aarjaneiro/planetmath huggingface dataset:
example of Nash equilibrium

example of Nash equilibrium

Consider the first two games given as

examples of normal form games

.

In Prisoner’s Dilemma the only

Nash equilibrium

is for both players to play

D

: it’s apparent that, no matter what player

1

plays, player

2

does better playing

D

, and vice-versa for

1

.

Battle of the Sexes has three Nash equilibria. Both

(

O

,

O

)

and

(

F

,

F

)

are Nash equilibria, since it should be clear that if player

2

expects player

1

to play

O

, player

2

does best by playing

O

, and vice-versa, while the same situation holds if player

2

expects player

1

to play

F

. The third is a mixed equilibrium; player

1

plays

O

with

2

3

probability and player

2

plays

O

with

1

3

probability. We confirm that these are equilibria by testing the

first derivatives

(if

0

then the

strategy

is either maximal or minimal). Technically we also need to check the second derivative to make sure that it is a

maximum

, but with simple games this is not really necessary.

Let player

1

play

O

with probability

p

and player

2

plays

O

with probability

q

.

u

1

⁢

(

p

,

q

)

=

2

⁢

p

⁢

q

+

(

1

-

p

)

⁢

(

1

-

q

)

=

2

⁢

p

⁢

q

-

p

-

q

+

p

⁢

q

=

3

⁢

p

⁢

q

-

p

-

q

u

2

⁢

(

p

,

q

)

=

p

⁢

q

+

2

⁢

(

1

-

p

)

⁢

(

1

-

q

)

=

3

⁢

p

⁢

q

-

2

⁢

p

-

2

⁢

q

∂

⁡

u

1

⁢

(

p

,

q

)

∂

⁡

p

=

3

⁢

q

-

1

∂

⁡

u

2

⁢

(

p

,

q

)

∂

⁡

q

=

3

⁢

p

-

2

And indeed the

derivatives

are

0

at

p

=

2

3

and

q

=

1

3

.

Title

example of Nash equilibrium

Canonical name

ExampleOfNashEquilibrium

Date of creation

2013-03-22 12:52:48

Last modified on

2013-03-22 12:52:48

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Example

Classification

msc 91A99

Generated on Sat Feb 10 13:37:40 2018 by

LaTeXML
examples of initial states in the Game of Life

examples of initial states in the Game of Life

Theoretically, the Game of Life is played on an

infinite

two-dimensional board. In practice, it is played on a finite board and only a small portion of the finite board is shown. Some implementations even use wrap-around (e.g., a spaceship

configuration

that moves horizontally to the right that hits

(

+

m

⁢

a

⁢

x

,

y

)

is moved to

(

-

m

⁢

a

⁢

x

,

y

)

).

In most of the examples here, whatever positions are not shown are assumed to be empty. Living cells will be represented by the letter “C” until I think of a better symbol or better graphics.

Another theoretical element of the game is that after the user sets the initial state of the game, no further input is accepted; the game proceeds entirely as a

consequence

of Conway’s rules. In practice, some implementations allow the user to stop a game in progress, add or delete some cells and resume the game.

The smallest possible ‘living’ configuration possible is a 2 by 2 square.

C

C

C

C

Since each of the four cells has exactly three neighbors, they live forever (any larger, filled-in square or rectangular configuration is doomed to extinction in two

iterations

). This might not seem terribly useful, even for modelling zero population growth, but for game aficionados, these static blocks can be useful as decorative

elements

or as “roadblocks” to keep a motile configuration from running outside of the visible board.

This configuration is slightly more interesting than the 2 by 2 square, but it still stays put in one place.

C

C

C

Initial state and even-numbered iterations (or odd-numbered iterations only)

C

C

C

Odd-numbered iterations (or initial state and even-numbered iterations)

Perhaps the most basic motile configuration is the glider. With the

orientation

shown below, it moves southwest indefinitely (unless it hits a block or a blinker that nudges it in a different direction or kills it altogether).

Figure 1:

The initial state of a glider.

There are many implementations of the Game of Life available in C++, Pascal,

Mathematica

(version 5.0 of which even provides a built-in command for it), etc. On the Web, Life is usually implemented in Java. It is possible to implement it in

JavaScript

of VBScript, but these days such implementations are likely to crash the browser.

Of the Java implementations I’ve seen, the one I like the best is

http://www.mirwoj.opus.chelm.pl/ca/mjcell/mjcell.html

Mirek’s Java Cellebration v.1.50. Select ”Life” from the top drop-down menu and ”Conway’s Life” from the second drop-down menu. You can draw your own configurations, start and stop, step through, etc. By default, the board is finite and wraps around.

Title

examples of initial states in the Game of Life

Canonical name

ExamplesOfInitialStatesInTheGameOfLife

Date of creation

2013-03-22 16:36:28

Last modified on

2013-03-22 16:36:28

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 91A99

Generated on Sat Feb 10 13:37:42 2018 by

LaTeXML
finite game

finite game

A

finite game

is a game in which each

instance

of the game is designed to reach a

conclusion

, either as a natural

consequence

of normal play, or due to the rules of the game.

For example, a game of Reversi is over when all the squares of the board have been filled up, if not sooner. Most chess games end with one side trapping the opponent’s king. Theoretically, certain scenarios could go on indefinitely if both sides lack the resources to trap the opponent’s king; for example, if White has only a king and a knight, and Black has only the king left, Black could theoretically evade indefinitely if it weren’t for the stalemate rule.

Title

finite game

Canonical name

FiniteGame

Date of creation

2013-03-22 16:35:08

Last modified on

2013-03-22 16:35:08

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 91A99

Generated on Sat Feb 10 13:37:44 2018 by

LaTeXML
game

game

In general, a

game

is a way of describing a situation in which

players

make choices with the intent of optimizing their

utility

. Formally, a game includes three features:

•

a set of pure strategies for each player (their strategy space)

•

a way of determining an

outcome

from the strategies selected by the players

•

a utility function for each player specifying their

payoff

for each outcome

Title

game

Canonical

name

Game

Date of creation

2013-03-22 12:51:19

Last modified on

2013-03-22 12:51:19

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

12

Author

Henry (455)

Entry type

Definition

Classification

msc 91A99

Related topic

CommonKnowledge

Related topic

NormalFormGame

Related topic

CompleteInformation

Related topic

Strategy2

Defines

outcome

Defines

game

Defines

player

Defines

utility function

Defines

payoff

Generated on Sat Feb 10 13:37:45 2018 by

LaTeXML
Game of Life

Game of Life

The

Game of Life

is a

cellular automaton

that models a population of living organisms living on a two-dimensional plane subdivided into squares. One cell may live in each square. John Horton Conway set down the rules of the game in

Scientific American

:

1.

If a cell has less than two neighbors alive in any of the eight adjacent squares (those immediately above and below, left and right, and those that touch corners diagonally), it dies.

2.

But if it has more than three live neighbors, it also dies.

3.

Having two or three neighbors, a cell lives on to the next generation.

4.

If an empty square has exactly three neighbors, a new cell is born there.

The rules are repeatedly applied, and one of two kinds of outcomes are possible: the entire population could die out, or the population settles into a periodic pattern that can go on infinitely.

Title

Game of Life

Canonical

name

GameOfLife

Date of creation

2013-03-22 16:35:11

Last modified on

2013-03-22 16:35:11

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 91A99

Synonym

Conway’s Game of Life

Generated on Sat Feb 10 13:37:47 2018 by

LaTeXML
game theory

game theory

Game theory

is the study of games in a formalized setting. Games are broken down into players and rules which define what the players can do and how much the players want each

outcome

.

Typically, game theory assumes that players are

rational

, a requirement which that players always make the decision which most benefits them based on the information available (as defined by that game), but also that players are always capable of making that decision (regardless of the amount of calculation which might be necessary in practice).

Branches of game theory include cooperative game theory, in which players can negotiate and enforce bargains and non-cooperative game theory, in which the only meaningful agreements are those which are “self-enforcing,” that is, which the players have an incentive not to break.

Many fields of mathematics (

set theory

, recursion theory, topology, and

combinatorics

, among others) apply game theory by representing problems as games and then use game theoretic techniques to find a solution. (To see how an application might work, consider that a proof can be viewed as a game between a ”prover” and a ”refuter,” where every

universal quantifier

represents a move by the refuter, and every existenial one a move by the prover; the proof is valid exactly when the prover can always win the corresponding game.)

Title

game theory

Canonical name

GameTheory

Date of creation

2013-03-22 12:51:55

Last modified on

2013-03-22 12:51:55

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Topic

Classification

msc 91A99

Generated on Sat Feb 10 13:37:48 2018 by

LaTeXML
Nash equilibrium

Nash equilibrium

A

Nash equilibrium

of a game is a set of (possibly mixed)

strategies

σ

=

(

σ

1

,

…

,

σ

n

)

such that, if each player

i

believes that that every other player

j

will play

σ

j

, then

i

should play

σ

i

. That is, when

u

i

is the

utility function

for the

i

-th player:

σ

i

≠

σ

i

′

→

u

i

⁢

(

σ

i

,

σ

-

1

)

>

u

i

⁢

(

σ

i

′

,

σ

-

1

)

∀

i

≤

n

⁢

and

⁢

∀

σ

i

′

∈

Σ

i

Translated, this says that if any player plays any strategy other than the one in the Nash equilibrium then that player would do worse than playing the Nash equilibrium.

Title

Nash equilibrium

Canonical name

NashEquilibrium

Date of creation

2013-03-22 12:51:35

Last modified on

2013-03-22 12:51:35

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 91A99

Generated on Sat Feb 10 13:37:54 2018 by

LaTeXML
Pareto dominant

Pareto dominant

An

outcome

s

*

strongly Pareto dominates

s

′

if:

∀

i

≤

n

[

u

i

(

s

*

)

>

u

i

(

s

′

)

]

An outcome

s

*

weakly Pareto dominates

s

′

if:

∀

i

≤

n

[

u

i

(

s

*

)

≥

u

i

(

s

′

)

]

s

*

is

strongly Pareto optimal

if whenever

s

′

weakly Pareto dominates

s

*

,

∀

i

≤

n

[

u

i

(

s

*

)

=

u

i

(

s

′

)

]

. That is, there is no

strategy

which provides at least as large a payoff to each player and a larger one to at least one.

s

*

is

weakly Pareto optimal

if there is no

s

′

such that

s

′

strongly Pareto dominates

s

*

.

Title

Pareto dominant

Canonical name

ParetoDominant

Date of creation

2013-03-22 12:51:32

Last modified on

2013-03-22 12:51:32

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 91A99

Defines

strongly Pareto optimal

Defines

weakly Pareto optimal

Defines

strongly Pareto dominates

Defines

strongly Pareto dominant

Defines

Pareto dominates

Defines

Pareto dominant

Defines

weakly Pareto dominates

Defines

weakly Pareto dominant

Generated on Sat Feb 10 13:37:56 2018 by

LaTeXML
strategy

strategy

A

pure

strategy

provides a definition for a way a player can play a game. In particular, it defines, for every possible choice a player might have to make, which option the player picks. A player’s strategy space is the set of pure strategies available to that player.

A

mixed strategy

is an assignment of a

probability

to each pure strategy. It defines a probability over the strategies, and reflect that, rather than choosing a particular pure strategy, the player will randomly select a pure strategy based on the

distribution

given by their mixed strategy. Of course, every pure strategy is a mixed strategy (the function which takes that strategy to

1

and every other one to

0

).

The following notation is often used:

•

S

i

for the strategy space of the

i

-th player

•

s

i

for a particular element of

S

i

; that is, a particular pure strategy

•

σ

i

for a mixed strategy. Note that

σ

i

∈

S

i

→

[

0

,

1

]

and

∑

s

i

∈

S

i

σ

i

⁢

(

s

i

)

=

1

.

•

Σ

i

for the set of all possible mixed strategies for the

i

-th player

•

S

for

∏

i

S

i

, the set of all possible of pure strategies (essentially the possible

outcomes

of the game)

•

Σ

for

∏

i

Σ

i

•

σ

for a

strategy profile

, a single element of

Σ

•

S

-

i

for

∏

j

≠

i

S

j

and

Σ

-

i

for

∏

j

≠

i

Σ

j

, the sets of possible pure and mixed strategies for all players other than

i

.

•

s

-

i

for an element of

S

-

i

and

σ

-

i

for an element of

Σ

-

i

.

Title

strategy

Canonical name

Strategy

Date of creation

2013-03-22 12:52:02

Last modified on

2013-03-22 12:52:02

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 91A99

Related topic

Game

Defines

strategy

Defines

pure strategy

Defines

mixed strategy

Defines

strategy space

Generated on Sat Feb 10 13:37:58 2018 by

LaTeXML
utility

utility

Utility

is taken to be an absolute, accurate measurement of how desirable something is; in particular, it differs from money in three key ways.

First, desire for it is linear (generally in economics and game a person with a lot of money receives less utility from an additional amount of money than someone with very little money).

Second, when modeling a situation, utility should include all external (the happiness received from doing a good deed, for instance).

Third, different people’s utility is incomparable. It is meaningless to ask whether one person gets more utility from a situation than another person does. Utilities for a given person can be compared only to other utilities for that person.

A

utility function

is a function which specifies how much utility a player gets for a particular outcome. It maps the space

S

of all possible

strategy

profiles to

ℝ

.

Title

utility

Canonical name

Utility

Date of creation

2013-03-22 12:51:59

Last modified on

2013-03-22 12:51:59

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 91A99

Defines

utility

Defines

utility function

Generated on Sat Feb 10 13:38:01 2018 by

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zero-sum game

zero-sum game

A

zero-sum game

is a game in which only one player can win (that is, achieve the goal of the game) and the losses (the failure to obtain a goal of the game) of any player are matched by gains by another player. A zero-sum game is a

finite game

(a game that eventually comes to an end), and though only one player can win, the game can also end in a draw (meaning that neither side can win).

For example, in chess, when a player loses a piece captured by another player, the other player gains more open avenues on which to attack the king of the opponent. In Reversi (or Othello), a player must capture at least one of the opponent’s pieces; if not, then the player must pass. In poker, all players must contribute to the pot; whoever has the best hand claims the entire pot and the losers lose everything they put in the pot.

Zero-sum games were extensively studied by

John von Neumann

.

Title

zero-sum game

Canonical

name

ZerosumGame

Date of creation

2013-03-22 16:32:52

Last modified on

2013-03-22 16:32:52

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 91A99

Synonym

zero sum game

Generated on Sat Feb 10 13:38:04 2018 by

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rate of return

rate of return

Suppose you invest

P

at time

0

and receive payments

P

1

,

…

,

P

n

at times

t

1

,

…

,

t

n

corresponding to

interest rates

(evaluated from

0

)

r

1

,

…

,

r

n

. The

net present value

of this investment is

N

⁢

P

⁢

V

=

-

P

+

P

1

1

+

r

1

+

P

2

1

+

r

2

+

⋯

+

P

n

1

+

r

n

.

The

rate of return

r

of this investment is a

compound interest

rate, compounded at every unit time period, such that the net present value of the investment is

0

. In other words, if

r

, as a real number, exists, it satisfies the following equation:

P

=

P

1

(

1

+

r

)

t

1

+

P

2

(

1

+

r

)

t

2

+

⋯

+

P

n

(

1

+

r

)

t

n

.

Remarks.

•

We typically assume that

t

1

≤

t

2

≤

⋯

≤

t

n

,  and, in most situations, that they are integers, so that the equation is a polynomial equation.

•

However, there is no guarantee that

r

exists, and if it exists, that it is unique.

•

Nevertheless, one can usually, by trial-and-error, determine if such an

r

exists. If

r

exists, and if

P

i

are all non-negative, then by Descartes’ rule of signs (

http://planetmath.org/DescartesRuleOfSigns

),

r

is always unique and

r

>

-

1

.

Title

rate of return

Canonical

name

RateOfReturn

Date of creation

2013-03-22 16:41:05

Last modified on

2013-03-22 16:41:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 91B28

Generated on Sat Feb 10 13:38:43 2018 by

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Lazear model of a private school

Lazear model of a private school

The economist Edward P. Lazear defined a theoretical of a (private) profit
maximizing school and was able to derive some interesting

conclusions

from
the model. Here we describe the model.

The school has a fixed number,

Z

of students.
The cost of a teacher is

W

>

0

.
Each student is assumed to be not disruptive with a probability

p

. So if
the class has

n

students the probability of ”peace in the classroom”
is

p

n

. Only when there is peace can any learning take place.
The school charges according the class size and all students in the class
of size

n

pay the same amount. Assuming that the value of a unit of
learning is

V

>

0

and that there are

m

classes, the average class
size is

n

=

Z

/

m

. The expected revenue then is

V

⁢

m

⁢

n

⁢

p

Z

/

m

=

V

⁢

Z

⁢

p

Z

/

m

. And the cost for these

m

classes is

W

⁢

m

.
Then the expected profit is

V

⁢

Z

⁢

p

Z

/

m

-

W

⁢

m

. This is a

function

to be maximized.

Using this theoretical

framework

Lazear was able to show:

1.

As

p

increases it is optimal to use fewer classes, so that class size increases.

2.

As

W

increases it is optimal to use fewer classes, so that class size increases.

Lazear briefly considered an

extension

of the model to the case where
there are two types of students, having different disruption probabilities.
He concluded that if the school wants to maximize revenue it would segregate the
students, that is, there would be no mixed classes.

References

LAZ

Edward P. Lazear

,

Educational

Production

, Quarterly Journal of Economics, CXVI(2001), 777-801.

Title

Lazear model of a private school

Canonical

name

LazearModelOfAPrivateSchool

Date of creation

2013-03-22 16:09:36

Last modified on

2013-03-22 16:09:36

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 91B38

Generated on Sat Feb 10 13:38:45 2018 by

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concepts of voting

concepts of voting

Various voting systems have been proposed and this entry lists the
possible systems.

1.

Single winner

2.

Range Voting

3.

Approval Voting

4.

Plurality Voting

5.

Strategic Voting

6.

Rank-order based

7.

Condorcet

8.

Instant Runoff

9.

Borda

In addition there are various Impossibilty Theorems:

1.

2.

Gibbard-Satterthwaite

And there are various that are important for understanding Voting systems:

1.

No show paradox

2.

Alabama paradox

3.

Bayesian regret

4.

clone of a candidate

Title

concepts of voting

Canonical name

ConceptsOfVoting

Date of creation

2013-03-22 16:10:22

Last modified on

2013-03-22 16:10:22

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 91F10

Generated on Sat Feb 10 13:38:46 2018 by

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range voting

range voting

Range voting

Warren D. Smith

Range voting

is a single-winner voting system in which each vote consists of one numerical score
from 0 to U awarded to each candidate,
where U is some positive number.
In

single-digit range voting

U is 9,
while

two-digit range voting

has U=99.
(One could also consider permitting the interval to be, e.g.

[

-

10

,

10

]

.)
The candidate with the highest average score wins.
In a common variant, “X” scores also are permitted for candidates
(meaning:“I wish to express no opinion about this candidate”).
For example (46,0,32,99) could be one vote in a 4-candidate two-digit range voting

election

; advise 99/0 for best/worst.

Approval voting

[

1

]

is the degenerate form of range voting in which
there are only two allowed scores: 0 and 1.

References

1

S.J. Brams & P. Fishburn: Approval voting, Birkhauser 1983.

Title

range voting

Canonical name

RangeVoting

Date of creation

2013-03-22 16:10:00

Last modified on

2013-03-22 16:10:00

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

51

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 91F10

Defines

approval voting

Defines

single-digit range voting

Defines

two-digit range voting

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bursting

bursting

In neuroscience,

bursting

denotes two or more action potentials (spikes) fired by a neuron, followed by a period of quiescence. A burst of two spikes is called a

doublet

, three spikes -

triplet

, four -

quadruplet

, etc.

Most mathematical models of bursting can be written in the singularly perturbed form

x

˙

=

f

⁢

(

x

,

y

)

(fast spiking)

y

˙

=

μ

⁢

g

⁢

(

x

,

y

)

(slow modulation)

where

x

∈

ℝ

n

is the fast

variable

that simulates fast spiking of the neuron, and

y

∈

ℝ

m

is the slow variable that modulates such spiking activity.

A topological classification of bursters relies on the

bifurcations

of the fast subsystem (variable

x

) when the slow subsystem (variable

y

) is treated as a

parameter

.

References

iz

Izhikevich E.M. (2007)

Dynamical Systems

in Neuroscience: The

Geometry

of Excitability and Bursting. The MIT Press.

izb

Eugene M. Izhikevich (2006) Bursting. Scholarpedia, p.1401 (available online at http://www.scholarpedia.org/article/Bursting).

Title

bursting

Canonical

name

Bursting

Date of creation

2013-03-22 16:28:51

Last modified on

2013-03-22 16:28:51

Owner

emi (15656)

Last modified by

emi (15656)

Numerical id

7

Author

emi (15656)

Entry type

Definition

Classification

msc 92B20

Classification

msc 92C20

Synonym

burst

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