The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity

Abstract

In this paper we examine the performance of theories of decision making under uncertainty/ambiguity from the perspective of their descriptive and predictive power. To this end, we employ an innovative experimental design which enables us to reproduce ambiguity in the laboratory in a transparent and non-probabilistic way. We find that judging theories on the basis of their theoretical appeal, or on their ability to do well in terms of estimation, is not the same as judging them on the basis of their predictive power. We find that the models that perform better in an aggregate sense are Gilboa and Schmeidler’s MaxMin and MaxMax Expected Utility Models, and Ghiradarto et al.’s Alpha Model, implying that more elegant theoretical models do not perform as well as relatively simple models. This suggests that decision-makers, when confronted with a difficult problem, try to simplify it, rather than apply a sophisticated decision rule.

Appendices

Appendix 1: the experimental instructions

Welcome to this experiment. MIUR (the Italian ministry for the universities) has provided the funds to finance this research. Depending on your decisions you may earn a considerable amount of money which will be paid to you in cash immediately after the end of the experiment. This sum will be composed of the £10 participation fee plus your ‘earnings’ from a lottery. This latter could be a loss of £10, a gain of £10 or gain of £100. You cannot walk away from this experiment with less money than that with which you arrived, though you might walk away with £20 more or with £110 more.

There are no right or wrong ways to complete the experiment, but the decisions that you take will have implications for what you are paid at the end of the experiment. This depends partly on the decisions that you take during the experiment and partly on chance. So you will need to read these instructions carefully.

At the end of the experiment you will be asked to complete a brief questionnaire and to sign a receipt for the payment that you received, and to acknowledge that you participated voluntarily in the experiment. The results of the experiment will be used for the purpose of academic research and will be published and used in such a way that your anonymity will be preserved.

1.1 Outline of the experiment

You will be asked 162 questions. Each will be of the same type. You will be presented with two lotteries and you will be asked which you prefer. After you have answered all 162 questions, one of them will be selected at random, the lottery that you said that you preferred on that question will be played out, and you will be paid the outcome: if the outcome is a loss of £10 you will leave the experiment with the same as when you came; if the outcome is a gain of £10 you will leave the experiment with £20 more than when you came; if the outcome is a gain of £100 you will leave the experiment with £110 more than when you came. If you did not express a preference on the selected question then one of the two lotteries will be selected at random and played out. It is clearly in your interests to answer each question as if that were the question to be played out.

1.2 The Bingo Blower

You will have noticed a Bingo Blower in the laboratory. In this Blower there are balls of three different colours: pink, blue and yellow. The balls are constantly being blown about in the Blower. At the end of the experiment, when we come to play out your preferred choice on one of the questions, we will use this Bingo Blower to determine a colour: we will allow you to open the exit chute — this will lead to one ball being expelled. Obviously this expulsion will be done at random as there is no way that you can control the colour of the ball that emerges. The colour of the ball and the lottery that you chose on the question that was selected will determine your payment.

1.3 The questions

A sample question is illustrated in the Figure attached to these Instructions. In this figure, there are two lotteries — that on the left and that on the right. The lottery on the left would lead to a loss of £10 if the ball expelled was yellow, to a gain of £10 if the ball expelled was blue and to a gain of £100 if the ball expelled was pink. The lottery on the right would lead to a loss of £10 if the ball expelled was pink or blue and to a gain of £10 if the ball expelled was yellow. You have to decide for each question whether you prefer the lottery on the left or that on the right. You should indicate your choice by clicking on the box below the appropriate lottery. You will be given at least 30 sec to make up your mind and you cannot proceed to the next question until these 30 sec have elapsed. The number of seconds left to make your decision will be indicated at the bottom of the screen. If you want more time, simply click on ‘STOP THE CLOCK’; then click on ‘RESTART THE CLOCK’ when you are happy to proceed. If the 30 sec have elapsed and you have not taken a decision then ‘no decision’ will be recorded for that question.

1.4 The end of the experiment

After you have answered all 162 questions you will be asked to call over an experimenter. In front of him or her you will choose at random one of the questions — by picking at random a ticket from a set of cloakroom tickets numbered from 1 to 162. The computer will recall that question and your answer to it, and then you will play out your preferred choice on that question — in the manner described above. If you did not take a decision on that question then you will toss a coin to determine which of the two lotteries will be played out. You will then be asked to fill in a short questionnaire. We will then pay you, you will sign a receipt and then you will be free to go. Note that the experiment will take at least 81 min of your time. You can take longer and it is clearly in your interests to be as careful as you can when you are answering the questions.

1.5 If you have any questions at any stage, please ask one of the experimenters

Appendix 2: the specifications

This appendix provides technical detail on the various specifications that we fitted to the data. We start with some notation. We denote the three possible outcomes in the experiment by x ₁ , x ₂ and x ₃ . Except for the CPT specification we normalise the highest to have a utility of 1 and the lowest to have a utility of 0; we denote the utility of the middle outcome by u. We denote the three colours by a, b and c. A lottery can be denoted by

$$ L = ({x_1},{S_1};{x_2},{S_2};{x_3},{S_3}) $$

(1)

Here S _i is the state (one of $\emptyset, \;a,\;b,\;c,\;a \cup b,\;a \cup c,\;b \cup c\;{\hbox{and}}\;a \cup b \cup c$) in which the lottery pays out x _i .

2.1 (Subjective) Expected Utility theory (EU)

In this subjects choose between lotteries on the basis of their (subjective) expected utility, calculated on the basis of the subject’s subjective probabilities attached to the various states. The (subjective) expected utility of the lottery L is given by

$$ SEU(L) = {p_2}u + {p_3} $$

(2)

where p _i is the (subjective) probability of state i. If we use the notation that p _i = P(S _i ) where S _i denotes the state in which the lottery pays out x _i , then we have

$$ \begin{array}{*{20}{c}} \begin{gathered} P(\emptyset ) = 0 \hfill \\ P(a) = {p_a}\quad P(b) = {p_b}\quad P(c) = {p_c} \hfill \\ \end{gathered} \hfill \\ {P(a \cup b) = {p_a} + {p_b}\quad P(a \cup c) = {p_a} + {p_c}\quad P(b \cup c) = {p_b} + {p_c}} \hfill \\ {P(a \cup b \cup c) = {p_a} + {p_b} + {p_c} = 1} \hfill \\ {} \hfill \\ \end{array} $$

(3)

where p _a , p _b and p _c are the subject’s subjective probabilities for the three colours. In this model we estimate u, p _a , p _b and p _c (subject to the constraint that ${p_a} + {p_b} + {p_c} = 1$).

2.2 Choquet Expected Utility theory (CEU)

Here the Choquet Expected Utility of the lottery L is given by

$$ CEU(L) = {w_{23}}u + {w_3}(1 - u) = ({w_{23}} - {w_3})u + {w_3} $$

(4)

where w _i is the Choquet capacity (or weight²⁴) of state i. If we use the notation that w _i = W(S _i ) where S _i denotes the state in which the lottery pays out x _i , then we have, in order to satisfy the Choquet conditions, that

$$ \begin{array}{*{20}{c}} {W(\emptyset ) = 0} \hfill \\{W(a) = {w_a}\quad W(b) = {w_b}\quad W(c) = {w_c}} \hfill \\{W(a \cup b) = {w_{ab}}\quad W(a \cup c) = {w_{ac}}\quad W(b \cup c) = {w_{bc}}} \hfill \\{W(a \cup b \cup c) = 1} \hfill \\\end{array} $$

(5)

Here w _a , w _b , w _c , w _ab , and w _bc are the subject’s Choquet capacities (or weights) for the various possible states. In this model we estimate u, w _a , w _b , w _c , w _ab , w _ac and w _bc . Note that there is no necessity that ${w_{de}} = {w_d} + {w_e}$ for any d or e. That is, there is no necessity that the weights are additive (probabilities). Indeed this is the main difference between (Subjective) Expected Utility theory and Choquet Expected Utility theory.

2.3 Prospect Theory (PT)

This is a preference functional ‘between’ that of EU and the Choquet Expected Utility functional. We should say at the outset that we are hesitant about the acceptability of this term being used in this context, but it seems appropriate. Prospect Theory (see Kahneman and Tversky 1979) envisages utilities being weighted by some function of the ‘true’ probabilities. If there are true probabilities of the various colours π _a , π _b and π _c then Prospect Theory envisages them being replaced by f(π _a ), f(π _b ) and f(π _c ). If we denote these respectively by p _a , p _b and p _c then we get this specification. It is precisely the same as the (Subjective) Expected Utility preference functional except for the fact that the ‘probabilities’ are not additive. In this model we estimate u, v, p _a , p _b and p _c (but no longer subject to the constraint that ${p_a} + {p_b} + {p_c} = 1$). We note that this preference functional may not satisfy dominance (though it does so in this context), unlike the Choquet preference functional, which does.

2.4 Cumulative Prospect Theory (CPT)

This is almost the same as Choquet Expected Utility theory except for the incorporation of a reference point. We take this to be a gain of £0 in the experiment. We normalise the utility function so that u(£0) = 0 and u(£100) = 1 and estimate u = u(£10). We also estimate v = -u(-£10). In other respects the theory is the same as CEU. We note that CEU is nested within EU.

2.5 Decision Field Theory (DFT)

The Decision Field Model, as proposed by Busemeyer and Townsend (1993) is similar to EU except insofar as the error term is heteroscedastic. So the difference between two lotteries is valued exactly as in EU but the error variance is not constant. To define it we have to introduce some extra notation. Consider a choice between a Left lottery which yields outcomes $O_a^L$, $O_b^L$ and $O_c^L$ in the states (colours) a, b and c respectively, and a Right lottery which yields outcomes $O_a^R$, $O_b^R$ and $O_c^R$ in these states (colours). Then the difference between the two lotteries is evaluated, as in EU, (using an obvious notation) by

$$ V(L) - V(R) + \varepsilon = {p_a}[U(S_a^L) - U(S_a^R)] + {p_b}[U(S_b^L) - U(S_b^R)] + {p_c}[U(S_c^L) - U(S_c^R)] + \varepsilon $$

where the error term ε has a normal distribution with mean 0 and variance given by:

$$ {s^2}\{ {p_a}{[U(S_a^L) - U(S_a^R)]^2} + {p_b}{[U(S_b^L) - U(S_b^R)]^2} + {p_c}{[U(S_c^L) - U(S_c^R)]^2} - {[V(L) - V(R)]^2}\} $$

Here we estimate u, the (additive) probabilities and s. Note that this model has exactly the same number of parameters as EU but neither is nested inside the other.

2.6 MaxMin expected utility theory (GS Min)

This is a special case of Alpha theory (described below) with the parameter α of that theory equal to 1.

2.7 MaxMax expected utility theory (GS Max)

This is a special case of Alpha theory (described below) with the parameter α of that theory equal to 0.

2.8 Alpha theory

In this theory, decision-makers are envisaged as thinking of the probabilities (of the various events) lying in some convex space. Denoting this convex space by P, each point in which represents a possible probability for each of the three colours (necessarily additive), then, according to this theory, the decision-maker chooses between lotteries on the basis of the maximum of

$$ \alpha \mathop {{\min }}\limits_{p \in P} [EU(L)] + (1 - \alpha )\mathop {{\max }}\limits_{p \in P} [EU(L)] $$

(6)

The convex set P is individual specific. We parameterise that and estimate these parameters, in addition to the utility parameter u. However, the theory does not specify how it should be parameterised. We assumed that this convex space can be represented as a convex area within the triangle defined by the vertices (0,0), (1,0) and (0,1) in a space with the probability of colour a on the horizontal axis and the probability of colour b on the vertical axis. In order to make the convex space symmetrical as far as the treatment of the three probabilities were concerned, we characterised it as bounded by a vertical line at ${\bar{p}_a}$ , a horizontal line at ${\bar{p}_b}$ and a line parallel to the hypoteneuse (with therefore a slope of −45°) such that $1-{p_a}-{p_b} = {\bar{p}_c}$ . We estimate ${\bar{p}_a}$, ${\bar{p}_b}$ and ${\bar{p}_c}$ along with the other parameters. We note that this convex space can take a variety of different forms. Clearly if it just consists of a single point then the Alpha model reduces to EU.

2.9 MaxMin

In this, the decision-maker is presumed to follow the rule of choosing the lottery for which the worst outcome is the best. We assume that the rule is followed lexicographically, so that we get the following rule, where l ₁ , l ₂ and l ₃ denote the three outcomes on one of the two lotteries, L, ordered from the worst to the best, and m ₁ , m ₂ and m ₃ denote the outcomes on the other lottery, M, also ordered from the worst to the best:

We note that there are no parameters to be estimated in this model, though we do assume that the decision-maker ranks £100 as the best outcome, £10 as the second best and −£10 as the worst.

2.10 MaxMax

In this the decision-maker is presumed to follow the rule of choosing the lottery for which the best outcome is the best. We assume that the rule is followed lexicographically, so that we get the following rule, using the same notation as above:

Again there are no parameters to estimate.

2.11 Minimax Regret²⁵ (MinReg)

With this preference functional, the decision-maker is envisaged as imagining each possible ball drawn, calculating the regret associated with choosing each of the two lotteries, and choosing the lottery for which the maximum regret is minimized. Again there are no parameters to estimate, though it is assumed that there is a larger regret associated with a larger difference between the outcome on the chosen lottery and the outcome on the non-chosen lottery.