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The relevance of irrelevant alternatives in lotteries under risk and ambiguity
Abstract:
The attraction effect violates the economic assumption that decisions should be independent of irrelevant alternatives. This violation is demonstrated in multiple studies, usually by using
hypothetical choice tasks. This online survey experiment is designed to test the attraction effect in situations where decision-makers face real economic consequences. The results replicate the finding that the attraction effect exists in risky lottery choices. Secondly, the results demonstrate that the attraction effect still exists in ambiguous lottery choices, where
decision-makers do not know the exact probabilities of the lotteries.
Ruben Smits 10632239
Master Economics; Track: Behavioral Economics and Game Theory Supervisor: Prof dr. C.M. van Veelen
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Statement of Originality This document is written by Student Ruben Smits (10632239) who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of
Economics and Business is responsible solely for the supervision of completion of the work, not for the contents
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1. Introduction
Imagine you are grocery shopping and you have to decide which toilet paper you buy. You have two options, sometimes you buy the cheaper paper and sometimes the paper which has a better quality. But now there is suddenly a third option which is more expensive then both original options but only better in quality then the cheap option. Obviously you will not buy this third option because it is worse on price and quality then the quality toilet paper.
Classical economics assumes choices to be consistent (Taversky & Kahneman, 1986) so this irrelevant alternative should not influence your decision. However Huber (1982) showed that adding a decoy option that is only dominated (worse on all dimensions) by one of the options will increase the probability of choosing that dominating option. This is called the attraction effect, asymmetric dominance effect or decoy effect..
The attraction effect is typically seen in options with two dimensions. One option is better than the other option in one of the dimensions but worse in the other. Preferences then should purely be based on the weights you assign to the differences in dimensions. The attraction effect then refers to the situation where adding an option X’ to a choice menu of X and Y increases the choices of X, in this example and in the rest of this paper X’ is dominated by X in both dimensions but only dominated by Y in one dimension, this notation will be used in the rest of this paper. X’ is called the decoy, X the target and Y the competitor.
This violation of rational choice has been studied in multiple fields such as consumer products (Gomez et al, 2016, Heath et al, 1995) , elections (O'Curry et al, 1959) or political issues (Herne, 1997). These experiments used hypothetical options in their experiments. As a result the found attraction effect might not be realistic because off a lack of motivation by participants. To create internal valid results options should have economic consequences (Lichters et al, 2015). So hypothetical options should not be used, also costly and difficult options to implement such as cars or washing machines which are previously used should be ruled out.
Therefore in this experiment real money is used to incentivize participants into making their real preferred choice. The research design is based on Herne’s (1999) research where she let students choose between binary lotteries, also called binary gambles. A binary lottery consists of two dimensions: probability and potential reward. An example of a lottery option is a 74% chance of winning 12 euro, in the rest of this paper this will be noted by
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(0.74, 12). The core options then are different in the way that one has a higher probability of winning but the other has a higher payoff if the lottery is won. One of the choices participants had to make in Herne’s experiment was between (0.4,75), (0.8,38) and (0.35,69). The third option here is the decoy and should increase the frequency the first option is chosen because it is dominated by the first option but not by the second. The benefit of using lotteries is that choices have real monetary value and so participants have a high motivation to concentrate and choose their preferred option.
Besides the benefit that lotteries are easy to incentivize there are more reasons that lottery options are a good tool for experiments. They are important to research because in your daily life you have to make choices that are based on probabilities and payoff often. For example in choosing what medical treatment you want to take. Some treatments are more expensive than others but also have a higher probability of healing. Also most investments options are risky, in deciding in where you want to invest your money in, you might have to choose between a risky option with a high potential payoff or a safer option. A third example is deciding where to buy your groceries. You could go to the store which always has relative cheap prices or to a more expensive supermarket but sometimes they have very good
discounts. Going there is risky but it might save you some money.
As seen above there are multiple situations where lotteries translate to the real world so it is interesting to see if there is an attraction effect in these lotteries. Herne (1999) found a significant attraction effect in lotteries. A downside of her experiment is that she used exact numerical values for the probabilities of the lottery options. As a result her results might not have much external validity. In the real world you usually do not know the exact probabilities of your options. Think of the examples of the medical treatments, investment options or grocery stores, you know which options are riskier, you can guess how large the probabilities are but you never know them exact.
For this reason the design of this experiment includes lotteries where exact
probabilities are unknown but participants have a rough guess. These lotteries will be called ambiguous lotteries or lotteries under ambiguity. The lotteries where probabilities are known will be called risky lotteries or lotteries under risk. So the experiment is designed to give an answer to the following questions: is there an attraction effect in risky lotteries? Is there an attraction effect in lotteries under ambiguity? Is the attraction effect bigger in risky lotteries compared to lotteries under ambiguity?
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Despite this violation of rational choice it has not been studied often in the Economic field. Herne (1999) was the first to build a simple economic design, however her research does not explain the decoy effect. Also, Herne used the same expected values in all her lotteries. This is not recommended by Grossetto and Gaudeul (2016), they showed in a perceptual choice task that it is important to use option with different expected values. Kroll and Vogt also used lotteries to research the attraction effect, but they used the attraction effect to obtain an increase in certainty equivalents and did not look at multiple lotteries as options.
Another utility that the attraction effect can have is that it can be a way to nudge people into making an optimal decision. Nudging is a concept in behavioral economics where behavior of individuals or groups is influenced by the government in a favorable direction without decreasing the freedom of these individuals or groups (Thaler & Sunstein, 2008). An example of nudging is in 2018 performed by the Dutch government who set the default that everybody is an organ donor until you opt-out instead of only being an organ donor if you opt-in. This does not decrease the freedom of citizens because it is still 100% their decision if they will be an organ donor or not, but by this new ruling more people will be an organ donor which is favorable for the people in need of an organ. This is parallel to the attraction effect because adding a decoy to a choice menu can influence the frequency of choices in a particular direction without decreasing the freedom of the decision makers.
This has become more important since the 2008 banking crisis. Since then retirement savings plans largely shifted from being managed by firms to the responsibility of the
individual workers (Kroll & Vogt, 2012). Consequently policymakers shifted their attention from the firms to giving assistance and information to individuals about their private financial investments.
The rest of this paper is organized as follows. The next part gives an overview of the relating literature. Part 3 explains the experimental design. Part 4 gives an overview of the first experiment, the second experiment is explained in part 5, finally part 6 gives a
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2. Related literature
2.1 History of the attraction effect
Traditional economic theories assume choices to be consistent and stable. But in practice choices are influenced by the context in which it is displayed. These are called context effects. An example is the compromise effect, this is the tendency of decision-makers to choose the option that they consider a compromise (Simenson, 1989). For example: if somebody has to choose between a 25cl-, 35cl- or 50cl- drink he will often choose the 35cl because it is a compromise. This is a problem of consistent choice if that same person chooses a 50cl drink if he has to choose between a 35cl-, 50cl- or 70cl drink.
Another context effect is the attraction effect, this is shown in figure 1. The Attraction effect is violating traditional economic theories because consistent choices assume choices to be independent of irrelevant alternatives. The decoy is dominated by one of the options so should be irrelevant. However the decoy is relevant because it influences the way decision makers come to their preferred choice.
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The attraction effect is has been a huge debate and it has been researched in multiple fields. The first who reported the asymmetric dominance effect were Huber et al in 1982. They did an experiment where students had to choose between two options in different product categories such as cars, restaurants, beers and films. Then they added an alternative options that was dominated by one but not by other. Huber et al indeed found that the decoy was rarely chosen, but they were the first who found that the proportion of the dominating target option increased by adding the ´irrelevant´ alternative.
In further research it is shown that there is also an effect if the decoy is not dominated by the target (Huber & Puto, 1982). If the decoy is clearly worse on one attribute and only a little better on the other attribute there is still a decoy effect. The problem is that this can be explained by both an asymmetric dominance effect, because the decoy is still perceived as dominated, and a compromise effect because now the target looks like a compromise between the competitor and the decoy.
2.2 Characterizations of the attraction effect
There are two ways how the attraction effect can be characterized. The first one violates the Weak Axiom of Revealed Preference also called WARP. WARP says that if X is chosen over y, while y also is available then in any situation where y is chosen, x should also be chosen. In the case of choosing between lotteries every option is available and only one can be chosen. So if x is chosen in the menu {X, Y, X’} and y is chosen in the menu {X, Y, Y’} this is a violation of the Weak Axiom of Revealed Preferences. So to test if this consistency condition is violated people should have to pick an option in both menus where the core options are the same but the decoy is different in which it is dominated by. If people switch between de core options this is a sign of the attraction effect. This is the design Herne (1999) used.
The second way of characterizing the attraction effect is a violation of the Chernoff Condition. This condition says that if x is chosen from a large menu then it should also be chosen from a small menu consisting of the same options (Chernoff, 1954). In words you can say, if the Netherlands are the favorite to win the World Cup football then they should also
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be the favorite to win the European Championship. So if x is chosen in the menu {X, Y, Y’} then x should also be chosen in the menu {X, Y}. If that is not the case this is a violation of the Chernoff Condition and can be explained by the attraction effect. This design is used in most marketing and consumer product experiments such as the first discovery of the effect by Huber et al in 1982. A downside to this is that there might be experimenter demand effects. If subjects first have to make a decision between two options, and later between these same options plus an extra option, they might suspect what the experiment is about and act accordingly.
2.3 Explanations of the attraction effect
There are multiple studies done trying to give an explanation of the attraction effect. These explanations can be summarized into two categories.
The first explanation that is given is the range explanation. This explanation argues that adding an asymmetric dominated decoy increases the range of one of the dimensions. Because of this range increase the weight people put on this dimension decreases (Wedell, 1991). In Figure 2 you can see that adding the decoy y’ increases the range of dimension 1. According to the weight explanation this decreases the weight decision-makers attach to dimension 1 so they will more often choose y compared to x because y has more of dimension 2 but less of dimension 1.
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The second category of explanations is a process explanation. This is more a
psychological explanation based on that decision makers select the alternative supported by the best reason (Simonson, 1989) . It comes from the fact that the decoy is asymmetric dominated, the decoy y’ is dominated by y but not by x, the decoy gives an extra argument to choose y because it is for sure not the worst option (Slaughter et al, 1999). Supporting this explanation Simonson found that the attraction effect was bigger among students who had to justify their decisions afterwards.
Subsequently Castillo (2018) designed an experiment to test these explanations. To distinguish the explanations he made some manipulations in the decoys, these manipulations are shown in figure 3. The core options are ωp and ωf, the standard decoys are ωp’ and ω£’, these decoys are comparable to Herne’s (1999) design. Castillo argued that if the range explanation is correct then increasing the range further should also increase the attraction effect even more, to test this he created the decoys ωp’’ and ω£’’. As you can see in the figure these decoy double the range and are still asymmetric dominated.
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Furthermore he made some manipulations in de decoys to test the process
explanation. The process explanation relies on the fact that a decoy is asymmetric dominated so this gives a reason to choose the dominating option. To test if this is correct he created the decoys ωp* and ω£*, these decoys have the same range increase as ωp’’ and ω£’’ but are now dominated by both core options so symmetrically dominated. The range explanation predicts that adding this decoy should not change the attraction effect, however according to the process explanation, the attraction effect should disappear because the decoy is not asymmetrical dominated. Finally, the decoy ω* was created to test if just adding an extra option also changes preferences. This decoy is not asymmetrically dominated, nor is
increasing the ranges, so both explanations of the attraction effect predict that this decoy will not change preferences.
As expected he found a significant attraction effect in the option sets were the standard decoys were used, this replicates the findings of Wedell (1991) and Herne (1999). With the decoys with larger ranges ωp’’ and ω£’’, there was also an attraction effect, but not larger than with the standard decoys, so this weakens the range explanation.
Surprisingly, in the sets where the decoy was symmetrically dominated but only increased the range of one dimension(ωp* and ω£*), the choices were in the opposite direction than what the range explanation predicts. So the range increase of a dimension makes people weight that range even more causing a positive range effect. Concluding, the process
explanation causes the Attraction effect and there is a positive range effect running against it (Castillo, 2018). Castillo is the first to study the attraction effect in a within-subjects- and a between-subjects design so his results are robust.
2.4 Critics on the attraction effect
Apart from the multiple papers showing and explaining the attraction effect there is also some critique. Yang and Lynn (2014) argue that the attraction effect is not as robust and useful as supposed. From the 91 attempts to find an attraction effect only 11 had a reliable effect. The result only occurred if the dimensions were numerical and not if it was a descriptive text or displayed as a picture. This decreases the practical use for using a decoy to increase the market share of a target product.
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Another critique is given by Huber et al (2014). They argue that the attraction effect is rarely seen because there are multiple factors that diminish the effect. They describe five situations where the decoy effect is not robust .
The first factor that mitigates the effect is if decision-makers have strong prior trade-offs. A decoy will only have effect if the decision-maker is close to indifferent between the target and competitor. If the decision maker already has a strong preference for the target or the competitor a decoy will be unlikely to have any effect.
Also if the dominance relation is not easily detectable it will reduce the decoy effect. If a graph is shown with both attributes on the axes it is easy to detect dominance. But if effort is needed and so decision makers need motivation to see the dominance relation, this will also diminish the attraction effect. This is supported by Simonson (1989) who found that less motivated participants were less sensitive for the attraction effect.
The third factor that decreases the attraction effect according to Huber et al is if there is a strong heterogeneity between respondents. First it was assumed that an equal proportion in the original choice shares was best to test the decoy effect. However if half of the people strongly prefers the target and the other half strongly prefers the competitor, a decoy will be unlikely to have any effect. If preferences differ a lot the effect might be strong for a part of the group but not for all, or how Huber states it: “The most critical condition is that people have either very weak or initially unformed preferences between the target and the
competitor”.
What also diminishes the asymmetric dominance effect is if the decoy is either really undesired or really desired. For example if the options are framed as losses instead of gains the attraction effect fails (Malkoc et al, 2013). This is because now the decoy looks really undesirable so rather shifts preferences to the competitor. The other way around is also a problem, in the studies Yang and lynn followed, 18% of choices were decoys. This is either due to a lack of attention or not noticing the dominance. Because the decoy is close to the target it is likely that it takes most of its share from the target. This is counterproductive because the decoy’s intention is to increase the share of the target.
As a consequence the attraction effect is not often happening in marketplace choices, this has two main reasons. First, products in marketplaces usually don't have two numerical attributes but multiple more complex attributes. People have different preferences for these attributes so the dominance relation might not apply to everybody. Secondly, it is costly to
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produce and distribute decoy options so they often do not last long. Despite these shortcomings Huber et al conclude that it is still important to teach the attraction effect because it is a good example of how much context matters.
This study is designed to test the attraction effect in a way that is has more external validity then previous studies but keeps its internal validity. Internal validity is achieved by using lottery options with real monetary incentives. These options have two dimensions: probability and money, preferences are clear because more money is always better and the same goes for the probability to win money. Compare this to the hypothetical consumer goods tasks where there are no incentives and a subject might choose the decoy option because he personally prefers the product with a lower quality.
External validity is achieved by studying lotteries under ambiguity. In most of the risky decisions in real life you don't know the exact probabilities of your options. By using lottery option where exact probabilities are also unknown the results can be used in situations with medical treatments or financial decisions.
3. Experimental Design
This study is split in two almost similar experiments. The first experiment is examining the attraction effect in risky lotteries and the second does the same for lotteries under ambiguity. So in the first experiments, lottery options are designed in a way that subjects know the probabilities of winning the corresponding money, in the second experiment the options are designed in a way that subjects do not know the probabilities but can form a guess.
The way the experiment is conducted is by an online survey spread over the internet to friends, family and acquaintances. The survey is anonymous but participants who want to be able to win money have to fill in their email address. Besides the gamble questions participants are asked to give their age and gender. This is to check if the attraction effect might be dependent of age or gender.
The way the attraction effect is tested is by violation of the Weak Axiom of Revealed Preference. So participants have to make choices between three options, one option is the target, one option is the competitor and one option is the decoy. According to standard decision theory the amount of people choosing x in {X, Y, X’} should be the same as in {X, Y, Y’}. If a person chooses x in {X, Y, X’} but switches to y in {X, Y, Y’} this could be due
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to the attraction effect. However it can also have different explanations, the participant might not understand the question or just filled in random answers. It could also be the case that this person is indifferent between the core options and so his change in preferred option is not irrational because they both give the decision maker the same expected utility. But note that in these explanations the chance that somebody changes from X to Y is the same as the chance that he changes from Y to X in {X, Y, X’} and {X, Y, Y’}. As a result this will cancel out if we calculate the attraction effect as the proportion of subjects who switch from X to Y minus the proportion of subjects who switch from y to x. In formula, the attraction effect will be characterized by:
Pr(X=c({X, Y, X’}) & (Y=c({Y, X, Y’}) - Pr(Y=c({X, Y, X’}) & X=c{X, Y, Y’})
In this formula c is the choice function and Pr is the proportion. This will be tested by a one sided McNemar test.
In creating the binary gamble options there are a few limitations. First of all,
probabilities should at least be 20% and maximally be 80%. If probabilities are outside these boundaries, they will be easy to estimate in the lotteries under ambiguity, consequently the option will not really be ambiguous anymore so this should be avoided.
Secondly, the attraction effect will only work if participants are close to indifferent between the core options (Huber et al, 2014). So the target and the competitor are created in a way where participants are expected to not have strong prior trade-offs between the options. To achieve this, it is not enough to make the options have the same expected value in
monetary value. This because utility function for money are not linear and the average person is risk averse (Pratt, 1975).
To create options which have similar expected utilities the Holt & Laury (2011) formula for risk aversion will be used. Their formula accounts for absolute risk aversion as well as relative risk aversion.
𝑈𝑈(𝑥𝑥) =1 − exp (−𝑎𝑎𝑥𝑥𝑎𝑎 1−𝑟𝑟)
In this formula U is the utility that is obtained from getting an amount of money x. The r in this formula stands for the individual level or relative risk aversion, the a stands for the absolute level of risk aversion. These parameters are different for each individual but Holt &
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Laury made predictions that would fit the aggregate average data. They predicted r = 0.027 and a = 0.0029. Accordingly, the following 5 lottery sets shown in table 1 are created.
Table 1: Lottery sets used in experiments 1 and 2
X Y X’ Y’ P € P € P € P € a 0.25 100 0.61 37 0.22 89 0.49 35 b 0.34 71 0.70 32 0.30 65 0.61 31 c 0.29 85 0.51 45 0.22 82 0.46 42 d 0.45 39 0.74 23 0.43 36 0.71 20 e 0.42 46 0.63 30 0.30 40 0.49 25
In these decision sets X is constantly the riskier option with a higher potential payoff and Y is the safer option. These options are predicted to be similar in expected utility. The option X’ is a decoy of X and is dominated by X but not by Y. The option Y’ is a decoy of Y and so is dominated by Y but not by X.
In order to find inconsistencies in the Weak Axiom of Revealed Preferences a subject has to change his preference between the core options, therefore a within-subjects design will be used. So in both experiments a respondent has to make a decision between the same core options (X and Y) twice, once with the decoy X’ as a third option and once with the decoy Y as a third option. For example for gamble set b, a subject once makes a decision in decision set {(0.34,71), (0.70,32), (0.30,65) } and once in {(0.34,71), (0.70,32), (0.61,31) }. So for both experiments a subjects has to choose an option in 10 decision sets, consequently the surveys exists of a total of 20 decision sets, all these 20 decision sets are randomly mixed to prevent any order effects. Besides the order of the decision sets, also the three lottery options in a decision set are randomly mixed for each subject. The survey will approximately take 10 minutes including reading the instructions.
3.2 Payout
After collecting the data, one respondent and one of the 20 decision sets will randomly be selected. The lottery this respondent chose in that decision set will then play out. So the
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payoff of this person will be between 0 and 100 euro. The way the lottery will be played out is by an online random number generator, then if for example the participant in this question has chosen the (0.25,100) option I will randomly select a number between 1 and 100, if this number is 25 or less the participant will receive the 100 euro. The average of the expected values is around 20 euro. Participants know only one of them will be randomly drawn to play out one of their options. Unfortunate the budget for this experiment is not high enough to create lotteries with higher expected values or to pay out more participants.
4. Experiment 1
4.1 Procedure
The following hypotheses are tested during this experiment:
H0: The effect of an asymmetrically dominated decoy in risky lottery choice sets is zero. H1: An asymmetrically dominated decoy increases the proportion of target choices in risky lottery decision sets.
In this experiment the attraction effect will be tested in binary lotteries under risk. To create lotteries under risk a different design is used then Herne used in 1999. The difference is in the way that the probabilities are presented. Herne used numbers but here the
probabilities are shown visually. Each option will contain a square field consisting of 400 blocks. Each block is either black or white. The probability is then the percentage of black blocks in the total field. To make it easy to estimate the probability, the black blocks are ordered in rows starting at the top left corner. An example of a choice task is shown in figure 4, here you see the decision set {X, Y, X’} where X = (0.25, 100), Y = (0.61, 37) and
X’=(0.22, 89). So in the first option 25% or 100 blocks are colored black. The reason that the probabilities are presented visual and not numerical is because this is necessary to create lotteries under ambiguity. To be able to compare the results in both experiments and to prevent framing effects this is also done in the lotteries under risk. In these risky lotteries exact probabilities are not known immediately, however by using heuristics it is easy to make a close approximation even without counting lines or blocks.
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Figure 4: Example of a decision set in experiment 1
4.2 Results
The data that is used contains all survey results from June 13th to June 25th. In this time period 127 people started the online survey. From these people 31 did not answer all the questions and are deleted from the analyses. From the respondents left, four were deleted from all analyses because they chose the decoy option in more than 33% of their decisions. It is likely that these subjects did not understand the questions or filled in random answers (Farmer et al, 2017). From the 92 respondents left, 40 are female and the average age is 25. It took on average seven minutes to read the instructions and answer all the questions.
Table 2 presents the contingency table for the results in the first experiment. Every lottery set (see table 1) is presented separately. The table shows the frequencies of the options chosen by subjects in each lottery set. For example in the top left cell you see that 17
participants chose option X in both choice sets of lottery set a. In the top-middle cell you see that in lottery set a 5 subjects chose Y in {X, Y, X’} and X in {X, Y, Y’}.
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Table 2: Contingency table risky gambles
X Y X’ a 17 5 (5.4%) 2 b 15 4 (4.3%) 1 X c 13 8 (8.7%) 1 d 16 6 (6.5%) 0 e 15 5 (5.4%) 2 a 34 (37.0%) 31 1 b 27 (29.3%) 43 2 Y c 28 (30.4%) 37 0 d 21 (22.8%) 41 3 e 29 (31.5%) 35 0 a 1 0 1 b 0 0 0 Y’ c 4 0 1 d 2 3 0 e 5 1 0
Overall the higher risk option X is chosen 36.6% of the time, the safer option Y is chosen 59.9% of the time and the dominated decoy option X’ or Y’ is chosen 3.5% of the time.
The top-left cell and the middle-middle cell are representing the amount of participants who had consistent choices, they chose the same option despite the different decoy. The middle-left and the top-middle cells represent the inconsistent choices, these respondents changed their preferences between the core options with the different decoy. According to the attraction effect, the decoy should shift preferences to the option that dominates the decoy. Recall that the Attraction effect is characterized by:
Pr(X=c({X,Y,X’}) & (Y=c({Y,X,Y’}) - Pr(Y=c({X,Y,X’}) & X=c{X,Y,Y’}).
In gamble set a, 37% chose X in {X,Y,X’} and Y in {Y,X,Y’}, 5.4% chose Y in {X,Y,X’} and X in {Y,X,Y’}. So in gamble set a the attraction effect is 37.0% - 5.4% = 31.6%. An overview of the results is presented in table 3.
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Table 3: Attraction effect in lotteries under risk
Lottery set Attraction effect p-value
a 31.6% 0.000
b 25.0% 0.002
c 21.7% 0.003
d 16.3% 0.080
e 26.1% 0.002
All the five gamble sets show an attraction effect in the expected positive direction. Four out of the 5 lottery sets are significant at the 5% level, these results show that the H0 can be rejected, an asymmetrically dominated decoy increases the proportion of target choices in risky lottery decision sets. With an average effect of 24.1% these results are in line with Herne (1999) who found an average effect of 24.2% in her numerical risky lottery sets. No significant differences are found with regard to gender or age.
5. Experiment 2
5.1 ProcedureThe following hypotheses are tested during this experiment:
H0: The effect of an asymmetric dominated decoy in ambiguous lottery choice sets is zero. H1: An asymmetric dominated decoy increases the proportion of target choices in ambiguous lottery decision sets.
In this experiment the attraction effect will be tested in binary lotteries under
ambiguity, it is in most ways similar to the first experiment. The difference is in the way that the probabilities are presented. The probability of winning the corresponding amount of money is still the percentage of black blocks in the square field consisting of 400 black or white blocks. The difference is that the black blocks are not ordered in lines starting from the top-left corner but they are now randomly placed in the field. As a result it is hard to guess the percentage of black blocks and thus are the lottery options ambiguous. Counting the black
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blocks is still possible because there is no time limit. However this is very time consuming and I assume participants realize it is not worth the possible gain.
Despite that the probabilities are not known it is still easy to see the dominance relation between the decoy and the target. This is because the target is created in a way that it has the same placed black blocks plus some extra. An example of a choice set is shown in figure 5, this is the set {X, Y, X’} where X = (0.25, 100), Y = (0.61, 37) and X’ = (0.22, 89). First the decoy X is created by randomly making 22% or 88 of all blocks black, then the target X’ is created by randomly adding 3% or 12 black boxes to this field, finally the
competitor Y’ is created by randomly adding 36% or 144 black blocks to the remainder of the field.
Figure 5: Example of a decision set in experiment 2
5.2 Results
The data that is used in the second experiment comes from the same 92 respondents as experiment 1 .
Table 4 presents a contingency table for the results in the second experiment. Every lottery set is presented separately. The table shows the frequencies of the options chosen by
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subjects in each lottery set. For example in the top left corner you see that 19 participants chose option X in both choice sets of lottery set a. In the top-middle cell you see that in lottery set a, 7 subjects chose Y in {X, Y, X’} and X in {X, Y, Y’}.
Table 4: Contingency table ambiguous gambles
Overall the higher risk option X is chosen 32% of the time, the safer option Y is chosen 65.8% of the time and the dominated decoy option X’ or Y’ is chosen 2.2% of the time.
The top-left cells and the middle cells are representing the amount of participants who had consistent choices, they chose the same option despite the different decoy. The middle-left and the top-middle cells represent the inconsistent choices, these respondents changed their preferences between the core options with the different decoy. Recall that the attraction effect is characterized by:
Pr(X=c({X,Y,X’}) & (Y=c({Y,X,Y’}) - Pr(Y=c({X,Y,X’}) & X=c{X,Y,Y’}).
So in gamble set a the attraction effect is 26.1% - 7.6% = 18.5%. An overview of the results is presented in table 5. X Y X’ a 19 7 (7.6%) 2 b 7 4 (4.3%) 0 X c 20 4 (4.3% 0 d 16 6 (6.5%) 0 e 10 4 (4.3%) 1 a 24 (26.1%) 39 0 b 23 (25.0%) 55 1 Y c 27 (29.3) 39 0 d 21 (22.8%) 41 3 e 25 (27.2%) 50 1 a 0 0 1 b 0 2 0 Y’ c 0 2 0 d 2 3 0 e 0 1 0
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Table 5: Attraction effect in lotteries under ambiguity
Lottery set Attraction effect p-value
a 18.5% 0.054
b 20.7% 0.010
c 25.0% 0.001
d 16.3% 0.083
e 22.9% 0.004
All the five gamble sets show an attraction effect in the expected positive direction. Three out of the five lottery sets are significant at the 5% level, these results show that the H0 can be rejected, an asymmetric dominated decoy increases the proportion of target choices in ambiguous lottery options. The average attraction effect found in this experiment is 20.7%. Note that this is slightly less than the effect in experiment 1, however the difference is not significant. Also, no significant relation is found regarding the attraction effect and age or gender.
6. Discussion and conclusion
The results of this study are significant but note that there are some potential shortcomings. Firstly, there might have been some experimenter demand effects. The subjects are mostly friends, family and acquaintances. Some of them already knew the topic so they might have had an idea what the purpose of the experiment was, if these subjects acted accordingly, the attraction effect found in both experiments is biased in positive direction.
Secondly the decoy options should have been more clearly dominated, they are now close to the target in expected utility, a little mistake in examining the probabilities could lead to choosing the decoy, this also decreases the attraction effect because these decoy choices are more likely to gain share from the target than from the competitor.
Another possible shortcoming of the experiment is the lack of budget. Participants knew only one of them would be picked randomly to play out one of their lottery choices. This might decrease their motivation to attach a good valuation to each option.
The attraction effect has been studied in multiple fields, mainly in consumer research. The effect violates the assumption that preferences should be independent of irrelevant
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alternatives. Even though this is one of the basic principles of economics, the attraction effect has not been studied often in the economic field. Just as the most economic studies on the attraction effect binary gambles are used as options to choose from. Choosing between lottery options has monetary consequences and results have high internal validity.
The first experiment of this study replicates the attraction effect in decisions between multiple binary risky lottery options. Adding an asymmetrically dominated decoy to a choice set of risky gambles increased the proportion of choices for the target option with 24.1%. This is in line with previous studies.
The second experiment showed that the attraction effect also translates to ambiguous lottery options. Adding an asymmetrically dominated decoy to a choice set of ambiguous lottery options increased the proportion of choices for the target with 20.7%. These results are in line with the first experiment and previous studies. The attraction effect is not significant higher in risky lottery decision sets than in ambiguous lottery decision sets. This experiment is designed to have a higher external validity than previous studies on the attraction effect with lotteries. Most real world decisions are ambiguous, in financial investment options or medical treatments probabilities are usually unknown, this experiment shows that in these situations the attraction effect can have an effect.
Note that in the examples of investments or medical treatments usually not only the probabilities are unknown, but also the potential payoff. In this experiment the probabilities were unknown but the potential payoff was given exact. Future research could test if the decoy effect still holds in decision sets were the lotteries have ambiguous probabilities and ambiguous potential payoffs.
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7. References
Bergheim, R., & Roos, M. (2013). Intuition and Reasoning in Choosing Ambiguous and Risky Lotteries.
Castillo, G. (2018). “Choice consistency and the attraction effect. URL http://geoffreycastillo. com/pdf/Castillo-Choice-consistency-and-the-attraction-effect. pdf.
Chernoff, H. (1954). Rational selection of decision functions. Econometrica: journal of the Econometric Society, 422-443.
Crosetto, P., & Gaudeul, A. (2016). A monetary measure of the strength and robustness of the attraction effect. Economics Letters, 149, 38-43.
Farmer, G. D., Warren, P. A., El‐Deredy, W., & Howes, A. (2017). The Effect of Expected Value on Attraction Effect Preference Reversals. Journal of behavioral decision
making, 30(4), 785-793.
Gomez, Y., Martínez-Molés, V., Urbano, A., & Vila, J. (2016). The attraction effect in mid-involvement categories: An experimental economics approach. Journal of Business Research, 69(11), 5082-5088.
Heath, T. B., & Chatterjee, S. (1995). Asymmetric decoy effects on lower-quality versus higher-quality brands: Meta-analytic and experimental evidence. Journal of Consumer Research, 22(3), 268-284.
Herne, K. (1997). Decoy alternatives in policy choices: Asymmetric domination and compromise effects. European Journal of Political Economy, 13(3), 575-589. Herne, K. (1999). The effects of decoy gambles on individual choice. Experimental Economics, 2(1), 31-40.
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Huber, J., & Puto, C. (1983). Market boundaries and product choice: Illustrating attraction and substitution effects. Journal of Consumer Research, 10(1), 31-44.
Huber, J., Payne, J. W., & Puto, C. P. (2014). Let's be honest about the attraction effect. Journal of Marketing Research, 51(4), 520-525.
Kroll, E. B., & Vogt, B. (2012). The relevance of irrelevant alternatives. Economics Letters, 115(3), 435-437.
Leonard, T. C. (2008). Richard H. Thaler, Cass R. Sunstein, Nudge: Improving decisions about health, wealth, and happiness.
Lichters, M., Sarstedt, M., & Vogt, B. (2015). On the practical relevance of the attraction effect: A cautionary note and guidelines for context effect experiments. AMS Review, 5(1-2), 1-19.
Malkoc, S. A., Hedgcock, W., & Hoeffler, S. (2013). Between a rock and a hard place: The failure of the attraction effect among unattractive alternatives. Journal of Consumer
Psychology, 23(3), 317-329.
O'Curry, S., Pan, Y., & Pitts, R. (1995). The attraction effect and political choice in two elections. Journal of Consumer Psychology, 4(1), 85-101.
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Slaughter, Jerel E., Evan F. Sinar, and Scott Highhouse (1999), “Decoy effects and attribute-level inferences.” Journal of Applied Psychology, 84, 823–828.
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Tversky, A., & Kahneman, D. (1986). Rational choice and the framing of decisions. Journal of business, S251-S278.
Wedell, D. H. (1991). Distinguishing among models of contextually induced preference reversals. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(4), 767.
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7. Appendix
7.1 Instructions of the experiment
Welcome to this anonymous online experiment about individual decision-making. This experiment takes less than 10 minutes and you can win up to 100 euro based on your decisions so please pay good attention!
Instructions
This experiment consists of 20 questions. In all of these 20 questions you have to make a choice between 3 options.
The options differ in the amount of money that can be won and the probability of winning the corresponding amount of money.
The probabilities will not be given as a number but visually. For each option you will see a square field consisting of black and white blocks. The probability of winning the corresponding money is based on the amount of black blocks in the field.
26 Examples:
In the field above 100% of the blocks are black so the probability of winning the corresponding amount of money will be 100%.
In the field above 50% of the blocks are black so the probability of winning the corresponding amount of money will be 50%.
In the field above 50% of the blocks are black so the probability of winning the corresponding amount of money will also be 50%.
On the 30th of June one participant and one question will randomly be selected for payout. If you are the one selected then your payoff depends on the option you chose in that question.
