A mathematical model of social conformity as a function of group size

Transcription

1 A mathematical model of social conformity as a function of group size To Chin, Yu April 20, 2013 This paper aims to model mathematically the dependence of conformity on majority size with the intention of attaining the following three goals: (1) The model should make psychological sense and should be (at least partially) based on exisiting psychological laws. (2) The model should be mathematically as simple as possible to reduce the number of free parameters which are usually very dicult to be estimated based on known information. (3) The model should behave correctly qualitatively and t the data well quantitatively. Since Asch published his experiments on conformity, many studies have been done to investigate the relationship between the size of the majority and the degree of conformity.[2] However, the results from dierent experiments are often inconsistent. In most experiments, only a few dierent group sizes are tested. These make mathematical modeling of the relationship between group size and conformity highly challenging. Therefore, only data from Asch's original experiments is used here to avoid the problem due to inconsistent and often conicting data. The validity of this model should thus be more carefully examined once the inconsistencies among the experimental ndings are resolved. Existing models such as the SIM model [2] often do not have clear psychological basis. Conformity (or social impact) is assumed to be some ad-hoc mathematical function of size of majority. In addition, there are often too many free parameters which often have little or none intuitive meaning. It also makes data tting more dicult as dierent combination of parameter values may generate similar behavior, making it hard to decide which value is the real value of the parameter in concern. In response to these problems, this paper aims to propose a new, mathematically minimalistic model with clear psychological basis (namely Luce's utility) and good t with (Asch's) experimental data. It is also hoped that this will generate motivation for new experiments that aims to resolve the inconsistencies between existing data. The reader is refered to Asch's paper [1] for a detailed description of the experimental setting and procedure. 1

2 1 Theory 1.1 Utility The model proposed in this paper is based on the following utility function derived by Luce and Fishburn: [3] { C(1 e bx ), x 0 U(x) = K(1 e dx ), x < 0 They have proven that this is the only possible function that is concave for gains, convex for loses, while satisfying the requirement of segregation and binary prospect theory. (Luce, 1997) Here the independent variable x could signies any psychologically attractive quantity, in the sense that an increase in such quantity will increase the chance for the subject to choose the corresponding option. In our case, the size of majority is obviously the attractive quantity, which is nonnegative. Thus the utility of conforming to majority opinion (yield utility) can be written generally as U yield (x) = C(1 e bx ) + U 0,where x is the majority size and U 0 is the base utility when there is no inuence from other people (x = 0). The utility of not yielding is assumed to be a constant U 1. 1 It is possible to introduce a cost term in the non-yielding utility such that U yield (x) = D(1 e bx ) + U 1 but this turns out to be unnecessary if Gaussian distributions are used(refer to section 1.3). 1.2 Probability distribution and the model Due to dierences among individuals, at any xed size of majority, the utilities are not single-valued but random variables that follow certain probability distributions. In Asch's experiments, the task itself is very unambiguous and allows very little room for individual variation [1] so we can assume that when x = 0, the base utilities follow a very shape distribution centered around the mean values U 0 and U 1. In fact, we assume they follow delta distributions for simplicity. If we introduce the eect of other people, however, the variance of the yielding utility will not be zero as the susceptibiliy to other people's opinion can be vastly dierent among individuals. The non-yielding utility still follows a delta distribution as we assumed no cost term hence the non-yielding utility is still the base utility which is assumed to be basically the same among all subjects. 1 In the current context, not yielding means not conforming to the majority's response and choose the correct response. There is a possibility that the subject will choose a response that is neither the majority's response nor the correct response but the utility of doing so is much much smaller than the others so such possibility is neglected here. In settings where the more than two choices are given and the correct answer is not as obvious and unambiguous however, such possibility cannot be ignored. Such matters will be discussed later in this paper. 2

3 We assume that the distribution of the yielding utility is Gaussian (or normal) with mean value at C(1 e bx )+U 0 and s.d. σ. This is the mathematically simplest choice that we can make. 2 Note that it should be clear by now this model is a probabilistic model of statistical behavior rather than a deterministic model of individual behavior. Calculating the yielding probabiliy (conformity) then becomes a fairly trivial task, since the subject will yield if yield utility is larger Conformity (x) = P ( U yield (x) > U yield (x) ) = P ( U yield (x) > U 1 ) = 1 2 (1 + erf(c(1 e bx ) + U 0 U 1 2σ ) = 1 2 (1 + erf(c(1 e bx ) U 2σ ) We can obtain a reasonably good t to Asch's data with this formula. 1.3 Scaling invariance If U 1 N(µ 1, σ 2 1) and U 2 N(µ 2, σ 2 2), where N(µ, σ 2 ) denotes normal distribution with mean µand variance σ 2, then,where z = µ1 µ2 σ 2 1 +σ2 2 P (U 1 > U 2 ) = P (U 1 U 2 > 0) = P (z > 0) is the usual z-score. Hence if the scale of utility is transformed linearly, i.e. the transformation u = au + b is carried out, then z = aµ 1 aµ 2 a σ 2 1 σ2 2 Therefore for this model, the value of conformity is unchanged under a linear transformation in utility. More importantly, if the means of the two distributions are shifted but their dierence remains the same, conformity will also be unvaried. This explains why it is unncessary to introduce a cost term into the expression for U yield. If there is a cost term such that U yield (x) = D(1 e bx )+U 1, then the dierence between U yield and U yield will be C(1 e bx ) + D(1 e bx ) + U 0 U 1 = C (1 e bx ) + U 0 U 1, which is mathematically identical to the expression without cost term. 2 Also, the central limit theorem seem to suggest that this assumption will be more accurate when we have a large sample size. = z 3

4 2 Parameter estimation and data tting Although four parameters appear in the equation, there are in fact only three degrees of freedom. This can be seen by dividing both the numerator and denominator inside the error function: Conformity = 1 2 (1 + erf(c(1 e bx ) U 2σ ) = 1 2 (1 + erf((1 e bx ) U C ) 2 σ C = 1 2 (1 + erf((1 e bx ) U ) 2σ This can be viewed as a change of unit of utility. Since there is no currently existing standard unit for utility, we need not bother ourselves with nding the value of C. Thanks to the form of our expression, only ratios of U and σ to C are important rather than U and σthemselves. This helps to further simplify the model and reduce the number of free parameters. 2.1 Estimating b The reciprocal of b is a measure of the saturation majority size (or size of consensus). If x = b 1, the utility will reach 63% (1 1 e ) of the maximum level. For x = 2b 1, it will be at 86%. For x = 3b 1, it will be at 95%, which is pretty close to maximum. Asch has suggested that 3 people seems to be the saturation level for conformity in his experiments, if so we can estimate b by setting 3b 1 3, thus b Estimating U/σ U depends on the base utilities. Hence it is reecting the intrinsic nature of the task involved in the experiment. In the current context, if the correct response is ambiguous, the utility of the options will be quite close to each other and result in a small value of U, vice versa. On the other hand, σ represents the variation of utility among the subjects, i.e. a measure of individual dierences. Since little is known about the quantitative aspects of utilities, it is not possible here to obtain an estimation on its value based solely on the quanlitative aspects of the setting. However, if the base probability 3 of choosing the yielding option is known, it is possible to calculate U /σ = U/σ using the equation: P base = 1 (1 + erf( U 2 ) 2σ U σ = 2erf 1 (1 2P base ) 3 Probability when group size x is zero. 4

5 Figure 1: Plot of dierent model of conformity and Asch's data (circle). No constraint, P (x) = 1 2 (1 + erf( (1 e 1.006x ) ); Constraint, P (n) = 1 2 (1 + erf( (1 e x ) ); SIM [2], P (n) = 0.33exp( 4exp( n )), where x is the size of majority and P (x) is the probability that the subject will conform to the majority (i.e. choose the wrong, majority answer). SIM model is plotted here for comparison. In the context of Asch's experiment, if the error rate is, for example, 0.1%, then P base = 0.1% and U σ Results The data from Asch's original experiment [1] is used to carry out the data tting. The data is tabulated in Table 1. The curve-tting app available in MATLAB R2012b is used to carry out the tting. R-squared values are calculated separately in Excel and graphs are generated by MATLAB. x P yield (Asch[1]) P yield (no constraint 4 ) P yield (with constraint 5 ) SSE = 0.006, b = 1.006, U = 1.093, σ = SSE = 0.012, σ =

6 References [1] S.E. Asch, Opinions and social pressure, Scientic American, 193 (1955) [2] R. Bond, Group Size and Conformity, Group Processes & Intergroup Relations, 2005 Vol 8(4) [3] R.D. Luce, P.C. Fishburn, A note on deriving rank-dependent utility using additive joint receipts, Journal of Risk and Uncertainty, 11:5-16 (1995). 6