How did Homo Heuristicus become ecologically rational? Maria Otworowska (m.otworowska@donders.ru.nl)a, Marieke Sweersa , Robin Wellnera , Marvin Uhlmannb , Todd Warehamc , Iris van Rooija a Radboud University Nijmegen, Donders Institute for Brain, Cognition, and Behaviour b Max Planck Institute for Psycholinguistics c Department of Computer Science, Memorial University of Newfoundland Abstract humans would have evolved adaptive toolboxes of heuristics seems to be so far unexplored. Instead, proponents of the Gigerenzer and colleagues have proposed the ‘adaptive tool- box of heuristics’ as an account of resource-bounded human account seem to take the evolutionary plausibility of their decision-making. According to these authors, evolution has cognitive explanation for granted. In this paper we show that endowed such toolboxes with ‘ecological rationality’, defined the account’s evolutionary plausibility is not self-evident, as the ability to make good quality decisions in their specific environments. Here we explore to what extent the mechanisms and even questionable. To see why this is so, we start by of evolution alone can produce ecologically rational toolboxes. considering the notion of ecological rationality as Gigerenzer We present a formal argument for why evolution is unlikely to and colleagues conceptualise it. Next, we explain why produce ecologically rational toolboxes given the astronomi- cally large space of possible toolboxes. The probability of find- evolution is unlikely to produce adaptive toolboxes with the ing one or more ecologically rational toolboxes in this space feature of ecological rationality so construed. is negligibly small, even granting an evolutionary time scale Unlike classical notions of rationality that are based on of searching for it. We furthermore present artificial evolu- tion simulations results that show that evolution can produce optimality and internal coherence of beliefs and inferences, toolboxes of heuristics that are ‘good enough’ to survive, but the adaptive toolbox account defines ecological rationality that those toolboxes are not ecologically rational (in agree- in terms of the fit between actions and the world. For in- ment with our formal argument). Our results do not rule out that ontogenetic adaptation processes (development and learn- stance, Gigerenzer & Todd (1999, p. 13) state it as follows: ing) may yield ecologically rational toolboxes, but it does put “A heuristic is ecologically rational to the degree that it is into question the idea that phylogenetic processes (evolution) adapted to the structure of an environment.” Here, ‘adapted’ could. We discuss the implications of our findings for future theoretical research on heuristic decision-making. refers both to the property of being able to produce actions Keywords: resource-bounded decision making; heuristics; that fit the environment (i.e., being adapted), and to the pro- ecological rationality; adaptive toolbox; evolution; computer cess by which the toolbox comes to have that property (i.e., an simulation adaptation process that leads to the property of being adapted to the structure of the environment). Introduction With respect to the fit between heuristics and the environ- We make decisions every day, ranging from selecting an out- ment, Gigerenzer and colleagues claim consistently that this fit or choosing groceries to deciding whom to marry. Even fit (adapted in the property sense) is so good that the quality though our decisions aren’t always optimal, they seem to of decisions is high, and even can outperform optimisation be more often right than wrong in everyday contexts. One methods (Todd, 2002, Todd & Gigerenzer, 1999, p. 361), prominent account of how we are able to make good qual- at least in those environments to which the heuristics have ity decisions, despite our bounded resources, is the adap- been adapted (in the process sense). It is because of this tive toolbox of heuristics account proposed by Gigerenzer good quality that adapted heuristics can be genuinely said to and colleagues (Gigerenzer, 2002, 2004, Gigerenzer & Todd, have ecological rationality. With respect to the nature of the 1999). According to this account, an adaptive toolbox is a process of adaptation, two general variants need to be distin- collection of specialized cognitive mechanisms—called fast guished: phylogenetic adaptation processes (evolution) and and frugal heuristics—that evolution has built into the human ontogenetic adaptation processes (development or learning). mind for purposes of decision making (Gigerenzer, 2001, Although both types of processes have been claimed to be Gigerenzer & Sturm, 2012, Gigerenzer & Todd, 1999, p. 30). able to produce adaptive toolboxes that are ecologically ra- The heuristics are called ‘fast’ because they can reach deci- tional, here we focus specifically on the (im)plausibility of sions with only a few computation steps, and ‘frugal’ because the idea that a phylogenetic adaptation process would do so. they use little information. Furthermore, the heuristics in Clearly, evolution can produce organisms with ecological the adaptive toolbox are believed to be ‘ecologically rational’ rationality. By a combination of random variation and selec- (Gigerenzer, 2002, Gigerenzer & Todd, 1999), i.e. tailored to tion, organisms can come into existence that have decision the contexts in which they are used. tendencies that are particularly tuned to particular environ- The adaptive toolbox account has had many em- ments. However, it is highly implausible, that organisms (es- pirical and explanatory successes in cognitive sci- pecially humans) would come to have such high degrees of ence (Bergert & Nosofsky, 2007, Bröder, 2000, ‘fitness’ if their decisions were based on toolboxes of heuris- Dieckmann & Rieskamp, 2007, Goldstein & Gigerenzer, tics and evolution was to set the parameters of these tool- 1999, Pohl, 2006). Yet, the plausibility of the claim that boxes directly. The reason is that toolboxes of heuristics have 324 an enormous amount of degrees of freedom: A toolbox can of truth-values. For every situation there is a certain favored vary in terms of the number of heuristics it contains, and each action a to perform, where a is an element of the set of all heuristic can vary in terms of both the possible environmental possible actions A = {a1 , a2 , . . . , am }. A function D : S → A cues to which it responds and the different possible actions it maps each situation s ∈ S to an action a ∈ A. can perform. Given that the number of possible cue-heuristic- action mappings grows exponentially in these parameters, the Heuristics number of distinct possible toolboxes does as well. Each heuristic in the toolbox is represented as a fast and fru- Given these considerations, what are the odds of evolution gal tree (Gigerenzer & Gaissmaier, 2011, Martignon et al., producing toolboxes that are ecologically rational? This de- 2003), a chain of cues with associated actions. Each cue is pends on how many toolboxes in the vast space of possible a boolean function, evaluating whether an event e ∈ E is true toolboxes are ecologically rational. As we will show, the in a given situation, c(e, s). When executing a heuristic, the vast majority of possible toolboxes aren’t ecologically ratio- tree is traversed starting at the top. Step by step the cue func- nal. Even though the mechanisms of natural selection are tions are passed, checking whether the cue holds. If the cue not random, the only evolutionary mechanisms that can pro- c(e, s) evaluates to true for event e is in situation s, then the duce different toolboxes—such as mutation and crossover— action a associated to that cue c is executed. If the cue is false are random. This means that the chance of creating, and the next cue is evaluated until the bottom cue is reached. If subsequently selecting, ecologically rational toolboxes is so this last cue is false, the last action in the tree is performed. nanoscopically small that even on an evolutionary time scale c1 it is extremely improbable that evolution would yield ecolog- ically rational toolboxes. In this paper, we elaborate on this c1 = =T c1 F argument both formally and using computer simulations. The remainder of this paper is organized as follows. We a1 c2 present a formalization of the notion of an adaptive toolbox, c2 = =T to be used both in our formal argument and our computer c2 F simulations. Next, we put forth a formal argument for the im- a2 a3 plausibility of the idea that evolution could produce ecolog- ically rational toolboxes based on illustrative numerical esti- Figure 1: A single heuristic represented as a fast and frugal mates for even small toolboxes. We then describe the setup of tree. The tree contains cues C = {c1 , c2 } and associated ac- an artificial evolution environment that we use to empirically tions {a1 , a2 , a3 }. Each cue c ∈ C is a simple boolean func- validate our argument. We present results of simulations for tion which evaluates whether an event e j ∈ E is true or false, three different setups, each demonstrating that even though depending on the situation c(e j , sk ). If the cue function re- evolution can produce toolboxes that are ‘good enough’ to turns ’true’, the tree traversal stops and the action associated survive, these toolboxes do not display any notable ecological with the cue is executed; otherwise the next cue function is rationality. We close by discussing the broader implications executed. For example, if c1 is false, but c2 is true, then the of our findings for research into resource-bounded decision action a2 will be executed. making. Formalizing the Adaptive Toolbox Selector In this section we will present a formalization of the adaptive toolbox account, which involves formalizing components of A selector determines which heuristic to use in a given situ- the adaptive toolbox (heuristics with a selector) as well as its ation. We represent the selector as a fast and frugal tree as environment. We represent each of the components as a fast well1 ; the internal nodes are cues associated with heuristics and frugal tree (see Figure 1). Each internal node in such a (see Figure 2). A heuristic is executed in the case a cue is tree stands for a boolean function; a tree evaluates only a lim- evaluated to be true. ited set of statements (cues; which can be either true or false) Mathematical analysis and a particular action is triggered by a particular sequence of cues progressing from the root-node to the leaf representing In this section we present a formal argument for the implau- that action. sibility of generating the ecologically rational adaptive tool- boxes by means of evolutionary processes alone. The ar- Environment gument is composed of three parts: search space argument, The environment consists of a set of events (environmental probability argument and time argument. cues) E = {e1 , e2 , . . . , en }, every event can be either true or 1 Hypotheses about the exact nature of the selector mechanism false. A truth assignment for each event is called a situation haven’t been developed to the same extent as hypotheses about the s. That is, a function s assigns truth values to each event in E, structure of individual heuristics. Nevertheless, the common idea seems to be that the selector, like the heuristics, is fast and frugal. s : E → {T, F}. We denote the set of all possible situations by For our purposes, and without loss of generality, we assume that the S = {T, F}n , where S is the set of all possible n-length vectors selector can be modelled by a fast and frugal tree as well. 325 c3 = F S1 S2 S3 H1:C1 H1:A1 H2:C1 H2:A1 H3:C1 H3:A1 (a) SELECTOR H1:C2 H1:A2 H2:C2 H2:A2 H3:C2 H3:A2 c4 = F c4 ¬c2 c1 c3 H1:C3 H1:A3 H2:C3 H2:A3 H3:C3 H3:A3 c4 = T 50% 25% 12.5% a12 c3 a2 c5 a5 a7 ¬c5 25% 25% 12.5% 12.5% 6.25% 6.25% (b) 12.5% 12.5% 6.25% 6.25% 3.125% 3.125% HEURISTIC 3 6.25% 6.25% 3.125% 3.125% 1.6% 1.6% a8 ¬c2 a1 ¬c1 a10 HEURISTIC 4 Table 1: (a) A schematic representation of a toolbox of size a9 ¬c3 a3 12. In this toolbox, S1 is the first selector cue, H1:X is the HEURISTIC 2 first heuristic, H1:C1 is the first heuristic cue and H1:A1 is a4 the first action in the first heuristic. (b) A representation of contribution of cues and actions to fitness depending on the HEURISTIC 1 their locations in a toolbox. The blue color indicates the min- imal requirement for an ecologically rational toolbox. Figure 2: The adaptive toolbox selector and heuristics as fast and frugal trees. The selector is represented by the or- ange nodes. The tree is traversed from left to right (select- If the first selector cue (S1 in Table 1a), the first heuristic cue ing a heuristic) and from top to bottom (executing a heuris- (H1:C1) and the first action of the first heuristic (H1:A1) are tic). For instance, let’s assume a situation such that c4 (“it correct,3 that already ensures performing a correct action in is sunny outside”) = F, ¬c2 (“I have not read any book in a 256 situations (25% of a total number of 1024 situations) and while”) = T , and c5 (‘my favorite book is on the shelf”) = T ; it is worth 25% of the overall fitness score (see Table 1b). in such a case the action a2 = “read the book” will be exe- Given these dependencies, it is enough for a toolbox to cuted. Note that when the last cue of the selector (c3 ) returns have three actions and five cues correct in order to reach false, the first heuristic is executed by default. the 0.5 score of fitness (see Table 1b). The search space for mapping three actions to five cues is of size 503 × 105 = 1010 . This number holds given the assumption of equally dis- Part 1: Search space and location-sensitivity tributed chances for a cue being true or false. In case one Let’s assume a simple environment (10 events, 50 actions).2 takes, say, a 1:10 ratio instead, the first action (H1:A1) is no For the purpose of the analysis we use the simplification that longer worth 25% of fitness, but only 1%, which makes the environments are structured such that at least one adaptive search space grow drastically. toolbox would be able to act perfectly in it. Then there are 210 = 1024 situations an individual may encounter dur- Part 2: Probabilities ing its lifetime (see section Environment). Further, let’s as- Given the size of the search space for adaptive toolboxes, sume a simple toolbox of a size 12 = 3 (number of selector what is the probability that a random process–à la mutation cues) + (3 (number of heuristics) × 3 (number of cue/action and crossover–generates a toolbox of a certain level of fit- pairs in each heuristic)). The number of all possible differ- ness? To estimate these probabilities, we considered the fit- ent toolboxes is 1012(cues)×509(actions) = 1027. Let’s con- ness scores of any toolbox with cues and actions at each po- sider a toolbox to be ecologically rational if it performs ac- sition of the toolbox being either correct or incorrect. Only a tions which are more often right than wrong. Given that we correct action can positively contribute to the overall fitness define the fitness score as the proportion of the number of sit- score of the toolbox. If all cues leading to this action are also uations in which a toolbox executes a correct action to the correct, it increases the fitness by the relative probability of total number of all possible situations, the fitness is in a range this action being executed. For example, if H1:A2 is correct 0 to 1 inclusive, and a score of ≥ 0.5 indicates ecological ra- and all of the cues S1, H1:C1 and H1:C2 are as well, the fit- tionality. ness of the toolbox is increased by the corresponding 12.5% Table 1a represents a toolbox of size 12. We set the prob- points (see Table 1). However, if one of the cues leading to ability of a given cue being true or false to 0.5. That means this action is incorrect, it will be executed in half of the cases. that for the first cue of the selector (S1 in the Table 1a) there If two cues are incorrect, only in a quarter of the the cases is a 50% chance that it will be true (and the first heuristic will will the action be executed, and so on. Given the total number be executed) and 50% chance that it will be false (and the of actions and cues, the correct actions only occur in 2%, and next selector (S2) cue will be evaluated). We can now esti- correct cues in 10% of all possible toolboxes. That means that mate the degree to which cues and actions contribute to the toolbox’s fitness as a function of their location in the toolbox. 3 Note that, if for instance, the first heuristic cue (H1:C1, Table 1a) is incorrect (e.g., instead of C1, there is C3; and they are both 2 Here, 50 actions may seem like a lot, but taking into account either true or false), then it can still lead to execution of the first, and the number of different things one can do e.g. with any given object say, correct action (H1:A1). However, in half of the cases, where (grasp it, throw it, squeeze, cut it, etc.) it is actually a moderate those cues are either true and false or false and true, that will not estimate. lead to execution of correct (H1:A1) action. 326 fitness ≥0.1 ≥0.2 ≥0.3 ≥0.4 ≥0.5 ≥0.6 ≥0.7 probability of a toolbox with a given fitness score 0.09 0.008 0.0002 1.2 × 10−6 1.9 × 10−9 2.4 × 10−13 2.6 × 10−18 number of toolboxes with a given fitness score 1.8 × 1026 1.6 × 1025 4.9 × 1023 2.3 × 1021 3.8 × 1018 4 × 1014 5 × 109 total number of possible toolboxes 1953125000000000000000000000 (1027 ) Table 2: Probabilities of randomly generating a toolbox with a certain fitness score. toolboxes with a larger number of incorrect actions and cues Simulations are much more likely to happen. Using these probabilities, we To support our theoretical point using computer simulations computed the probabilities of randomly generating a toolbox we designed an evolutionary algorithm. In our setup, we ran- with a certain level of fitness. For example, the probability of domly generate environments. As in our formal argument, we generating an ecologically rational toolbox (fitness ≥ 0.5) is use the simplification that environments are structured such 1.9 × 10−9 and the probabilities decline super-exponentially that at least one adaptive toolbox would be able to act per- for higher fitness scores (see Table 2). fectly in it. We achieve this by generating the environment with a toolbox. The size of that toolbox is always constant. Part 3: Time The number of selector cues (5), the number of heuristics (5) Evolution operates on a time scale of billions of years. To and the number of cue/action pairs in each heuristic (5) gives estimate how long it would take to generate a toolbox with the total size of the environment 5 + 5 × 5 = 30. Each individ- a certain level of fitness, we assume that the environment is ual in a population is represented as a toolbox as well (the size constant and the average size of the population is 500. Fur- of an individual may vary from generation to generation and thermore, the duration of one generation is assumed to be 15 it is not restricted to ≤ 30). The first generation of individu- years, and mutations happen for almost all individuals in ev- als are randomly generated simple toolboxes. More detailed ery generation. With these values, the expected time to evolve description of our setup is available in online supplementary a toolbox with a 0.5 level of fitness is: materials.4 generation length 15y time0.5 = = ≈ 107 y Results prob × population size 1.9 × 10−9 × 500 We designed three different conditions and ran 20 simulations Here, prob is the probability of generating a toolbox with a for each one. In the first, baseline condition we set the param- certain level of fitness in one generation. Time grows super- eter ‘death rate’ based on evolution science literature (normal exponentially for higher scores of fitness (see Figure 3). This death rate condition). In the second condition (higher death means that given the odds of randomly generating an ecolog- rate), the death rate was increased relative to the normal death ically rational toolbox, a random process is expected to take rate condition. Finally, for the third condition (higher chances on the order of 10 million years to, by accident, produce a of offspring), the death rate was normal, but the growth rate single ecologically rational individual. was increased. Other parameters (e.g., size of the world gen- erating toolbox, mutation rate) are always constant. 1023 Condition 1: normal death rate Time (years) The initial size of a population was 500 and the death rate was 0.0004. The chances of dying was a function of both 107 death rate and fitness. For instance, individuals with a fitness 10−1 score 0 (no correct decisions) had 65% chance of survival and reproduction, individuals with a fitness score 0.2 had 73% 0.1 0.3 0.5 0.7 chance of survival, and individuals with a fitness score 0.5 Fitness had 81% chance of survival (for details, see supplementary materials4 ). Each of the parents always generates at least one Figure 3: Time (in years) required to generate toolboxes with child, and the probability of getting a second child is 33.3% a certain level of fitness. per individual. This number creates the minimal conditions for a population to be able to grow. With this numerical examples we wish to illustrate the im- Under this condition 0% of the populations survived. Ta- plausibility that evolution would generate ecologically ratio- ble 3 represents an overview of all results, and Figure 4 shows nal toolboxes. Even though adaptive toolboxes have appar- the variation in fitness of populations of toolboxes throughout ently simple structures, they are still characterized by ex- the different generations. As the Table 3 shows, fitness of the tremely many degrees of freedom. As we have shown, this populations is overall remarkably poor. The average fitness makes it highly improbable that an evolutionary adaption pro- cess would endow them with ecological rationality. 4 http://www.dcc.ru.nl/ irisvr/papers/suppl15.pdf ˜ 327 was 0.028, which is considerably lower than the 0.5 thresh- Condition 1 Condition 2 Condition 3 old that we defined for ecologically rational toolboxes. The % of Survival: 0% 20% 80% fitness of the ‘best toolbox (from each generation) oscillates Average FitS : – 0.071 0.044 in the range [0.1, 0.4]. Average FitD : 0.028 0.031 0.027 All simulations ended far before one thousand generations, Total average: 0.028 0.041 0.041 often even before a hundred. All of the above indicate, that toolboxes perform poorly and do not improve with time. We Table 3: Results from the simulations for the three differ- explored two parameters which potentially could have influ- ent conditions (1: normal death rate; 2: higher death rate; 3: ence the results. First, we reasoned that this effect might be higher chances of offspring). Starting from the top, the rows due to a relatively low death rate. Such a low death rate (i) show: percentage of surviving populations for every condi- may ensure the survival and possibility of reproduction of in- tion; the average fitness score (FitS ) for a set of surviving dividuals with lower fitness and (ii) imposes a lower pressure populations per condition; the average fitness score (FitD ) for to select better toolboxes. Second, we explored the possibility a set of dying out populations per condition; the total average of giving toolboxes more offspring. This change may lead to of a fitness score per condition. more populations surviving but we would not expect it to im- prove the overall individuals fitness. To test these predictions we ran two simulation studies, Conditions 2 and 3. ‘ecological rationality’ (defined as the ability to make choices that are more often right than wrong; i.e. ≥50% correct). Condition 2: higher death rate In our simulation maximum fitness of populations hovered In this condition the death rate was increased (p = 0.00045; around 0.2 (20% correct decisions) and never got anywhere we opted for this relatively small increase in death rate, be- close to 0.5, let alone anything higher than that. The simu- cause a higher death rate would not afford successful runs, lation results align well with our formal derivations: the ex- because none of individuals would survive the first survival- pected number of generations needed to produce a toolbox selection phase). In total, 20% of the simulations ended with grows exponentially. That means that even for only 10 pos- a surviving population (Figure 4). The average performance sible cues and 50 possible actions the expected number of of the surviving populations is 0.071, and the average perfor- generations needed to produce at least one toolbox in the en- mance for the dying out populations is 0.031. In order to cal- tire population with a fitness of at least 0.5 is 2,000,000 gen- culate the average performance scores, we considered results erations. For more possible cues or actions, the number of from all the runs of simulations for surviving populations and expected generations needed to produce at least one ecologi- all for the dying out populations separately (for a given con- cally rational toolbox is even vastly larger. dition), taking into account all the possible individual scores Crucially, we refer here to the expected number of genera- per every generation. Comparing the fitness in this Condi- tions for producing a single toolbox with the feature of ‘eco- tion 2 with the fitness from Condition 1, it becomes clear that logical rationality’. Even if evolution would beat all odds even if the higher pressure does improve performance of the and such an individual would be generated, the changes of toolboxes, as we had expected, the improvement is of a very its existence leading to a population with that feature are small magnitude and does not bring the toolboxes anywhere nanoscopically small. The reason is that toolboxes can sur- closer to the 0.5 fitness. vive with much lower fitness, and the chances of mutation and crossover leading to fitness below 0.2 is very high. With Condition 3: higher chances of offspring every new generation mutation and crossover occur, leading In this condition, the probability of generating a second child to a high probability that even if there is one ecologically ra- was increased to 47.4% per an individual. In total, 80% of tional individual in the pool that its offspring will be non- the populations survived. As expected this survival rate was ecologically rational individuals that can again survive and higher than in Condition 1 and 2. The average performance procreate. of the surviving populations is 0.044, and the average perfor- Does this mean that the adaptive toolbox account is im- mance for the dying out populations is 0.027. In sum, the sim- plausible as an account of resource-bounded (human) deci- ulations in Condition 3 show that a larger growth rate leads to sion making? Certainly not. Our findings do not rule out larger populations, but it does not make the individuals more that adaptive toolboxes could be produced by ontogenic pro- ecologically rational. cesses (learning and development), or even ontogenetic and phylogenetic processes combined (i.e., evolution could have Discussion produced those learning mechanisms that can produce adap- Using both formal argument and computer simulation, we tive toolboxes on a developmental time scale). After all, on- have demonstrated the implausibility that phylogenetic pro- togenetic processes–unlike phylogenetic processes–are able cesses (i.e., evolution) alone would ever produce ecologi- to more actively search the space of possible parameters set- cally rational adaptive toolboxes. Our simulations showed tings, e.g. by building a model of the environment and using that populations of toolboxes that are ‘good enough’ to sur- that model to guide the search in a way that ensures ecologi- vive can evolve without these toolboxes showing any signs of cally rationality. However, in such a case it seems that one has 328 0.4 0.4 0.4 Fitness Fitness Fitness 0.2 0.2 0.2 0 0 0 50 50 100 150 200 150 250 350 20 40 60 80 Generation Generation Generation (a) Example of dying out population: (b) Example of dying out population: (c) Example of dying out population: normal death rate condition higher death rate condition higher chances of offspring condition 0.4 0.4 Fitness Fitness 0.2 0.2 0 0 50 100 150 300 600 900 Generation Generation (d) Example of a surviving population: (e) Example of a surviving population: higher death rate condition higher chances of offspring condition Figure 4: Examples of the simulations of dying out populations (a-c) and surviving populations (d-e). The plots show the changes of fitness over the time of many generations, including scores from the best ( ), worst ( ) and average ( ) fitness per generation. For the normal death rate condition there is no surviving population. to use a non-frugal learning mechanism to explain the emer- Gigerenzer, G. (2004). Striking a blow for sanity in theo- gences of adaptive toolboxes of fast and frugal heuristics. Re- ries of rationality. Models of a man: Essays in memory of solving this tension seems an important target for future re- Herbert A. Simon, 389–409. search in the area of resource-bounded decision making. Gigerenzer, G., Gaissmaier, W. (2011). Heuristic decision making. Annual Review of Psychology, 62:451–482. Acknowledgments Gigerenzer, G., Sturm, T. (2012). How (far) can rationality We would like to thank the Computational Cognitive Sci- be naturalized? Synthese, 187(1):243–268. ence group and in particular Mark Blokpoel for helpful in- Gigerenzer, G., Todd, P. M. (1999). 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