# 0.18 Game theory  (Page 3/4)

 Page 3 / 4

Suppose in [link] , Robert decides to show a dime with $\text{.}\text{20}$ probability and a quarter with $\text{.}\text{80}$ probability, and Carol decides to show a dime with $\text{.}\text{70}$ probability and a quarter with $\text{.}\text{30}$ probability. What is the expected payoff for Robert?

Let $R$ denote Robert's strategy and $C$ denote Carol's strategy.

Since Robert is a row player and Carol is a column player, their strategies are written as follows:

$R=\left[\begin{array}{cc}\text{.}\text{20}& \text{.}\text{80}\end{array}\right]$ and $C=\left[\begin{array}{c}\text{.}\text{70}\\ \text{.}\text{30}\end{array}\right]$ .

To find the expected payoff, we use the following reasoning.

Since Robert chooses to play row 1 with $\text{.}\text{20}$ probability and Carol chooses to play column 1 with $\text{.}\text{70}$ probability, the move row 1, column 1 will be chosen with $\left(\text{.}\text{20}\right)\left(\text{.}\text{70}\right)=\text{.}\text{14}$ probability. The fact that this move has a payoff of 10 cents for Robert, Robert's expected payoff for this move is $\left(\text{.}\text{14}\right)\left(\text{.}\text{10}\right)=\text{.}\text{014}$ cents. Similarly, we compute Robert's expected payoffs for the other cases. The table below lists expected payoffs for all four cases.

 Move Probability Payoff Expected Payoff Row 1, Column 1 $\left(\text{.}\text{20}\right)\left(\text{.}\text{70}\right)=\text{.}\text{14}$ 10 cents 1.4 cents Row 1, Column 2 $\left(\text{.}\text{20}\right)\left(\text{.}\text{30}\right)=\text{.}\text{06}$ -10 cents -.6 cents Row 2, Column 1 $\left(\text{.}\text{80}\right)\left(\text{.}\text{70}\right)=\text{.}\text{56}$ -25 cents -14 cents Row 2, Column 2 $\left(\text{.}\text{80}\right)\left(\text{.}\text{30}\right)=\text{.}\text{24}$ 25 cents 6.0 cents Totals 1 -7.2 cents

The above table shows that if Robert plays the game with the strategy $R=\left[\begin{array}{cc}\text{.}\text{20}& \text{.}\text{80}\end{array}\right]$ and Carol plays with the strategy $C=\left[\begin{array}{c}\text{.}\text{70}\\ \text{.}\text{30}\end{array}\right]$ , Robert can expect to lose 7.2 cents for every game.

Alternatively, if we call the game matrix $G$ , then the expected payoff for the row player can be determined by multiplying matrices $R$ , $G$ and $C$ . Thus, the expected payoff $P$ for Robert is as follows:

Which is the same as the one obtained from the table.

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

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

Let us suppose that the row player uses the strategy $R=\left[\begin{array}{cc}r& 1-r\end{array}\right]$ . Now if the column player plays column 1, the expected payoff $P$ for the row player is

$P\left(r\right)=1\left(r\right)+\left(-3\right)\left(1-r\right)=4r-3$ .

Which can also be computed as follows:

$P\left(r\right)=\left[\begin{array}{cc}r& 1-r\end{array}\right]\left[\begin{array}{c}1\\ -3\end{array}\right]$ or $4r-3$ .

If the row player plays the strategy $\left[\begin{array}{cc}r& 1-r\end{array}\right]$ and the column player plays column 2, the expected payoff $P$ for the row player is

$P\left(r\right)=\left[\begin{array}{cc}r& 1-r\end{array}\right]\left[\begin{array}{c}-2\\ 4\end{array}\right]=-6r+4$ .

We have two equations

$P\left(r\right)=4r-3$ and $P\left(r\right)=-6r+4$

The row player is trying to improve upon his worst scenario, and that only happens when the two lines intersect. Any point other than the point of intersection will not result in optimal strategy as one of the expectations will fall short.

Solving for $r$ algebraically, we get

$4r-3=-6r+4$

$r=7/\text{10}$ .

Therefore, the optimal strategy for the row player is $\left[\begin{array}{cc}\text{.}7& \text{.}3\end{array}\right]$ .

Alternatively, we can find the optimal strategy for the row player by, first, multiplying the row matrix with the game matrix as shown below.

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

And then by equating the two entries in the product matrix. Again, we get $r=\text{.}7$ , which gives us the optimal strategy $\left[\begin{array}{cc}\text{.}7& \text{.}3\end{array}\right]$ .

We use the same technique to find the optimal strategy for the column player.

Suppose the column player's optimal strategy is represented by $\left[\begin{array}{c}c\\ 1-c\end{array}\right]$ . We, first, multiply the game matrix by the column matrix as shown below.

$\left[\begin{array}{cc}1& -2\\ -3& 4\end{array}\right]\left[\begin{array}{c}c\\ 1-c\end{array}\right]=\left[\begin{array}{c}3c-2\\ -7c+4\end{array}\right]$

And then equate the entries in the product matrix. We get

$3c-2=-7c+4$
$c=\text{.}6$

Therefore, the column player's optimal strategy is $\left[\begin{array}{c}\text{.}6\\ \text{.}4\end{array}\right]$ .

To find the expected value, $V$ , of the game, we find the product of the matrices $R$ , $G$ and $C$ .

$\begin{array}{c}V=\left[\begin{array}{cc}\text{.}7& \text{.}3\end{array}\right]\left[\begin{array}{cc}1& -2\\ -3& 4\end{array}\right]\left[\begin{array}{c}\text{.}6\\ \text{.}4\end{array}\right]\\ -\text{.}2\end{array}$

That is, if both players play their optimal strategies, the row player can expect to lose $\text{.}2$ units for every game.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
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?