Academic journal article Fuzzy Economic Review

Fuzzy Grouping Variables in Economic Analysis. a Pilot Study of a Verification of a Normative Model for R&d Alliances

Academic journal article Fuzzy Economic Review

Fuzzy Grouping Variables in Economic Analysis. a Pilot Study of a Verification of a Normative Model for R&d Alliances

Article excerpt

(ProQuest: ... denotes formulae omitted.)

1.INTRODUCTION

Many investments decisions in presence of uncertainty can be characterized as real options problems [9, 19]. Consequently, in the last decade, the development of normative techniques to evaluate real options investments, summarized by the seminal book by Dixit and Pyndyck [11], has significantly shaped the research on sequential investments and created a fruitful paradigm for its treatment [17].

More recently real options analysis (ROA) has been used also to evaluate R&D alliances established between firms [8,15,16,17,25]. These agreements not only generate stochastic benefits (which suggest the existence of uncertainty) but also bring sunk costs (which imply irreversibility). In addition, a key element in these agreements is flexibility: firms have the opportunity, but not the obligation to sign an alliance, or sometimes the right to renew an existing one. In other words, they can postpone their decisions to form an alliance when more information is available. Therefore, the study of R&D alliances in a real options framework per sé includes these important three aspects [23]. Despite the normative work on real option valuation, only recently some researches have started analysing behavioral aspects [19]. These studies highlight that individuals exhibit systematic deviations from the predictions derived using normative models [22].

This research attempts to extend the boundaries of real options analysis to environments where people have biases in their decision-making, especially in the context of R&D alliances and merger and acquisitions as well as to provide tools for appropriate validation of the relevant models. Consistent with neoclassical rational theory, prior studies show that mergers are driven by rational expectations of growth options, synergies or reallocation of assets in a response to industry shocks. The rational view also includes real options models, where decisions such as acquisitions waves are driven by growth opportunities [30]. However, contrary to rational theory, decision makers may act irrationally when making acquisition choices under uncertainty [18,28]. Individuals exhibit systematic deviations from the predictions derived using normative models - i.e. models assuming that individuals are risk neutral expected value maximizing agents [22]. Investors' exuberance, positive sentiments of boards as well as cognitive biases - such as overconfidence - influence companies' acquisition behavior under uncertainty. To use Smit and Lovallo's [28] words, "acquisition strategy is vulnerable to the way managers perceive risk and losses, judgment biases in their strategy, the bidding behavior of rivals and mispricing in financial markets". Note, that these assumptions need to be reflected also in the design of the validation instruments and in the collection of validation data. Such data need to be gathered in a way that allows for the identification of specific subgroups in the validation sample that meet the assumptions of the models. Standard grouping variables may not be applied in the analysis of the sample, since the definition of the subgroups may not be crisp (e.g. rationality, risk-aversion, etc. can manifest themselves in a particular respondent to a certain degree, not necessarily fully). The requirement of full satisfaction of certain assumptions (full rationality, full information, etc.) may render the applicable part of the validation sample too small to be still relevant. On the other hand the use of the full sample without any knowledge concerning their fulfilment of the assumptions of the model that is being validated is methodologically incorrect. This calls for two things - first, the tools for the assessment of the degree of fulfilment of the assumptions of the model in the participants (and the data they provide) need to be available (or at least built-in the data gathering procedure), and second, data analysis tools capable of dealing with fuzzy subsets of the validation sample need to be used. Both these issues are discussed in this paper.

The controlled environment of a laboratory setting may allow us to take these important aspects into consideration and to get a better idea on how subjects make risky decisions in the particular context of R&D alliances and mergers and acquisitions. With this aim, starting from a real options model available in literature (developed by Lo Nigro et al. [15]) that deals with R&D alliance timing decisions in a stochastic environment, we propose a design of experiment that can be used to test if people make decisions conforming to the normative model. We also conducted a pilot that illustrates how the design could be implemented. Specifically, we presented the decision makers with risky choices formulated as abstract gambling decisions in order to validate the normative predictions of the model, which assumes decision makers to be fully rational. Notice that when rationality is assumed in terms of (expected) payoff maximization, risk neutrality is implied in situations when people are faced with a decision between a sure outcome and a risky alternative with the same expected payoff. The verification of normative models under this assumption should be performed on a risk-neutral sample to comply with the assumptions of the model. A risk-neutral subset of representative sample of the population can also be used. The question is, however, how to introduce risk-neutrality as a grouping variable for further analysis. In this paper we deal with these issues in two steps - first by designing a survey with an in-built risk-neutrality measure and second by introducing a fuzzy grouping variable of risk-attitude that defines a fuzzy partition of the sample. We define a fuzzy set of risk-neutral respondents and validate the model on this fuzzy subset of the original sample that complies with the assumptions of the model under investigation. By doing so we introduce the novel concept of a fuzzy grouping variable to economic analysis. On a practical example we show its performance and discuss the differences from the analysis (model validation) on an "unfocused" sample. This can be considered as an important step in the direction of more relevant analysis of economic data and a first step towards a wider use of fuzzy partitions of experimental data in economics and social sciences.

The remainder of the paper is organized as follows. In the next section we discuss the relevant literature. In section 3 we present the theoretical model, while in section 4 we describe the design of the conducted survey as well as the sample used to obtain data and we introduce the basic concepts of fuzzy grouping variables. In section 5 we present the results of our assessment of the model performance both for the full-sample case and for the case focused on the fuzzy set of risk-neutral respondents and comment on the relevance and practical impact of fuzzy grouping variables. In section 6, conclusions are drawn and further developments are anticipated.

2.RELATED LITERATURE

As Moel and Tufano [20] note, "empirical research on real options has lagged considerably behind the conceptual and theoretical contribution". This is primarily due to the problems that researchers face in obtaining "key variables" in real options, i.e. reliable data on such components of the real options approach as the current and future value of the underlying asset [32]. As a matter of fact, such information is very difficult to estimate from the field data. Conversely, such data can be easily generated in a laboratory experiment. In addition, even if data are available, very often they are not available in the form that would respect the assumptions of the theoretical models. Moreover, subsample of the data that respects the model assumptions cannot be identified. As a consequence, in the last decade, researchers have been using laboratory experiments to study real option theory empirically.

In Miller and Shapira's [19] work, decision makers are presented with simple binary lotteries and asked to specify the price for selling or buying a call or a put option for the gambles. The results show that the value of the price specified for selling and buying the derivate do not coincide, suggesting inadequacy with the normative model's descriptive power [22]. Yavas and Sirmanas [32] investigate the option "wait and see" in the laboratory. The results of their experiments highlight that fundamental insights of real options theory are not so evident to individual investors. As a matter of fact, the majority of subjects tend to invest too early compared to the optimal timing suggested by the theoretical model and thus fail to realize the benefits of waiting. Close to the spirit of this paper, Oprea et al. [24] investigate behaviour in uncertain investment opportunities governed by Brownian motion. Their results indicate that people can closely approximate optimal exercise of wait options if they have decent chance to learn from personal experience [24]. In fact, while at the beginning investors tend to exercise the option prematurely, over time their average behaviour converges close to the optimum. In Murphy and Knaus's [22] work, a decision maker must choose how much to invest in a risky environment that evolves over time. Their experimental results contrast predictions from theory.

Overall, these findings suggest innate behavioural tendencies that are contrary to the normative dictates: several cognitive aspects influence people's decisions in risky contexts. Our research contributes to this research area. We propose the fuzzy grouping variables as tool for focusing research to a specific subset of the sample which does not violate the assumptions of the normative model, even in cases where the assumptions are met partially.

3.THE THEORETICAL MODEL

3.1Problem description and assumptions

As noted previously, we refer to a real options model developed by Lo Nigro et al. [15], which deals with alliance timing decisions in a stochastic environment. To better understand the design of the survey we propose, let us briefly recall the theoretical model in the monopoly case.1

Consider a firm, named firm A, which is working on a new project. Also, consider a simple setup in which there are two stages of the development process. The first stage, named R&D stage, includes the research phases as well as the development phase. In the second and final stage the project is approaching commercialization. The first stage is performed during a limited interval [0, T], whereas the manufacturing and commercialization phase starts at T. The market value of the project is uncertain and it can change during the first stage. In particular, let the market value projection, V(t), t e [0, T], be represented by a non-negative random process, specifically by a geometric Brownian motion (gBm). At time 0, beginning of the R&D phase, the firm A buys the right - paying an option price I1 - of developing and manufacturing the project by investing an amount I2 at T. At each stage, the firm will have the option to form an alliance with firm B. Firm A can also choose to forgo the alliance and choose to enter the market alone.

The sequence of the game is as follows. At the beginning of the first stage (t=0), firm A's project value is given by V0. Firm B offers an alliance contract to Firm A consisting of an ex-ante payment P1 and percentage of royalties retained by firm A is equal to 1-α, (with 0<α<1). If firm A rejects the offer, the decision game is repeated. In the second stage (t=T), Firm B offers a different alliance contract consisting of an ex-ante payment P2 and a percentage of royalties retained by firm A is equal to 1-α, (with 0<α<1). If firm A rejects this offer, the firm proceeds to the final market unassisted. If an alliance is formed at the first or second stage, the size of the project's market increases relative to the case of no alliance by an amplification factor δ>1. This factor reflects the value added to the project by the synergies derived from the alliance. Thus, if an alliance is signed, the value of the project will be multiplied by k, with k=δ(1-α). Figure 1 depicts the extensive form of the choices firm A can take at t=0 and t=T.

The game can be solved via backward induction procedure. Therefore, we start from the second stage where firm A has to decide if ally or not with firm B and go back to the first stage. It is straightforward to see that we can obtain three possible scenarios j=1,2,3 of equilibrium for the game. These are (see Table 1): Firm A's alliance at stage 1 (equilibrium Qi), Firm A's alliance at stage 2 (Q2), and no Firm A's alliance (Q3).

3.2Computing payoffs

As mentioned, the development process is split in two stages: the R&D stage and the commercialization stage, with the market value of the project that is uncertain and able to change during the first stage. As a consequence, the R&D stage is an option for the second and final stage. This means that the process can be seen as a 1-fold option, i.e. a simple option. Accordingly, we can model it with the Black and Sholes approach, which ensures the flexibility offered by the option to decide further investments when more information is available. Adopting such an approach means assuming that the value of the project V0 at the beginning of the first stage follows a geometric Brownian motion which results in a lognormal distributed random payoff (project value, see Figure 2) at the maturity T (i.e. at the end of the first stage or alternatively at the beginning of the second stage), with mean and variance equal respectively to [11]:

...

Where:

V0= value of the project at t=0;

r = the riskless interest rate;

T= the time to maturity(length of the first stage);

σ = the standard deviation of the project's returns.

On the right side the probability density function of the lognormal distribution of the value of the project at maturity obtained with 10.000 replications.

As a consequence, the second stage payoffs are function of the known realization of the value of the project at maturity (Vt). In other words, after the R&D stage, the company has more information about the value of cash flows coming from commercialization. Therefore, the simple NPV (Net present value) methodology is more appropriate to compute firms' second stage payoff expressions [1]. Table 2 shows the second stage payoffs of firm A (and the elements necessary to compute them), under the two possible scenarios, namely, the alliance is formed with B at the second stage and no alliance is formed. These payoffs are simply computed as the difference between the project value realized at maturity Vt (or kVr + P2 in case of alliance) and the net investment I2 in second stage.

As far as the first stage payoffs are concerned, they are computed as call European options in order to take into account uncertainty in this stage. Specifically, first stage payoffs are computed as the difference between the European call option Cj (with underlying value Sj and exercise price I2, in case of scenario j), which represents the gross payoff in the first stage, and the net investments in the same stage, NIJ (see Table 3). The latter is actually computed as the difference between the investment needed in the first stage I1 and the exante payment in case of alliance at the first stage, P1. Of course, in case of no alliance or an alliance at the second stage, the net investment in the first stage is simply the investment required in the first stage I1. Note that both I1 and P1 do not affect the option value CJ.

Table 3 reports first stage payoffs and all the elements necessary to compute them under all possible scenarios of equilibrium (Q1, Q2, Q3 respectively).

For the sake of clarity we report the Black and Scholes formula [6], below:

...

Where:

...

and SJ is the underlying of the project, i.e. expected net cash flows arising after the commercialization at the beginning of the first stage under the scenario J, t is the current time (i.e., time t=0), N(.) is the cumulative normal density function, and l2, r,T, σ assume the same meaning as described earlier.

3.3Game solutions

The game is solved by backward induction. Therefore, starting from the second stage (see Figure 3), it is possible to find the payment P2 which makes the "alliance" payoff (W1+P2-I2) equal to the "no alliance payoff" (VT-I2).

Referring to Table 3, the minimum value of P2 which satisfies this condition, i.e. the value that makes firm A indifferent between allying at the second stage or not allying at all, is given by the expression P2= kVT-VT=VT(1-k). Then, if P2>VT(1-k), the solution of the sub-game in the second stage is "Alliance at second stage". Note that this payment is a linear function of the value of the project Vt. Clearly, in order to solve the model and find theoretical solutions at t=0, we need to find expected values of Vt and, consequently, of P2. Particularly, at the beginning of the second stage (i.e. at t=T) the value of the project Vt is log-normally distributed, with expected value E(Vi)= VoerT. As far as the expected second stage payment E(P2) is concerned, since P2 is linear a function of Vt, the value that makes firm A indifferent between allying at the second stage or not allying at all, is given by the formulation

...

Then, if E(P2)> VoerT(1-k), firm A should prefer an alliance at second stage. Backing to the first stage (please refer to Figure 4), we can find the payment P1 which makes firm A indifferent between the first stage alliance (C1-NI1) and the second stage alliance evaluated at t=0, computed as a call option (C2-NI2). Referring to Table 3, the minimum value of P, which satisfies this condition is given by the expression P1=C2- C1.

Therefore, if P1 is higher than this threshold, the solution of the game will be the first stage alliance, otherwise, i.e. for values of P1 lower than the same threshold, the solution of the game will be the second stage alliance (see Table 4).

Conversely if E(P2)

Table 4 summarizes the threshold payments, i.e., P1 and E(P2) which help understand whether the alliance will be signed in the first stage, in the second stage, or no alliance will be established.

Specifically, the normative model suggests that, if firm B does not offer a considerable amount of payment in the initial stage, i.e., P1 is low, firm A might profit more from waiting until the second stage to possibly obtain better payment conditions. In the second stage, in fact, firm A will ally with firm B only under favourable expected payment conditions, i.e., high values of E(P2). Otherwise, firm A should continue the R&D process on her own. Interestingly, if in the first stage the payment conditions are sufficiently high, firm A will sign an early alliance independently of any expected value of P2.

To better visualize these results and provide a practical application of the relative insights, we complement the analytical derivation with a numerical analysis, assuming high values of E(P2).

Given this assumption, depending on the value assumed by P1, the equilibrium suggested by the model would be an alliance at the first or at the second stage. Figure 5 identifies the region of alliance as a function of P1 and α (considering the following set of parameters:

...

Therefore, assuming a payment P1 equal to 17,8 and α=0.6, the theoretical equilibrium suggested by the model is the alliance at the second stage.

It is straightforward to see that the game-solutions (equilibria) strongly depend on both the expected value of the second stage payment E(P2) and first stage payment P1.

Since the payment in the second stage is a linear function of the value of the project, if Firm A decides to wait and sign an alliance at the second stage, she will receive a payment P2, which is a realization from a log-normal distribution with mean and variance equal to:

...

Conversely, if firm A signs an alliance at the first stage, she will receive a sure payment P1 computed as a difference of two calls options2.

4.METHODOLOGY, SAMPLE AND DESIGN OF THE SURVEY

4.1 The theoretical model for the survey

Consistent with past research on decision-making processes (see, e.g., [10, 19, 27]) we analyzed students' responses to a survey3, explicitly designed to analyze if people make decisions according to a particular configuration of the above described theoretical model. In order to keep our option pricing problems as simple as possible, we maintain several assumptions throughout our presentation. Following Miller and Shapira (2004) [19], we consider options with zero exercise prices (i.e. I2=0). In this way, the market value is risky but it can assume only positive values. It is straightforward to see that when V0 is much higher than I2, the sure P1, considered at its threshold value, tends to be almost equal to the discounted expected value of the risky payment E(P2). In fact, the higher the underlying value compared to the exercise price, the higher the probability to exercise the call option (N(d1) tends to 1). Specifically, assuming I2 equal to zero, the call option Cj assumes the same value of its Underlying Sj - being N(d1) equal to 1. Then, the sure payment P1 - for example considered at its threshold value when E(P2) is high - and the discounted expected value of the risky payment E(P2)e-rT assume the same value.

In fact:

...

(please refer to Tables 3 and 4).

Also, we consider a particular scenario: we assume E(P2) to be high. Given this assumption (please refer to Table 4), depending on the value assumed by P1, the equilibrium suggested by the model would be an alliance at the first stage (if P1C2-C1). Specifically, we set P1=C2-C1, i.e. equal to the threshold payment. Under this particular setting, the normative model suggests indifference between an alliance at the first stage and an alliance at the second stage.

Through these assumptions at the end subjects were provided with a sure payoff P1 - meaning an alliance at first stage - and a risky one, whose expected value (discounted at t=0) 4 E(P2)e-rT was equal to the sure one meaning an alliance at the second stage. Therefore, according to [26], we consider a risk-neutral decision-maker as an individual who, when faced with a choice between a certain P1 and a risky alternative (P2) with the same expected value, will be indifferent between accepting the sure payoff P1 or choosing the risky payoff P2. When "being indifferent" is the optimal strategy suggested by the model, do investors really exhibit such behaviour? In other words do people make decisions conforming to the normative model? We aim at empirically testing this particular situation.

4.2 Basic notation of fuzzy sets

Let U be a nonempty set (the universe of discourse). A fuzzy set A on U is defined by the mapping A: U^-[0,1]. For each x e U the value A(x) is called a membership degree of the element x in the fuzzy set A and A(.) is called a membership function of the fuzzy set A. When the universe U is discrete,

i.e. U=(x1,...,xn],

then the fuzzy set A on U is denoted

A={A(x1)|l×x1...A(x(n)|xn}

The family of all fuzzy set on U is denoted F(U). We refer readers to [12] or [14] for more details on the basic notions of fuzzy sets. In this paper we will restrict ourselves just to fuzzy sets defined on discrete universes. Similar to the traditional notion of sets, a fuzzy set can be defined by (can represent) a characteristic feature, which can be possessed by the elements of the universe to various extent. This extent is for each x e U represented by A(x)e [0,1]. The set of all the elements of U that possess the given characteristic feature completely is called the kernel of A and is denoted as Ker (A) = {x e U|A(x) =1}. Aa={x e U|A(x)>a} denotes an α-cut of A for any ae[0,1] and represent all the elements of U that possess the given feature at least to the extent a. Supp(A)={x e U|A(x)>0} denotes a support of A, i.e. the elements of U that possess the given feature at least to some extent. Let A and B be fuzzy sets on the same universe U. We can define the intersection5 of A and B as a fuzzy set (AHB) on U with the membership function (AAB)(x)= min{A(x),B(x)} ; for any x e U the value of (AHB)(x) represents the degree to which x possesses the characteristic features of both A and B at the same time. The cardinality of fuzzy set A on U is a real number Card (A) defined as follows: Card(A)= Σ|=1 A(xi). Cardinality can be understood as the amount of support for the characteristic feature represented by A that can be found in the universal set U, the relative cardinality Card'(A)= Σ|=1 A(xi) /n represents the proportion of the universal set that possesses the given characteristic feature.

4.3Sample and Questionnaire

We elicited responses from 58 Master Industrial Management students at University of Palermo (Italy), who agreed to participate in a research project on decision-making. The mean age was 23.7 years (SD= 1.75 years). Moreover 35% percent were female and 65% were male. This sample of students was selected mainly for their focus on management as a reasonable proxy of managerial decision making (obviously practical experience cannot be compared between students and real-life managers, but the theoretical background, know-how and education gives reasonable grounds for a careful extension of our results to the management practice). According to our assumptions, the survey consisted of several items where students had to choose between a sure gain A (i.e. P1 referring to the theoretical model) and a risky one B (i.e. P2 referring to the theoretical model), whose expected value was the same as the sure payoff in option A. In addition, they were given the opportunity to be indifferent between these two options. These risk neutral answers are further used in the definition of the risk-neutrality measure.

The survey itself comprised of 8 items presenting the respondents with a choice between a sure payment and a risky alternative with the same expected payment. All the items used very similar amounts of money as payoffs and can thus were considered to be different representations of the same decision problem. Two sets of these eight items were provided to each respondent - one with low payoffs and one with higher payoffs - thus obtaining 16 choice items in total. For the purpose of verification of the theoretical model described above, we will focus on the responses provided by the respondents in four items relevant for our research question (both low and high payoffs will be considered). These chosen items will be referred to as item (hist), item (continuous), item (mean and hist) and item (MEAN AND CONTINUOUS).

Specifically, the risky option in item (hist) was represented by a histogram which approximates a lognormal distribution; the risky option in item (mean and hist) provided also the information concerning the mean value in addition to the histogram, the risky option item (continuous) was represented by a probability density function of a lognormal distribution, and the option item (mean and continuous) by the probability density function and its mean value. Both histograms and probability density functions were truncated at a preselected value for presentation purposes (the truncation was made always at the value of the 99th percentile of the respective distributions).

See the appendix where the items we have provided to the students are shown in more details (we considered the following set of parameters of the theoretical model: =58,8%, r=5%, T=2, k=0,5 and V0 equal to 40, 58, 50 and 466 EUR in item (HIST), ITEM (CONTINUOUS), ITEM (MEAN AND HIST) AND ITEM (MEAN AND CONTINUOUS) respectively for the low-payoff items, and 438, 558, 598 and 458 eur in the highpayoff items).

The Items were presented to the students in a randomized order (6 versions of the questionnaire with a different order of items were used). As can be seen in the appendix, the respondents were asked to choose one of the four alternatives in each item - alternative A, i.e. a sure gain (this answer is denoted A), alternative B, i.e. a risky alternative with the same expected value (this answer is denoted B), and two options for expressing initial indifference (I cannot decide, but if I had to choose, it would be A/B). These answers are denoted AA and BB respectively. Although this design is not very frequent, it allows us to harness the powers of fuzzy sets to summarize the composition of the sample.

Since the theoretical model assumes rational decision makers (which in turn implies risk neutrality), it might be interesting to decompose the sample into risk averse, risk neutral and risk seeking respondents. Let us denote the set of respondents as H=(h1,_,hn], in our case n=58, and the set of items (questions) in the survey Q={qr...,qm}, in our case m=16 and qr...,q8 were items considering low payoffs and q9,...,q16 items offering the respondents high payoffs.

First we need to define the data set available after the survey. All the answers of the n respondents to the m items can be summarized by an nxm data matrix

D = {dij}

Where:

dj∈{A,AA,BB,B} Represents the answer of respondent i to item j

i=1,...,n

j=1,...,m.

Now we can define three distinct n x m matrices:

A matrix of risk-neutral answers D0={d0ij}

Matrix of risk-averse answers DA={dA}

Matrix of risk-seeking answers DB={dB}.

According to our design, we have (d0=l) - (dij∈{AA,BB}) and dj-=0 otherwise. We define data matrices DA and DB of risk-averse and risk-seeking answers so that they do not overlap with the data matrix of risk-neutral answers D0: we define (dA=l) o(dy=A) and dA=0 otherwise and analogically (dB=l) ^(dj=B) and diB=0 otherwise.

We can now consider the relative amount of risk-neutral answers (AA or BB) to the 16 items to be a general measure of risk neutrality for each decision maker, more specifically the risk-neutrality rate of a respondent i can be defined as:

...

Analogically a measure of risk-aversion can be defined for each respondent as:

...

And a measure of risk-seeking as:

...

Using these measures7, we can now define the fuzzy set of risk-neutral respondents H0e F (H) as:

...

The fuzzy set of risk-averse respondents Ha ∈ F (H) as:

...

And the fuzzy set of risk-seeking respondents Hb ∈ F (H) as:

...

The results of this process are summarized in Figure 6, where the membership degrees of the respondents to the respective fuzzy sets are represented by the heights of the bars for better visibility. This way a fuzzy partition of the set of respondents into three fuzzy sets H0, HA and HB is introduced.

Note that the full membership degree of each respondent hi is divided between the three fuzzy sets H0,HA and HB, i.e. H0(hi)+HA(hi)+HB(hi)=r0+rA+rB=1 for all i=1 ,...,n. By doing so, we have introduced a fuzzy grouping variable "risk propensity" with values "risk-neutral", "risk-averse" and "risk-seeking", which defines a fuzzy partition of the set H into the fuzzy sets H0,HA and HB respectively. Note, that if r0, if, rBe{0,1} for all i=1 ,...,n, we would actually have a crisp grouping variable (analogical to e.g. sex). In this sense the methodology proposed in this paper is a generalization of the usage of crisp grouping variables in data analysis, since crisp grouping variables are a special case of the proposed fuzzy grouping variables.

The data sets D0,DA and DB are defined as nonoverlapping. The sum of the membership degrees of each respondent to these three fuzzy sets is 1, i.e. H0(hi)+HA(hi)+HB(hi)=1 for all i=1 n.

It is also possible to define the elements of matrices DA and DB in a different way, e.g. to reflect the finally chosen answer despite the fact that the "indifferent" answer was chosen, since the design of the instrument allows us to still say whether the sure or risky alternative was preferred if indifference was not an option. In this case we would get the alternative definitions (dA=l) ^(djE{A,AA}) and dA=0 otherwise and dj could be defined analogically.

Overlaps of data matrices DA and D'B with the data matrix D0 sets are all allowed, i.e. using r0, r'A and r'B we get H0(hi)+H'A(hi)+H'B(hi)>1 for all i=1.n.

Clearly, the answers AA and BB have lower information power than A and B concerning risk-aversion or risk-preference respectively. Yet if a decision would be required in a real life situation, the decision maker would be required to side with the risk-averse or risk-seeking alternative anyway. Although the choice in these cases can be done based on additional criteria or even by chance, we can assume that a similar process that was used in answering the question that would be used in practice as well and hence that the choice itself tells us something about the risk-attitude of the decision-maker. Such decision (AA or BB) also still qualifies the decision maker in the set of risk-neutrals.

The alternative definitions of dA and dB thus allow for the partial overlap of data matrices - the answers AA and BB can be under the above mentioned definitions of diA and dB an indication of both risk-neutrality (since the response starts "I cannot decide") and of risk-aversion or risk-seeking respectively (since the decision makers were effectively forced to choose an alternative in the end ".but if I had to choose, it would be A/B"). If a normative model was designed e.g. for risk-averse decision makers, even those that answered AA would be considered as viable respondents for its verification under this setting. Thus defined fuzzy sets H0,H'a and HB are presented in Figure 7.

H0 is presented in this figure again for better comparability. Note that under d0, d* and dB we now have H0(hi)+HA(hi)+HB(hi)=r0+r'iA+r'iB>1. If our model assumes e.g. risk neutrality, we should be validating it on the data provided by risk-neutral decision makers. We can utilize the fuzzy sets of risk-neutral for this purpose.

Let us now consider item q in the questionnaire and let us consider that a risk-neutral answer to this item complies with the results of our theoretical model (remember that the model assumes rationality). We can easily define the fuzzy set of risk neutral answers provided by the n decision makers Q°eF(H) as

...

Note, that at this point Q° is a fuzzy-set-representation of a crisp set {hi|di0=1}, which is the set of respondents that provided a risk-neutral answer to item q. The cardinality of Q°, i.e. Card(Q°), is equal to the number of risk neutral answers to item q provided by the n respondents, that is Card(Q°)=Card({hi |d°=1})= Σ=ι d°. This is, however, still not the correct piece of information we are interested in, because although we can now compute e.g. the proportion of answers to q that comply with the outputs of our normative model, this still involves not only risk-neutral respondents, but also those that are riskaverse and risk-seeking, for which the model is not intended. In this case we should focus on a subset of the respondents - the risk-neutral respondents - represented by the fuzzy set H°.

Since we require two conditions to be met at the same time (two characteristic features) - the answers to q need to be risk neutral and they need to be provided by risk-neutral respondents, we can define the fuzzy set of risk-neutral answers to item q provided by risk-neutral respondents as:

...

The proportion of risk-neutral answers to item q provided by risk-neutral respondents can thus be defined as

...

The computations would be analogous for any type of answers to q in combination with any type of risk-propensity.

5.RESULTS AND THEIR DISCUSSION

The standard approach to model validation when no risk-aversion measure is available would be to use the whole sample. We would compute how high are the proportions of risk-neutral answers in each item (or overall), since riskneutrality is what is expected by the normative model presented in section 3 in its specific setting that is investigated by the proposed empirical study. The results of such an analysis are summarized in Figure 8.

It is evident from the results summarized in Figure 8, that in the low-payoff setting, about 1/3 of the respondents behaved according to the normative model, in the high-payoff setting this proportion drops to about 1/6. This suggests that the majority of the decisions taken by the respondents were not in accordance with the expectations of the normative model under investigation. Using this standard approach, we could conclude that there is no evidence in the data supporting the above mentioned model as a normative model for decision making in R&D alliances. However, this would be a misleading conclusion.

The fuzzy set of risk-neutral respondents H0 was used to compute the depicted percentages (...).

From Figures 6 and 7 it is clear that there are some (at least partially) risk-averse and risk-seeking respondents in the sample (see e.g. the cardinalities of HA and Hb, and Ha and h'b). Note, that Card(H)=58, but Card(H0)=17.125, Card(HA )=25.75 and Card(HB)=15.125. Clearly once we introduce the measure of risk-neutrality (r°), risk-aversion (rf) and risk-seeking (rB) and partition the sample into three fuzzy sets of risk-neutral, risk-averse and risk-seeking respondents, we realize that a significant part of the sample is not risk neutral. Notice that Card(HA)+Card(HB)=40.775, that is much more than the cardinality of H0, which can be interpreted as risk-neutrality not being the majority feature in the sample.

Also Figure 6 illustrates this quite well. Since our theoretical model assumes riskneutrality, we should, however, focus our analysis on the risk-neutral respondents defined by H0 and on the proportion of risk-neutral answers provided by these respondents. Obviously the proportion of risk-neutral responses r0 is not the only possible measure of risk-neutrality that can be used. If another proxy on risk neutrality (or risk-aversion or risk-seeking) is available, it can be used to define the membership of hi to H0,HA and HB respectively. If we define H0 using r0, we obtain the results summarized in Figure 9. We can clearly see that when we "focus" the investigation on the fuzzy set of risk-neutral participants, the prediction of behavior as suggested by the theoretical model improves and is valid in about 1/2 of the cases for the low-payoff context and in about 1/3 of the cases in the high-payoff context. This is a much more realistic assessment of the validity of the model, since it draws information only from the sample that does not violate the basic assumptions of the model.

6.CONCLUSIONS AND FURTHER DEVELOPMENTS

There is a growing awareness that a large proportion of acquisition strategies fail to deliver the expected value. Too often managers make risky investments decisions based on their intuition or on their cognitive biases - such as overconfidence - exposing their companies to potentially costly pitfalls [29]. This important consideration calls for integrating mathematical models used to evaluate investments under uncertainty (i.e. real option models) with experimental economics methods in order to predict and explain the decision process, preferences and cognitive limitations that the real decision makers exhibit when deliberating over complex options. Starting from a real options model dealing with alliance timing decisions, we propose a simple design that can be used to test some of the fundamental insights of real options theory in the context of R&D alliances. With this aim, we designed a survey with an in-built risk-neutrality measure and introduced a fuzzy grouping variable for riskneutrality. Specifically, we defined a fuzzy set of risk-neutral respondents and validated the model on this fuzzy subset of the original sample, which complies with the assumptions of the model under investigation. We compare the results (in terms of risk-neutrality) we obtained using the afore-mentioned subset (i.e. adopting our proposed methodology) and the results we obtained using the whole sample (i.e. adopting the standard approach, where no risk-neutrality measure is available).

Adopting a fuzzy set of risk-neutral respondents we found that an higher number of the decisions taken by the respondents were in accordance with the expectations of the normative model under investigation both in the low-payoff setting and in the high-payoff setting. The logic behind is that it draws information only from the sample that does not violate the basic assumptions of the model. After all, the adoption of the proposed approach is a more realistic assessment of the validity of the model and can be considered as an important step to build a new methodology for the assessment of the economic models (and the validity of their normative outputs) in the experimental and behavioural setting. The use of fuzzy grouping variables not only makes perfect sense in the management and social science research. The presence of a measure capable of defining the membership of the respondents (data instances) to the fuzzy subsets of the sample can also help focus research on such groups that are truly relevant for the validation of the models and theoretical findings. Therefore, fuzzy grouping variables and the analysis of fuzzy subsets of data seem to be a promising course of future research and development of new methods in economic modelling.

There are other several directions to build upon this work for future research. First, the possibility of adjusting the normative models for risk-averse and riskseeking decision makers (or "shifting" the results of models designed for riskneutral decision makers) is definitely an interesting topic for our future research. Also, a comprehensive work should aim at testing the theoretical model by including other possible paths of the model itself and use the proposed methodology in order to infer whether people make decisions according to the solutions suggested by the normative model. In addition, an interesting extension of the present research could focus on the more complex duopoly case (please refer to Lo Nigro et al. [15]), in which people should make decisions regarding the optimal time considering not only uncertainty on the project value due to the nature of the setup (such that in the monopoly case), but also uncertainty about reactions of competitors. How does the presence of competition influence people decisions? For instance, it is recognized in option pricing literature that in absence of competition, an incumbent firm would delay project initiation.

Conversely, ceteris paribus, the presence of competition may speed up a firm's planned investment [7]. Under particular values of the input parameters, we find the same situation in the alliance timing problem under uncertainty (see Figure 10). In fact, given the same input parameters, the optimal time to sign the alliance is the second stage when the monopoly case is considered, whereas it is the first stage when the duopoly case is considered.

Given the same payment conditions in the duopoly structure, does the first mover anticipate the alliance at the first stage in order to pre-empt the follower? It would be very interesting to study empirically what can happen in such a situation.

Finally, our study builds also on an active literature of risk choice and individuals' risk preference [3,31]. More recent writings include Miller and Shapira [19] and Andreoni [2]. In these studies, risky choices involve probabilistic and discrete outcomes. Since we use the well-known Black and Sholes formula (which assumes that the distribution of possible stock price at the end of any finite interval is lognormal), our first goal is to study individuals' risk attitudes when facing a choice between a sure outcome and a risk alternative coming from a continuous probability distribution. Actually, recent contributions [4,5,13] show a change in risk taking behaviour depending on whether the risk is simply described by just a number, or by a more complete format (like histograms and continuous distributions). Therefore, it would be interesting to understand whether the amount and the form of information presented to individuals influence their decision-making process and thus their abilities to follow the dictates of normative decision policies.

Acknowledgments

The research presented in this paper was partially supported by the grant IGA_FF_2015_014 of the internal grant agency of the Palacky University Olomouc.

[Footnote]

1 The authors introduced and analyzed the effect of competition on alliance timing decisions in a duopoly case. In order to highlight the role of competition, they also considered an environment where just one firm operates in the market, i.e. the monopoly case.

2 Readers can refer to Lo Nigro et al. [15] for an exhaustive description of the model.

3 The survey has been used for a larger research project, aiming at increasing our knowledge on different aspects related to the proposed topic [21 ]

4 Similarly to Miller and Shapira [19], in order to avoid confounding discounting effects, we considered discounted options.

5 Note that in general other t-norms can be used instead of min to define the intersection of fuzzy sets. A t-norm is a commutative, associative, monotonic function T: [0,1]x[0,1]^[0,1] that satisfies T(a,1)=a for any ae[0,1]. Since in our application discussed further in this section one of the fuzzy sets is always such, that the membership degrees of the elements of U to this fuzzy set are either 0 or 1, the choice of the t-norm has no actual effect on the result.

6Note that, given these input parameters, the threshold payment - discounted at t=0 - that makes firm A indifferent between allying at the second stage or not allying at all, i.e E(P2)e"rT=V°(1-k), is equal to 20, 29, 25, 23 IN ITEM (HIST), ITEM (CONTINUOUS), ITEM (MEAN AND HIST) AND ITEM (MEAN AND continuous) respectively for the low-payoff items. According, to our assumptions, in order to have E(P2)e"rT high, i.e. E(P2)e"rT>V0(1-k), we chose the risky payment to be 21, 30, 26, 24 respectively (see Appendix).

7 Obviously, if needed, the risk-propensity can be assessed separately in the domain of low payoffs and in the high-payoff domain.

[Reference]

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[Author Affiliation]

A. Morrealea, J. Stoklasab, T. Talásekc

a School of Business and Management, Lappeenranta University of Technology Skinnarilankatu 34, 53850 Lappeenranta, Finland azzurra.morreale@lut.fi

b,c Palackÿ University, Olomouc, Faculty of Arts, Department of Applied Economics Krizkovského 8, 771 47 Olomouc, Czech Republic {jan.stoklasa, tomas.talasek}@upol.cz

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