on a single technology. Because of the strong nonlinearities, there may be a very large
number of local maxima.
The problem is also complicated by additional issues that arise in realistic situations:
- The noise (i.e. the uncertainty in future costs if parameters are known) has heavy
tails and possibly long memory. - The response of technologies to investment may be correlated across
technologies, e.g. an improvement in the structural material for one technology
may also help another technology. - Such models are difficult to calibrate. Even worse, there may be Knightian
uncertainty (ambiguity), i.e. we may not know the correct probability distributions
and we may want to understand the robustness under variations in the
assumptions.
A few interesting questions: - What are the best numerical methods for finding good solutions?
- How does the number of technologies one should invest in depend on the
properties of the problem (e.g. total number of technologies, level of uncertainty,
learning rates, nature of the utility function, correlations, heavy tails)? - Which parameter regimes have robust solutions, and which have unstable
solutions?
Uncertainty and Climate Change
Economists, following Frank Knight, distinguish between risk and uncertainty.
Risk occurs when we have a stochastic outcome following a known probability
distribution (the toss of a fair coin): uncertainty occurs when we don’t know the
distribution (rolling an unfair dice when we have no historical record of its outcomes).
The word ambiguity is now used to refer to these situations of uncertainty, i.e.
stochasticity without a known probability density function (pdf). Take climate change:
there is a wide range of estimates of key parameters such as the climate sensitivity s
(the equilibrium temperature response to a doubling of CO 2 ). Such a diversity raises the
question: what assumption about the probability density function over outcomes should a
decision-maker maintain, if any? There are several competing approaches.
One is to combine the probability density functions from the various underlying
models following a Bayesian approach.
A competing approach is to recognize that there is no single distribution over outcomes
and to work with a second order probability distribution over the different models, so that
pi is the probability that the i-th model is the correct model. In this approach it is assumed
consistently with many experimental studies of human behavior that decision-makers are
ambiguity-averse so that their payoff is the expectation according to the second order
probabilities of a concave function of the expected outcomes of the various models.
There are several competing axiom sets that seek to provide a basis for this type of
approach to decision-making under ambiguity, following a generalization of the approach
in Savage’s Foundation of Statistics (see e.g. Klibanoff et.al. (2005) or Schmeidler
(1989).
This issue occurs not only with climate change, but also in a wide range of
situations of relevance to sustainability. We frequently face stochastic outcomes without
well-defined pdfs, and often have competing models with divergent predictions. This is
true for the impact of genetically modified organisms, the availability and costs of