Help with game theory (specific knowledge in game theory probably not required)


by WillJ
Tags: game, knowledge, required, specific, theory
WillJ
WillJ is offline
#1
Sep17-06, 02:53 PM
P: 13
Note: At the bottom of this post is a scan of the problem from the textbook, and then in the next post are scans of previous pages from the textbook, but (unless you're familiar with game theory) you probably should read my own words before you read the textbook scans.

I have an equation:

u1 = w + c2 + (w - c1)(c1 + c2)

My goal is to find an equation that tells me what value of c1 maximizes u1, in terms of w and c2. (If you can go ahead and do this yourself, please do so and see if you get the same answer as me, and tell me what you got. If you can't do it,, read through my logic.)

So what I first did was take the case of c2 = 0. The equation then becomes:

u1 = w + (w - c1)(c1)

This is a quadratic equation, which is clearly seen when you multiply the two last terms and rewrite the equation like so:

u1 = w + wc1 -(c1)^2

Re-ordering the terms to make it look nice and pretty, like a standard quadratic equation:

u1 = -(c1)^2 + wc1 + w

To find the roots of this quadratic equation, I used the quadratic formula, keeping in mind that the first coefficient is -1, the second is w, and the third is w.

This gave me the two roots, one negative and one positive. I found the maximum value by finding the difference of these two and dividing by 2.

The maximum value is [sqrt(w^2 + 4w)] / 2 , meaning that u1 is maximized if that's what c1 equals (if c2 = 0).

Of course, c2 might not be 0. I now have to generalize, with whatever c2 may be.

If c2 goes from 0 to something else, the equation changes from

u1 = w + (w - c1)(c1)

to

u1 = w + c2 + (w - c1)(c1 + c2)

meaning that the graph of u1 as a function of c1 shifts upwards by c2 units (since c2 is being added) and it shifts to the left by c2 units (since c2 is being added within the parantheses that enclose c1).

Right? I might be wrong there, so please tell me if I'm right or wrong. If I'm right, that means that, whatever c2 may be, u1 is maximized when c1 equals

( [sqrt(w^2 + 4w)] / 2 ) - c2

(Note that this maximizer is the same as before, except c2 is subtracted, since the graph shifted to the left, so we can imagine shifting it back to the right.) Right?

Now, then, we have the following equation:

b1 = ( [sqrt(w^2 + 4w)] / 2 ) - c2

"b1" in this case stands for "player 1's best response" to whatever c2 is doing. ("u1," by the way, meant "player 1's utility" or "player 1's payoff," which player 1 tries to maximize.) At the same time, player 2 is trying to maximize his payoff, so at the same time we have the following similar equation:

b2 = ( [sqrt(w^2 + 4w)] / 2 ) - c1

To find the "Nash equilibrium," we make "c2" at the end of the b1 equation equal "b2," and we make "c1" at the end of the b2 equation equal "b1," and then find the intersection of these two equations (which implies that player 1 and player 2 are each taking an action, player 1's action being the best response to player 2's action, and player 2's action being the best response to player 1's action). We can do this by substituting one equation into another. That is, you can take the b1 = ... equation and when you get to "c2," substitute that with what "b2" equals in the second equation.

However, if you do that, b1 just cancels itself out and you end up with the following:

0 = ( [sqrt(w^2 + 4w)] / 2 ) - ( [sqrt(w^2 + 4w)] / 2 )

Which is a true statement, but an utterly useless one. I must have done something wrong. So can someone help me here?

Attached at the bottom is a scan of the question that all this work is derived from. Using the vocabulary of game theory, what I'm having trouble doing in this problem is correctly calculating the best response functions and then calculating the Nash equilibrium.

In the next post I'll post scans of some of the textbook's previous pages that relate to this problem.
Attached Thumbnails
gt044-045_c.JPG  
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HallsofIvy
HallsofIvy is online now
#2
Sep17-06, 03:01 PM
Math
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Quote Quote by WillJ
Note: At the bottom of this post are scans of the problem from the textbook, and then in the next post are scans of previous pages from the textbook, but (unless you're familiar with game theory) you probably should read my own words before you read the textbook scans.

I have an equation:

u1 = w + c2 + (w - c1)(c1 + c2)

My goal is to find an equation that tells me what value of c1 maximizes u1, in terms of w and c2. (If you can go ahead and do this yourself, please do so and see if you get the same answer as me, and tell me what you got. If you can't do it,, read through my logic.)

So what I first did was take the case of c2 = 0. The equation then becomes:

u1 = w + (w - c1)(c1)

This is a quadratic equation, which is clearly seen when you multiply the two last terms and rewrite the equation like so:

u1 = w + wc1 -(c1)^2

Re-ordering the terms to make it look nice and pretty, like a standard quadratic equation:

u1 = -(c1)^2 + wc1 + w

To find the roots of this quadratic equation, I used the quadratic formula, keeping in mind that the first coefficient is -1, the second is w, and the third is w.

This gave me the two roots, one negative and one positive. I found the maximum value by finding the difference of these two and dividing by 2.

The maximum value is [sqrt(w^2 + 4w)] / 2 , meaning that u1 is maximized if that's what c1 equals (if c2 = 0).

Of course, c2 might not be 0. I now have to generalize, with whatever c2 may be.
Why not just do the problem with c2 left in?

If c2 goes from 0 to something else, the equation changes from

u1 = w + (w - c1)(c1)

to

u1 = w + c2 + (w - c1)(c1 + c2)
Okay, that's he general equation for any c2.


meaning that the graph of u1 as a function of c1 shifts upwards by c2 units (since c2 is being added) and it shifts to the left by c2 units (since c2 is being added within the parantheses that enclose c1).
More importantly, meaning that u1 is a quadratic function of c1 depending on the parameter c2- and, as before, its maximum value occurs a the vertex.

[/quote]Right? I might be wrong there, so please tell me if I'm right or wrong. If I'm right, that means that, whatever c2 may be, u1 is maximized when c1 equals

( [sqrt(w^2 + 4w)] / 2 ) - c2[/quote]
How did you get this? Complete the square in q1 and see what you get.

at this maximizer is the same as before, except c2 is subtracted, since the graph shifted to the left, so we can imagine shifting it back to the right.) Right?

Now, then, we have the following equation:

b1 = ( [sqrt(w^2 + 4w)] / 2 ) - c2

"b1" in this case stands for "player 1's best response" to whatever c2 is doing. ("u1," by the way, meant "player 1's utility" or "player 1's payoff," which player 1 tries to maximize.) At the same time, player 2 is trying to maximize his payoff, so at the same time we have the following similar equation:

b2 = ( [sqrt(w^2 + 4w)] / 2 ) - c1

To find the "Nash equilibrium," we make "c2" at the end of the b1 equation equal "b2," and we make "c1" at the end of the b2 equation equal "b1," and then find the intersection of these two equations (which implies that player 1 and player 2 are each taking an action, player 1's action being the best response to player 2's action, and player 2's action being the best response to player 1's action). We can do this by substituting one equation into another. That is, you can take the b1 = ... equation and when you get to "c2," substitute that with what "b2" equals in the second equation.

However, if you do that, b1 just cancels itself out and you end up with the following:

0 = ( [sqrt(w^2 + 4w)] / 2 ) - ( [sqrt(w^2 + 4w)] / 2 )

Which is a true statement, but an utterly useless one. I must have done something wrong. So can someone help me here?

Here is the question that all this work is derived from (the first scan is of one page in the textbook, and the problem continues into the next page/scan). Using the vocabulary of game theory, what I'm having trouble doing in this problem is correctly calculating the best response functions and then calculating the Nash equilibrium.

In the next post I'll post scans of some of the textbook's previous pages that relate to this problem.
WillJ
WillJ is offline
#3
Sep17-06, 03:05 PM
P: 13
Scans of the previous pages from the textbook:

http://forums.civfanatics.com/uploads/20424/gt042_b.JPG

http://forums.civfanatics.com/uploads/20424/gt043_c.JPG

http://forums.civfanatics.com/uploads/20424/gt044_b.JPG

WillJ
WillJ is offline
#4
Sep17-06, 03:29 PM
P: 13

Help with game theory (specific knowledge in game theory probably not required)


Quote Quote by HallsofIvy
More importantly, meaning that u1 is a quadratic function of c1 depending on the parameter c2- and, as before, its maximum value occurs a the vertex.

Right? I might be wrong there, so please tell me if I'm right or wrong. If I'm right, that means that, whatever c2 may be, u1 is maximized when c1 equals

( [sqrt(w^2 + 4w)] / 2 ) - c2
How did you get this? Complete the square in q1 and see what you get.
I got that by taking the value of c1 at the vertex of the function when c2 = 0, and then subtracting c2, since as c2 increases, the best response for player 1 (that is, the maximizer of u1 as a function of c1) decreases by c2. I know my thinking must be off here, though. Iím trying to sort through whatís going on here. At any rate, I'm not sure what you mean by "complete the square in q1" (I'm assuming q1 is a typo, but I still can't see what square I'm supposed to complete or why I should complete it --- I could perhaps complete the square w^2 + 4w ... + 4, but why would I do that?)
nerdyjoe
nerdyjoe is offline
#5
Dec6-08, 01:51 PM
P: 1
the answer is (w-c2)/2
expand the original equation:
u1 = -c1^2 + (w-c2)c1+w+c2
it's a upside down parabola, with constant w and c2
max of a parabola is : -b/2a
b=(w-c2)
a=-1
so -b/2a=(w-c2)/2


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