The Dynamics of Trustworthiness Among the Few

The Japanese Economic Review Vol. 53, No. 4, December 2002 THE DYNAMICS OF TRUSTWORTHINESS AMONG THE FEW* By SANDRA GÜTH , WERNER GÜTHà and HARTMUT KLIEMTx University of Bielefeld àMax Planck Institute xUniversity of Duisburg Conventional stochastic models of evolutionary processes with infinitely many agents are deterministic models in disguise. Only finite population models become truly stochastic. Therefore this paper focuses on an indirect evolutionary model of pairwise interaction in a pool of three (corresponding to analysing oligopolies in terms of duopoly markets). The outcomes of the process over the long haul are characterized by the stationary distribution of the underlying Markov process. Our example indicates that intermediate cases cannot be seen as convex combinations of the two polar non-stochastic cases of two or infinitely many individuals. JEL Classification Numbers: C72, C73. 1. Introduction In advanced species who command some foresight and understanding, evolution seems indirect in its operation. In such species genotypes do not directly determine behaviour. Preference-based conscious decision-making intervenes and influences behavioural choices. A complex interaction between a ‘‘surface structure’’ of rational decision-making and a ‘‘deep structure’’ of genotypical evolution emerges. We believe that the dynamics of the process can best be understood in terms of an ‘‘indirect evolutionary’’ approach that takes into account the interaction between surface and deep structure. (For a more detailed general characterization of the indirect evolutionary approach, see Güth and Kliemt, 1998.) According to the indirect evolutionary approach, subjective motivational factors determine the choice of actions. The relative success of the actions chosen and thus of the individuals choosing them is measured in objective terms. Objective success influences the distribution of motivational types (which need not be genotypically fixed but could also be determined by adaptation in a learning process). This in turn feeds back on the subjective decision situation or on the decision situation as perceived by the decision-makers, and thereby influences their rational decision-making. In this paper’s evolutionary analysis of a basic trust game, we shall deviate from conventional direct evolutionary approaches (for a survey see Hammerstein and Selten, 1994), not only by adopting an indirect evolutionary perspective, but also by giving up the conventional assumption of an infinite pool of interaction partners who are randomly matched. (For another exception, see e.g. Huck and Oechsler, 1998.) Within an infinite pool, knowledge of a type of individual cannot influence estimates of the probability of encountering any such type. Assuming that there are only finitely many individuals whose knowledge of their own type influences their expectations about their co-players’ types, the problem becomes truly stochastic. * We gratefully acknowledge helpful and constructive comments of an anonymous referee. – 369 –
Japanese Economic Association 2002. The Japanese Economic Review Subsequently we shall apply the indirect evolutionary approach to a situation involving just three individuals. We choose a pool of three because this is the smallest pool in which individuals can still form different pairs of players interacting in the dyade. In Section 2 the basic dyadic trust interactions are characterized. Section 3 describes and solves the evolutionary game of trust in groups of three, while Section 4 discusses aspects of the dynamics of the process. Section 5 concludes. 2. The basic interactions In this section we introduce the basic game of trust when type information is common knowledge (Section 2.1). We then turn to the polar case in which the second mover’s type is determined in a random move whose probability distribution is commonly known while its result is revealed only to the second mover (Section 2.2). Finally, we sketch the most interesting intermediate case, in which costly but not completely reliable type signals are available (Section 2.3). 2.1 The game model of the basic interaction The trust predicament cannot be captured completely in objective terms by the material payoffs (i.e. monetary income, number of offspring, etc.) accruing to the actors in a specific situation: subjective perceptions of the objective action situation must also be taken into account. To incorporate the subjective and the objective levels in the most simple way in one model, we assume (see Figure 1) that the players’ preferences can be represented by the objective payoffs of their choices. Assuming risk neutrality, we can use a utility function representing preferences by adopting the same numerical values as the objective payoff function over the set of outcomes and all their convex combinations. By construction, all these subjective utilities correspond to objective payoffs. But they are modified by an additional, purely subjective payoff component ‘‘m’’ which is not related to some objective payoff. The relation between the purely subjective and the objective dimension (which is also mirrored in the subjective payoffs) is essential for an adequate model of trust problems. The resulting simple game of trust is shown in Figure 1. Player 1 decides between nontrusting (N) and trusting (T) behaviour. If player 1 decides on T, player 2 can respond by Figure 1. – 370 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few either exploitation (E) or reward (R). If e þ m > 1, strategic choice after T is dictated by the extrinsic motivational factors. Overt behaviour unfolds as if no factor m were present. Only if e þ m < 1 will the intrinsic motivational factor m influence choice behaviour. For obvious reasons, we say that a player is endowed with a conscience if m < 1  e and that she lacks a conscience if m > 1  e. Since the strategic properties of the game depend solely on which of the inequalities applies, we merely consider two specific parameter values. In case m > 1  e (when a conscience is lacking) we focus on m ¼ 0. In case 1  e > m (when a conscience affects behaviour), the focus is on some m with ð1Þ ð0 >Þ 1  e > m: For m > 1  e, the choice of E will be dominant for player 2, and for 1  e > m the choice of R will be dominant for her. Since, in the game of Figure 1, the value of m is common knowledge, rationality dictates (N, E) for m ¼ 0 and (T, R) for m ¼ m. 2.2 The basic game when type information is completely private If the first-mover merely knows the general probability p that his co-player is an m-type and the probability ð1  pÞ that she is a 0-type, the following game emerges (see Figure 2). The first-mover anticipates that the second-mover decides according to her type, i.e. the m-type chooses R and the 0-type E. But when making his own decision, the first-mover does not know his co-player’s type. The initial fictitious chance move captures this uncertainty of the first-mover about the second-mover’s type. The first-mover’s own type is irrelevant for his decision. Therefore the first-moving player will choose to act the following way: if p þ ð1  pÞ0 > s then T is chosen; if p þ ð1  pÞ0 < s, then N is chosen. In sum: • • If the second-mover is a trustworthy m-type, then for p > s the solution is (T, R) while it is (N, R) for p < s. If the second-mover is an untrustworthy 0-type, then the solution is (T, E) for p > s and (N, E) for p < s. m m Figure 2. – 371 –
Japanese Economic Association 2002. The Japanese Economic Review 2.3 The basic game when some type information is revealed So far the reasoning has been grounded on the premise that no specific information about the outcome of the chance move is available. This is merely a limiting or extreme case of a situation in which the first-mover can acquire some type information at some cost. The essential characteristics of such a more general situation are easily modelled. Assume that before deciding between N and T the first-mover must choose between S and S¢. Choosing S¢ amounts to playing the game in which information about the secondmover’s type is purely private. Choosing S amounts to investing cost C in a detection technology which provides a stochastic signal x. A trustworthy m-type is indicated by x ¼ ‘‘m’’. Applied to a trustworthy type, the detection technology signals the truth by x ¼ ‘‘m’’ with reliability l while leading astray by x ¼ ‘‘0’’ with probability 1  l. Similarly an untrustworthy 0-type will be indicated correctly by x ¼ ‘‘0’’ with reliability l while the misleading signal x ¼ ‘‘m’’ is provided with probability 1  l if a 0-type is encountered. In sum, the signal x ¼ ‘‘m’’ occurs with probability  ð1  l Þ if m ¼ 0 ; ð2Þ l; if m ¼ m the signal x ¼ ‘‘0’’ occurs with probability   l if m ¼ 0 ð1  lÞ if m ¼ m ð3Þ with 1  l > 1=2 and 1  l > 1=2. After receiving the signal x ¼ ‘‘m’’, the first-mover’s conditional probability of dealing with a trustworthy m-type player in the second-mover role is pðm=‘m’Þ ¼ pl : pl þ ð1  pÞð1  l Þ ð4Þ After receiving the signal ‘‘0’’ the conditional probability of dealing with a trustworthy m-type second-mover is pðm=‘0’Þ ¼ pð1  lÞ : pð1  lÞ þ ð1  pÞl ð5Þ For all p 2 ð0; 1Þ, i.e. whenever type information matters, both conditional probabilities are well defined. Assuming that the second-mover will always choose according to type, we can leave out of account the dominated moves E of the m-type as well as R of the 0-type of player 2 and represent the game with detection technology as shown in Figure 3. 3. The evolutionary game in the triad In this section we introduce the basic evolutionary game for three players (Section 3.1) and determine its solutions (Section 3.2). On the basis of the first two steps we then characterize reproductive success (Section 3.3). – 372 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few m m Figure 3. 3.1 The evolutionary game In previous analyses (see e.g. Güth and Kliemt, 1994) we imagined an infinite number of matches in which individuals would play the game of Figure 3. The game was to be played after random assignment of co-players from a pool of potential interaction partners as well as a random assignment of first- and second-mover roles, respectively. Owing to the infinite number of individuals involved, information about his own type could not influence any player’s assessment of probabilities. Therefore players in first-mover roles made their decisions to invest or not to invest in detection technology (the choice between S and S¢, respectively), and whether or not to show trust (the choice between T and N), independently of their own type on the basis of rational expectations. Accordingly, all subjective probabilities for facing a trustworthy second-mover were given by the true population share p of trustworthy individuals in the population. For anonymous interaction in a very large population, it would seem quite natural to assume that players are aware of the general share of trustworthy individuals but cannot judge the trustworthiness of any specific individual. In case of an infinite number of matches, although for 0 < p < 1 the outcome of any individual play of the game depended on chance, the average success of alternative player types did not. The assumption of an infinite population rendered the evolutionary dynamics basically non-stochastic. In the present paper we give up the conventional assumption and consider a finite population. In fact, as has already been indicated in the Introduction, we shall go to the other extreme and focus on populations of three individuals, the minimal size in which alternative matches are still possible. Denoting the individuals by 1, 2 and 3, we get three matches f1; 2g; f2; 3g and f1; 3g, which in turn lead each to two assignments of players to the roles of first- and second-mover, respectively. Assuming an infinite number of rounds of play of the game of Figure 3, on each round of play each individual is going to play with each other individual in both roles. That is, on – 373 –
Japanese Economic Association 2002. The Japanese Economic Review each round of play, each player will simultaneously play four of the games depicted in Figure 3. Moreover, in each of the six plays of the game of Figure 3, the number M O 3 of trustworthy individuals in the population of three individuals is assumed to be common knowledge among the players. For a finite population, this corresponds to the (rational expectations) assumption for the infinite population. Every player’s prior probability for encountering a trustworthy second-mover is determined by the true population share p of trustworthy types in the population. Starting from a true assessment p of the share of trustworthy individuals, and aware of their own type, players in the special case N ¼ 3 have to distinguish situations • • • where others can be trusted since both are of the trustworthy type; where others cannot be trusted since both are untrustworthy; where both are possible since one of both types is present. These three possibilities represent the main qualitative aspects of a social environment which can be completely reliable, completely unreliable, or mixed. Finally, on each round of play, a second-mover always sends the same signal to both firstmovers with whom she is matched; i.e., randomness results from a noisy sender technology. Otherwise, in each of her interactions, each of the three players knows only her own type. 3.2 Solution of the basic game Consider the game of Figure 3 under the conditions described in Section 3.1. In view of their knowledge of M, in each of the six simultaneous games players in the first-mover role know the probability p of being matched with a trustworthy individual. Knowing M and her own type, a player who is assigned the first-mover role faces one of the following three cases: • • • Case 1: both other players are of type m; Case 2: both other players are of type 0; Case 3: one other player is of type m and one of 0. Whichever of the two other individuals is assigned to the second-mover role is trustworthy in the first and untrustworthy in the second case. This is known among the players. Therefore in cases 1 and 2 a first-mover will not invest C in a technology for detecting the type of his co-player. Only in the third case does the first-mover not know his co-player’s type. He knows that in each of the simultaneous plays, with probability 1/2, the secondmover will be of the trustworthy m-type and with probability 1/2 she will be of the untrustworthy 0-type. Obviously, only in the ‘‘mixed’’ case 3 can the information technology be useful at all for the first-mover. Unless otherwise indicated, in this section we shall focus solely on this case. In comparing the expectations after S and S0 , recall first that by assumption l > 1=2 and  l > 1=2. From this, along with (4) and (5), one gets, for all p 2 ð0; 1Þ, pðm=‘m’Þ > p > pðm=‘0’Þ; ð6Þ or, since we are dealing with the mixed case with some private information only, 1 pðm=‘m’Þ > > pðm=‘0’Þ: ð60 Þ 2 Moreover, since 12 > s, we must have either pðm=‘m’Þ > 12 > s  pðm=‘0’Þ, or pðm=‘m’Þ > 12 > pðm=‘0’Þ > s. – 374 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few The payoff expectation after the choice of S0 and a subsequent choice of T in the tree of Figure 3 is p þ ð1  pÞ0 ¼ p. The choice of T is therefore rational after S0 if p > s: ð7Þ Since by assumption s < 12 and p ¼ 12, relation (7) always holds good in the mixed case. The potential gains from cooperation outweigh its risks even if, in the absence of type detection, exploitation occurs frequently. Therefore after S0 the alternative T will be chosen and yield p. The payoff expectation after S depends on the anticipated choices and expectations after alternative signals. After receiving the signal x ¼ ‘‘m’’ the first-mover’s expectation is pðm=‘m’Þ  C after playing T and s  C after choosing N. From ð60 Þ and s < 12 ð8Þ pðm=‘m’Þ > s: Consequently T is chosen after x ¼ ‘‘m’’. Neglecting the limiting cases of indifference, after x ¼ ‘‘0’’ the first-mover will choose T if pðm=‘0’Þ > s ð9Þ and the first-mover will choose N after x ¼ ‘‘0’’ if s > pðm=‘0’Þ: ð10Þ From the reasoning leading to (8), (9) and (10) we derive the payoff expectation after S which is  pC if pðm=‘0’Þ > s ð11Þ pl þ pð1  lÞs þ ð1  pÞl s  C if pðm=‘m’Þ > s > pðm=‘0’Þ If pðm=‘0’Þ > s applies the first-mover will not decide on S. For in this case he would choose T regardless. The information technology would therefore be useless. Thus, it can only be rational to decide on S if pðm=‘m’Þ > s > pðm=‘0’Þ. This necessary condition will become sufficient if the expectation pl þ pð1  lÞs þ ð1  pÞl s  C after S is larger than the expectation p ¼ 12 after S0 as determined in (7), that is, if   1 1 1  1 ð12Þ l þ ð1  lÞs þ 1  l s  C > 2 2 2 2 or s> 2C þ 1  l : 1 þ l  l s> 2C þ 1  l 1 þ l  l ð120 Þ Since 12 > s > 0, the relation implies 1 2C þ 1  l > ; 2 1 þ l  l – 375 –
Japanese Economic Association 2002. The Japanese Economic Review which in turn is equivalent to l þ l  1 > C: ð13Þ 4 Owing to l; l > 1=2, the left-hand side of (13) is positive. Whenever (13) is fulfilled, there is a generic interval of payoff parameters s satisfying 1=2 > s > 0 and inequality (12). Subsequently we assume throughout that ð120 Þ (which is equivalent to (12)) applies, and therefore that the condition rendering the choice of S rational is fulfilled. 3.3 Reproductive success In evolutionary biology it is quite common to identify material with reproductive success. The number of offspring is assumed to be positively and directly correlated with the availability of resources like food. Clearly, such an argument did apply to humankind in very early history. However, in our world the correlation between measures of material success and the number of offspring is clearly not so close that all interesting evolutionary issues can be dealt with in terms of offspring. Even if, eventually, relative numbers of biologically defined types (biological offspring or shares in a gene pool) may be all that matters, the dynamics of interaction of culturally defined types should be studied at least as an intervening process in its own right. Since there is an intervening process that may not be directly reduced to an underlying biological one an indirect approach seems promising. Still, whether such indirect evolutionary approaches to cultural dynamics are appropriate and go beyond spurious analogies clearly depends on how close the analogies between the biological and cultural spheres are. Not every dynamic process in which some phenomenon or other enfolds through time may properly be called ‘‘evolutionary’’. Some form of material success that feeds back on relative shares of some fixed entity or cultural type must be present. As far as the latter is concerned, the assumption that diffusion of cultural types depends on their material success makes good sense. For instance, market participants will tend to copy the policy of firms with higher material success or, for that matter, with higher profits. Moreover, more profitable firms will be comparatively more successful in founding new firms or in taking over established ones. It is not far fetched to assume that such forms of material success will determine the relative market share of alternative ‘‘types’’ of firms. These cultural types of firms are characterized by their internal procedures, internal constitution, routines, their corporate culture, etc. Such internal ‘‘cultural programmes’’ form an analogue to fixed biological ‘‘programmes’’ on which comparative success as measured in material terms can feed back selectively (see Nelson and Winter, 1982). This suggests that an evolutionary analysis of the dynamics of trustworthiness among the few need not be confined to studying the population dynamics of offspring of biologically defined types: it may also be based on culturally defined types. For reducing the number of individuals so as to create a truly stochastic setting, it does not matter whether we think of the underlying process in terms of cultural or biological evolution. As long as there are sufficiently close analogues to differential reproductive success of some ‘‘fixed entity’’, the argument should provide some insights into the differences between the dynamics of trustworthiness between few as opposed to many interacting individuals. In view of the solution behaviour derived in the previous section, we can now define a measure of relative or differential reproductive success D. Alternative values of the number – 376 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few M 2 f0; 1; 2; 3g of trustworthy individuals in the population DðMÞ measure the differential reproductive success of an m-type compared with a 0-type. Since, for the two possible monomorphic populations characterized by M ¼ 0 and M ¼ 3, the notion of differences in reproductive success between alternative types does not make sense, we can restrict our attention to the cases M ¼ 1 and M ¼ 2. If M ¼ 1, there exists exactly one m-type individual. The difference between the material success of the trustworthy individual and the mean material success of the two untrustworthy individuals is Dð1Þ. If M ¼ 2 there exists exactly one untrustworthy individual. The difference between the mean material payoffs of the two trustworthy players minus the material success of the untrustworthy player is defined as Dð2Þ. In both cases M ¼ 1 and M ¼ 2, whether eventually DðMÞ > 0 or DðMÞ < 0 applies depends stochastically on the chance events that determine the signal combination prevailing among the players. For this reason, the stochastic adjustment of M can be characterized only by a somewhat lengthy analysis involving several case distinctions. We next turn to this task for M ¼ 1, while relegating the analogous analysis for M ¼ 2 to an appendix. Since all players know their own type, the single trustworthy m-type player, to whom we shall refer as m1 , can infer, from the private information about his own type along with the commonly known population composition, that he will be paired with an untrustworthy second-mover. Therefore for M ¼ 1 a trustworthy player in the first-mover role will decide on S0 and N. Obviously, the material success of trustworthy m1 will be s and that of his untrustworthy co-players 0 in the two interactions in which the single trustworthy player is the first-mover. We can characterize the success of untrustworthy players in the first-mover role as well. Both of the untrustworthy players know that, with probability 1/2, a trustworthy co-player is assigned to them. Since we assume that ð120 Þ applies, both of the 0-type first-movers choose S and rational behaviour will vary according to the signal received; i.e., after a trustworthy second-mover has been signalled, the first-mover will choose T and he will choose N after receiving the signal x ¼ ‘‘0’’. The objective payoffs that accrue to the players are contingent on the alternative signals indicating the second-mover type. Referring to the two 0-type players as player 01 and player 02, respectively, we can sum up the preceding considerations for the six matches (‘‘each against each in each role’’) simultaneously played on each round of the overall evolutionary game in two tables. Table 1 lists objective payoffs, signals and signal probabilities. Table 2 shows the success of the players aggregated over the six matches on each round of play as a function of the three signals sent by each of the three players on each round of play. In Table 2 the signal originating from the one trustworthy individual is listed first, that from the first untrustworthy second and that from the second untrustworthy third. In interpreting the entries of Table 2, it should be recalled first that signals are relevant only for 0-type players in first-mover roles. Under our parameter constellation, 0-type players invest in the detection technology at objective cost C. They follow the signal by choosing T if a trustworthy type is signalled and N if not. Players of the trustworthy m-type choose N without investing in detection technology. With respect to signals, therefore, only the matches ð01 ; m1 Þ; ð01 ; 02 Þ and ð02 ; m1 Þ; ð02 ; 01 Þ must be considered. Signals can originate either from an m-type or from a 0-type player, and they can be correct or incorrect. For each 0-type player in the first-mover role, the probability of receiving two correct signals is ll ; the probability that the signal from the m-type is correct while the signal from the 0-type is deceptive is lð1  l Þ; the probability that the signal from the m-type is deceptive while that from the 0-type is correct is ð1  lÞl ; and, finally, the probability of two faulty signals is ð1  lÞð1  l Þ. – 377 –
Japanese Economic Association 2002. The Japanese Economic Review Table 1 The Objective Success of the Three Individuals in the Six Matches of the Game in Case M ¼ 1 Matches and information about the second-mover Objective payoff for players of both types First, second-mover in matches Prob., signal for second-mover type m1 01 02 1, ‘‘0’’ 1, ‘‘0’’ l; ‘‘m’’ 1  l; ‘‘0’’ l ; ‘‘0’’ 1  l ; ‘‘m’’ l; ‘‘m’’ 1  l; ‘‘0’’ l ; ‘‘0’’ 1  l ; ‘‘m’’ s s 1 0 0 0 1 0 0 0 0 0 1C sC sC C 0 0 0 e 0 0 0 0 0 e 1C sC sC C m1 ; 0 1 m1 ; 0 2 01 ; m1 01 ; m1 01 ; 02 01 ; 02 02 ; m1 02 ; m1 02 ; 01 02 ; 01 The eight possible signal combinations emerge with probabilities as indicated in Table 2. To characterize the stochastic adjustment of M ¼ 1, we must derive the probabilities P ½Dð1Þ > 0 for an increase and P ½Dð1Þ < 0 for a decrease of M depending on the probability of each of the eight signal combinations listed in Table 2. These combinations can be grouped and analysed as follows: ‘‘m’’; ‘‘0’’; ‘‘0’’ and ‘‘0’’; ‘‘0’’; ‘‘0’’; C > 0 ) Dð1Þ > 0 Moreover, for ‘‘0’’; ‘‘m’’; ‘‘m’’; Dð1Þ < 0. To see this, note first that l þ l < 2 and relation ðl þ l  1Þ=4 > C (which follows from the assumption that inequality ð120 Þ holds good) together imply that 1=4 > C. Since by assumption s < 1=2, we also can immediately infer from e > 1 that e  s > 2C and that Dð1Þ ¼ 0 does not apply for ‘‘0’’; ‘‘m’’; ‘‘m’’. In the remaining cases Dð1Þ > 0 amounts to es 1þs  ; for ‘m’; ‘m’; ‘0’ and ‘m’; ‘0’; ‘m’; 4 2 es 1þs C>  ; for ‘m’; ‘m’; ‘m’; 2 2 es ; for ‘0’; ‘0’; ‘m’ and ‘0’; ‘m’; ‘0’: C> 4 C> Obviously (15) ) (14), and (16) ) (14). Moreover, es 1þs es  > ; if e > 3s þ 2 2 4 2 4 and thus es 1þs es  < 4 2 4 es 1þs es 1þs  <  e < 3s þ 2 ) 4 2 2 2 e > 3s þ 2 ) – 378 –
Japanese Economic Association 2002. es 1þs  2 2 es < 4 < es 2 es < : 2 < ð14Þ ð15Þ ð16Þ Signal comb. and probability of combination – 379 – ‘‘m’’; ‘‘0’’; ‘‘0’’ ll l ‘‘m’’; ‘‘0’’; ‘‘m’’ ll ð1  l Þ ‘‘m’’; ‘‘m’’; ‘‘0’’ lð1  l Þl ‘‘m’’; ‘‘m’’; ‘‘m’’ lð1  l Þð1  l Þ ‘‘0’’; ‘‘0’’; ‘‘0’’ ð1  lÞl l ‘‘0’’; ‘‘0’’; ‘‘m’’ ð1  lÞl ð1  l Þ ‘‘0’’; ‘‘m’’; ‘‘0’’ ð1  lÞð1  l Þl ‘‘0’’; ‘‘m’’; ‘‘m’’ ð1  lÞð1  l Þð1  l Þ m1 01 02 Mean 0 Dð1Þ 2ðs þ 1Þ 2ðs þ 1Þ 2ðs þ 1Þ 2ðs þ 1Þ 2s 2s 2s 2s s þ 1  2C 1  2C 1 þ s þ e  2C 1  2C þ e 2ðs  CÞ s  2C 2ðs  CÞ þ e s  2C þ e s þ 1  2C 1 þ s þ e  2C 1  2C 1  2C þ e 2ðs  CÞ 2ðs  CÞ þ e s  2C s  2C þ e s þ 1  2C 1 þ eþs 2  2C 1 þ eþs 2  2C 1  2C þ e 2ðs  CÞ eþ3s 2  2C eþ3s 2  2C s  2C þ e s þ 1 þ 2C þ 1 þ 2C þ 1 þ 2C 2s þ 1  e þ 2C 2C se 2 þ 2C se 2 þ 2C s þ 2C  e 3se 2 3se 2
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few Table 2 The Objective Success of the Three Individuals in the Six Matches of the Game for All Chance Events (Signals Received) in Case M ¼ 1, and the Differential Success Dð1Þ; Signals Ordered as in ðm1 ; 01 ; 02 Þ The Japanese Economic Review Since we know already that C  ðe  sÞ=2 cannot be fulfilled we have to consider five conceivable locations for C: es 1þs  4  2  es es 1þs ;  Case 2: C > max 4 2 2 Case 1: C    es es 1þs ;  4 2 2 es es 1þs Case 4a: e > 3s þ 2 and <C  4 2 2 es 1þs es  <C Case 4b: e < 3s þ 2 and 2 2 4 In Table 2 a positive value Dð1Þ > 0 indicates that, under the specific constellation, the mean fitness of m-types is higher than that of 0-types and vice versa for Dð1Þ < 0. Bearing in mind that for the signal constellations ‘‘m’’; ‘‘0’’; ‘‘0’’ and ‘‘0’’, ‘‘0’’, ‘‘0’’ we have Dð1Þ > 0, while for ‘‘0’’; ‘‘m’’; ‘‘m’’ the relation Dð1Þ < 0 always holds good, the probabilities P ½Dð1Þ > 0 can be easily determined for all five cases. 1þs Case 3: C > es 4  2 and C  min Case 1: C is so low that the 0-types, even though investing in specific type information at cost C, out-compete the non-investing m-type for all signal constellations except those where the trustworthy has the edge over the untrustworthy type for any positive C: P ½Dð1Þ > 0 ¼ ll l þ ð1  lÞl l ¼ l l : Case 2: C is so high that the investing 0-types are at a disadvantage compared with the m-type, except for the signal constellation in which the 0-types fare better than the m-type regardless: P ½Dð1Þ > 0 ¼ 1  ð1  lÞð1  l Þð1  l Þ: Case 3: The probability that the m-type fares better is basically that of case 1 plus the probability of the signal constellations ‘‘m’’; ‘‘m’’; ‘‘0’’ and ‘‘m’’; ‘‘0’’; ‘‘m’’ (as in (14)): P ½Dð1Þ > 0 ¼ l l þ 2lð1  l Þl : Case 4a: The probability that the m-type fares better is basically that of case 2 minus the probability of the signal constellation ‘‘m’’; ‘‘m’’; ‘‘m’’ (since (15) is violated): P ½Dð1Þ > 0 ¼ 1  ð1  lÞð1  l Þð1  l Þ  lð1  l Þð1  l Þ ¼ 1  ð1  l Þð1  l Þ: Case 4b: The probability that the m-type fares better is basically that of case 2 minus the probability of the signal constellations ‘‘0’’, ‘‘0’’, ‘‘m’’ and ‘‘0’’, ‘‘m’’, ‘‘0’’ (since (16) is violated): P ½Dð1Þ > 0 ¼ 1  ð1  lÞð1  l Þð1  l Þ  2ð1  lÞð1  l Þl ¼ 1  ð1  lÞð1  l l Þ: – 380 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few 4. Evolutionary dynamics Evolution takes place whenever the population composition changes. This can occur either as a change in the relative shares of previously existing traits or through the emergence of an entirely new trait by a mutation or innovation. A trait can be introduced into the population by a ‘‘mutation as innovation’’ if either M ¼ 0 or M ¼ 3 obtains. If M ¼ 1 or M ¼ 2, evolution can proceed by transition to another population composition without ‘‘mutation as innovation’’. Of course, only in the latter case can comparative advantage and selection influence the evolutionary process. In conventional models of evolutionary processes the focus is on changes on the level of individuals. It is assumed in general that the random variables describing the probability of mutations in individuals are independent and identically distributed. This implies that the population composition may change simultaneously by any number of individuals at any given time. (For an instructive example of the approach, see Kandori et al., 1993.) Since every population composition can emerge from any other one – though, perhaps, with an extremely low probability – the existence of a unique stationary equilibrium is guaranteed. It is this, rather than systematic reasoning to the effect that models which allow any number of mutations are superior representations of the facts, that seems to be responsible for the popularity of models in which all population compositions can emerge from any existing one. We shall adopt a different approach here. We focus on the population composition as such and thereby on the collective as opposed to the individual level. We assume that at each stage of the evolutionary process the number of trustworthy individuals in the population can change by only one individual at a time. This seems justified as an approximately realistic abstraction if the time period is so short that the probability of two or more simultaneous changes in the population composition can be neglected. (For a similar view, see Güth and Ludwig, 1998.) To speak of variables that describe properties of a whole group may seem a dubious approach to the true methodological individualist. However, relating the collective level at which the population composition is considered to the level of single individuals is fairly straightforward. ‘‘Reduction’’ of the higher to the lower level is possible in principle if we imagine a two-stage process in which, first, the individual who is to change is randomly chosen, and then, with some probability, the change that is to take place determined. We shall first specify the transition matrix under the assumption that not more than one change is possible (Section 4.1). We then take a somewhat brief general look at the evolutionary process at large (Section 4.2) and discuss a specific example as an illustration of what may happen over the long haul (Section 4.3). 4.1 Specifying the transition matrix Let e; e > 0, denote the probability that exactly one mutant emerges in a monomorphic population and thereby transforms it into a bimorphic one. If M ¼ 0, then the population share of trustworthy m individuals increases with probability e. If M ¼ 3, the population share M of trustworthy m individuals can only decrease. This will happen with the same probability e. If M ¼ 1 or M ¼ 2, the number M of trustworthy m individuals can either increase or decrease by one. With probability d, 1 > d > e, the bimorphic population composition will change, and with ð1  dÞ it will remain unchanged. The direction of change depends on the – 381 –
Japanese Economic Association 2002. The Japanese Economic Review Table 3 Transition Matrix Based on the Mutation Probabilities e for Monomorphic and d for Bimorphic Populations where 1 ¼ dþ ðMÞ þ d ðMÞ 0 1 2 3 1e dd ð1Þ 0 0 e 1d dd ð2Þ 0 0 ddþ ð1Þ 1d e 0 0 ddþ ð2Þ 1e M tþ1 Mt 0 1 2 3 signal combination that emerges. With signal-dependent probabilities the trustworthy will fare better, DðMÞ > 0, or no better, DðMÞ ¼ 0, than the untrustworthy. Neglecting its size, we take only the sign of DðMÞ into account. For convenience, we set dþ ðMÞ :¼ P ½DðMÞ > 0 and d ðMÞ :¼ P ½DðMÞ ¼ 0 for M ¼ 1; 2. Assuming stochastic independence yields overall probabilities ddþ ðMÞ for an increase and dd ðMÞ for a decrease of M. Let M t 2 f0; 1; 2; 3g denote the number of trustworthy m-individuals at time t  0. (M t Þ, t 2 N, defines a Markov process with transition matrix (see Table 3). 4.2 Some global properties of the evolutionary process Let T denote the 4  4 general transition matrix of Table 3. Let q ¼ ða; b; c; dÞ denote a probability vector. In the case at hand q is stationary if qT ¼ q: ð17Þ ð1  eÞa þ dd ð1Þb ¼ a; ð18Þ ea þ ð1  dÞb þ dd ð2Þc ¼ b; ð19Þ ddþ ð1Þb þ ð1  dÞc þ ed ¼ c; ð20Þ ddþ ð2Þc þ ð1  eÞd ¼ d: ð21Þ From Table 3, one derives The system (18)–(21) is of rank 3. Expressing the variables a, c, d as functions of b and adding to the set of linearly dependent equations (18)–(21) the condition that q as a probability vector must sum to unity, a þ b þ c þ d ¼ 1; ð22Þ a¼ dd ð1Þd ð2Þ ; D ð23Þ b¼ ed ð2Þ ; D ð24Þ we arrive at – 382 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few c¼ edþ ð1Þ ; D ð25Þ d¼ ddþ ð1Þdþ ð2Þ ; D ð26Þ with D ¼ d½d ð1Þd ð2Þ þ dþ ð1Þdþ ð2Þ þ e½d ð2Þ þ dþ ð1Þ. Over the long haul, the system will be in state M ¼ 0 with probability a, in M ¼ 1 with probability b, in M ¼ 2 with probability c and in M ¼ 3 with probability d. The relative value of a and d, i.e. a d ð1Þd ð2Þ ½1  dþ ð1Þ½1  dþ ð2Þ ¼ ¼ ; d dþ ð1Þdþ ð2Þ dþ ð1Þdþ ð2Þ is independent of d. As the second part shows, it is in fact a function of dþ ð1Þ, dþ ð2Þ. If for M ¼ 1; 2 the value of dþ ðMÞ increases, then the relative likelihood of populations being formed exclusively by trustworthy types increases while it decreases if either dþ ð1Þ or dþ ð2Þ decrease. The relative likelihood b=c ¼ d ð2Þ=dþ ð1Þ is independent of e. The combined likelihood a þ d that the system will adopt one of the monomorphic states increases if d increases and e decreases. 4.3 A specific illustration We can go beyond these general observations if some additional specific assumptions are made. To illustrate how the analysis could proceed in principle, let us look at two specific and, in a way, polar examples: Case a: dþ ð1Þ and dþ ð2Þ maximal; Case b: dþ ð1Þ and dþ ð2Þ minimal. Case a We assume dþ ð1Þ ¼ P ½Dð1Þ > 0 and dþ ð2Þ ¼ P ½Dð2Þ > 0 both to be maximal. The es 1þs probabilities P ½DðMÞ > 0, M ¼ 1; 2, are maximal if C > max es 4 ; 2  2 , leading to   P ½Dð1Þ > 0 ¼ 1  ð1  lÞð1  l Þð1  l Þ, and ð3=4Þs > C, leading to P ½Dð2Þ > 0 ¼ ll ð2  lÞ. In view of preceding definitions, we get dþ ð1Þ ¼ 1  ð1  lÞð1  l Þð1  l Þ d ð1Þ ¼ ð1  lÞð1  l Þð1  l Þ dþ ð2Þ ¼ ll ð2  lÞ d ð2Þ ¼ 1  ll ð2  lÞ: Inserting into D ¼ d½d ð1Þd ð2Þ þ dþ ð1Þdþ ð2Þ þ e½d ð2Þ þ dþ ð1Þ yields D ¼ dð½ð1  lÞð1  l Þð1  l Þ½1  ll ð2  lÞ þ ½1  ð1  lÞð1  l Þð1  l Þ½ll ð2  lÞÞ þ eð½1  ll ð2  lÞ þ ½1  ð1  lÞð1  l Þð1  l ÞÞ: – 383 –
Japanese Economic Association 2002. The Japanese Economic Review a¼ d½ð1  lÞð1  l Þð1  l Þ½1  ll ð2  lÞ D b¼ e½1  ll ð2  lÞ D c¼ e½1  ð1  lÞð1  l Þð1  l Þ D d¼ d½1  ð1  lÞð1  l Þð1  l Þ½ll ð2  lÞ D For l ¼ l ¼ 1, we obtain a ¼ b ¼ 0 and c ¼ e=ðd þ eÞ, d ¼ d=ðd þ eÞ. For l < 1 and l < 1, the probability distribution over states M ¼ 0; 1; 2; 3, is fully mixed if both e > 0 and d > 0. In that case all states are adopted with positive probability. Case b Assume that both dþ ð1Þ ¼ P ½Dð1Þ > 0 and dþ ð2Þ ¼ P ½Dð2Þ > 0 are minimal. The probabilities P ½DðMÞ > 0, M ¼ 1; 2, are minimal if C es 1þs  4 2 leading to P ½Dð1Þ > 0 ¼ l l , and ð3=4Þs  C, leading to P ½Dð2Þ > 0 ¼ l ll . We get dþ ð1Þ ¼ l l ; d ð1Þ ¼ 1  l l ; dþ ð2Þ ¼ l ll and d ð2Þ ¼ 1  l ll : Inserting into D ¼ d½d ð1Þd ð2Þ þ dþ ð1Þdþ ð2Þ þ e½d ð2Þ þ dþ ð1Þ dð½1  l l ½1  l ll  þ ½l l ½l ll Þ þ eð½1  l ll  þ ½l l Þ. a¼ d½1  l l ½1  l ll  D b¼ e½1  l ll  D c¼ e½l l  D d¼ d½l ll ½l l  : D yields D¼ Again, in view of the specifications of case b, some further conclusions can be drawn. If l ¼ 1 and l ¼ 1, the probability is concentrated on states M ¼ 2 and M ¼ 3, since in that case, c þ d ¼ e=ðd þ eÞ þ d=ðd þ eÞ ¼ 1. Thus, as in case a, perfect signals can minimize the probability that more than one individual is untrustworthy. – 384 –
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few With both, l < 1 and l < 1, the probability distribution over states M ¼ 0; 1; 2; 3 is fully mixed if e > 0 and d > 0. Thus, again, all states are adopted with positive probability. So much for a specification of the process that allows merely for one change at a time. If more than one change in the population composition were allowed on each round of the process, the matrix would merely contain more non-zero entries. If, for instance, the conventional assumption is made that, at all times t  0, from any M t 2 f0; 1; 2; 3g one can reach any state M tþ1 2 f0; 1; 2; 3g with positive probability, then the transition matrix will contain no zero entries. Such modifications would leave the basic analysis unaltered, though with somewhat different expressions. Since it is obvious how such an analysis would proceed once parameter values in the transition matrix are specified, we are content to let it rest with the previous illustration and instead turn to a somewhat more general discussion of the approach so illustrated. 5. Putting formal models into perspective Once the twin standard economic modelling assumptions of zero transaction costs and perfectly enforcable contracts are relaxed, a whole cluster of trust problems emerges. Giving up these two while basically leaving in place other standard assumptions, such as large numbers of participants and anonymity, one can study the trust predicament within a framework reasonably close to the classic models of perfectly competitive markets. But clearly, the conditions of large numbers, in fact N ¼ 1, and anonymity that are used to formulate conventional stylized models of perfect competition do not apply in most instances of real-world market interactions. This suggests relaxing the latter assumptions, too. As in oligopoly theory, where duopoly serves as a model for studying the interaction and strategic considerations of finitely many sellers (typically contrasted with the extreme cases of either monopoly or perfect competition: Tirole, 1988), our model with N ¼ 3 can be used to illustrate the reasoning of rational players who know their own type and know how this information, along with additional information about the type composition, affects their inferences and choices. Contingent on stochastic signals, this can yield as many as six different matches and even more different plays. In intermediate cases of 3  N < 1, not only is the interaction between the players influenced by stochastic elements on each round of play, but overall changes in the population composition are stochastic as well. As our discussion of the example of N ¼ 3 shows in a proto-typical way, intermediate cases with N ; 2 < N < 1, have characteristics categorically distinct from the two extreme cases of N ¼ 2 and N ¼ 1. Studying extreme cases does not yield insights into the intermediate ones. The latter cannot be viewed as resulting from a kind of convex combination of the former. This is a methodological message of some importance. One might want to compare our basic conclusions with experimental results of playing the trust game (e.g. Güth et al. 1993; Snijders and Keren, 1994; a study allowing one to play the trust game repeatedly with experimentally controlled incomplete information is Anderhub et al., 2001). For rather large exploitation incentives e  1 (e.g. 45 German marks), Güth et al. hardly ever observed trust (T) and never a reward of trust (R), in spite of the substantial efficiency increase by trustful cooperation (1  s, resp. 1). Snijders and Keren (1994) obtained partly more encouraging results, but only for less substantial monetary incentives. Repeated interaction allows for reputation formation (Kreps et al., 1982) what Anderhub et al. (2001) could check even for individual participants, since they – 385 –
Japanese Economic Association 2002. The Japanese Economic Review were allowed to use mixed strategies. Except for the experimentally induced trustworthy second-movers, only few participants always rewarded trust (even in the last round of repeated interaction). Thus, the population share p of intrinsically trustworthy participants seems to be rather small. We are not aware of experiments where more or less reliable signals regarding other’s type could be bought. Since Anderhub et al. allowed for the repetition of repeated trust games, their results throw some light on the ‘‘evolution’’ of trust (in reciprocity). After experiencing quite systematic exploitation in the last round of repeated interaction, first-movers have learned to trust less when the end is near. Similar effects are observed by Selten and Stöcker (1986), who also studied repeated play of repeated games. Appendix: Stochastic adjustment of M in case M = 2 If M ¼ 2, the single 0-type player 01 can infer from her knowledge of the population composition that both of the other individuals are trustworthy m-types. Therefore, if 01 plays in the first-mover role, her a priori beliefs are described by p ¼ 1. It does not make sense for such a fully informed 0-type player to choose S at cost C. Contrary to the 0-type, the two m-type individuals, m1 , m2 , cannot infer from their information about their own type or their partner’s type in a match. Like the two 0-type individuals in case M ¼ 1, they merely know that they encounter a trustworthy individual with probability p ¼ 12. Since in the first-mover role rational behaviour is type-independent, the reasoning of a trustworthy first-mover in case M ¼ 2 corresponds to that of an untrustworthy first-mover in case M ¼ 1. Taking this into account along with the different pattern of matches, we can derive the Tables A1 and A2 in full analogy to the previous Tables 1 and 2, respectively. In the same way that we were able to derive Table 2 from Table 1, we can now derive Table A2 from Table A1. In Table A2 the signal originating from the one untrustworthy individual is listed first, that from the first trustworthy second and that from the second trustworthy, third. In Table A2, as in Table 2, a positive value Dð2Þ > 0 indicates that, under the specific constellation, the mean fitness of m-types is higher than that of the 0-type. Recalling that Table A1 The Objective Success of the Three Individuals in the Six Matches of the Game in Case M ¼ 2 Matches and information about the second-mover First, second-mover in matches 01 ; m1 01 ; m2 m1 ; m2 m1 ; m2 m1 ; 0 1 m1 ; 0 1 m2 ; m1 m2 ; m1 m2 ; 0 1 m2 ; 0 1 Objective payoff for players of both types Prob., signal for second-mover type 01 m1 m2 1; ‘‘m’’ 1; ‘‘m’’ l ‘‘m’’ 1  l; ‘‘0’’ l ; ‘‘0’’ 1  l ; ‘‘m’’ l; ‘‘m’’ 1  l; ‘‘0’’ l ; ‘‘0’’ 1  l ; ‘‘m’’ 1 1 0 0 0 e 0 0 0 e 1 0 1C sC sC C 1 0 0 0 0 1 1 0 0 0 1C sC sC C – 386 –
Japanese Economic Association 2002. Signal comb. and probability of combination – 387 – ‘‘m’’; ‘‘0’’; ‘‘0’’ ð1  l Þð1  lÞð1  lÞ ‘‘m’’; ‘‘0’’; ‘‘m’’ ð1  l Þð1  lÞl ‘‘m’’; ‘‘m’’; ‘‘0’’ ð1  l Þlð1  lÞ ‘‘m’’; ‘‘m’’; ‘‘m’’ ð1  l Þll ‘‘0’’; ‘‘0’’; ‘‘0’’ l ð1  lÞð1  lÞ ‘‘0’’; ‘‘0’’; ‘‘m’’ l ð1  lÞl ‘‘0’’; ‘‘m’’; ‘‘0’’ l lð1  lÞ ‘‘0’’; ‘‘m’’; ‘‘m’’ l ll 01 m1 m2 Mean m Dð2Þ 2 þ 2e 2 þ 2e 2 þ 2e 2 þ 2e 2 2 2 2 1 þ s  2C 2  2C 2  2C þ s 3  2C 1 þ 2ðs  CÞ 2 þ s  2C 2ð1 þ s  CÞ 3 þ s  2C 1 þ s  2C 2  2C þ s 2  2C 3  2C 1 þ 2ðs  CÞ 2ð1 þ s  CÞ 2 þ s  2C 3 þ s  2C 1 þ s  2C 2  2C þ 2s 2  2C þ 2s 3  2C 1 þ 2ðs  CÞ 3 2 s þ 2  2C 3 2 s þ 2  2C 3 þ s  2C s  2e  2C  1 s 2  2C  2e s 2  2C  2e 1  2e  2C 2ðs  CÞ  1 3 2 s  2C 3 2 s  2C 1 þ s  2C
Japanese Economic Association 2002. S. Güth, W. Güth, H. Kliemt: Trustworthiness among the Few Table A2 The Objective Success of the Three Individuals in the Six Matches of the Game for All Chance Events (Signals Received) in Case M ¼ 2, and the Differential Success Dð2Þ; Signals Ordered as in ð01 ; m1 ; m2 Þ The Japanese Economic Review l þ l ¼ 2 and relation ðl þ l  1Þ=4 > C together imply that 1=4 > C, it is obvious that 1 þ s  2C > 0: Except for the two constellations ‘‘0’’, ‘‘0’’, ‘‘m’’ and ‘‘0’’, ‘‘m’’, ‘‘0’’, where Dð2Þ ¼ ð3=2Þs  2C, in all other constellations Dð2Þ < 0 will apply. Now, ð3=2Þs  2C > 0 amounts to ð3=4Þs > C. In view of this, we can infer that P ½Dð2Þ > 0 ¼ lll þ 2ðl ð1  lÞlÞ ¼ ll ð2  lÞ; for ð3=4Þs > C: P ½Dð2Þ > 0 ¼ lll ; for ð3=4Þs  C: Final version accepted 5 November 2001. REFERENCES Alchian, A. A. (1950) ‘‘Uncertainty, Evolution and Economic Theory’’, Journal of Political Economy, Vol. 58, pp. 211ff. Anderhub, V., D. Engelmann and W. Güth (2002) ‘‘An Experimental Study of the Repeated Trust Game with Incomplete Information’’, Journal of Economic Behavior and Organization, Vol. 48, pp. 197–216. Güth, S. and H. Ludwig (1998) The Evolutionary Emergence and Vanishing of Assets in Incomplete Financial Markets. Bielefeld: University of Bielefeld. Güth, W. and H. Kliemt (1994) ‘‘Competition or Co-operation: On the Evolutionary Economics of Trust, Exploitation and Moral Attitudes’’, Metroeconomica, Vol. 45, pp. 155–187. —— and H. Kliemt (1998) ‘‘Towards a Fully Indirect Evolutionary Approach’’, Rationality and Society, Vol. 10, pp. 377–399. ——, P. Ockenfels and M. Wendel (1993) ‘‘Efficiency by Trust in Fairness? Multi-period Ultimatum Bargaining Experiments with an Increasing Cake’’, International Journal of Game Theory, Vol. 22, pp. 51–73. Hammerstein, P. and R. Selten (1994) ‘‘Game Theory and Evolutionary Biology’’, in R. Aumann and S. Hart (eds), Handbook of Game Theory, Amsterdam: Elsevier/North-Holland. Huck, S. and J. Oechsler (1998) ‘‘The Indirect Evolutionary Approach to Explaining Fair Allocations’’, Games and Economic Behaviour, Vol. 28, pp. 13–24. Kandori, M., G. J. Malaith and R. Rob (1993) ‘‘Learning, Mutation, and Long Run Equilibria in Games’’, Econometrica, Vol. 61, pp. 29–56. Kreps, D., P. Milgrom, J. Roberts and R. Wilson (1982) ‘‘Rational Cooperation in the Finitely Repeated Prisoners’ Dilemma’’, Journal of Economic Theory, Vol. 27, pp. 245–252. Nelson, R. R. and S. G. Winter (1982) An Evolutionary Theory of Economic Change. Cambridge, Mass.: Harvard University Press. Selten, R. and R. Stöcker (1986) ‘‘End Behaviour in Sequences of Finite Prisoner’s Dilemma Supergames: A Learning Theory Approach’’. Journal of Economic Behavior and Organization, Vol. 7, pp. 47–70. Snijders, C. and G. Keren (1994) ‘‘Giving Trust and Honoring It: An Experimental Test of Two Utility Transformations’’, Working Paper, University of Utrecht. Tirole, J. (1988) The Theory of Industrial Organization, Cambridge, Mass.: MIT Press. – 388 –
Japanese Economic Association 2002.