# 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.

#### 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?