工作記憶與統計推理及決策判斷: 工作記憶如何影響有限
理性?
研究成果報告(精簡版)
計 畫 類 別 : 個別型
計 畫 編 號 : NSC 98-2628-H-004-001-
執 行 期 間 : 98 年 08 月 01 日至 99 年 10 月 31 日
執 行 單 位 : 國立政治大學心智、大腦與學習研究中心
計 畫 主 持 人 : 楊立行
計畫參與人員: 碩士級-專任助理人員:鄭惟尹
碩士級-專任助理人員:陳宜家
碩士班研究生-兼任助理人員:柯佳妏
碩士班研究生-兼任助理人員:楊格政
碩士班研究生-兼任助理人員:莊子怡
報 告 附 件 : 國外研究心得報告
出席國際會議研究心得報告及發表論文
處 理 方 式 : 本計畫可公開查詢
中 華 民 國 100 年 01 月 31 日
Running head: DBS MODEL
NSC-Report-98-2628-H-004-001-The Evaluation of NSC-Report-98-2628-H-004-001-The NSC-Report-98-2628-H-004-001-Theory of Decision Making by Sampling with Taiwanese Participants
Lee-Xieng Yang Department of Psychology National Chengchi University
Lee-Xieng Yang
Department of Psychology National Chengchi University
NO.64, Sec.2, ZhiNan Rd., Wenshan District, Taipei City 11605, Taiwan
NSC-Report-98-2628-H-004-001-The Evaluation of NSC-Report-98-2628-H-004-001-The NSC-Report-98-2628-H-004-001-Theory of Decision Making by Sampling with Taiwanese Participants
The Concept of Utility
Economists have spent more than 200 years to understand the concept of utility. One self-evident presumption is that people want to keep in hand the objects of high utility. For example, given two lotteries, A and B, to be chosen, one to gain 1 dollar in 50% of chance and the other to lose 1 dollar in 80% of chance, one should always choose lottery A rather than B. In this case, the expected rewards of A and B are 1 × 0.5 = 0.5 for A and −1 × 0.8 = −0.8 for B. However, the expected reward of an object is not its expected utility. Bernoulli (1738) asked people how much bet they would like to pay for playing the game called St. Petersburg paradox, in which people toss a fair coin and if the result was tail, they could toss the coin again; otherwise, they could get 2n dollars back, n =tossing times, and the game stopped. The expected value of this game is P 0.5n2n for n times of tossing, which should be infinitely large as n increases, lim
n→∞P 0.5
n2n= ∞. Thus, people should be willing to pay whatever bet to play this game. However, Bernoulli thought most gamblers would not do so. It was implied that the gamblers was thinking what is worth to him while deciding the bet to pay for this game instead the objective expected value.
Although economists tried to describe utility in terms of some kind of psychological quantity, such as happiness (see Bentham, 1970, original work published, 1789), the mainstream economical theories take preference to represent for utility (see Samuelson, 1937) instead. That is, choosing A rather than B means A is more preferred, namely higher utility. Therefore, the expected utility of a game becomes EU = P
i=1
PiU (Xi), Pi: probability of event Xi and U (Xi): the utility of Xi. With many experiments, Friedman
and Savage (1948) showed that the utility is a convex function of value following the power law, that is, the marginal utility of a value decreases as the value increases. As shown in Figure 1, with the tendency of risk aversion, people become less willing to pay infinite bets in this game.
However, psychologists like Kahneman and Tversky (1979) reported a number of results inconsistent with the prediction of the expected utility theory. For instance, a same option for people to choose would be regarded as different when it is described in different frames. As a result, Kahneman and Tversky proposed their famous prospect theory, which different from the expected utility theory includes the problem frame as a factor to explain how people make decisions. In a loss context, people tend to take a risky choice (risk seeking) and in a gain context, people tend to be conservative and would not take a risky choice (risk aversion). Following their assertion, not only the utility of an event but also the probability of the presence of it is subjective. Although the prospect theory provides a plausible explanation to human’s decision in different contexts, we still want to know the psychological reality behind it. The DbS (Decision by Sampling) theory is the one developed in this attempt and also the focus of this study. Therefore, it will be introduced in more detail here.
Decision by Sampling
Recently, Stewart, Chater, and Brown (2006) tried to apply the basic
psychophysical principle in cognition to understanding how preference is generated. In their DbS theory, it is assumed that utility is generated by comparison among choices. For instance, the utility of a certain wage of NT$k is determined by its rank order among all wages in a sample temporarily stored in our STM (Short-Term Memory). It has been acknowledged that comparison is a very fundamental mental function of cognition. We can always tell which object is heavier, which tone is higher, which size is bigger, and so
on. Therefore, we should have no problem to tell a wage is larger/smaller than the other. The question here is how these wages look to us, for the utility is our subjective preference to the choice. According to the expected utility theory and the prospect theory, the increased amount of utility decreases as the value gets large. In DbS, this utility function is thought to reflect the relative rank Ri of any event xi in a sample of events
[x1, x2, ...xi..., xn], where Ri = n−1i−1 and i is the rank of event xi and n is the total number of events. In the study of Stewart et al. (2006), these authors exploited the database containing more than 1000 pieces of data and revealed that the relative rank, Ri, follows the convex function of the materials in a common life, such as credits and debits in bank account, bread prices, coffee prices, and so on. See the examples for credits and debits in bank account in Figure 2. What is more than the precedent theories in the DbS theory is that it provides an explanation of how and why utility is shown like that.
Range-Frequency Theory
The DbS theory is not the only theory which takes event rank as the reference point for making judgements. The RFT (Range-Frequency Theory) is obviously the other, regarding as the source for comparison the frequency value, Ti = xxni−x−x11 instead of the relative rank, where xi is the value of event xi and xnand x1 mean the maximum and minimum values in the full range of values (see Parducci, 1965, 1995). Brown, Gardner, Oswald, and Qian (2008a) conducted a series of experiments to examine whether the rank information or the range value is more referenced for people to make judgements. These authors gave the participants a list of 11 salaries to learn and told that these were the salaries of their classmates. Following that, the participants were asked to rate how satisfied they would be with 10 hypothetical salaries. The main manipulation was the distribution over the salaries for learning. Figure 3 showed two of their distributions with a unit of 10 thousand NT dollars: positive and negative skew distributions over salary. It
is apparent that the same values in both distributions have different rank orders. Thus, if people were more sensitive to the rank information the same salary would induce different degrees of satisfaction and vise versa. Take value 3 as an example. If the rank information is more important, it should be judged as satisfied in the positive skew distribution (3 is the second highest salary) more than in the negative skew distribution (3 is the fifth highest salary). The participants’ rating data were fit by a hybrid model consisting of both the DbS theory and RFT computations for utility, Mi = WDbSRi+ (1 − WDbS)Ti, with as the goodness of fit the proportion of the variance of participant’s rating data accounted for by Mi and WDbS as the only freely estimated parameter. The larger WDbS the more influential the rank information is. The results showed that on average the rank information was more important than the range information with WDbS = 0.64,
0 ≤ WDbS ≤ 11. Therefore, the sense of utility of an event is more determined by its rank in a reference sample, as predicted by the DbS theory.
Experiment 1A
As revealed in the previous discussion, the rank information seems to be more relevant to the sense of utility. One strategy to further evaluate this account is to explore the application of the DbS theory for other cultures. Therefore, in this study, I would like to first replicate the Brown et al. (2008a) study and see if the DbS theory can provide a good account for the results. Similar to their study, the negative and positive skew salary distribution are used seen in the above panel in Figure 3 with the salaries shown to participants shown underneath. Different from their study, the participants are asked to rate for 7 hypothetical salaries in a questionnaire after reading the learning list.
Method Participants and Apparatus
Ninety-nine undergraduate students in National Chengchi University participated in this study, who were the students in the Psychology 101 class (54 in the positive skew condition and 45 in the negative skew condition). The 3-pages questionnaire containing the autobiographical questions and sample items in page 1, the 11 salaries for learning in page 2, and the 7 test salaries in page 3.
Stimuli and Procedure
All participants were asked to fill in the questionnaires at the same time. They were also told not to return to the previous pages during rating. The participants spent about 15 minutes to finish the questions on average.
Results
The mean rating scores for the 7 choices are plotted in the panels respectively for the positive and negative skew conditions in Figure 4. In order to evaluate how well the DbS theory can capture the observed data pattern, the observed data were fit by the hybrid model, Mi= WDbSRi+ (1 − WDbS)Ti, with the weight WDbS freely estimated. The R2 on fit to the data in the positive skew condition is 0.87 and W
DbS = 0.85. This means that the rank information is far more important than the range information to account for people’s satisfaction rating for the salaries of the positive skew distribution. On the contrary, although there is no prior expectation, WDbS = 0 on fit to the data in the negative skew condition with R2 = 0.81. This means that the rank information has nothing to do with the satisfaction judgement with the salaries of the negative skew distribution. This is inconsistent with the result of Brown, Gardner, Oswald, and Qian (2008b) that the rank information is more important than the range information with the
salaries of no matter which distribution. Before we draw any conclusion for this discrepancy, it is worth checking if the result is stable or just some coincidence.
Experiment 1B
If the DbS theory is correct, the rating pattern in both conditions should be accounted for well by it. However, the results in Experiment 1A show that only the data in the positive skew condition can be fit well. Together with the result of Brown et al. (2008b), the participants’ data in the positive skew condition is consistent with theirs. Thus, I put my focus on the negative skew condition and re-conducted the experiment but with undergraduate students in another university. If the same pattern as shown in Experiment 1A occurs again, then the pattern should be reliable.
Method Participants and Apparatus
In this experiment, the negative skew condition in Experiment 1A is replicated with 81 undergraduate students in Taughai University. Thus, the same questionnaire was used as the measuring instrument in this experiment.
Stimuli and Procedure
Same as in Experiment 1A. Every participant took about 15 minutes to finish the questionnaire averagely.
Results
The rating pattern is quite close to that in the same condition in Experiment 1A, shown in Figure 5. In fact, the correlation between these two experiments on the same choice points is r = .99, p < .001, even though they are made by different participants. Therefore, there is no reason to regard the result in the negative skew condition is
unreliable. The possible reason for getting this result is now turned to the experimental material and design.
Experiment 2
The current findings are half consistent and half inconsistent with the result of Brown et al. (2008b). Could this be a result of inappropriate use of experimental materials? The value range in the distribution in Experiment 1 is shorter than that in Brown et al. (2008b)’s study. Perhaps, the relative rank of the salary in the negative skew condition is not obvious enough. Also, when answering the questionnaire, could it be possible that the participants take the other alternative test salaries as reference instead of the 11 salaries in the preceding page? In addition, participants might as well use their knowledge about the current salary level to make their judgement. All these can be potential confounding factors in Experiment 1. Therefore, in this experiment, the identical distributions in the positive and negative skew conditions in Brown et al. (2008b) is used. Further, in order to prevent the involvement of background knowledge to wellbeing rating, in stead of salary, the participants are asked to rate their performance in a category learning task which they have never seen before.
Method Participants and Apparatus
Eighty undergraduate students in National Chengchi University participated in this experiment, half in the positive skew condition and half in the other condition.
Stimuli and Procedure
In the category learning task, the stimulus consisted of a rectangle and a vertical little bar slighting attached above the bottom of the rectangle. The category structure is depicted in Figure 6. The ascending diagonal is the categorization rule which divides the
category space to two regions above and below the boundary, each of which corresponds to a category marked by color blue or red. The abscissa corresponds to the bar position in the rectangle and the ordinate corresponds to the rectangle height. There are 160 training stimuli in total, presented to participants in 5 blocks with no repetition. There are 36 test stimuli which were presented once in the transfer phase. During the training phase, the participant would get a corrective feedback (”correct” or ”wrong”) immediately after they made the A or B response. However, during the transfer phase, there was no feedback. After taking the category learning task, the participants were given 11 accuracies and told that these were the other students’ learning performance. The learning accuracy can be seen in Figure 7. The numbers in the table below show the learning accuracies in percentage. Afterward, they were shown the same 11 accuracies one after one and instructed to rate how satisfied they would feel if this number was their learning performance by choosing one of the 7 points (1 7, the larger, the more satisfied) on computer screen. The options were arranged in a random order. The rating data are the focus of this study. Therefore, the following analysis does not include the category learning data.
Results
After excluding the outliers in both conditions, there are 38 and 36 participants’ data left for further analysis. The ratings are plotted separately in two panels for two conditions. See Figure 8. The observed data are denoted by the solid line. Apparently, the rating pattern in the positive skew condition is like a power law, which is very similar to what Brown et al. (2008b) found. Once again, the rating pattern in the negative skew condition is not a concave function but closer to a linear line as shown in the previous experiments.
very well, R2 = 0.91 and R2= 0.95 for the positive and negative skew conditions. Same as the previous experiments, the weight for the rank information is extremely high,
WDbS = 1, for the fit to the data in the positive skew condition, while it is extremely low, WDbS = 0 for the fit to the data in the negative skew distribution. The dashed line with a legend of rank effect denotes the prediction of DbS alone; whereas the other dashed line with a legend of range effect denotes the prediction of range value alone. Thus, after all these attempts, it is clear that the linear rating pattern found in the negative skew condition is reliable and needs theoretical interpretation.
Subjective Income Rank Model
The previous results consistently show that the pure DbS model can only account for half of the conditions. Although we can introduce RFT model for accounting for the observed pattern in the negative skew condition, this may give birth to another problem why/when people know which information, rank or range, should be used. Therefore, it is not a good solution to the inconsistency reported in these experiments. In the study of Brown et al. (2008b), they provided an upgraded equation to compute the rank,
SRi = 0.5 +2[(i−1)+η(n−i)](i−1)−η(n−i) , which is named subjective income rank model (SR model in abbreviation) for the case of wage experiment. In this equation, the only parameter is η. If η = 1, the comparison between xi and those higher or lower than it is equally important. If η > 1, the comparison between xi and those higher than it is more emphasized; whereas if η <, the comparison between xi and those lower than it is more emphasized otherwise.
Therefore, the SR model is fit to the data in Experiment 2. The results are pretty good, R2 = 0.96 for the positive skew condition and R2= 0.98 for the negative skew condition. What is interesting is the best estimated value of η is less than 1, η = 0.97 for the positive skew condition and η = 0.39 for the negative skew condition. As a result, the SR model can handle both conditions in these experiments. Since it is the upgraded
version of the DbS theory, at least we can draw a conclusion that the utility of an event results from the comparison between the events in the sample and the relative rank of an event can represent its utility. Comparing with the results in the Brown et al. (2008b)’s study, in which they found η > 1 in their experiments, the current η values are less than 1 or close to 1 but never larger than 1. Can this discrepancy be regarded as cultural
difference? At the moment, there is no answer but it is worth investigating in the future study.
References
Bentham, J. (1970). An introduction to the principles of morals and legislation. London: The Athlone Press (Original work published, 1789).
Bernoulli, D. (1738). ”specimen theoriae novae de mensara sortis”. Commentarii Academiae Scientiarum Imperialis Petropolitanae.
Brown, G. D. A., Gardner, J., Oswald, A. J., & Qian, J. (2008a). Does wage rank affect employees’ well-being? Industrial Relations, 47 , 355-389.
Brown, G. D. A., Gardner, J., Oswald, A. J., & Qian, J. (2008b). Does wage rank affect employeeswell-being? Under preparation for submission.
Daneman, M., & Carpenter, P. A. (1980). Individual differences in working memory and reading. Journal of Verbal Learning & Verbal Behavior , 19 , 450-466.
DeCaro, M. S., Thomas, R. D., & Beilock, S. L. (2008). Individual differences in category learning: Sometimes less working memory capacity is better than more. Cognition, 107 , 284-294.
Friedman, M., & Savage, L. J. (1948). The utility analysis of choices involving risk. The Journtal of Political Economy, 56 , 279-304.
Kahneman, D., & Tversky, A. (1979). Prospect theory. Econometrica, 47 , 263-292. Parducci, A. (1965). Category judgment: A range-frequency theory. Psychological
Review , 72 , 407-418.
Parducci, A. (1995). Happiness, pleasure, and judgment: The context theory and its application. Mahwah, NJ: Erlbaum.
Samuelson, P. A. (1937). A note of measurement of utility. The Review of Economic Studies, 4 , 155-161.
Stewart, N., Chater, N., & Brown, G. D. A. (2006). Decision by sampling. Cognitive Psychology, 53 , 1-26.
Footnotes 1Namely, W
Figure Captions Figure 1. Expected utility is a convex function of value.
Figure 2. Relative ranks of different credits and debits.
Figure 3. Positive and Negative Skew Distributions over Salary.
Figure 4. The Observed and Predicted Rating Scores in Experiment 1A. The DbS prediction is denoted by dashed line.
Figure 5. The Observed Rating Scores in Experiment 1B.
Figure 6. The Category Structure in Experiment 2.
Figure 7. The Accuracy Distribution in Experiment 2.
Positive skew 2.12 2.16 2.21 2.27 2.35 2.43 2.54 2.67 2.83 3.01 3.24 Negative skew 2.12 2.35 2.53 2.69 2.82 2.93 3.01 3.09 3.15 3.20 3.24
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 x 104 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 wages sa ti sf a c ti o n positive distribution pos observ DbS 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 x 104 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 wages sa ti sfa c ti o n negative distribution neg observ DbS
2 2.2 2.4 2.6 2.8 3 3.2 3.4 x 104 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 wages sa ti sfa c ti o n 1a 1b
Positive skew 50 51.8 54 57 60.3 63.8 68.8 74.6 81.7 89.7 100 Negative skew 50 60.3 68.3 75.4 81.3 86.2 89.7 93.3 96 98.2 100
50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 performance sat isf ac ti on
Satisfaction in Positive Distribution
observed rank effect range effect 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 performance sat isf a c ti on
Satisfaction in Negative Distribution
observed rank effect range effect
1
日期: 年 月 日
一、國外(大陸)研究過程
本次參加澳洲數學心理學會舉辦之貝氏統計研習工作坊,席中發現澳洲有數個重點研究中心專門
研究貝氏統計在心理學上的應用,像是阿德雷德大學(University of Adelaide)及新堡大學
(University of New Castel)都有專人在研究,同時也以驚人的速度發表相關論文。參與工作
坊的成員來自澳洲本地、美國和英國,前一半課程著重於講授相關理論與背景知識,後一半課程
則為實際上機演練。一邊學習如何使用 WinBugs 這套軟體,一邊試著將課程範例的資料進行演算
檢驗。
二、研究成果
本人目前正應用習得之貝氏統計觀念與方法,發展類別學習的模型。主要採取貝氏統計中建構模
型的方法,以階層式貝式模型,將人類分類的歷程切分為知道該以何種方法分類,以及確認方法
後實際所得的分類結果。此一模型目前尚在發展中,預計今年會投稿數學心理學期刊(Journal of
Mathematical Psychology)
。
三、建議
有鑑於國外研討會中常有各類工作坊介紹目前最新的研究方法,建議國內主辦相關研討會的主辦
單位,能多邀請國外學者來台進行工作坊,也建議國科會能增加這類補助。
四、其他
無其他事項。
計畫編號
NSC98-2628-H-004-001-
計畫名稱
工作記憶與統計推理及決策判斷: 工作記憶如何影響有限理性?
出國人員
姓名
楊立行
服務機構
及職稱
政治大學心理學系
出國時間
99 年 2 月 13 日至
99 年 2 月 19 日
出國地點
澳洲伯斯
1 日期: 年 月 日
一、參加會議經過
為期三天的會議於羅馬近郊舉行,舉會者多半來自歐洲各國,以工作記憶為主要研究對象的學者。
每一天會議分成上午和下午兩大場次,每一場次中各有三到四場論文發表。本人於第一場次的第
一場發表,主要提出數據說明工作記憶有助於分類學習歷程,且不分各種類型的分類學習。其它
場次的主題多數圍繞於工作記憶的模型,以及儲存的資訊是否會隨時間而消褪。每場發表都有很
多討論,加上中午休息時間,各有兩場的壁報發表,與會人數雖少,卻相當熱絡。
二、與會心得
國內對於國際會議的印象多數以美國為主,少數則為英國或澳洲,相對地,對於歐洲方面的學術
發展並不太熟悉。此次會議發現歐洲的心理學發展比之前期望地還要更精進,除了有許多實驗均
使用 fMRI 技術進行外,更有法國學者提出工作記憶的模型,也引起極大迴響。另一項特別的心得
是很多歐洲人的英文其實沒有想像中好,但也都有能在英文期刊上發表的寫作實力,這點對於同
屬於非英語系國家的我們而言,不啻為重要提醒與警惕。
三、考察參觀活動(無是項活動者略)
四、建議
建議鼓勵國內學者多多與歐洲學者或大學交流,歐洲方面的學術底子深厚,觀點也有別於美國,
對於長久以來就仰賴美國的我們而言,是一項值得投資的嘗試。
計畫編號
NSC98-2628-H-004-001-
計畫名稱
工作記憶與統計推理及決策判斷: 工作記憶如何影響有限理性?
出國人員
姓名
楊立行
服務機構
及職稱
政治大學心理系
會議時間
99 年 8 月 30 日至
99 年 9 月 5 日
會議地點
義大利羅馬
會議名稱
(中文)第五屆歐洲工作記憶工作坊
(英文)European Working Memory Workshop V
發表論文
題目
(中文)工作記憶與分類:單一歷程證據
2
日期:2011/01/29