‘Elimination by aspects’ and probabilistic choice Reynald-Alexandre LAURENT∗† 9 juin 2006 Short version

Résumé “Elimination by aspects” (EBA) is a heuristic followed by decision makers during a process of sequential choice and which constitutes a good balance between the cost of a decision and its quality. At each stage of decision, the individuals eliminate all the options not having an expected given attribute, until only one option remains. This heuristic was used by Tversky (1972) to construct a discrete choice model having a degree of flexibility comparable to the nested logit. This paper expose the conditions of application of the EBA model to the theoretical economics. Rotondo (1986) proposed to integrate price differences as attributes in the EBA model. This property was used by Laurent (2006) to work out demands of a duopoly with bounded rationality of consumers and comprising a double differentiation of products, horizontal and vertical. Properties of equilibrium differ from those obtained with the logit model and the success of this application underlines the productivity of links between psychology and economics.

classification jel : D11, D43, D89. keywords : "elimination by aspects", decision heuristics, product differentiation, discrete choice.

∗ PSE, Paris-Jourdan Sciences Economiques (CNRS, EHESS, ENS, ENPC). Contact : [email protected] † I wish to thank my PhD Advisor, Jacques Thisse, for its multiple councils. I also thank the participants of CID working group and of PSE and CREST-LEI lunch seminars, for their helpful comments. The usual disclaimers applies.

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1

Introduction

Discrete choice models characterize the behavior of an individual whose choice among a finite set of mutually exclusive options is described by a probability. The existence of such probabilities supposes that choices of the individuals are subjected to a particular random term. Since Block and Marschak (1960), two families of models are traditionally distinguished according to the nature of this random term : models with random utility (RU) and models with random decision rule (RDR). For the former, utility assigned to the options by decision makers can fluctuate, whereas for the latter, the decision rule used by the agents has a probabilistic form. The purpose of this paper is to underline the interest of RDR models for economics, while focusing on one of them : the "elimination by aspects" (or "EBA") model proposed by Tversky (1972a, b). As McFadden (2001) noted it, random utility models hold a dominating place in the economic analysis, both at theoretical and econometric level. The existence of a random term can come from changing state of minds (cognitive interpretation) or incapacity of the modeler to apprehend individual behaviors (econometric interpretation). The second interpretation is frequently retained : economic agents would be perfectly rational and the observed choice variations would come from an imperfect modeling of their behavior, because of a lack of information. In this case, the behavior of an individual can, at best, be predicted by using a probability function. Thus, the random utility of an option comprises a deterministic utility, corresponding to the observable characteristics of the option, and a random factor, corresponding to the uncertainty supported by the modeler. These models fit with the econometric tradition, which is a first reason for their success. Another reason is the proximity with the standard approach of maximization of the deterministic utility. However, such a perfect rationality is continually called into question in decision science. In this context, the RDR models can provide a credible alternative to the RU models because they clearly fit with the cognitive interpretation : the random variable comes from the individual mental states. During the decision-making process, the agent can occult some aspects of an alternative or be mistaken in evaluating its characteristics. As Billot and Thisse (1995, p 922-923) note it, several explanations are consistent with this cognitive interpretation : evolution of the individual state of mind at the passage of time (for example, focusing on specific aspects of an alternative, following a mode phenomenon or an advertising), imperfect knowledge of its preference by the decision maker (or fuzzy preferences), mistake during the evaluation of the possible options (cognitive capacities are limited, which reminds Simon’s (1957) concept of "bounded rationality"). On the other hand, the RDR models are not compatible with the econometric interpretation since the origin of probabilities explicitly comes from the specified decision process. The RDR models thus seem less ambiguous than the RU models to take into account imperfect rationality, because they allow to certify the cognitive origin of the probabilities. However these RDR models have been rarely explored by the economists up to now, whereas they are widely diffused in psychology or marketing1 . The EBA heuristic supposes that the agents follow a process of sequential choice and eliminate at each stage all the options not having an expected given attribute, until only one option remains. Payne and Bettman (2001) showed that this heuristic is efficient within the "adaptive toolbox", because it carries out a good balance between the cognitive cost and the quality of decision. Starting from this heuristic, the probabilistic model of choice worked out by Tversky (1972) is introduced. Its degree of flexibility is comparable 1

Look at Batsell and Seetharaman, 2005, for instance

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with those of nested logit and probit models, very widespread in economics. However, difficulty of prices integration in the EBA model probably delayed its diffusion in economic theory. This problem is now overcome : Rotondo (1986) suggested that price differences between goods should be integrated like attributes in the model. The author also shows that a linear price difference constitutes a good approximation of individual behaviors. This modeling of prices is consistent with the probability formula of Tversky but break the equivalence with the RU models, in which prices are a component of each option utility. Consequently, demands constructed with the EBA model have different properties that demands constructed, for instance, with the probabilities of the logit. When these EBA demands are used in a framework of duopoly, the existence of a double differentiation and a "reference firm" in tariffing are highlighted. This paper is structured in the following way. The "elimination by aspects" heuristic of Tversky and its positioning among the other decision rules is presented in section 2. The resulting formula of probability is introduced in section 3. The section 4 focus on the mode of price integration in the model. Finally, the section 5 present the economic properties of a differentiated duopoly in which individuals follow the EBA heuristic. Our conclusions are presented in section 6.

2

"Elimination By Aspects" heuristic and its positioning

The EBA heuristic at the basis of the model of Tversky is presented here and we show how it can be integrated in the paradigm of the "adaptive toolbox".

2.1

Presentation of the EBA decision strategy

In complex or uncertain choice situations, in which the maximization of utility is hard to implement, individuals frequently use heuristics, as Kahneman and Tversky (1972) underlined it. They are simple principles of reasoning allowing to reduce the complexity of a decision task. The EBA procedure of Tversky is one of these heuristics. Tversky (1972a, b) defines a choice option as a set of characteristics : for instance, if the option is a good, this last is described by the set of attributes it has, an approach which recalls that of Lancaster (1966). From this definition of the options, Tversky proposes a procedure of selection functioning as an algorithm : (a) the common characteristics of the considered choice set are eliminated, as any discriminating choice cannot be based on them ; (b) a characteristic is randomly selected and all the options not having this characteristic are eliminated. The higher the utility of a characteristic is, the larger the probability of selecting this characteristic is ; (c) if remaining options still have specific characteristics, one turns over at the first stage. In the contrary, if the residual choices have the same characteristics, the procedure ends. If only one option remains, it is selected. In the contrary, all remaining options have the same probability to be selected. Finally, the links between the EBA strategy and lexicographic or "satisficing" procedures, which all belongs to "non-compensatory" models, are exposed by Payne and Bettman (2001, p 128). 3

2.2

An efficient strategy in the "adaptive toolbox"

Gigerenzer, a psychologist, and Selten, Nobel Prize in economics, proposed in 2001 to model the bounded rationality through an "adaptive toolbox". According to them, individuals use various fast and frugal decision rules, forming a "toolbox", when they are confronted to decision problems, often approached in situation of uncertainty, limited resources and constrained time. The EBA heuristic is one of these "tools". This "toolbox" is "adaptive" because heuristics used depend on context, decision parameters, its importance... Decision strategies differ according to some criteria, and are therefore more or less adapted to a specific context. Once these criteria given, we compare the EBA process with other decision strategies. 2.2.1

Evaluation criteria for heuristics

The question of heuristic selection by the individuals is essential : how could a decision rule be judged enough effective to belong to the "toolbox" ? An answer is provided by Payne and Bettman (2001), for who heuristics are evaluated according to two criteria : the weakness of the cognitive cost required for their use and the relevance of the decision to which they lead. Thus, if a heuristic is more expensive than another at the cognitive level, then it must lead to a better result to be considered as effective. However, the use of such criteria supposes the definition of adapted measuring instruments. Newell and Simon (1972) showed that the cognitive effort could be measured by the number of elementary operations of information processing required to achieve a given decision task, with a specific decision strategy. Indeed, a decision strategy can be broken up in elementary information processes (EIP), like "read an information element", "compare two elements", "multiply or add elements", and so on. Thus, faced with a decision problem, it is possible to determine how much EIP are necessary according to the strategy selected. This approach was validated empirically by showing that the weighted sum of EIP for a decision strategy is a good estimator of the time spent to make the decision and the intensity of effort felt by the individual (Bettman et al., 1990). Then, to measure the relevance or the quality of a decision, Payne and Bettman (2001) propose to follow a criterion of "consistence" by putting in parallel the result of the heuristic studied and a theoretical standard of rational answer. The decision strategy leading to the best result (the most consistent with the preferences of the agents) is "weighting adding" strategy (WADD) which supposes that the utility of an option is equal to the weighted sum of utility of its components. Such a process supposes to consider one option at the same time, to examine each of its attributes and to determine their subjective value, then to calculate the total value of this option. Finally, the option with the greatest value will be selected. The use of this rule imposes that the decision maker knows or can estimate the relative importance of each attribute and can assign a subjective value to each possible level of attribute. Moreover, this strategy is consistent with the maximization of utility of the standard theory. The quality of the decision resulting from a heuristic will thus be measured by the percentage of the result of the WADD strategy obtained by this heuristic (%UM)

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2.2.2

The positioning of the EBA heuristic

It is now possible to determine, for a given environment, the number of EIP and the solution adopted according to various strategies. Figure 1 shows one of the results obtained by Payne and Bettman (2001, p 135) : the EBA heuristic is evaluated according to the cognitive effort and of the quality of decision obtained. By comparison, points corresponding to the utility maximization "UM" and to the random choice "RC" (random choice) are also represented.

Fig. 1 – Representation of decision strategies according to the effort and the quality of decision This graph shows that the use of heuristics is relevant since the individual takes into account the cognitive cost of the decision in his objective function. Payne and al. (1996) showed empirically that this cognitive effort was an essential parameter during the decision. Thus, when the objective function of an individual incorporates two factors, quality and effort of decision, with convex preferences on these factors, then the EBA heuristic can seem more "rational" than utility maximization ! Moreover, for more complex tasks, the effort required for the use of heuristics increases less quickly than the effort required for the maximization of utility. Finally, the quality of decisions resulting from heuristics is rather robust to the increase in complexity : thus, the use of heuristics is more effective since the decision becomes complex, which is the case of many economic problems.

3

The "Elimination by aspects" model of Tversky

The decision process developed in the previous section can now be formalized. The version of Tversky’s model used here is very close to that adopted by Batsell and al. (2003). Let T = {i, j, k...} be the set of choice options and T 0 = {α, β, γ...} the set of attributes (or characteristics2 ) of the options in T. These characteristics can belong to one or more options. Suppose that there is a scale (or utility) function u : T 0 → R such that ∀α ∈ T 0 , u(α) is the scale of α. This scale measures the satisfaction felt by the individual having 2

The term "aspect" was initially used by Tversky

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the attribute α. An EBA model can thus be defined by a triplet {T,T’,u}. Scale functions are supposed to be additives between the attributes : u(α + β) = u(α) + u(β). According to the decision process, an individual faced with a set of options A ⊂ T chooses one of the characteristics in A’ and eliminates all the alternatives not having this characteristic. Let us retain the following definitions : - The set of attributes of a given option k is : k 0 = {α ∈ T 0 /k has characteristic α} - The set of options in T having a given attribute α is : Tα = {k ∈ T /α ∈ k 0 } - The set of attributes of the options available for the choice (gathered in A) is : [ A0 = {α ∈ T 0 /∃k ∈ A/α ∈ k 0 } = k0 k∈A

- The set of characteristics shared by all the options in A is noted : \ k0 A0 = {α ∈ T 0 /α ∈ k 0 ∀k ∈ A} = k∈A

- The set of options in A having a given attribute α is : Aα = {k ∈ A/α ∈ k 0 } = Tα ∩ A - The set of attributes shared by all the options in A, but not by at least an other option out of A, is noted : A = {α ∈ T 0 /α ∈ k 0 ∀k ∈ A et @l ∈ / A/α ∈ l0 } One can check that A ⊆ A0 . Let us see now how these sets can be used to model the EBA process. Initially, the individual selects a discriminating attribute and eliminates all the options not having this attribute. For a given set A, an attribute is discriminating if it belongs at least to an alternative of A but is not common to all the options. Formally, the set of the discriminating attributes is given by A0 \A0 . Among all the discriminating attributes, the probability of retaining a given attribute α in order to discriminate is equal to the ratio of the utility of α on the total sum of utilities of the attributes in A0 \A0 . Mathematically, this probability is given by : P (α) =

u(α) X u(β)

(3.1)

β∈A0 \A0

The choice probability of an option k among A (noted PkA ) is then equal to the sum, for each discriminating attribute of k, of the probability of selecting this attribute as discriminating criterion (given by the preceding formula) multiplied by the probability of choosing k among all the other options having this attribute (which can be written PkAα ).

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Formally, if A0 \A0 6= ∅, this process may be represented by the following recursive formula :   PkA =

X α∈k0 \A0

P (α).PkAα =

 X  u(α)   Aα X   .Pk .   u(β) α∈k0 \A0

(3.2)

β∈A0 \A0

If the remaining options in set A share all the same attributes (A0 \A0 = ∅), then / k 0 , the equation (3.2) probability of choosing k becomes PkA = 1/|A|. As PkAα = 0 if α ∈ may be rewritten in the form :   PkA

 X  u(α)   Aα =  X  .Pk .  u(β)  α∈T 0 \A0

(3.3)

β∈A0 \A0

If the utility is initially defined on the set of relevant attributes in the EBA model, a definition of the options relatively to the attributes is however not necessary to the application of the model. Indeed, the sets T and T’ being superimposable, the selection of an attribute during a stage is equivalent to retain a set of options in T. The EBA model can thus be formulated only in terms of elimination of options, in relation to the subsets of T. That gave rise to the "abstract" version of the EBA model, in which the set of attributes is replaced by 2T , the set of the subsets of T. An "abstract" EBA model is defined by a new triplet {T, 2T ,U }. The equivalence between the standard version and the abstract version was established rigorously by a theorem of Tversky (1972a, p 287-288). This version of the EBA model can also be seen as the result of a Markovian process and probabilities as transition probabilities between the states (Tversky 1972b, p 347). Let us underline that a generalization of the EBA model was proposed by Billot and Thisse (1999). For these authors, the individual valuates the possibility of choosing between several options : the choice sets corresponding to a positive context are thus affected of an additional positive utility and vice versa. Thus, for Tversky (1972), the specific utility of a choice set can be higher than the sum of the specific utilities of the choices because of the existence of common characteristics. For Billot and Thisse, the utility of a choice set can be lower or higher than the sum of the utilities of different choices because of the impact of contexts. Many links exist between the EBA model and RU models.3 Firstly, when characteristics are either specific to a single option, or shared by all the options, the EBA model is equivalent to the Luce model (1959). Thus, the EBA model allows to overcome the limits of the empirical properties of the Luce model and in particular allows to solve the the red bus-blue bus paradox (Debreu, 1960). Then, Batley and Daly (2003) identified conditions under which a hierarchical EBA model4 is equivalent to a nested logit model. Lastly, at a higher degree of generality, equivalences were realized between these structures. Since Block and Marschak (1960), it was established that a system of choice probabilities could be represented like a RU model if and only if it verified a "stochastic rationality". The verifying conditions of this property were studied by Falmagne (1978) 3

A synthesis of these links is proposed by Laurent, 2006a A variant of the EBA model suggested by Tversky and Sattath (1979) in which options are organized in form of a tree according to their attributes 4

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and Billot and Thisse (2005). Thus, some axiomatic structures make it possible to show if a system of choice probabilities is consistent with RU models. As Tversky (1972b, theorem 7) showed it, the probabilities of the EBA model verify stochastic rationality : thus the EBA model can be reformulated like a general RU model. This type of relation also exists in the opposite direction : indeed, McFadden (1981, p 268-269) established, in a little diffused demonstration, that every RU model could be reformulated like a model of elimination of the options belonging to a broader class than the EBA model. Models of this class are described as "elimination by strategy" (EBS). Thus, any system of probabilities verifying stochastic rationality is also consistent with the models of elimination type.

4

Conditions of application to economics

If there are many theoretical links between the EBA model and the class of RU models, they differ in their methods of application to economics. Firstly, in order to allow the comparison of predictions between these structures, econometric methods to estimate the parameters of the EBA model were developed. Two methods were proposed recently and provide significant progress since the representation of options in a tree, initially proposed by Tversky and Sattath (1979). On the one hand, Batsell and al. (2003) suggest to estimate the probability differences, which are linear with the parameters of the model : this method requires however to be able to observe the choice probabilities for all possible sets of options. On the other hand, Gilbride and Allenby (2005) propose to use the Markov Chain Monte Carlo method to estimate the parameters, but this technique requires to identify the structure of the attributes as a preliminary. The existence of such methods incites to carry out in the future a systematic comparison between the predictions of the EBA model and those of classical RU models. Then, the equivalence between RU models and the EBA model cease to be true since prices are introduced, which opens theoretical prospects. The initial version of the EBA model did not permit to take into account continuous characteristics5 , which limited the practical applications of the model, and in particular impeded price integration. This limitation probably contributed to the disinterest of economists for this model. However, this difficulty was overcome by Rotondo (1986) who proposes a method of price integration consistent with the model of Tversky and which we present here.

4.1

The question of prices integration

Rotondo (1986) suggests an extension of the formula of the EBA probabilities taking into account continuous characteristics, and in particular the prices (to which this section will focus). Prices are perceived as characteristics of the product and will thus be the subject of a process of elimination, like the other non-price characteristics. More precisely, the relative advantage of an alternative on an other, in term of price, will be given by a function of the price difference between these alternatives. Suppose that the notations introduced in section 3 apply to the set of non-price characteristics and that |A| = n without loss of generality. New definitions are then proposed : 5

An aspiration threshold was associated to each continuous characteristic, allowing to transform it in binary characteristic. For example, the characteristic "power" of a car was associated to an aspiration threshold "100hp". Thus, a car of 101hp was endowed with a "powerful" characteristic but not a car of 99hp, which can seem a bit paradoxical

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- The options, noted i = {1...n}, are ranked from the cheapest to the more expensive p1 ≤ ... ≤ pi ≤ ... ≤ pn . - Denote Ai = {i ∈ A/pi < pi+1 } the set of the i cheapest options. - A scale function on price differences is provided by w : R → R. Rp - The price advantage of option i on option i+1 is then given by v(i) = pii+1 w(α)dα. With these concepts, consider an extension of the formula (3.3) integrating the prices. In this case, probability of choosing k among A, which is the j st cheapest alternative among N, is given by : X PkA =

u(α)PkAα

+

α∈k0 \A0

X

n−1 X

v(i)PkAi

i=j

u(β) +

β∈A0 \A0

n−1 X

(4.1) v(i)

i=1

By isolating price characteristics, the choice probability of an alternative k depends on two probabilities6 : - the probability of eliminating all the options having a higher price than an option i, n−1 X v(i) avec v = v(i). given by Pi = v i=1 - the probability of choosing option k among the i cheapest options : PkAi . Now the option k, ranked in the j st position of price, could be selected at this decision stage only if elimination is carried out from a price equal to or higher than that of k (the elimination concern the n − 1 − j at least as expensive options). Consequently, the n−1 X A probability of selecting k is Pk = Pi PkAi . i=j

Let us note that if the options are identical in price, choice probabilities depend only on the scale values of non-price attributes which are not shared by all the alternatives. Moreover, if options are identical, except in price, the least expensive alternative is selected with a probability 1. In the case with n > 2 goods, price advantage of the most expensive option over the second most expensive is inevitably shared between the n − 1 least expensive options : the particular case of the Luce model (and logit model) cannot thus be obtained any more starting from an EBA model with prices. Indeed, in the multinomial logit model applied to the markets of differentiated products (Anderson, De Palma and Thisse, 1992), price is integrated as a component of the utility (for instance, for a good i, Ui = f (pi ) with f 0 (pi ) < 0) and the probability of choosing a good depends on this utility. The EBA and logit models have thus distinct structures with different properties : since applying to economics the EBA model, the equivalence with classical RU models ceases being true.

4.2

Form of prices difference

Prices differences can take several forms according to the specification retained for the function w(α) : for instance, if w = 1, prices difference has a linear form whereas if w = α, it is of a quadratic type. Which form is it preferable to choose ? 6

In a similar way to the version without price

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To answer at this question, Xan interesting property of the EBA X Rotondo highlighted Aα u(β). In this case, if px ≤ py and model. Let us note a = u(α)Pk and b = β∈A0 \A0

α∈k0 \A0

(b − a) 6= 0 then : P (x > y) a 1 = + P (y > x) b−a b−a

Z

py

w(α)dα

(4.2)

px

Thus, the ratio of the pairwise choice probabilities is linear with prices difference, which makes it possible to test the form of price differences with consumers’ behavior. To carry out this test, a study using data on demands of 3 phone operators was undertaken. Triplets of alternatives are presented to the decision-makers which must establish a ranking. The pairwise choice probabilities are then obtained by calculating the proportion in which an option dominates the other. Rotondo shows that the coefficient of correlation between the ratio of choice probabilities and a specific prices difference is maximum when prices difference has a linear form. The use, by a similar method, of pairwise choice probabilities resulting from the logit and probit models lead to the same result : the coefficient of correlation is also higher for a linear price difference. With such results, the assumption of a linear price difference seems acceptable, which corresponds to : vi = θ(pi+1 − pi )

(4.3)

where θ can be interpreted as a measurement of the relative importance of the saving made, compared to the other non-price characteristics7 .

5

Economic properties of a duopoly with EBA

After having noted that prices integration in the EBA model gave rise to a distinct structure from RU models, let us show now that the EBA model applied to the analysis of differentiated products provides many new results compared to the logit model. This section comments on the main results obtained by Laurent (2006b). First of all, the framework of the duopoly model is introduced and then the economic properties of price equilibrium are identified. Finally, the cycle of price obtained when no equilibrium exists is analyzed.

5.1

Equilibrium of the EBA duopoly

The growing number of differentiated products together with the multiplication of their attributes makes consumers’ choice more and more difficult. Thus, as their cognitive capacities are limited, it is not illogical to suppose that these decision makers uses heuristics. As the EBA heuristic fits particularly with this context, the probabilities of this model are used to establish demand functions. Consider a duopoly in which each firm sells exactly one differentiated product i at a price pi .8 The market is composed of N representative consumers buying exactly one unit of one of the products. Since the consumers follow the EBA heuristic, the final selection of an option depends on its price but also on the utility of the set of the specific attributes, 7 8

In the paper of Rotondo (1986, p 394), this parameter θ is called "c" The firm producing the good i will be described as "firm i"

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noted ui for product i, i ∈ A. As i0 \A0Pindicates the set of specific attributes of i in the previous notations, this utility is ui = α∈i0 \A0 u(α). In accordance with the analysis of Rotondo (1986), prices are seen like attributes in this model and the advantage in price of an option takes the form of a linear price difference. We can now write the choice probabilities corresponding to the specifications previously mentionned. The probability of choosing a product i depends on the level of price pi relatively to the price pj , j 6= i of the other firm : - if pi ≥ pj , Pi =

ui . ui + uj + θ(pi − pj )

(5.1)

Pi =

ui + θ(pj − pi ) . ui + uj + θ(pj − pi )

(5.2)

- if pj ≥ pi ,

The demand of product i is a continuous function taking the form Xi = N Pi . This function has a kink implying its non-concavity9 and therefore the existence of a price equilibrium is not guaranteed a priori. Let us suppose that each firm bears an unit cost ci and a fixed cost Fi and chooses the price maximizing its profit. We use the concept of Nash equilibrium in pure strategies to determine the outcome values and the conditions of existence of such equilibrium. Proposition 5.1 (Laurent, 2006b, p 13) Necessary and sufficient conditions of Nash equilibrium existence in pi ≥ pj , with i, j ∈ {1, 2} and i 6= j, are ui ≥ uj

(5.3)

and √ ci − cj ≥

ui u j − ui θ

(5.4)

If this equilibrium exists, then it is unique. Let us note that the two necessary and sufficient conditions do not guarantee the existence of an equilibrium for all the values of cost or utility parameters. The case with no equilibrium will be evoked in section 5.3.

5.2

Properties of the equilibrium

When an equilibrium exists, the couple of equilibrium price in pi > pj is given by : √ ∆ u + i p∗i = + ci (5.5) 2θ p∗j =

ui + u j + ci θ

where ∆ = u2i + 4ui (ui + uj + θ(ci − cj )) 9

In a similar way to the "switching costs" models, see for instance Klemperer, 1984

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(5.6)

Thus, firm i, which product is the most appreciated by consumers, sets a higher price than its rival. This equilibrium exists when the unit cost of i is higher than that of j, what √ seems intuitive. However, firm j may also have a higher unit cost than i since ui uj −ui < 0 but the gap of unit costs must necessarily be weak. Let us see now how to characterize the differentiation in this model by considering several cases : - When u1 = u2 = u > 0, the specific attributes of each good are appreciated in the same way within the population of consumers. If c1 = c2 = c, then pi = pj > c, which is the sign of a differentiation. Such a configuration, in which all varieties have a positive demand when they are sold at same price, refers to a pure horizontal differentiation, as in the Kaldorian model of Hotelling (1929) or in the Chamberlinian model of Dixit and Stiglitz (1977). - When u1 > 0 and u2 = 0, only one of the goods has additional specific attributes. Thus, at equal price, all consumers would prefer having good 1 rather than good 2 : existence of such a preference hierarchy is the sign of a pure vertical differentiation, as in models of Mussa and Rosen (1978), Gabszewicz and Thisse (1979) and Shaked and Sutton (1982). - Thus, the general case with ui > uj > 0 can be interpreted as double differentiation : differentiation is horizontal up to the level uj , goods offering to the consumer comparable levels of services, then vertical for a level ui − uj , product i proposing also additional attributes. Literature on models with double differentiation is quite little. Neven and Thisse (1990) were the first to propose a duopoly allowing the triple analysis of equilibrium in price, quality and variety competition, but with a rather heavy formalism. - Finally, the Bertrand case is obtained when u1 = u2 = 0 and c1 = c2 = c : prices fall at the level pi = pj = c. In the logit model, differentiation is intrinsically horizontal : it is possible to take into account aspects of vertical differentiation but, in this case, the expression of equilibrium prices becomes implicit. The way of representation of differentiation is also very distinct between these models. Note now that each price increases both with u1 and u2 . That means that price is increasing with the global degree of differentiation on the market : an effort of differentiation from a competitor will profit to all the protagonists. Moreover, price of j increases with ci which can seem counter-intuitive : this relation reminds practices of tariff imitation, like those described by Lazer (1957 p. 130-131), and in particular the case where the firm selling the "best quality" good sets a reference price on the market (or "focal price"). In this case, the other firms tariff at reference price minus a specific amount, which is function of the quality gap with the reference firm. The introduction of bounded rationality on the demand side can then imply a modification of relations between firms. Let us compare now the equilibrium profits when fixed costs are identical F1 = F2 . We find that Π∗i ≥ Π∗j if θ(ci − cj ) ≤ ui − uj . If the valuation gap of products between firms is higher than the weighted cost gap, then the firm selling the "more appreciated" product (such that ui > uj ) obtains a larger profit. By referring again to the assumption c1 = c2 , frequent in models of product differentiation, one finds thus the standard result : firm selling the "more appreciated" good always makes a higher profit than its rival. Moreover, when this condition is verified, firm i also has the larger market share. Indeed, a too strong difference in costs would generate a too high price for firm i and a transfer on demand of firm j. The omnipresence of this condition reveals a convergence of firms’ goals in this model : maximization of profit, short term objective, is consistent with maximization of market share, long term objective. 12

5.3

Non-existence of an equilibrium and Edgeworth cycle

Let us analyse now the EBA duopoly when one condition of existence of price equilibrium holds but not the other10 Consider a sequential infinite-horizon game with alternate moves of firms in which these firms make a choice in the space of pure price strategies. Suppose that each firm is unaware of the reaction function of its rival, which limits its temporal horizon of profit maximization to one period. Thus, each time t its "turn" to play occurs, firm i observes pt−1 and chooses the price pti maximizing its profit for the current period. Prices evolution j is thus described by successive and "naive" interactions of the firms’ reaction functions. In this framework, we show (Laurent, 2006b, p 18) that the strategic interaction of firms generates a price cycle a la Edgeworth (1925) instead of the market exit of a firm (as in the logit model). Let us recall that such a cycle, studied in particular by Maskin and Tirole (1988), comprises a long price war phase followed by an abrupt increase in prices when firms’ margins become too low. Noel (2004) recently showed that such a cycle can be a Markov Perfect Equilibrium when products are horizontally differentiated. Without using this particular concept of equilibrium, our analysis reveals that such cycles can exist with less sophisticated strategies of the producers and with a more general form of differentiation.

6

Conclusion

The "Elimination by aspects" model constitutes a good example of the productivity of the links between economics and psychology. Indeed, the subjacent heuristic carries out a good balance between the cost and the quality of a decision. It also permit the construction of a discrete choices model having a degree of flexibility comparable with probit or nested logit models. This EBA model opens interesting prospects for research in economics, both on theoretical and empirical levels. At the theoretical level, the application of the EBA model to the industrial organization made it possible to construct a duopoly with a double differentiation, horizontal and vertical, of products. This kind of models is not very frequent in the literature and often of a more complex formalism. This model also makes it possible to take into account a bounded rationality of the individuals and reveals the existence of a "reference" firm in tariffing. The application of the model to other fields of economic theory, as voting theory for instance, remains nevertheless to do. At the empirical level, the question of estimation of the parameters in the EBA model knows a revival in research, after ten years of silence on this subject. This estimation is made possible by the development, on the one hand, of scanner databases and of the methods which exploit them (like the probability differences) and, on the other hand, of the Monte Carlo Markov Chain simulations. The links between random utility models and models with elimination of the options, but also the existence of different properties, suggest that these models should be used and compared more systematically during the analysis of the economic sectors. 10

If the two conditions were simultaneously violated, another equilibrium configuration would be obtained

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