EFFICIENCY & EQUILIBRIUM Lecture 2 : Games in Economics and

Evolution

Wayne LawtonDepartment of Mathematics

National University of Singaporematwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwml

SPS2171 Presentation 28/01/2008

Industrialization and Economic Theory

1871 William Stanley Jevon - Theory of Political Economy 1871 Carl Menger - Principles of Economics1874 Leon Walrus - Elements of Pure Economics

Classical – Value and Distribution 1776 Adam Smith - The Wealth of

NationsNeoclassical – Equilibrium

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.

http://www.rjc.edu.sg/subjects/economics/Economists/adam%20smith.htm

Quantifying Demand

constant utility curves

Consumers : Maximize Utility within Cost Constraint

increasing utility

constant cost line

Quantifying Supply

Producers : Maximize Output within Cost Constraint

constant output curves

200 units of output

constant cost line

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 andOskar 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 andchooses 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 FormPlayer 2chooses Left

Player 2chooses Right

Player 1chooses Up

4, 3 –1, –1

Player 1chooses Down

0, 0 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

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 zero-sum 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.

Player 2chooses Left

Player 2chooses Right

Player 1chooses Up

-1, 1 3, –3

Player 1chooses Down

0, 0 -2, 2

Mixed Strategy

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.

Player 2chooses Left

Player 2chooses Right

Player 1chooses Up

-1, 1 3, –3

Player 1chooses Down

0, 0 -2, 2

Question : What values should player 1 choose for U and D ?

Expected Payoffs

The expected payoffs depend on the column that player 2 chooses. Note that in our derivations we use the fact that U + D = 1.

Player 2chooses Left

Player 2chooses Right

U = probability that Player 1

chooses Up

-1, 1 3, –3

D = probability that Player 1

chooses Down

0, 0 -2, 2

Payoff to player 1Payoff to player 2

P1L(U)= -U P2L(U)= U

P1R(U)= 5U-2 P2R(U)= 2-5U

Question : what column will player 2 choose (she / he knows U) ?

Expected Payoffs P2L and P2R to Player 2

Answer : player 2 chooses Left if

U

and Right if,31U ,3

1U31U

Player 1 minimizes31U

P2L(U) = U

P2R(U) = 2-5U

to obtain a payoff }52,max{)(2 UUUP )(2 UP by choosing so

31

31)(1 P

Expected Payoffs P2L and P2R to Player 2

Answer : player 2 chooses Left if

U

and Right if,31U ,3

1U31U

Player 1 minimizes31U

P2L(U) = U

P2R(U) = 2-5U

to obtain a payoff }52,max{)(2 UUUP )(2 UP by choosing so

31

31)(1 P

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 !

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.

This amazing result is not an coincidence but rather a consequence of :

Prisoner’s DilemmaThe 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 B Betrays

Prisoner A Stays Silent Each serves six months Prisoner A serves ten yearsPrisoner B goes free

Prisoner A Betrays Prisoner A goes freePrisoner 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, self-interested play results in each prisoner being worse off than if they had stayed silent.

Nash EquilibriumIn 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.

EvolutionZhuangzi 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 aninnate 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.

EvolutionCharles 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.

EvolutionAlfred 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 StrategiesUnlike 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).

Hawk Dove

Hawk v−c, v−c 2v, 0

Dove 0, 2v v, v

Biologists have used the hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.

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.

Battle of the SexesDawkins (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 alsocare 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 Problems1. 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