分类:Measurement of risk attitude

来自Big Physics


The core question and some background

Risk attitude is quite often a relevant factor in many decision making games/situations. Thus often researchers need to use some kind of standard means to measure of the risk attitude of test subjects. One of these typical methods is the famous Holt&Larry 10 lottery experiment [1][2]: a test subject is asked to choose option A or option B in each of the ten lottery choices, which are shown in more details in the following table.

Option A Option B Expected payoff difference Variation difference estimated range of [math]\displaystyle{ r }[/math][3]
1/10 of $2.00, 9/10 of $1.60 1/10 of $3.85, 9/10 of $0.10 $1.17 -1.25 r<-1.71
2/10 of $2.00, 8/10 of $1.60 2/10 of $3.85, 8/10 of $0.10 $0.83 -2.22 -1.71<r<-0.95
3/10 of $2.00, 7/10 of $1.60 3/10 of $3.85, 7/10 of $0.10 $0.50 -2.92 -0.95<r<-0.49
4/10 of $2.00, 6/10 of $1.60 4/10 of $3.85, 6/10 of $0.10 $0.16 -3.34 -0.49<r-0.14
5/10 of $2.00, 5/10 of $1.60 5/10 of $3.85, 5/10 of $0.10 -$0.18 -3.48 -0.14<r<0.15
6/10 of $2.00, 4/10 of $1.60 6/10 of $3.85, 4/10 of $0.10 -$0.51 -3.34 0.15<r<0.41
7/10 of $2.00, 3/10 of $1.60 7/10 of $3.85, 3/10 of $0.10 -$0.85 -2.92 0.41<r<0.68
8/10 of $2.00, 2/10 of $1.60 8/10 of $3.85, 2/10 of $0.10 -$1.18 -2.22 0.68<r<0.97
9/10 of $2.00, 1/10 of $1.60 9/10 of $3.85, 1/10 of $0.10 -$1.52 -1.25 0.97<r<1.37
10/10 of $2.00, 0/10 of $1.60 10/10 of $3.85, 0/10 of $0.10 -$1.85 0 r>1.37

Option B always has larger variation then option A, however, starting with lower average payoff at the first lottery, average payoffs from option B gradually become higher than that of option A. To see this, we have added each column for difference between average payoffs of option A and option B and also for difference between variations of option A and option B.

For a risk neutral player, who cares only about the average payoff, it is clear that he/she will choose option A for the first 4 lotteries and will choose option B for the latter ones. That is transition point is likely at 4-5 for risk neutral players. Risk aversion players is likely willing to trade some payoff for less variation, thus seems to choose more secure options. In turn, this will make the transition point later than 4-5. Likely, risk seeking players will like to get a high risk option thus start to choose option with larger variations earlier than 4-5. In this sense, the transition point can be used as an indicator of risk attitude.

The underlying theory of Holt&Larry: Expected Utility Theory

This is a very good idea in general to use a set of lotteries to reveal risk attitude. However, there are quite several problems in the details of the above setup. For example, values of the probabilities, might be further adjusted to reach a better measurement. The current setup is related to an underlying theory, so-called Measures of risk aversion under expected utility theory, and in particular constant relative risk aversion (CRRA), where the utility of mixed options are [math]\displaystyle{ U\left(p_{A}|A\rangle \langle A|+p_{B}|B\rangle \langle B|\right)=p_{A}U\left(A\right)+p_{B}U\left(B\right) }[/math] while the utility of a pure option is [math]\displaystyle{ U\left(A\right)=\left[E\left(A\right)\right]^{1-r} }[/math]. Here [math]\displaystyle{ E\left(A\right) }[/math] is the monetary payoff of the pure option A. Note that, ideally, [math]\displaystyle{ r }[/math] should be less than 1 since overall larger monetary payoff often means larger utility too. One can also define risk aversion, neutral and seeking according to this parameter [math]\displaystyle{ r }[/math]: [math]\displaystyle{ r=0 }[/math] for neutral, [math]\displaystyle{ r\gt 0 }[/math] for aversion and [math]\displaystyle{ r\lt 0 }[/math] for seeking.

With this underlying theory, given the transition point [math]\displaystyle{ j }[/math], one can calculate value of r. [math]\displaystyle{ P_{j-1}U(A_{j-1,1})+\left(1-P_{j-1}\right)U(A_{j-1,2})\geq P_{j-1}U(B_{j-1,1})+\left(1-P_{j-1}\right)U(B_{j-1,2}) }[/math] and [math]\displaystyle{ P_{j}U(A_{j,1})+\left(1-P_{j}\right)U(A_{j,2})\leq P_{j}U(B_{j,1})+\left(1-P_{j}\right)U(B_{j,2}) }[/math]. With these two equations but only one unknown variable [math]\displaystyle{ r }[/math], it is easy to estimate the unknown. It can be seen clearly that in some cases, [math]\displaystyle{ r }[/math] is required to be much over 1. This makes not too much sense.


Besides the linearity[4], expected utility theory when applied to risk attitude, also assumes the following, [math]\displaystyle{ U^{rn}\left(A\right)=E\left(A\right) }[/math], [math]\displaystyle{ U^{ra}\left(A\right) \lt U^{rn}\left(A\right)=E\left(A\right) }[/math] and [math]\displaystyle{ U^{rs}\left(A\right) \gt U^{rn}\left(A\right)=E\left(A\right) }[/math]. Discussions on specific forms of [math]\displaystyle{ U^{ra} }[/math] and [math]\displaystyle{ U^{rs} }[/math] can be found from again Measures of risk aversion under expected utility theory.

However, this is not the most general form of utility function of mixed options. There are other issues with the Holt&Larry method, like applicability of the measured attitude from one situation towards a different situation, see for example [5] and [2].

A More general theory is the prospect theory, [math]\displaystyle{ U^{PT}\left(A,P_{A}|B, P_{B}\right)=W\left(P_{A}\right)E\left(A\right)+W\left(P_{B}\right)E\left(B\right) }[/math].

The most general utility function of mixed options

In principle, for a mixed option [math]\displaystyle{ \rho }[/math], for example [math]\displaystyle{ \rho=P_{A}\left|A\right\rangle\left\langle A\right|+P_{B}\left|B\right\rangle\left\langle B\right| }[/math], [math]\displaystyle{ U=U\left(\rho\right) }[/math] should be a function as [math]\displaystyle{ U=U\left(E\left(\rho\right),\left(\Delta\right)^{\frac{1}{2}}\left(\rho\right),\left(\eta\right)^{\frac{1}{3}}\left(\rho\right),\cdots \right) }[/math], where [math]\displaystyle{ E\left(\rho\right), \Delta\left(\rho\right), \eta\left(\rho\right), \cdots }[/math] is respectively the first, second, third and so on moment of the distribution function [math]\displaystyle{ \rho }[/math]. This is because of the fact that a distribution is equivalent to all of its moments, [math]\displaystyle{ \vec{m}=\left[E\left(\rho\right), \Delta\left(\rho\right), \eta\left(\rho\right), \cdots\right]^{T} }[/math].

Following this line of thinking, it should be interesting to know that when human makes decisions, how much of those moments are really relevant? Clearly for risk neutral players, only the average is relevant, that is if we denote the utility function as [math]\displaystyle{ U\left(\rho\right)=U\left(\vec{m}; \vec{\theta}\right)=\sum_{j}\left(m_{j}\right)^{\frac{1}{j}}\theta_{j} }[/math], then [math]\displaystyle{ \theta^{rn}_{1}=1, \theta^{rn}_{j\gt 1}=0 }[/math].

In this sense, what about risk seeking and risk aversion players? How many [math]\displaystyle{ \theta_{j} }[/math] is required?

If we can find such a description, then can find device some measurement to estimate this parameter [math]\displaystyle{ \vec{\theta} }[/math] for each player? If we can, then we will have a good theoretical description and a standard experimental device of risk attitude.

Asking question in this way, we avoid to take the expected utility assumption, while of course, it is much easier to start from the expected utility theory, which is at least to me quite questionable.

Experimental Setup

In order to have a better measurement of risk attitude, and also to look into the utility function of mixed options, we propose to do the following:

  1. options with the same average payoff but different variation, or higher-order moments
  2. options with the same variation or the same higher-order moments but different average payoff
  3. other combinations

Plan for the next step

  1. literature review on measurement of risk attitude and also on expected utility theory
  2. trial experiments
  3. real experiments

References

  1. Holt, C. A., & Laury, S. K. (2002). Risk aversion and incentive effects. American Economic Review, 92(5), 1644-1655.
  2. 2.0 2.1 Gary Charness, Uri Gneezy and Alex Imas, Experimental methods: Eliciting risk preferences, Journal of Economic Behavior & Organization, 87(2013), 43-51, https://doi.org/10.1016/j.jebo.2012.12.023.
  3. Dave C, Eckel C C, Johnson C A, et al. Eliciting risk preferences: When is simple better?[J]. Journal of Risk & Uncertainty, 2010, 41(3):219-243.
  4. Kahneman, Daniel, Tversky, Amos (1979). "Prospect theory: An analysis of decision under risk". Econometrica: Journal of the econometric society. 47 (2): 263–291.https://www.jstor.org/stable/1914185
  5. Habib, S., Friedman, D., Crockett, S. et al. J Econ Sci Assoc (2017). https://doi.org/10.1007/s40881-016-0032-8.

There is a Chinese version of this page (这是这个页面的中文版): 风险态度的测量.

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