分类:Measurement of risk attitude

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 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^[3], 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 ^[4] and ^[2].

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}=E\left(\rho\right), \Delta\left(\rho\right), \eta\left(\rho\right), \cdots }[/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].

两个理论基础的结合

现在如何把期望效用理论和分布函数的函数（其实是泛函）的数学形式相结合，以及从各阶矩的函数的角度来检验一下，到底函数形式如何，或者至少有哪些变量？这是一个有深远意义的问题。

广义地来看，如果我们允许分布函数的泛函包含货币收益的线性和非线性函数当作效用，则实际上，我们的框架包含了期望收益理论——期望收益相当于仅仅考虑一阶矩[math]\displaystyle{ E\left(\rho\right)=P_{A}U\left(A\right)+P_{B}U\left(B\right) }[/math]。因此，分布函数的泛函的讨论是更加基本的问题。

同时，许彬等人的预实验发现，风险态度——假设可以用一个参数[math]\displaystyle{ \theta }[/math]来描述的话，受到被试的认知负担的影响。因此，实际上，我们是在做关于[math]\displaystyle{ U=U\left(\left. E\left(\rho\right),\Delta\left(\rho\right),\eta\left(\rho\right),\cdots \right|\theta\right) }[/math]，或者说[math]\displaystyle{ U=U\left(\left. \rho \right|\theta\right) }[/math]的函数形式的研究。

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