The Random Preference Model

This provides a comparison to the RP model (Loomes and Sugden, 1995), that can use the same model for the same pair of bets. This is possible by making the risk

aversion parameterr stochastic:

u(x) =x1 r where r_⇠(beta(3,3)_· + (↵ /2)) where ↵= 0.185 = 1.0

Here, abetadistributionbeta(3,3) creates a random symmetrical bell-shaped r distribution; is equal to its range and↵ determines where the midpoint of the range lies. beta(3,3) creates random values over [0; 1] with mean = median = 0.5. This distribution is then centred around ↵ with range . This distribution is identical to the one used for the novel valuation model described in chapter 4, so it is useful to explain its properties in detail.

Choices: For each decision task, a DM then samples a single r parameter to use in the utility function. This r parameter then gives a CE, e.g. for a choice between the described P-Bet and $-Bet or for a CE elicitation. As the median of r parameters equalsmedian(r) = 0.185, half of the possiblervalues are either below or above it. Since a lowerrparameter results in lower risk aversion, this means that the $-Bet is preferred forr parameters below the median, i.e. in half of the cases. For the other half of cases with higher risk aversion, the $-Bet is rejected instead. Therefore, the P-Bet is preferred over the$-Bet with a probability of 50%.

Choices between bets and sure amounts can be inferred from the distribution of CE values from the lotteries, shown in figure 3.1. The P-Bet’s CE range is [£5.95;£10.13] and the$-Bet’s CE range is [£0.64;£17.40]. This means that the P- Bet is always preferred over£5.95 but never preferred over£10.13 (£0.64 and£17.40 for the $-Bet). Replicating Mosteller and Nogee’s observation, the probability of choosing a lottery increases gradually from 0% at the lower bound of the range and until it reaches 100% at the upper end of the range. The median CE of both bets lies at£9.13. Therefore both bets will be preferred over£9.13 exactly 50% of times, meaning that they have the same SI point at£9.13.

Figure 3.1: RP Model: Underlying CE sample distributions

Valuations: Figure 3.1 shows the distribution of lottery CEs for the P-Bet and the $-Bet (see section A.1 in the Appendix for an explanation why the CE distri- butions are not symmetric). These are the same CEs that govern choice behaviour.

$-Bet CEs are more spread out because of the higher variance of the $-Bet. But picking a P-Bet and a$-Bet CE at random leads to a probability of either CE being

higher at exactly 50% because the medians of both distributions are equal. So the RP model does not predict the preference reversal phenomenon. Although prefer- ence reversals occur, the asymmetric e↵ect of a majority preference for the P-Bet in choices to a majority preference for the $-Bet in valuations does not occur as observed by Grether and Plott (1979).

Adjustments of Sure Amounts: To illustrate MSoP values that are elicited after a choice, first consider a choice between a lottery and a sure amount at the SI point. By definition, the lottery is chosen 50% of times. Assume an occasion where the lottery is chosen based on a momentary rparameter. But based on that momentary preference, what will the MSoP be? If the DM uses the samerparameter for the MSoP elicitation, the MSoP value must equal the distance between the lottery’s momentary CE and the sure amount because this is precisely what the di↵erence in utility is worth in money to the DM. Also, the lottery CEmustbe higher than the SI point because the DM would have chosen the sure amount otherwise. So it is possible to infer from the lottery choice that the lottery CE must be within a subset of the CE distributionabove the SI point. This is illustrated in figure 3.2 for both lotteries. P-Bet choices result in a mean MSoP mismatch of £0.39 (sd=0.23).

$-Bet choices result in a far larger mean MSoP mismatch of£2.82 (sd=1.84) because the $-Bet CE distribution is larger.

Figure 3.2: RP Model: MSoP Values after Lottery Choices

This e↵ect goes into the opposite direction for sure amount choices. MSoP values for sure amount choices must be based on CEs below the SI point and are shown in figure 3.3. Here, P-Bet MSoPs overshoot the SI point with a mean of

£0.64 (sd=0.54) after sure amount choices. Again, the mean mismatch is larger at

Figure 3.3: RP Model: MSoP Values after Lottery Choices

These predictions come from the assumption that a choice preference from the RP model spills over to an MSoP task. But if an MSoP value is elicited later after a choice between a lottery and sure amount, it is not sensible to assume that the DM would use the RP parameter from the past choice again for the MSoP. So if no spill-over e↵ect is present, or if the DM starts the MSoP task with a “fresh” preference, DMs would report CEs from the distribution in figure 3.1. The result

after a choice between a lottery and a sure amount at the SI point would be a distribution of MSoP values that do not show an MSoP mismatch because half of the corresponding CEs would be either above or below the SI point.

So the RP model with assumption of spill-over e↵ects can predict an MSoP mismatch and probabilistic choice and valuation behaviour. But the RP model does not predict the preference reversal phenomenon.