Mixing
Create the game matrix
Clash Royale CLAN TAG#URR8PPP
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Each of two players shows one or two fingers (simultaneously) and $C$ pays to $R$ a sum equal to the total number of fingers shown. Write the game matrix. Show that the game is strictly determined and find the value and optimal strategies.
I'm brand new at game theory and just learned how to tell if a game is fair or strictly determined. I think I'm thinking about this wrong but if there are only two options and I need the game to be strictly determined wouldn't the answer be a matrix with all one's or all two's?
game-theory
edited Jul 19 at 1:04
RayDansh
882214
asked Jul 19 at 0:20
Lil
94431935
add a comment |Â
up vote
0
down vote
favorite
Each of two players shows one or two fingers (simultaneously) and $C$ pays to $R$ a sum equal to the total number of fingers shown. Write the game matrix. Show that the game is strictly determined and find the value and optimal strategies.
I'm brand new at game theory and just learned how to tell if a game is fair or strictly determined. I think I'm thinking about this wrong but if there are only two options and I need the game to be strictly determined wouldn't the answer be a matrix with all one's or all two's?
game-theory
edited Jul 19 at 1:04
RayDansh
882214
asked Jul 19 at 0:20
Lil
94431935
The game would be fair if the expected outcome would be that neither player gains or loses any money. $C$ always pays money in every outcome, so it is hardly a fair game. "Hey, fred, how about we play a game where you pay me $ 1 if I flip a coin and its heads and pay me $ 2 if it was tails" "Heck no"
– JMoravitz
Jul 19 at 0:25
Makes perfect sense- I missed the part that said simultaneously... Thank you again!
– Lil
Jul 19 at 12:35
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Each of two players shows one or two fingers (simultaneously) and $C$ pays to $R$ a sum equal to the total number of fingers shown. Write the game matrix. Show that the game is strictly determined and find the value and optimal strategies.
I'm brand new at game theory and just learned how to tell if a game is fair or strictly determined. I think I'm thinking about this wrong but if there are only two options and I need the game to be strictly determined wouldn't the answer be a matrix with all one's or all two's?
game-theory
edited Jul 19 at 1:04
RayDansh
882214
asked Jul 19 at 0:20
Lil
94431935
Each of two players shows one or two fingers (simultaneously) and $C$ pays to $R$ a sum equal to the total number of fingers shown. Write the game matrix. Show that the game is strictly determined and find the value and optimal strategies.
I'm brand new at game theory and just learned how to tell if a game is fair or strictly determined. I think I'm thinking about this wrong but if there are only two options and I need the game to be strictly determined wouldn't the answer be a matrix with all one's or all two's?
game-theory
edited Jul 19 at 1:04
RayDansh
882214
asked Jul 19 at 0:20
Lil
94431935
edited Jul 19 at 1:04
RayDansh
882214
edited Jul 19 at 1:04
RayDansh
882214
edited Jul 19 at 1:04
RayDansh
882214
882214
asked Jul 19 at 0:20
Lil
94431935
asked Jul 19 at 0:20
Lil
94431935
asked Jul 19 at 0:20
Lil
94431935
94431935
The game would be fair if the expected outcome would be that neither player gains or loses any money. $C$ always pays money in every outcome, so it is hardly a fair game. "Hey, fred, how about we play a game where you pay me $ 1 if I flip a coin and its heads and pay me $ 2 if it was tails" "Heck no"
– JMoravitz
Jul 19 at 0:25
Makes perfect sense- I missed the part that said simultaneously... Thank you again!
– Lil
Jul 19 at 12:35
add a comment |Â
The game would be fair if the expected outcome would be that neither player gains or loses any money. $C$ always pays money in every outcome, so it is hardly a fair game. "Hey, fred, how about we play a game where you pay me $ 1 if I flip a coin and its heads and pay me $ 2 if it was tails" "Heck no"
– JMoravitz
Jul 19 at 0:25
Makes perfect sense- I missed the part that said simultaneously... Thank you again!
– Lil
Jul 19 at 12:35
The game would be fair if the expected outcome would be that neither player gains or loses any money. $C$ always pays money in every outcome, so it is hardly a fair game. "Hey, fred, how about we play a game where you pay me $ 1 if I flip a coin and its heads and pay me $ 2 if it was tails" "Heck no"
– JMoravitz
Jul 19 at 0:25
The game would be fair if the expected outcome would be that neither player gains or loses any money. $C$ always pays money in every outcome, so it is hardly a fair game. "Hey, fred, how about we play a game where you pay me $ 1 if I flip a coin and its heads and pay me $ 2 if it was tails" "Heck no"
– JMoravitz
Jul 19 at 0:25
Makes perfect sense- I missed the part that said simultaneously... Thank you again!
– Lil
Jul 19 at 12:35
Makes perfect sense- I missed the part that said simultaneously... Thank you again!
– Lil
Jul 19 at 12:35
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
0
down vote
The specific arrangement of how you insert the information varies based on textbook, but a common way to write it out would be such as the following:
$beginarray&C~textshows&C~textshows\&text1 finger&text2 fingers\hline R~textshows 1 finger\
hline R~textshows 2 fingers\hline endarray$
writing values into the table according to how much money $R$ would win (which is the same amount that $C$ would lose) if those selections were simultaneously made.
For example, in the top left entry with $R$ showing one finger and $C$ showing one finger, there would be a total of two fingers shown implying that $C$ would need to pay $R$ two dollars (or whatever currency they happen to be using).
As for deciding strategies... $C$ knows that if he chooses to show more fingers he'll lose more money than if he chose to show only one finger... so $C$'s strategy will be to _______.
Similarly, $R$ wants to make as much money as possible so the more fingers $R$ shows the more money he'll stand to make so $R$'s strategy will be to ______.
Since both players are dead-set on what strategies they'll be following, we can expect that every time they play this game the outcome will be identical every time... i.e. it is indeed a strictly determined game, and the outcome every time they play will be that $R$ wins ______ dollars from $C$.
Since the expected outcome of the game is not zero, the game is not fair.
answered Jul 19 at 0:36
JMoravitz
44.3k33481
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
add a comment |Â
up vote
0
down vote
This is what we called as Expected Payoff for mixed strategies:
Here you have $2$ players $R$ and $C$, playing a
zero-sum, simultaneous move game. It is implicitly assumed that the players make their choice of strategy independently of each other. Lets assume that $R$ has two strategies $R_1$ and $R_2$ and $C$ also has $2$ strategies $C_1$ and $C_2$ and that the payoff matrix for R is given by
We assume that $R$ is playing the mixed strategy $(p1, p2)$ and $C$ is playing the mixed strategy $dbinomq_1q_2$.Because the players choose their strategies independently the probability that $R$ will choose $R_1$ and $C$ will
choose $C_2$ is $p_1q_2$ (from our formula for independent events $P(A ∩ B) = P(A)P(B))$. Notice that the payoff for $R$ is a random variable, $X$, and its value depends on which of the four situations occurs. We can find its
probability distribution using the fact that the decisions on strategy are made independently:
beginarrayc
mbox Choice&mbox $X=$Pay-off for $R$&mboxProbability&mbox$XP(X)$\hline
R_1C_1&a&(p_1)(q_1)&a(p_1)(q_1)\
R_1C_2&b&(p_1)(q_2)&b(p_1)(q_2)\
R_2C_1&c&(p_2)(q_1)&c(p_2)(q_1)\
R_2C_2&d&(p_2)(q_2)&d(p_2)(q_2)\
\
\
endarray
answered Jul 19 at 0:39
Key Flex
4,346425
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The specific arrangement of how you insert the information varies based on textbook, but a common way to write it out would be such as the following:
$beginarray&C~textshows&C~textshows\&text1 finger&text2 fingers\hline R~textshows 1 finger\
hline R~textshows 2 fingers\hline endarray$
writing values into the table according to how much money $R$ would win (which is the same amount that $C$ would lose) if those selections were simultaneously made.
For example, in the top left entry with $R$ showing one finger and $C$ showing one finger, there would be a total of two fingers shown implying that $C$ would need to pay $R$ two dollars (or whatever currency they happen to be using).
As for deciding strategies... $C$ knows that if he chooses to show more fingers he'll lose more money than if he chose to show only one finger... so $C$'s strategy will be to _______.
Similarly, $R$ wants to make as much money as possible so the more fingers $R$ shows the more money he'll stand to make so $R$'s strategy will be to ______.
Since both players are dead-set on what strategies they'll be following, we can expect that every time they play this game the outcome will be identical every time... i.e. it is indeed a strictly determined game, and the outcome every time they play will be that $R$ wins ______ dollars from $C$.
Since the expected outcome of the game is not zero, the game is not fair.
answered Jul 19 at 0:36
JMoravitz
44.3k33481
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
add a comment |Â
up vote
0
down vote
The specific arrangement of how you insert the information varies based on textbook, but a common way to write it out would be such as the following:
$beginarray&C~textshows&C~textshows\&text1 finger&text2 fingers\hline R~textshows 1 finger\
hline R~textshows 2 fingers\hline endarray$
writing values into the table according to how much money $R$ would win (which is the same amount that $C$ would lose) if those selections were simultaneously made.
For example, in the top left entry with $R$ showing one finger and $C$ showing one finger, there would be a total of two fingers shown implying that $C$ would need to pay $R$ two dollars (or whatever currency they happen to be using).
As for deciding strategies... $C$ knows that if he chooses to show more fingers he'll lose more money than if he chose to show only one finger... so $C$'s strategy will be to _______.
Similarly, $R$ wants to make as much money as possible so the more fingers $R$ shows the more money he'll stand to make so $R$'s strategy will be to ______.
Since both players are dead-set on what strategies they'll be following, we can expect that every time they play this game the outcome will be identical every time... i.e. it is indeed a strictly determined game, and the outcome every time they play will be that $R$ wins ______ dollars from $C$.
Since the expected outcome of the game is not zero, the game is not fair.
answered Jul 19 at 0:36
JMoravitz
44.3k33481
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The specific arrangement of how you insert the information varies based on textbook, but a common way to write it out would be such as the following:
$beginarray&C~textshows&C~textshows\&text1 finger&text2 fingers\hline R~textshows 1 finger\
hline R~textshows 2 fingers\hline endarray$
writing values into the table according to how much money $R$ would win (which is the same amount that $C$ would lose) if those selections were simultaneously made.
For example, in the top left entry with $R$ showing one finger and $C$ showing one finger, there would be a total of two fingers shown implying that $C$ would need to pay $R$ two dollars (or whatever currency they happen to be using).
As for deciding strategies... $C$ knows that if he chooses to show more fingers he'll lose more money than if he chose to show only one finger... so $C$'s strategy will be to _______.
Similarly, $R$ wants to make as much money as possible so the more fingers $R$ shows the more money he'll stand to make so $R$'s strategy will be to ______.
Since both players are dead-set on what strategies they'll be following, we can expect that every time they play this game the outcome will be identical every time... i.e. it is indeed a strictly determined game, and the outcome every time they play will be that $R$ wins ______ dollars from $C$.
Since the expected outcome of the game is not zero, the game is not fair.
answered Jul 19 at 0:36
JMoravitz
44.3k33481
The specific arrangement of how you insert the information varies based on textbook, but a common way to write it out would be such as the following:
$beginarray&C~textshows&C~textshows\&text1 finger&text2 fingers\hline R~textshows 1 finger\
hline R~textshows 2 fingers\hline endarray$
writing values into the table according to how much money $R$ would win (which is the same amount that $C$ would lose) if those selections were simultaneously made.
For example, in the top left entry with $R$ showing one finger and $C$ showing one finger, there would be a total of two fingers shown implying that $C$ would need to pay $R$ two dollars (or whatever currency they happen to be using).
As for deciding strategies... $C$ knows that if he chooses to show more fingers he'll lose more money than if he chose to show only one finger... so $C$'s strategy will be to _______.
Similarly, $R$ wants to make as much money as possible so the more fingers $R$ shows the more money he'll stand to make so $R$'s strategy will be to ______.
Since both players are dead-set on what strategies they'll be following, we can expect that every time they play this game the outcome will be identical every time... i.e. it is indeed a strictly determined game, and the outcome every time they play will be that $R$ wins ______ dollars from $C$.
Since the expected outcome of the game is not zero, the game is not fair.
answered Jul 19 at 0:36
JMoravitz
44.3k33481
answered Jul 19 at 0:36
JMoravitz
44.3k33481
answered Jul 19 at 0:36
JMoravitz
44.3k33481
answered Jul 19 at 0:36
JMoravitz
44.3k33481
44.3k33481
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
add a comment |Â
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
makes perfect sense now. Thank you!
– Lil
Jul 19 at 12:34
add a comment |Â
up vote
0
down vote
This is what we called as Expected Payoff for mixed strategies:
Here you have $2$ players $R$ and $C$, playing a
zero-sum, simultaneous move game. It is implicitly assumed that the players make their choice of strategy independently of each other. Lets assume that $R$ has two strategies $R_1$ and $R_2$ and $C$ also has $2$ strategies $C_1$ and $C_2$ and that the payoff matrix for R is given by
We assume that $R$ is playing the mixed strategy $(p1, p2)$ and $C$ is playing the mixed strategy $dbinomq_1q_2$.Because the players choose their strategies independently the probability that $R$ will choose $R_1$ and $C$ will
choose $C_2$ is $p_1q_2$ (from our formula for independent events $P(A ∩ B) = P(A)P(B))$. Notice that the payoff for $R$ is a random variable, $X$, and its value depends on which of the four situations occurs. We can find its
probability distribution using the fact that the decisions on strategy are made independently:
beginarrayc
mbox Choice&mbox $X=$Pay-off for $R$&mboxProbability&mbox$XP(X)$\hline
R_1C_1&a&(p_1)(q_1)&a(p_1)(q_1)\
R_1C_2&b&(p_1)(q_2)&b(p_1)(q_2)\
R_2C_1&c&(p_2)(q_1)&c(p_2)(q_1)\
R_2C_2&d&(p_2)(q_2)&d(p_2)(q_2)\
\
\
endarray
answered Jul 19 at 0:39
Key Flex
4,346425
add a comment |Â
up vote
0
down vote
This is what we called as Expected Payoff for mixed strategies:
Here you have $2$ players $R$ and $C$, playing a
zero-sum, simultaneous move game. It is implicitly assumed that the players make their choice of strategy independently of each other. Lets assume that $R$ has two strategies $R_1$ and $R_2$ and $C$ also has $2$ strategies $C_1$ and $C_2$ and that the payoff matrix for R is given by
We assume that $R$ is playing the mixed strategy $(p1, p2)$ and $C$ is playing the mixed strategy $dbinomq_1q_2$.Because the players choose their strategies independently the probability that $R$ will choose $R_1$ and $C$ will
choose $C_2$ is $p_1q_2$ (from our formula for independent events $P(A ∩ B) = P(A)P(B))$. Notice that the payoff for $R$ is a random variable, $X$, and its value depends on which of the four situations occurs. We can find its
probability distribution using the fact that the decisions on strategy are made independently:
beginarrayc
mbox Choice&mbox $X=$Pay-off for $R$&mboxProbability&mbox$XP(X)$\hline
R_1C_1&a&(p_1)(q_1)&a(p_1)(q_1)\
R_1C_2&b&(p_1)(q_2)&b(p_1)(q_2)\
R_2C_1&c&(p_2)(q_1)&c(p_2)(q_1)\
R_2C_2&d&(p_2)(q_2)&d(p_2)(q_2)\
\
\
endarray
answered Jul 19 at 0:39
Key Flex
4,346425
add a comment |Â
up vote
0
down vote
up vote
0
down vote
This is what we called as Expected Payoff for mixed strategies:
Here you have $2$ players $R$ and $C$, playing a
zero-sum, simultaneous move game. It is implicitly assumed that the players make their choice of strategy independently of each other. Lets assume that $R$ has two strategies $R_1$ and $R_2$ and $C$ also has $2$ strategies $C_1$ and $C_2$ and that the payoff matrix for R is given by
We assume that $R$ is playing the mixed strategy $(p1, p2)$ and $C$ is playing the mixed strategy $dbinomq_1q_2$.Because the players choose their strategies independently the probability that $R$ will choose $R_1$ and $C$ will
choose $C_2$ is $p_1q_2$ (from our formula for independent events $P(A ∩ B) = P(A)P(B))$. Notice that the payoff for $R$ is a random variable, $X$, and its value depends on which of the four situations occurs. We can find its
probability distribution using the fact that the decisions on strategy are made independently:
beginarrayc
mbox Choice&mbox $X=$Pay-off for $R$&mboxProbability&mbox$XP(X)$\hline
R_1C_1&a&(p_1)(q_1)&a(p_1)(q_1)\
R_1C_2&b&(p_1)(q_2)&b(p_1)(q_2)\
R_2C_1&c&(p_2)(q_1)&c(p_2)(q_1)\
R_2C_2&d&(p_2)(q_2)&d(p_2)(q_2)\
\
\
endarray
answered Jul 19 at 0:39
Key Flex
4,346425
This is what we called as Expected Payoff for mixed strategies:
Here you have $2$ players $R$ and $C$, playing a
zero-sum, simultaneous move game. It is implicitly assumed that the players make their choice of strategy independently of each other. Lets assume that $R$ has two strategies $R_1$ and $R_2$ and $C$ also has $2$ strategies $C_1$ and $C_2$ and that the payoff matrix for R is given by
We assume that $R$ is playing the mixed strategy $(p1, p2)$ and $C$ is playing the mixed strategy $dbinomq_1q_2$.Because the players choose their strategies independently the probability that $R$ will choose $R_1$ and $C$ will
choose $C_2$ is $p_1q_2$ (from our formula for independent events $P(A ∩ B) = P(A)P(B))$. Notice that the payoff for $R$ is a random variable, $X$, and its value depends on which of the four situations occurs. We can find its
probability distribution using the fact that the decisions on strategy are made independently:
beginarrayc
mbox Choice&mbox $X=$Pay-off for $R$&mboxProbability&mbox$XP(X)$\hline
R_1C_1&a&(p_1)(q_1)&a(p_1)(q_1)\
R_1C_2&b&(p_1)(q_2)&b(p_1)(q_2)\
R_2C_1&c&(p_2)(q_1)&c(p_2)(q_1)\
R_2C_2&d&(p_2)(q_2)&d(p_2)(q_2)\
\
\
endarray
answered Jul 19 at 0:39
Key Flex
4,346425
answered Jul 19 at 0:39
Key Flex
4,346425
answered Jul 19 at 0:39
Key Flex
4,346425
answered Jul 19 at 0:39
Key Flex
4,346425
4,346425
add a comment |Â
add a comment |Â
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The game would be fair if the expected outcome would be that neither player gains or loses any money. $C$ always pays money in every outcome, so it is hardly a fair game. "Hey, fred, how about we play a game where you pay me $ 1 if I flip a coin and its heads and pay me $ 2 if it was tails" "Heck no"
– JMoravitz
Jul 19 at 0:25
Makes perfect sense- I missed the part that said simultaneously... Thank you again!
– Lil
Jul 19 at 12:35