Interval and Symmetry Approaches to Uncertainty – Pioneered by Wiener – Help Explain Seemingly Irrational Human Behavior: A Case Study

It has been observed that in many cases, when we present a user with three selections od different price (and, correspondingly, different quality), then the user selects the middle selection. This empirical fact – known as a compromise effect – seems to contradicts common sense. Indeed, when a rational decision-maker selects one of the two alternatives, and then we add an additional option, then the user will either keep the previous selection or switch to a new option, but he/she will not select a previously rejected option. However, this is exactly what happens under the compromise effect. If we present the user with three options a < a′ < a′′, then, according to the compromise effect, the user will select the middle option a′, meaning that between a′ and a′′, the user will select a′. However, if instead we present the user with three options a′ < a′′ < a′′′, then, according to the same compromise effect, the use will select a previously rejected option a′′. In this paper, we show that this seemingly irrational behavior actually makes sense: it can be explained by an application of a symmetry approach, an approach whose application to uncertainty was pioneered by N. Wiener (together with interval approach to uncertainty). I. COMPROMISE EFFECT: A PARTICULAR CASE OF SEEMINGLY IRRATIONAL HUMAN BEHAVIOR Customers make decisions. A customer shopping for an item usually has several choices. Some of these choices have better quality, lead to more possibilities, etc. – but are, on the other hand, more expensive. For example, a customer shopping for a photo camera has plenty of choices ranging from the cheapest one whose photos are good to very professional cameras enabling the user to make highest-quality photos even under complex circumstances. A person planning to spend a night at a different city has a choice from the cheapest motels which provide a place to sleep to luxurious hotels providing all kinds of comfort, etc. A customer selects one of the alternatives by taking into account the additional advantages of more expensive choices versus the need to pay more money for these choices. It is important to understand customer’s decisions.Whether we are motivated by a noble goal of providing alternatives which are the best for the customers – or whether a company wants to make more money by providing what is wanted by the customers – it is important to understand how customers make decisions. Experimental studies. In many real-life situations, customers face numerous choices. As usual in science, a good way to understand complex phenomena is to start by analyzing the simplest cases. In line with this reasoning, researchers provided customers with two alternatives and recorded which of these two alternatives a customer selected. In many particular cases, these experiments helped better understand the customer’s selections – and sometimes even predict customer selections. At first glance, it seems like such pair-wise comparisons are all we need to know: if a customer faces several choices a1, a2, . . . , an, then a customer will select an alternative ai if and only if this alternative is better in pair-wise comparisons that all other possible choices. To confirm this common-sense idea, in the 1990s, several researchers asked the customers to select one of the three randomly selected alternatives. What was expected. The experimenters expected that since the three alternatives were selected at random, a customers would: • sometimes select the cheapest of the three alternative (of lowest quality of all three), • sometimes select the intermediate alternative (or intermediate quality), and • sometimes select the most expensive of the three alternatives (of highest quality of all three). What was observed. Contrary to the expectations, the experimenters observed that in the overwhelming majority of cases, customers selected the intermediate alternative; see, e.g., [17], [18], [21]. In all these cases, the customer selected an alternative which provided a compromise between the quality and cost; because of this, this phenomenon was named compromise effect. Why is this irrational? At first glance, selecting the middle alternative is reasonable. However, it is not. For example, let us assume that we have four alternative a1 < a2 < a3 < a4 ordered in the increasing order of price and at the same time, increasing order of quality. Then: • if we present the user with three choices a1 < a2 < a3, in most cases, the user will select the middle 978-1-4799-4562-7/14/$31.00 c ⃝2014 IEEE choice a2; this means, in particular, that, to the user, a2 better than the alternative a3; • on the other hand, if we present the user with three other choices a2 < a3 < a4, in most cases, the same user will select the middle choice a3; but this means that, to the user, the alternative a3 better than the alternative a2. If in a pair-wise comparison, a2 is better, then the second choice is wrong. If in a par-wise comparison, the alternative a3 is better, then the first choice is wrong. In both cases, one of the two choices is irrational. This is not just an experimental curiosity, customers’ decisions have been manipulated this way. At first glance, the above phenomena may seem like one of optical illusions or logical paradoxes: interesting but not that critically important. Actually, it is serious and important, since, according to anecdotal evidence, many companies have tried to use this phenomenon to manipulate the customer’s choices: to make the customer buy a more expensive product. For example, if there are two possible types of a certain product, a company can make sure that most customers select the most expensive type – simply by offering, as the third option, an even more expensive type of the same product. Manipulation possibility has been exaggerated. Recent research shows that manipulation is not very easy: the compromise effect only happens when a customer has no additional information – and no time (or no desire) to collect such information. In situations when customers were given access to additional information, they selected – as expected from rational folks – one of the three alternatives with almost equal frequency, and their pairwise selections, in most cases, did not depend on the presence of any other alternatives; see, e.g., [20]. Compromise effect: mystery remains. The new experiment shows that the compromise effect is not as critical and not as wide-spread as it was previously believed. However, in situation when decisions need to be made under major uncertainty, this effect is clearly present – and its seemingly counterintuitive, inconsistent nature is puzzling. How can we explain such a seemingly irrational behavior? What we do in this paper. In this paper, we show that it is possible to find a rational explanation for such a behavior. Interesting, this explanations is related to two ideas promoted by N. Wiener – interval uncertainty (which later encouraged fuzzy uncertainty) and symmetry. Comment. This paper focuses on one specific example of a seemingly irrational behavior. We should emphasize, however, that there are many well-known examples of such behavior; see, e.g., [5] and references therein. These examples cover both seemingly irrational individual choices and seemingly irrational group choices. In some cases, a seemingly irrational behavior can be explained in rational terms – sometimes, by using fuzzy techniques; see, e.g., [7], [8]. In particular, many seeming paradoxes related to group decision making – paradoxes related to Arrow’s impossibility result – can be explained if instead of simply recording which participant prefers which alternative, we also take into account the degree to which to each participants prefer one alternative to another (see, e.g., [11]) – in line with Zadeh’s idea pioneered in [1], [2], [26], [27], [28], [29]. It should be also mentioned that many instances of human behaviors are indeed irrational – in the sense that humans sometimes select actions which are detrimental to the decision maker’s own interests. There are many examples of such actions, from irrational decision making in simple economic situations to self-damaging behavior related to drug and alcohol addiction. In this paper, we concentrate only on one specific case of seemingly irrational human behavior. II. BEFORE WE EXPLAIN THE MYSTERY OF SEEMINGLY IRRATIONAL BEHAVIOR, LET US FIRST RECALL HOW RATIONAL BEHAVIOR IS USUALLY DESCRIBED Traditional decision theory is based on the assumption that a decision maker can always make a definite choice. Traditional decision theory (see, e.g., [3], [9], [13], [16]) is based on the assumption that if we present a decision maker with two alternatives A and A′, then the decision maker will always make a definition decision about which of this alternatives is better for him/her. In other words, the decision maker will always select one of the following three options: • the alternative A is better than the alternative A′; we will denote this option by A > A′; • the alternative A′ is better than the alternative A; we will denote this option by A < A′; • the alternatives A and A′ are of equal value; we will denote this by A = A′. Resulting numerical description of preferences. The above assumption enables us to provide a numerical scale for describing quality of different alternatives. To describe this scale, let us select two fixed outcomes: • we select a very bad outcome, which is worse than any of the alternatives that we encounter in decision making; we will denote this situation by A0; • we also select a very good outcome, which is better than any of the alternatives that we encounter in decision making; we will denote this situation by A1. Then, for each number p from the interval [0, 1], we can form a lottery in which we get A1 with probability p and A0 with the remaining probability 1 − p. This lottery will be denoted by L(p). • When p = 1, the corresponding lottery L(1) means that we select a very good outcome with probability 1, i.e., we have L(1) = A1. • When p = 0, the corresponding lottery L(0) means that we select a very bad outcome with probability 1, i.e., we have L(0) = A0. When the probability p in strictly between 0 and 1, the resulting lottery is better than A0 but worse than A1: