# 0.18 Game theory  (Page 4/4)

 Page 4 / 4

For the game in [link] , determine the optimal strategy for both Robert and Carol, and find the value of the game.

Since we have already determined that the game is non-strictly determined, we proceed to determine the optimal strategy for the game. We rewrite the game matrix.

$G=\left[\begin{array}{cc}\text{10}& -\text{10}\\ -\text{25}& \text{25}\end{array}\right]$

Let $R=\left[\begin{array}{cc}r& 1-r\end{array}\right]$ be Robert's strategy, and $C=\left[\begin{array}{c}c\\ 1-c\end{array}\right]$ be Carol's strategy.

To find the optimal strategy for Robert, we, first, find the product $\text{RG}$ as below.

$\left[\begin{array}{cc}r& 1-r\end{array}\right]\left[\begin{array}{cc}\text{10}& -\text{10}\\ -\text{25}& \text{25}\end{array}\right]=\left[\begin{array}{cc}\text{35}r-\text{25}& -\text{35}r+\text{25}\end{array}\right]$

By setting the entries equal, we get

$\text{35}r-\text{25}=-\text{35}r+\text{25}$
or $r=5/7$ .

Therefore, the optimal strategy for Robert is $\left[\begin{array}{cc}5/7& 2/7\end{array}\right]$ .

To find the optimal strategy for Carol, we, first, find the following product.

$\left[\begin{array}{cc}\text{10}& -\text{10}\\ -\text{25}& \text{25}\end{array}\right]\left[\begin{array}{c}c\\ 1-c\end{array}\right]=\left[\begin{array}{c}\text{20}c-\text{10}\\ -\text{50}c+\text{25}\end{array}\right]$

We now set the entries equal to each other, and we get,

$\text{20}c-\text{10}=-\text{50}c+\text{25}$

or $c=1/2$

Therefore, the optimal strategy for Carol is $\left[\begin{array}{c}1/2\\ 1/2\end{array}\right]$ .

To find the expected value, $V$ , of the game, we find the product $\text{RGC}$ .

$\begin{array}{lll}V& =& \left[\begin{array}{cc}5/7& 2/7\end{array}\right]\left[\begin{array}{cc}\text{10}& -\text{10}\\ -\text{25}& \text{25}\end{array}\right]\left[\begin{array}{c}1/2\\ 1/2\end{array}\right]\\ & =& \left[0\right]\end{array}$

If both players play their optimal strategy, the value of the game is zero. In such case, the game is called fair .

## Reduction by dominance

Sometimes an $m×n$ game matrix can be reduced to a $2×2$ matrix by deleting certain rows and columns. A row can be deleted if there exists another row that will produce a payoff of an equal or better value. Similarly, a column can be deleted if there is another column that will produce a payoff of an equal or better value for the column player. The row or column that produces a better payoff for its corresponding player is said to dominate the row or column with the lesser payoff.

For the following game, determine the optimal strategy for both the row player and the column player, and find the value of the game.

$G=\left[\begin{array}{ccc}-2& 6& 4\\ -1& -2& -3\\ 1& 2& -2\end{array}\right]$

We first look for a saddle point and determine that none exist. Next, we try to reduce the matrix to a $2×2$ matrix by eliminating the dominated row.

Since every entry in row 3 is larger than the corresponding entry in row 2, row 3 dominates row 2. Therefore, a rational row player will never play row 2, and we eliminate row 2. We get

$\left[\begin{array}{ccc}-2& 6& 4\\ 1& 2& -2\end{array}\right]$

Now we try to eliminate a column. Remember that the game matrix represents the payoffs for the row player and not the column player; therefore, the larger the number in the column, the smaller the payoff for the column player.

The column player will never play column 2, because it is dominated by both column 1 and column 3. Therefore, we eliminate column 2 and get the modified matrix, $M$ , below.

$M=\left[\begin{array}{cc}-2& 4\\ 1& -2\end{array}\right]$

To find the optimal strategy for both the row player and the column player, we use the method learned in the [link] .

Let the row player's strategy be $R=\left[\begin{array}{cc}r& 1-r\end{array}\right]$ , and the column player's be strategy be $C=\left[\begin{array}{c}c\\ 1-c\end{array}\right]$ .

To find the optimal strategy for the row player, we, first, find the product $\text{RM}$ as below.

$\left[\begin{array}{cc}r& 1-r\end{array}\right]\left[\begin{array}{cc}-2& 4\\ 1& -2\end{array}\right]=\left[\begin{array}{cc}-3r+1& 6r-2\end{array}\right]$

By setting the entries equal, we get

$-3r+1=6r-2$

or $r=1/3$ .

Therefore, the optimal strategy for the row player is $\left[\begin{array}{cc}1/3& 2/3\end{array}\right]$ , but relative to the original game matrix it is $\left[\begin{array}{ccc}1/3& 0& 2/3\end{array}\right]$ .

To find the optimal strategy for the column player we, first, find the following product.

$\left[\begin{array}{cc}-2& 4\\ 1& -2\end{array}\right]\left[\begin{array}{c}c\\ 1-c\end{array}\right]=\left[\begin{array}{c}-6c+4\\ 3c-2\end{array}\right]$

We set the entries in the product matrix equal to each other, and we get,

$-6c+4=3c-2$

or $c=2/3$

Therefore, the optimal strategy for the column player is $\left[\begin{array}{c}2/3\\ 1/3\end{array}\right]$ , but relative to the original game matrix, the strategy for the column player is $\left[\begin{array}{c}2/3\\ 0\\ 1/3\end{array}\right]$ .

To find the expected value, $V$ , of the game, we have two choices: either to find the product of matrices $R$ , $M$ and $C$ , or multiply the optimal strategies relative to the original matrix to the original matrix. We choose the first, and get

$\begin{array}{lll}V& =& \left[\begin{array}{cc}1/3& 2/3\end{array}\right]\left[\begin{array}{cc}-2& 4\\ 1& -2\end{array}\right]\left[\begin{array}{c}2/3\\ 1/3\end{array}\right]\\ & =& \left[0\right]\end{array}$

Therefore, if both players play their optimal strategy, the value of the game is zero.

We summarize as follows:

## Reduction by dominance

1. Sometimes an $m×n$ game matrix can be reduced to a $2×2$ matrix by deleting dominated rows and columns.
2. A row is called a dominated row if there exists another row that will produce a payoff of an equal or better value. That happens when there exists a row whose every entry is larger than the corresponding entry of the dominated row.
3. A column is called a dominated column if there exists another column that will produce a payoff of an equal or better value. This happens when there exists a column whose every entry is smaller than the corresponding entry of the dominated row.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges$12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?       By Qqq Qqq   