Mixing
Why does finding the partial derivative solve this payoff matrix?
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So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never took a formal course in multi-variable calculus, but I learned on my own about partial derivatives and I get the gist of it. I got to this question on the homework:
Question
Well in the textbook, it calls for setting up the rows choices as [x, 1-x] and having the column's choices being [0,1] and then repeating the process with [1,0] to find the optimum row strategy, and then doing the inverse with columns. When I saw how tedious this would be, I noticed that there're only two unknowns, so I could just have an x and y and use some multi-variable calculus.
I simply did this:
matmul
and did dp/dx and dp/dy and set each equal to 0 and solved for x and y. I plugged these into the problem and it was the right answer.
Now looking back on this, I can't seem to figure out why this actually solves the problem. I think I don't have a very good understanding of partial derivatives, but from standard Calc 1, I know you would set the derivative to 0 and that would give you the min/max of the function. But translating this to 2 variables is confusing to me, and consequently I've spent the past hour trying to understand why this solves the problem in the first place. Could somebody explain to me very simply how this is actually working and why this finds the optimum strategies?
Thanks!!
matrices multivariable-calculus optimization partial-derivative game-theory
asked Jan 17 at 5:05
Carson PCarson P
11
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add a comment |
$begingroup$
So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never took a formal course in multi-variable calculus, but I learned on my own about partial derivatives and I get the gist of it. I got to this question on the homework:
Question
Well in the textbook, it calls for setting up the rows choices as [x, 1-x] and having the column's choices being [0,1] and then repeating the process with [1,0] to find the optimum row strategy, and then doing the inverse with columns. When I saw how tedious this would be, I noticed that there're only two unknowns, so I could just have an x and y and use some multi-variable calculus.
I simply did this:
matmul
and did dp/dx and dp/dy and set each equal to 0 and solved for x and y. I plugged these into the problem and it was the right answer.
Now looking back on this, I can't seem to figure out why this actually solves the problem. I think I don't have a very good understanding of partial derivatives, but from standard Calc 1, I know you would set the derivative to 0 and that would give you the min/max of the function. But translating this to 2 variables is confusing to me, and consequently I've spent the past hour trying to understand why this solves the problem in the first place. Could somebody explain to me very simply how this is actually working and why this finds the optimum strategies?
Thanks!!
matrices multivariable-calculus optimization partial-derivative game-theory
asked Jan 17 at 5:05
Carson PCarson P
11
$endgroup$
add a comment |
$begingroup$
So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never took a formal course in multi-variable calculus, but I learned on my own about partial derivatives and I get the gist of it. I got to this question on the homework:
Question
Well in the textbook, it calls for setting up the rows choices as [x, 1-x] and having the column's choices being [0,1] and then repeating the process with [1,0] to find the optimum row strategy, and then doing the inverse with columns. When I saw how tedious this would be, I noticed that there're only two unknowns, so I could just have an x and y and use some multi-variable calculus.
I simply did this:
matmul
and did dp/dx and dp/dy and set each equal to 0 and solved for x and y. I plugged these into the problem and it was the right answer.
Now looking back on this, I can't seem to figure out why this actually solves the problem. I think I don't have a very good understanding of partial derivatives, but from standard Calc 1, I know you would set the derivative to 0 and that would give you the min/max of the function. But translating this to 2 variables is confusing to me, and consequently I've spent the past hour trying to understand why this solves the problem in the first place. Could somebody explain to me very simply how this is actually working and why this finds the optimum strategies?
Thanks!!
matrices multivariable-calculus optimization partial-derivative game-theory
asked Jan 17 at 5:05
Carson PCarson P
11
$endgroup$
So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never took a formal course in multi-variable calculus, but I learned on my own about partial derivatives and I get the gist of it. I got to this question on the homework:
Question
Well in the textbook, it calls for setting up the rows choices as [x, 1-x] and having the column's choices being [0,1] and then repeating the process with [1,0] to find the optimum row strategy, and then doing the inverse with columns. When I saw how tedious this would be, I noticed that there're only two unknowns, so I could just have an x and y and use some multi-variable calculus.
I simply did this:
matmul
and did dp/dx and dp/dy and set each equal to 0 and solved for x and y. I plugged these into the problem and it was the right answer.
Now looking back on this, I can't seem to figure out why this actually solves the problem. I think I don't have a very good understanding of partial derivatives, but from standard Calc 1, I know you would set the derivative to 0 and that would give you the min/max of the function. But translating this to 2 variables is confusing to me, and consequently I've spent the past hour trying to understand why this solves the problem in the first place. Could somebody explain to me very simply how this is actually working and why this finds the optimum strategies?
Thanks!!
matrices multivariable-calculus optimization partial-derivative game-theory
matrices multivariable-calculus optimization partial-derivative game-theory
asked Jan 17 at 5:05
Carson PCarson P
11
asked Jan 17 at 5:05
Carson PCarson P
11
asked Jan 17 at 5:05
Carson PCarson P
11
asked Jan 17 at 5:05
Carson PCarson P
11
asked Jan 17 at 5:05
Carson PCarson P
11
11
add a comment |
add a comment |
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