930s and 1940s. Major figures in
the development of the modern synthesis include R. A. Fisher,
Theodosius Dobzhansky, J.B.S. Haldane, Sewall Wright, Julian Huxley,
Ernst Mayr, Bernhard Rensch, Sergei Chetverikov, George Gaylord
Simpson, and G. Ledyard Stebbins.
Evolutionary Stable Strategies
Unlike economics, the payoffs for games in biology are often interpreted
as corresponding to fitness. In addition, the focus has been less on
equilibria that correspond to a notion of rationality, but rather on ones
that would be maintained by evolutionary forces. The best known
equilibrium in biology is known as the Evolutionary stable strategy or
(ESS), and was first introduced by John Maynard Smith (described in his
1982 book). Every ESS is a Nash equilibrium (but not the converse).
Game theory was first used to explain the evolution (and stability) of the
approximate 1:1 sex ratios. Ronald Fisher (1930) suggested that the 1:1
sex ratios are a result of evolutionary forces acting on individuals who
could be seen as trying to maximize their number of grandchildren.
Biologists have used the hawk-dove game (also known as chicken) to
analyze fighting behavior and territoriality.
Hawk
Hawk v−c, v−c
Dove 0, 2v
Dove
2v, 0
v, v
Battle of the Sexes
Dawkins (1976) considers the following imaginary game. Suppose that
the successful raising of an offspring is worth +15 to each parent. The
cost of raising an offspring is -20, which can be borne one parent only,
or shared equally between two. The cost of a long courtship is -3 to both
participants. Females can be 'coy' or 'fast'; males can be 'faithful' or
'philanderer'. Coy females insist on a long courtship, whereas fast
females do not; all females care for offspring they produce. Faithful
males are willing, if necessary, to engage in a long courtship, and also
care for the offspring.
Philanderers are not
prepared to engage in a
long courtship, and do not
care for their offspring.
With this assumption, the
payoff matrix is shown in
Table 23.
Battle of the Sexes
The characteristic feature of this matrix is its cyclical character. That is:
If females are coy, it pays males to be faithful.
If males are faithful, it pays females to be fast.
If females are fast, it pays males to philander.
If males philander, it pays females to be coy.
Thus we have come full circle.
Oscillations are certain, but whether they are divergent or convergent
will depend on details of the genetics. Dawkins’ game was an imaginary
one. Parker (1979), using explicit genetic models, has suggested that
similar cycles could arise from parent-offspring conflict.
Tutorial Problems
1. Describe the extensive form for the game Tic-tac-toe.
Hint: this will involve an inverted tree with 9 branches for the 1st
player etc resulting in huge tree with 9 factorial nodes, you can use
some symmetry to significantly simplify the tree however.
2. Derive the assertions in the top paragraph in slide #14.
3. Explain what is meant by tit-for-tat and discuss its relationship
to reciprocal altruism in biology.
4. * Compute the fractions of faithful and philanders in a
population of men as a function of the fractions of coy and fast
females in the population.
Hint: treat the fractions as probabilities in a mixed strategy and then
compute a Nash equilibria
* Warning : this problem may induce hyper-cognitive activity