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instantaneous-device-utility-curve-analysis.md

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Instantaneous Utility Function Analysis

Instantaneous means utility is obtained from instaneous consumption. Example: dimmable ligthing. What are the high level and necessary requirements and desirata for instantaneous utility curve, for a device class like dimmable lighting?:

  • The curve must be concave.
  • The curve only has to be defined over [q_min(t), q_max(t)]
  • For the devices we are trying to model here, the maximum utility should coincide with q_max(t) and decrease towards q_min(t). It doesn't make much sense for the curve to be eventually decreasing.

In practice a human user is going to define the parameters to the curve, so what they mean has to be intuitive. In general, a user cares about 3 things:

  • Where the peak occurs.
  • Where the curve intersects the q_{t,min} and q_{t,max}.
  • How the curvy-ness matches their understanding of their own utility.

Alt 1 (IDevice)

IDevice instantaneous utility function tries to capture these things with 4 parameters. Refer to notes in source. This figure shows some example curves.

Alt 2 (IDevice2)

User specifies q_{t,min}, q_{t,max} hard bounds. Consumption at q_{t,max} is utility maximizing and utility decreases towards q_{t,min}. It is unreasonable to expect the user to completely define the mathematical parameters to the utility function. An intuitive way for the user to specify flexibility is to specify at what price they wish to start compromising (from q_{t,max}) and at what price they are willing to stop compromising (at q_{t,min}). This corresponds to the derivative of the utility function at q_{t,max} and q_{t,min} respectively.

To acheive this, define a reference quadratic section and scale / fit it accordingly. Etc etc.

u = np.poly1d([-1/2, 1, 0])
print(u)
u(0)
u(1)
u.deriv()(1)
u.deriv()(0)

Reference Instantaneous Utility Function

The device modeling this paper defined a utility function for two devices whose utility is defined independently at each time-slot as follows:

U(q(t)) = a - (b - q(t)/q_bar)**(-3/2)

Where:

  • a, b are "positive constants".
  • q_bar is not defined, but we assume they mean be q_max(t).

The above utility function didn't make sense to me so build IDevice, IDevice2 instead.