Please explain this equasion from a Game Theory lecture
February 11, 2010 12:03 PM Subscribe
I am watching this yale lecture on game theory and the Nash equilibrium. Please explain why the cost coefficient is exponential.
at 36:00, the professor creates this formula
Payoff(player 1) = 1/2 [ 4 * (S1 + S2) + B(S1 S2) ] - S1^2
My question is, why is the cost coefficient squared? It does not make sense to me that the effort should exponentially increase, except perhaps that as time available to do things other than what this equation models decreases, that time becomes more expensive. However, since the number of hours used is held between zero and four, I don't think this is a good reason.
I am not an economist, this is the first class I have had with these types of things, but I did take several calculus and stats classes in college so I think I am pretty good at math.
at 36:00, the professor creates this formula
Payoff(player 1) = 1/2 [ 4 * (S1 + S2) + B(S1 S2) ] - S1^2
My question is, why is the cost coefficient squared? It does not make sense to me that the effort should exponentially increase, except perhaps that as time available to do things other than what this equation models decreases, that time becomes more expensive. However, since the number of hours used is held between zero and four, I don't think this is a good reason.
I am not an economist, this is the first class I have had with these types of things, but I did take several calculus and stats classes in college so I think I am pretty good at math.
Response by poster: My bad. But in any case, why is is squared?
posted by rebent at 12:48 PM on February 11, 2010
posted by rebent at 12:48 PM on February 11, 2010
Without looking at the lecture, my guess is that this particular functional form for the disutility of effort is chosen to obtain a nice, closed form solution. The player is going to maximization his payoff over effort -- a linear cost wouldn't given you a solution in terms of S1.
posted by diftb at 12:50 PM on February 11, 2010
posted by diftb at 12:50 PM on February 11, 2010
Response by poster: Diftb, are you saying that the cost coefficient is the way it is so the professor can produce a nice answer?
posted by rebent at 1:08 PM on February 11, 2010
posted by rebent at 1:08 PM on February 11, 2010
Best answer: I just watched that lecture! I agree, the model certainly wasn't super-intuive or anything. As diftb says (sort of) - it's just a model for prof Polak to play around with, chosen so that the math can work out nice. In the context of the lecture, I don't think that's particularly a criticism- this is just presented as a game that happens to follow a certain function, not as a realistic model of how partnerships work.
That said, if you want to hang *some* sense on it: Think about how overtime (sort of) works. I might be willing to work 40 hours a week this week at $20/hour, but to work an extra 10 hours, I might want to be paid $40/hour for the remaining time, because it gets harder to work more hours. If you wanted be to work 80 hours, I might not be willing to do that unless you *really* increased the pay- maybe even to the extent that you'd need to quadruple the pay to make it worth my while. In that case - if I need you to quadruple the pay to make it worth my while to double my hours, you have something (at least for two data points) roughly like "the cost (in dollars) of my time is proportional to the square of how many hours of my time you need".
There are a million ways wher that's still not intuitive (ie: if that function went all the way down, the cost of just one hour of my time would be very low) but maybe that's a start.
Mostly - it's just a model....
posted by ManInSuit at 1:19 PM on February 11, 2010
That said, if you want to hang *some* sense on it: Think about how overtime (sort of) works. I might be willing to work 40 hours a week this week at $20/hour, but to work an extra 10 hours, I might want to be paid $40/hour for the remaining time, because it gets harder to work more hours. If you wanted be to work 80 hours, I might not be willing to do that unless you *really* increased the pay- maybe even to the extent that you'd need to quadruple the pay to make it worth my while. In that case - if I need you to quadruple the pay to make it worth my while to double my hours, you have something (at least for two data points) roughly like "the cost (in dollars) of my time is proportional to the square of how many hours of my time you need".
There are a million ways wher that's still not intuitive (ie: if that function went all the way down, the cost of just one hour of my time would be very low) but maybe that's a start.
Mostly - it's just a model....
posted by ManInSuit at 1:19 PM on February 11, 2010
Best answer: When the exponent on effort is greater than 1, the marginal disutility of effort is increasing, which ManInSuit explains intuitively. Picking the exponent = 2 makes the derivation of the NE. Given players 2's effort S2, player 1's best response is to choose S1 to solve
max 1/2 [ 4 * (S1 + S2) + B(S1 S2) ] - S1^2
or (assuming interior solution)
S1 = ( 2 + B * S2 ) / 2.
Here's it very easy to see a NE by plotting the S1's best response vs S2's best response because both of these response functions are linear. Were the exponent not equal to 2, the best response would be a ugly mess of a function (or maybe not even a function at all)! This way it's easy to teach NE on a blackboard.
posted by diftb at 2:06 PM on February 11, 2010 [1 favorite]
max 1/2 [ 4 * (S1 + S2) + B(S1 S2) ] - S1^2
or (assuming interior solution)
S1 = ( 2 + B * S2 ) / 2.
Here's it very easy to see a NE by plotting the S1's best response vs S2's best response because both of these response functions are linear. Were the exponent not equal to 2, the best response would be a ugly mess of a function (or maybe not even a function at all)! This way it's easy to teach NE on a blackboard.
posted by diftb at 2:06 PM on February 11, 2010 [1 favorite]
This thread is closed to new comments.
posted by hattifattener at 12:40 PM on February 11, 2010