認知能力與認知階層 : 選美賽局的實驗分析 - 政大學術集成

全文

(1)國立政治大學經濟學研究所博士論文指導教授: 陳樹衡博士. 認知能力與認知階層: 選美賽局的實驗分析 Cognitive Capacity and Cognitive Hierarchy: 治政 Experimental Evidence from 大 Keynes’s Beauty 立 Contest. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 研究生: 杜業榮中華民國一零二年七月.

(2) 謝詞在你看來, 千年如已過的昨日, 又如夜間的一更。 — 聖經詩篇. 十五年悄然過, 在道南橋畔的日子。. 政治大. 感謝陳樹衡老師, 楊立行老師。正式非正式的指導與切磋。感謝父母。有形無形的支持與鼓勵。感謝老婆與女兒。有聲無聲的激勵與鞭策。. 立. ‧. ‧ 國. 學. 學術之路是否有續章, 已過的昨日可否視為沉沒成本, 關鍵是經濟人 (Homo Economicus) 是否回歸智人 (Homo Sapiens)!. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 杜業榮 2013 年 7 月.

(3) 摘要晚近行為實驗的發展, 特別是在賽局實驗的研究中, 認知所扮演的角色漸受重視。認知階層與認知能力, 是文獻上兩個相關並且廣被討論的概念。雖然這兩者往往出現在同樣的實驗中, 但仍少有研究正式地探討兩者之間的關係。在本研究中, 我們透過 15 至 20 人為一組的重複選美賽局觀察受試者的認知階層, 並以工作記憶測驗測量其認知能力, 試圖檢驗認知能力對於認知階層的影響。總的來說, 我們發現認知能力對於認知階層有正向的影響, 即認知能力較高的受試者, 所觀察到的認知階層也較高。在最初幾個回合中, 認知能力的影響顯著。接下來的回合中雖然效果漸. 政治大. 弱, 但並不會完全消失。這意謂著認知能力可能進一步影響其學習行為, 因此透過認知階層的馬可夫轉移動態與經驗加權吸引力學習模型, 我們檢驗此一可能性。證. 立. 據顯示認知能力不同反映學習行為的差異, 尤其相較於強化學習, 認知能力較高的. ‧. ‧ 國. 學. 受試者可能更傾向信念學習。. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(4) Abstract Recent developments in behavioral experiments, in particular game experiments, have placed human cognition in a pivotal place. Two related ideas are proposed and are popularly used in the literature, namely, cognitive hierarchy and cognitive capacity. While these two often meet in the same set of experiments and observations, few studies have formally addressed their relationship. In this study, based on six series of 15- to 20-person beauty contest experiments and the associated working memory tests, we examine the effect of cognitive capacity on the observed cognitive hierarchy. It is found that cognitive capacity has a positive effect on the observed cognitive hierarchy. This effect is strong in the initial rounds, and may become weaker, but without disappearing, in subsequent rounds, which suggests the possibility that cognitive capacity may further impact learning. We examine this possibility using the Markov transition dynamics of cognitive hierarchy and experience-weighted attraction learning. There is evidence to show that subjects with different cognitive capacities may learn differently, which may cause strong convergence to be difficult to observe.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(5) Contents 1 Introduction 9 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. 政治大. 2 Literature Review 2.1 BCG and Cognitive Hierarchy . . . . . . . . . . . . 2.2 BCG and Cognitive Capacity . . . . . . . . . . . . 2.3 Cognitive Capacity and Cognitive Hierarchy . . . . 2.4 Cognitive Capacity and Learning . . . . . . . . . . 2.4.1 Cognitive Capacity, Cognitive Hierarchy and 2.4.2 Cognitive Capacity and EWA Learning . . 2.5 A Summary of Literature . . . . . . . . . . . . . .. 立. n. Ch. engchi. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. y. . . . . . .. . . . . . .. i n U. . . . . . .. . . . . . .. sit. . . . . . .. er. io. al. ‧. ‧ 國. 學. Nat. 3 Method 3.1 Beauty Contest Experiment 3.2 Working Memory Task . . . 3.3 Cognitive Hierarchy . . . . . 3.4 EWA Learning Model . . . . 3.4.1 The Model . . . . . . 3.4.2 Estimation Strategy. v. . . . . . .. . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . Learning . . . . . . . . . . . .. 13 13 14 17 19 19 20 22. . . . . . .. 24 24 25 26 28 28 31. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 4 Results 4.1 WMC and Guessing Performance . . . . . . . . . . . . . . . 4.2 WMC and Cognitive Hierarchy . . . . . . . . . . . . . . . . 4.2.1 Static Analysis: Conditional Distribution and Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Dynamic Analysis: Evolution of Distributions . . . . 4.2.3 Dynamic Analysis: Markov Transition Matrix . . . . 4.3 Guessing Performances and Cognitive Hierarchy . . . . . . . 4.4 WMC and EWA Learning . . . . . . . . . . . . . . . . . . . 4.4.1 Camerer’s EWA Learning . . . . . . . . . . . . . . . 4. . . . . . .. 33 . 33 . 37 . . . . . .. 38 43 46 51 53 53.

(6) CONTENTS 4.4.2. EWA Rule Learning . . . . . . . . . . . . . . . . . . . 57. 5 Discussion and Conclusions. 64. A Instructions of BCG. 74. B Instructions of WMC Task. 76. C Level-k Distribution Conditional on WMC Percentiles: Period 3 to 10 82. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 5. i n U. v.

(7) List of Figures 1.1 An Overview of the Research Backgrounds . . . . . . . . . . . 11 3.1 Level classification procedure for guesses in period 1 . . . . . . 27 3.2 Reinforcement and the corresponding interval for winner and loser in beauty contest experiments . . . . . . . . . . . . . . . 30 4.1. 政治大 Guessing Errors between the High WMC groups and the Low 立 WMC groups . . . . . . . . . . . . . . . . . . . . . . . . . . .. n. Instruction Instruction Instruction Instruction Instruction. y. . 47. sit. io. B.1 B.2 B.3 B.4 B.5. Ch. for for for for for. engchi. i n U. . 44 . 45. . 48. er. Nat. al. 36 . 39. ‧. ‧ 國. 學. 4.2 Level-k distribution conditional on WMC percentiles . . . . 4.3 Level-k distribution conditional on WMC percentiles in period 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Level-k distribution conditional on WMC percentiles in period 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Markov Transition Matrix of the State Space of Reasoning Levels, Estimated Using the Pool of 10 Periods . . . . . . . . 4.6 Markov Transition Matrix of the State Space of Reasoning Levels, Estimated Using the Pool of the Initial 5 Periods . . 4.7 Markov Transition Matrix of the State Space of Reasoning Levels, Estimated Using the Pool of the Last 5 Periods . . . 4.8 Guess distribution and predicted guess distribution . . . . . 4.9 Level distribution and predicted level distribution . . . . . .. v. backward digit span task(Dspan) . . . spatial short-term memory test(SSTM) memory updating task(MU) . . . . . . sentence-span task(SentSpan) . . . . . operation-span task(OpsSpan) . . . . .. . . . . .. . . . . .. . . . . .. . 49 . 56 . 60 . . . . .. 77 78 79 80 81. C.1 Level-k distribution conditional on WMC percentiles in period 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 C.2 Level-k distribution conditional on WMC percentiles in period 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.

(8) LIST OF FIGURES C.3 Level-k distribution conditional on WMC percentiles in period 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Level-k distribution conditional on WMC percentiles in period 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Level-k distribution conditional on WMC percentiles in period 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 Level-k distribution conditional on WMC percentiles in period 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7 Level-k distribution conditional on WMC percentiles in period 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.8 Level-k distribution conditional on WMC percentiles in period 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 7. i n U. v. . 83 . 84 . 84 . 85 . 85 . 86.

(9) List of Tables 2.1 Literature on the Effect of Cognitive Capacity on Strategic Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 Summary of Estimated Learning Models . . . . . . . . . . . . 31. 政治大. 4.1 Correlation coefficients between WMC scores and guessing differences in the beauty contest experiment . . . . . . . . . . . 4.2 Level-k thinking and working memory capacity . . . . . . . . 4.3 Do High and Low WMC subjects exhibit similar level/guess distributions? . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Target-d distribution . . . . . . . . . . . . . . . . . . . . . . . 4.5 Model Parameter Estimates of EWA Following Camerer and Ho (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Model Parameter Estimates of EWA Following Camerer, Ho and Chong (2002) . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Model Parameter Estimates of EWA Rule Learning: Model IV 4.8 Model Parameter Estimates of EWA Rule Learning: Model V 4.9 LR Test for the Significance of Difference in Parameter Estimates: Model V . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 8. i n U. v. 34 41 43 52 54 55 58 59 62.

(10) Chapter 1 Introduction 1.1. Background 政. 治. 立. 大. ‧. ‧ 國. 學. This study attempts to investigate the relation between cognitive hierarchy, recently formulated in behavioral game theory, and cognitive capacity, long studied in cognitive psychology. These two concepts, while being quite closely related, belong to two different bodies of knowledge, one in economics and one in psychology. Cognitive hierarchy is a concept developed from the recent behavioral experimental games which involve iterative reasoning or depth of reasoning. Among many such games, one of the most famous is the Keynes’ beauty contest game (Nagel, 1995, 1998), which is also known as the outguessing game. Cognitive hierarchy is a core concept, shared by a family of models, which aims to explain subjects’ behavior observed in this kind of experiment. Cognitive capacity, as is well known, is a concept related to the psychological study of human intelligence. As we can see from Figure 1.1, attempts to connect the two intuitively close concepts have not been frequently made by either economists or psychologists. In this study, we shall formally examine the relation between the two; in particular, we would like to see whether cognitive capacity as a psychological trait of a subject can help predict his/her revealed cognitive hierarchy in game experiments which explicitly involve depth of reasoning, such as the beauty contest game (BCG). Therefore, this study naturally makes economics meet psychology. While the possible cooperation between the two disciplines has been long ignored, it has picked up significant momentum in recent years (Earl, 1990; Frey and Stutzer, 2007; DellaVigna, 2009), and this research is just one of many on-going interactions between the two. Let us start from the side of economics, the cognitive hierarchy. There are two possible ways to think of the observed cognitive hierarchy from ex-. n. er. io. sit. y. Nat. al. Ch. engchi. 9. i n U. v.

(11) 1.1. BACKGROUND perimental games (to be briefly reviewed in Section 2.1). One way is to think that it is only an ex post representation of our experimental results; as to who correspond to the upper levels of the hierarchy and who correspond to the lower levels, that may just be stochastically determined by subjects’ choices based on their own experiences and beliefs.1 In this case, the cognitive hierarchy is emerged from the interaction of agents, or, simply stated, it is endogenously formed. If this is so, the cognitive hierarchy of a group of subjects observed in one game may have limited applicability to the cognitive hierarchy of the same group of subjects in the other game. Subjects whose behavior was regarded as having a high cognitive hierarchy may switch to a lower one in different experiments, and vice versa. Alternatively, one may think that cognitive hierarchy is exogenously determined by some fundamental attributes of the participating subjects. In other words, the cognitive hierarchy with which the subjects are associated has a functional relation with these fundamental attributes. If so, then a further inquiry into what these fundamental attributes are and whether cognitive capacity is one of the them define the research question of this research. In sum, the scope of this research is, given the success of the cognitive hierarchy model in experimental games, to decide whether we should include personal traits as part of the given conditions (the exogenous variables) of the game so that it can help us either better predict or better explain the strategic sophistication observed in the game. Figure 1.1, using a blue triangle, shows the main backgrounds of the research question and the related literature. The three vertices of the triangle are the Keynes’ beauty contest game (top left), cognitive hierarchy (top right) and cognitive capacity (bottom). The three arrows which connect the three vertices correspond to three strands of the literature. The arrow number one shows the research that applies the cognitive hierarchy models to explain the behavior observed in the beauty contest game. This belongs to the typical game-theoretic literature. The arrow number two shows the research trying to examine the significance of cognitive capacity in subjects’ performance in beauty contest games, and the arrow number three shows the research trying to examine the effects of cognitive capacity upon cognitive hierarchy. These latter two belong to the literature on psychological studies of game. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 1. i n U. v. For example, McCain (2010) used the Brock-Hommes adaptive belief system (Brock and Hommes, 1998) to build an agent-based model to simulate the evolution of the distribution over different depths of reasoning. In this model, agents can in principle choose whatever depth of reasoning they prefer; however, based on the experienced rewards and costs they may choose different depths of reasoning at different points in time. The point is that all depths are available for all agents, regardless of their personal attributes, not to mention cognitive capacity.. 10.

(12) 1.1. BACKGROUND. 立. 政治大. ‧ 國. 學. Figure 1.1: An Overview of the Research Backgrounds. ‧. al. er. io. sit. y. Nat. Due to the space limitation, three arrows are only numbered without related literature. They are given as follows. Direction 4 includes Nagel (1995); Stahl (1996); Duffy and Nagel (1997); Stahl (1998); Ho, Camerer, and Weigelt (1998); Camerer and Ho (1999). Both direction 5 and 6 include only Gill and Prowse (2012). v. n. experiments, which are also known as cognitive game experiments. As we can see from the figure, studies distributed over these three arrows are not even. There have been numerous studies in the literature addressing the usefulness of the cognitive hierarchy models in BCG (arrow number one, colored in black) since 1995. In addition, a growing number of researches are now available to shed some light on the the significance of cognitive capacity on BCG (arrow number two, colored in black). However, few studies on arrow number three (colored in white), until very recently, complete the circle. A complete circle means that the beauty contest game specifically and the iterated dominance game in general may be given a unified structure overarching psychology and economics. This is exactly what this study aims to do. If we successfully complete the circle, a next legitimate question would be what role does learning play? In particular, given the fact that the relation. Ch. engchi. 11. i n U.

(13) 1.2. OVERVIEW between cognitive capacity and cognitive hierarchy has been established in a repeated environment, we may implicitly infer distinct learning processes behind. Then it is natural to explicitly investigate the significance of cognitive capacity on learning process (arrow number five) revealed in BCG (arrow number four) and how these learning behavior may lead to the evolution of cognitive hierarchy (arrow number six). Similarly, studies distributed over these three arrows are not equal, too. As we can see from the figure, there has been considerable attentions on the learning behavior revealed in BCG (arrow number four, colored in black). Few studies, however, have reported the effects on both arrows number five and six (colored in white). The evidences on three inner arrows (number four to six) help us to better understand and examine why cognitive capacity may relate to cognitive hierarchy in a repeated environment.. 1.2. Overview 政. 治. 立. 大. ‧. ‧ 國. 學. The rest of the theis is organized as follows. In Section 2, we review the literature about each of the arrow described before: BCG and cognitive hierarchy (Section 2.1); BCG and cognitive capacity (Section 2.2); cognitive capacity and cognitive hierarchy (Section 2.3); cognitive capacity and learning (Section 2.4); we finally provide a summary of literature (Section 2.5) In Section 3, we shall first provide an overview of the beauty contest experiments conducted (Section 3.1) as well as the measurement of working memory capacity applied in this study (Section 3.2), while the detailed instructions of the BC experiment and WMC task are given in the appendix (Appendix A and B). We then describe how to collect repeated behavioral entries and infer the subjects’ level of reasoning (cognitive hierarchy) from them (Section 3.3). Section 4 examines possible effects of working memory capacity on the individual’s performance (Section 4.1) and the level of reasoning (Section 4.2), in view of the total experiment (Section 4.2.1), every single period and the corresponding dynamics (Section 4.2.2), including the transition of the reasoning levels (Section 4.2.3). In light of these results, we also study the relation between cognitive hierarchy and contest performance (Section 4.3). We finally presents the parameterized learning behavior and explore its possible correspondence to cognitive capacity. (Section 4.4) Section 5 gives concluding remarks and directions for further study.. n. er. io. sit. y. Nat. al. Ch. engchi. 12. i n U. v.

(14) Chapter 2 Literature Review 2.1. Beauty Contest Game and Cognitive Hi政治大 erarchy. 立. ‧. ‧ 國. 學. The beauty contest game, or the guessing game, was first presented by Herve Moulin (Moulin, 1986), and was first studied experimentally by Rosemarie Nagel (Nagel, 1995).1 The game is called the beauty contest game because of the following frequently-cited phrases from Keynes’s General Theory.. n. al. er. io. sit. y. Nat. It is not a case of choosing those [faces] that, to the best of one’s judgment, are really the prettiest, nor even those that average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practice the fourth, fifth and higher degrees. (Ibid, p.156). Ch. engchi. i n U. v. Keynes (1936) used the beauty contest as an analogy to describe the strategic thinking behind the investment behavior. Nagel (1995) grasped the essence of the previous message and redefined Keynes’s beauty contest as a guessing game. Players in the beauty contest game compete with each other to win a prize by selecting a number between [0, 100]. The prize is given to the player whose guessed number is closest to the target number, which is calculated by averaging all guesses then post-multiplied by a factor p, for example, say, p = 2/3. With this parameter, the game is called the p-beauty contest game. 1. Moulin (1986), p.72, introduces the game called “guess the average” as an example to illustrate the idea of the successive elimination of dominated strategies. The beauty contest game has an interesting history. Its origin can be more complex than what one generally thought. The interested reader is referred to Buhren, Frank, and Nagel (2012).. 13.

(15) 2.2. BCG AND COGNITIVE CAPACITY This game implicitly requires each player to form his/her expectations of other players’ expectations. If other players are doing the same thing, the game then suffers from the familiar infinite regress problem. Under the homogeneous rational expectations hypothesis, a Nash equilibrium will be reached where everyone chooses an equilibrium of zero, which is the result of 50 (the middle point between 0 and 100) post-multiplied by pk when k goes to infinity. However, the resultant beauty contest experiments have demonstrated great deviations from this game-theoretic prediction (Nagel, 1998, 2008). For example, the experiment run by the Financial Times in 1997 in collaboration with Richard Tayler showed that the most popular guesses are the ones close to the above multiplication with k being two or three, far away from being infinity (Thaler, 2000). This has motivated some recent progress in cognitive economics, such as Crawford’s level-k reasoning and Camerer’s cognitive hierarchies (Camerer, Ho and Chong, 2004). These two models exhibit similar features in that they reveal the nature of the step-by-step elimination process, but they are different based on the assumptions of human cognitive ability. Crawford’s level-k model assumes that different “k” arise from non-standard beliefs, rather than irrationality (Costa-Gomes, Crawford and Broseta, 2001; CostaGomes and Crawford, 2006). On the contrary, Camerer’s cognitive hierarchy (CH) model assumes that the hierarchy probably arises from the players’ inability to realize the existence of higher level players and probably attribute this to the brain’s limits, such as working memory constraint (Camerer, Ho and Chong, 2004).2 The CH model also presumes that the difference between the perceived and actual level distribution shrinks as the “k” increases. This implies that smarter people are endowed by nature with a better model of others’ thinking and are more capable of making guesses near the target.. 立. 政治大. ‧. ‧ 國. 學. er. io. sit. y. Nat. al. n. v i n Ch 2.2 Beauty Contest e n g cGame h i U and Cognitive Capacity. A number of studies have investigated the relevance of cognitive capacity or intelligence in BCG, but the results are mixed (see the summary in Table 2. Devetag and Warglien (2003) also notice these two different interpretations of the observed cognitive hierarchies or level-k reasoning. They cite Costa-Gomes, Crawford and Broseta (2001) as an example of the interpretation which attributes observed behavioral heterogeneity to the differences in preferences, decision rules and beliefs, while they themselves are inclined to consider the alternative, which attributes the observed behavioral heterogeneity to computational limits. Other work that also addresses this difference includes Grosskopf and Nagel (2008).. 14.

(16) Table 2.1: Literature on the Effect of Cognitive Capacity on Strategic Thinking Games. Measures. Relevance. Camerer (1997). Beauty contest. SAT math. Numbers: null. Ohtsubo (2002). Beauty contest. Imposing memory task (a Theory-of-Mind test). Numbers:↓; Performances:↑. Devetag and Warglien (2003). Normal-form game solvable by iterated dominance; Dirty faces; Extensive-form game solvable by backward induction. Wechsler digit span task. Performances:↑. Beauty contest. Standard psychometric test of general intelligence. Numbers:↓; Performances:↑. Beauty contest. Mathematical calculation task. Performances: null. Simplified beauty contest. Operation span task. Dominance:↑. 2-person guessing game; Undercutting game. General intelligence test; Eye gaze test (a Theory-of-Mind test); Wechsler digit span task; Cognitive reflection test (CRT); One-player takeover game. Level-k: null; Earnings:↑ (only Eye gaze test and CRT). y. sit. io. al. n. Schnusenberg and Gallo (2011). Beauty contest. Gill and Prowse (2012). Beauty contest. Branas-Garza et al. (2012a). Beauty contest. Ch. CRT. engchi U. er. 15. Nat. Georganas et al. (2010). ‧ 國. Rydval et al. (2009). ‧. Coricelli and Nagel (2009). 學. Burnham et al. (2009). 立. 政治大. v i n. Numbers:↓; Clustering:↑ (only matters for initial responses in both measures). Raven test. Numbers:↓ (Own-matched groups); Level-k:↑; Earnings:↑. Raven test; CRT. Numbers:↓(CRT); Dominance:↑(CRT); Level-k:↑(CRT). Due the space limitations, the last column is written in a very compact manner. What is written before the colon includes the key behavioral variables examined and tested by the respective paper. After the colon is the relation found between the cognitive ability and the variable under examination. The positive relation is represented by an upward arrow, whereas the negative relation is represented by a negative arrow, or “null”, if there is no relation found. Taking the first row as an example (Camerer, 1997), “numbers: null” means that the effect of cognitive ability on guessed numbers in the selected game is found to be insignificant.. 2.2. BCG AND COGNITIVE CAPACITY. Authors.

(17) 2.2. BCG AND COGNITIVE CAPACITY 2.1). Camerer (1997), the first study in this direction, did not find its relevance. He demonstrated that Caltech undergraduates, of which half scored the maximum in SAT math, did not choose numbers much closer to the Nash equilibrium than average people, although they did exhibit the lowest median and mean guesses among all subjects in the pool. Coricelli and Nagel (2009) found similar results in an fMRI (functional magnetic resonance imaging) study of BCG demonstrating that mathematical proficiency, defined as the accuracy in calculation task, was unrelated to the ability to match the right guess.3 On the other hand, Burnham et al. (2009) demonstrated the significance of cognitive capacity. In his study subjects were given a standard psychometric test of cognitive ability. It was then found that subjects with higher cognitive ability exhibited significantly lower beauty contest entries, and the average guesses of the smartest group turned out to be the closest to the target number. Schnusenberg and Gallo (2011) also showed that cognitive ability, measured by the cognitive reflection test (Frederick, 2005), contributes to lower and more clustering beauty contest entries in the first round, although the effect will not be sustained in the later periods. These mixed results may not necessarily be conflicting because different measures of cognitive capacity are employed by these studies. In Camerer (1997) and Coricelli and Nagel (2009), the measure is mainly limited to mathematical capability, whereas in Burnham et al. (2009) and Schnusenberg and Gallo (2011), it is based on various kinds of psychological IQ tests. Hence, one possible reconciliation is that, while the BCG is formed as a problem involving numerical calculations, it may rest little on the specific mathematical ability; instead, some general components of intelligence, for example, the ability to suppress an intuitive and spontaneous incorrect answer so as to leave room for a reflective and deliberative one, as Schnusenberg and Gallo (2011) suggest, are what count. Motivated by this initial evidence, this study will take a further examination along this research line to corroborate on the earlier findings. What, however, distinguishes this paper from the earlier ones is the measure of cognitive capacity. We consider a measure that is not exactly the same but closely related to IQ or general intelligence, namely, working memory capacity (WMC).4. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 3 Coricelli and Nagel (2009) actually introduce a measure for subjects’ capability to guess a number that could potentially win against a large population of opponents. They even invent a new term for this measure, called strategic IQ. 4 For the studies showing the strong connection between working memory capacity and general intelligence, the interested reader is referred to Kyllonen and Christal (1990), Engle et al. (1999), and Conway et al. (2002).. 16.

(18) 2.3. COGNITIVE CAPACITY AND COGNITIVE HIERARCHY. 2.3. Cognitive Capacity and Cognitive Hierarchy. WMC, or, sometimes, narrowly known as short-term memory (STM), is one of the most important behavioral invariants, accounting for many cognitive phenomena observed in thinking and learning (Simon, 1990).5 It is a physiological constant that determines the feasible computation required for human information processing, and it is also considered as the main processing component that supports general intelligence (Kyllonen, 1996). On learning, WMC not only helps agents to learn from novel experiences through the construction of mental representation (Cantor and Engle, 1993), but also enables them to practice what they learned in applicable contexts (Daily, 2001; Hambrick and Engle, 2002). Economists’ interest in working memory capacity or short-term memory is rather recent but is growing. It was first applied to study economic experimental games, which normally involve interactive strategic thinking. The first study which relates the working memory to experimental games is Devetag and Warglien (2003). In a number of dominance-iterated games, they showed that the subjects’ depth of reasoning or steps of iterated thinking can be affected by their short memory capacity. Also, Rydval et al. (2009) considered a much more simplified version of the beauty contest game, which involves only two persons but has a noticeable dominant strategy. They found that subjects who did not adopt the dominant strategy were more likely to be those subjects with lower working memory. However, due to their simpler designs, the above two studies did not involve the use of level-k reasoning, and hence did not study the relation between working memory capacity and the parameter k. Georganas et al. (2010), Branas-Garza et al. (2012a), Gill and Prowse (2012) and this study are probably the only four which place this potential relation under examination. Georganas et al. (2010) consider two types of two-person games, namely, a novel undercutting game and a beauty contest game. 174 subjects are classified by their behavior according to the level-k reasoning model, and their cognitive capacities are measured by five quizzes, and one of the five,. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 5. i n U. v. Although there has been great confusion about short-term memory and working memory in the literature of psychology, the distinction between them may be only a matter of semantics (Cowan, 2008). Cowan argues that working memory has been conceived and defined in three slightly discrepant ways: as short-term memory applied to cognitive tasks, as a multi-component system that holds and manipulates information in short-term memory, and as the use of attention to manage short-term memory. In short, working memory contains short-term memory and other processing mechanisms that help to make use of short-term memory.. 17.

(19) 2.3. COGNITIVE CAPACITY AND COGNITIVE HIERARCHY i.e., a digital span memory test, is a direct measure of short memory used by Devetag and Warglien (2003). They found that cognitive capacity in general, including working memory specifically, has limited prediction power for subjects’ levels. In fact, the relative ordering of subjects’ levels is not even stable over different guessing games. The positive relation between cognitive capacity and cognitive hierarchy is found in both Branas-Garza et al. (2012a) and Gill and Prowse (2012). Branas-Garza et al. (2012a) run the 24-person p-beauty contest game, with six different versions of ps, each with one round only. A total of 191 subjects were involved. They found that subjects with high cognitive capacity, measured by the cognitive reflection test (Frederick, 2005), are significantly associated with a high level of reasoning. Gill and Prowse (2012) considered a 3-person 10-round version of the beauty contest game. A total of 510 subjects were involved in their study. The subjects’ cognitive ability was measured by the Raven test, one of the most prominent tests for intelligence, and their behavior in games was estimated and analyzed using a level-k reasoning model with the inclusion of Stahl’s rule learning (Stahl, 1996). With these measurements and estimations, they found that subjects with higher cognitive capacity, on average, had a higher reasoning level than the subjects with lower cognitive capacity. However, the Raven test was also used by Branas-Garza et al. (2012a) as a proxy for cognitive capacity, but was not found significant in its relation to the cognitive hierarchy. This study is complementary to the above three existing studies in the sense that it also directly deals with the relation between subjects’ cognitive capacities and their reasoning levels, but differs from theirs in one of the following two essential aspects, which, we believe, have their significance and can contribute to this body of knowledge. First, instead of running twoperson or three-person beauty games, we run the games on a much larger scale, up to 15 to 20 persons, which is closer to Branas-Garza et al. (2012a). After all, many beauty contest experiments are conducted using a larger size of subjects6 , and so the two-person or the three-person version, while being simple and convenient for analysis, does have its limits since deliberatethinking behavior may change when the environment is complexified by a large number of subjects and interactions.7 Therefore, it is useful to explore. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 6. i n U. v. Among a total of 20 papers included in the survey by Nagel (2008) or in this study, the number of subjects in the beauty contest experiment runs from a minimum of 2 to a maximum of 3,696. Barely half involve less than 10 subjects; for the others the number of subjects is either greater or much greater than 10. 7 The number of agents and their heterogeneities can contribute to the complexity of games and may have a further impact on the outcomes of the game (Ho, Camerer, and Weigelt, 1998; Guth, Kocher, and Sutter, 2002; Kovac, Ortmann, and Vojtek, 2007).. 18.

(20) 2.4. COGNITIVE CAPACITY AND LEARNING the relation between subjects’ cognitive capacity and their cognitive hierarchy in a scaling-up environment, rather than being limited to the analysis of twoor three-person games. In addition to this distinction, our paper and Gill and Prowse (2012) are also distinguished from Georganas et al. (2010) and Branas-Garza et al. (2012a) in that we both have multiple rounds of the experiment, instead of a one-shot game. This design serves the purpose of studying learning well, and is an important subject to which we now turn.8. 2.4 2.4.1. Cognitive Capacity and Learning Cognitive Capacity, Cognitive Hierarchy and Learning. 政治大. Learning is normally not assumed in level-k reasoning, as if the model can best apply or only apply to the situation when the environment presented to the subject is novel. Once agents have experiences and have learned, their level of reasoning becomes irrelevant since they may all be updated. This concern implicitly assumes that learning can eliminate the initially observed heterogeneity in subjects’ levels of reasoning, and therefore, subjects’ cognitive capacity should be irrelevant when learning takes effect. For example, the main feature found in repeated BCG is that behavior converges to equilibrium over time. If all subjects converge to choose equilibrium zero, there will be no behavior heterogeneity. In this case, we should not expect a cognitive hierarchy revealed and the effect of cognitive capacity. This “convergence hypothesis” has been well supported by Schnusenberg and Gallo (2011) in their repeated beauty contest game, while their analysis did not bring in the element of cognitive hierarchy. This results implies that learning dominated, or even stronger, that learning is independent of cognitive capacity in the repeated environment. However, there are also experimental studies showing that learning is not independent of cognitive capacity. By that, subjects’ learning capability or dynamics can be affected by their cognitive capacity. For example, Casari, Ham and Kagel (2007) showed that subjects with lower composite SAT/ACT scores were more likely to suffer from a winner’s curse. Even if they are re-. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 8. i n U. v. Branas-Garza et al. (2012a) have only one round for each of their six different versions of the beauty contest game, but the use of this multiple-version design may still allow subjects to learn from their experiences such as the learning observed in the multiple-stage game. Nevertheless, the learning dynamics, i.e., how learning can happen by carrying over the experience from one version of the game to other subsequent versions, is not the focus of their analysis.. 19.

(21) 2.4. COGNITIVE CAPACITY AND LEARNING cruited as experienced subjects participating in similar experiments, they can still not avoid the “curse”. This implies that the effect of cognitive capacity persists even after learning is taken into account. Gill and Prowse (2012) also found that the performance gap caused by cognitive capacity will not decay away simply due to learning. In fact, their study shows that cognitive capacity may positively affect learning in the way that subjects with higher cognitive capacity may learn more actively than subjects with lower cognitive capacity and hence, in the end, their performance gap will become even more significant than the initial time.9 In addition, they also applied a level-k learning model to understand the behavioral mechanism that leads to these differences. Gill and Prowse (2012) found that high cognitive ability subjects follow level-k choice type that are significantly higher than low ability subjects, either in own-matched group or in cross-matched group. In our case, the issue is then whether subjects of different working memory capacity may actually learn in the way that their initial levels of reasoning, absolutely or relatively, remains unchanged, or, alternatively, in the way that the initial gaps among subjects can be filled. In other words, our study examines, to what extent, the level-k reasoning holds for not just a one-shot game, but may in effect work also in repeated games. More generally, this study makes a fundamental inquiry into the effect of cognitive capacity on cognitive hierarchy in contexts with experiences and without experiences.. 立. ‧. ‧ 國. 學. 2.4.2. 政治大. Cognitive Capacity and EWA Learning. y. Nat. sit. n. al. er. io. Learning behavior in BCG has been widely studied by applying various learning models (Nagel, 1995; Stahl, 1996; Duffy and Nagel, 1997; Stahl, 1998; Ho, Camerer, and Weigelt, 1998; Camerer and Ho, 1999). Among them, a generalized reinforcement learning model, said, experience-weighted attraction (EWA) learning was proposed to combine reinforcement and belief learning, two seemingly unrelated mechanisms, and to include them as special cases. EWA learning was created by Camerer and Ho (1998, 1999) to model the learning behavior in several repeated games. BCG is one of them. The formulation of EWA learning model is described as follows. In general, the choice probabilities of strategies are determined by the logit transformation of their attractions, which reflect initial propensity and are updated according to payoff experience. There are two variables, attractions Aji (t) and an experience weight N(t) in EWA model. Both variables are updated. Ch. engchi. 9. i n U. v. Gill and Prowse (2012) found that, while in the first five rounds there is no significant difference in earnings between high and low cognitive ability subjects, their earnings difference becomes bigger in the last five rounds, and the difference is statistically significant.. 20.

(22) 2.4. COGNITIVE CAPACITY AND LEARNING according to the experience after each period. The first variable N(t) begin with an initial value N(0) and is updated according to N(t) = ρN(t − 1) + 1, t ≥ 1 The parameter ρ is a discount factor that captures decay in the strength of prior beliefs. The second variable Aji (t) denote player i’s attraction of strategy j. It starts with some prior values Aji (0) and is updated according to. Aji (t) =. φ · N(t − 1) · Aji (t − 1) + [δ + (1 − δ) · I(sji , si (t))] · πi (sji , s−i (t)) N(t). where I(x, y) is an indicator function that equals 1 if x = y and 0 if x 6= y, sji denotes player i’s strategy j, si (t) is player i’s choice at time t, s−i (t) = (s1 , ..., si−1 , si+1 , ..., sn ) is a strategy combination of all other subjects at time t. In general, the attraction Ai (t) is the running total of past attractions, which are constituted by a depreciated experienced-weighted past attraction Aji (t) plus the payoff yielded from period t. The key component of this updating rule is the weighted payoff term [δ + (1 − δ) · I(sji , si (t))] · πi (sji , s−i (t)), which captures two basic principles of learning. First, the attractions of chosen strategies si (t) are updated by actual payoff, which means that successful strategies are given more reinforcement and are more likely to be repeated subsequently. This is so called the ’law of effect’ (Thorndike, 1911; Herrnstein, 1970) in the literature of learning by behavioral psychologist. Second, the attractions of unchosen strategies are updated by forgone and hypothetical payoff with a weight δ (0 ≤ δ ≤ 1). Camerer and Ho (1999) introduced this effect and called it the law of simulated effect and renamed the former one as the law of actual effect. In this setting, both chosen and unchosen strategies are ’reinforced’ by the payoff that strategy either yielded or would have yielded. EWA model can be reduced to choice reinforcement or belief-based models under different parameters configurations. In other words, reinforcement learning and belief learning are special cases of EWA learning. Define κ = (φ − ρ)/φ. The family of choice reinforcement model corresponds to δ = 0. With further restriction of parameter κ, the EWA learning can be either cumulative choice reinforcement when κ = 1, or averaged-reinforcement when κ = 0. The family of belief learning model corresponds to δ = 1. Similarly, further restricting some parameters derive specific belief learning models such as Cournot best-response dynamics when φ = κ = 0, standard fictitious play when κ = 0 and φ = 1, weighted fictitious play when κ = 0.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 21. i n U. v.

(23) 2.5. A SUMMARY OF LITERATURE Note that δ is the most important parameter in EWA because it shows the way that EWA captures two learning principles, the law of actual effect and the law of simulated effect, is different from reinforcement and belief models. In fact, EWA require that the strength of these two effects is only a matter of degree, while reinforcement assumes that only actual effect matters (δ = 0) and belief model supposes that actual and simulated effects are equally strong (δ = 1). By investigating the empirical δ, we can identify the subjects as being reinforcement (δ = 0), belief learning (δ = 1) or in between (0 < δ < 1). Would this heterogeneity in learning behavior corresponds to heterogeneity in cognitive capacity? In other words, would belief learning require more capacity than reinforcement learning? To distinguish the cognitive requirement of learning models designed for calibrating heterogeneous agents, Chen (2012) has proposed three cognitive elements, including memory, consciousness and reasoning. Reinforcement learning agents requires some capacity of memory to retrieve their own past experience of success. However, they are not conscious about the existence of other players and it’s impossible for them to reason based on the entire environment they are embedded. On the other hand, belief learning agents are endowed with additional cognitive ability, said, some degree of consciousness and reasoning. They are aware of the presence of their opponents and also taking their opponents’ decisions into account. In fact, belief learning agents count the frequencies of their opponents’ past actions, calculate the foregone payoffs on which their future decisions are based. EWA allows the possibility that the agent’s cognitive ability to imagine all forgone payoffs is partial by introducing δ, ranging from 0 to 1. Chen (2012) actually gives us a correspondence between the spectrum of various learning models and the levels of cognitive capacity.. 立. 政治大. ‧. ‧ 國. 學. er. io. sit. y. Nat. al. n. v i n C h of Literature 2.5 A Summary engchi U. Few studies investigated the relationship between cognitive capacity and cognitive hierarchy, two closely related concepts, one is from psychology and the other is from economics. Recent works began to address this issue, yet all of them applied at least one simplification of game structure. They either performed one-shot game to exclude the effect of learning or conducted twoperson or three-person game to reduce game complexity. However, these two features reflect real economic decisions, in particular, investment decisions. Investors compete with thousands of participants in the financial market. They also make a sequence of decisions based on the experiences of previous trades. Our study was the first to investigate how cognitive capacity and 22.

(24) 2.5. A SUMMARY OF LITERATURE cognitive hierarchy may related in a larger scale environment when learning takes effect. The primary research questions to be addressed in this study are as follows: • First, would cognitive capacity have a positive effect on subjects’ revealed cognitive hierarchies? • Second, would this effect endure when subjects become more experienced? • Third, would cognitive capacity affect the way in which subjects learn? Our predictions are as follows. According to the results of current studies, cognitive capacity should be related to cognitive hierarchy at least in initial rounds. Both the second and the third ones are open questions because many aspects of experimental design, such as time horizon and provided feedback information, would also affect the results. There is no clue from previous literature and we didn’t address these issues in this study. Our specific experimental design serves as initial attempts to answer the later two questions. In considering the the third question, in particular, we hope to related working memory capacity to the counterfactual simulation of forgone payoff, the parameter δ in EWA.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 23. i n U. v.

(25) Chapter 3 Method 3.1. Beauty Contest Experiment 政治. 大. 立. ‧. ‧ 國. 學. The experiment consisted of 6 sessions and there were 15-20 subjects in each session, there being a total of 108 subjects involved. The subjects were required to complete both a repeated beauty contest game and then a working memory test. All experiments were conducted in the Experimental Economics Laboratory (EEL) of National Chengchi University from October 2009 to August 2010. Experiments were announced on the NCCU EEL web site1 and on the part-time job board in PPT, one of the most popular bulletin board systems in Taiwan. Subjects were required to register through the NCCU EEL Registration System2 and sign up for our experiments. After signing up for one experiment, subjects immediately received an e-mail for confirmation. The BCG was conducted by means of a z-tree (Fischbacher, 2007). For each period, subjects were required to select an integer number between [0, 100] and competed with all of the others in the session. The prize was given to the one whose guess number was closest to the target number, denoted by τ , which was calculated by averaging all guesses and then multiplying the result by a factor p = 2/3. After collecting all subjects’ guesses, the screen would display feedback information regarding the target number, the subject’s chosen number, profit and cumulated profit. The BCG was repeated 10 times and it took about 60 minutes to finish. In addition to a fixed showup fee of NT$125, we also provided a prize for the winner in each period of NT$100, and the prize was to be evenly split if there was more than one winner. The details of the instructions for the subjects are given in Appendix. n. er. io. sit. y. Nat. al. 1 2. Ch. engchi. see http://eel.nccu.edu.tw/ see http://eel.nccu.edu.tw/Registration/. 24. i n U. v.

(26) 3.2. WORKING MEMORY TASK A.. 3.2. Working Memory Task. The task we used for eliciting working memory capacity was developed by Lewandowsky et al. (2010). This task includes 5 tests, a backward digitspan task (Dspan), a spatial short-term memory test (SSTM), a memory updating task (MU), a sentence-span task (SentSpan), and an operationspan task (OpsSpan). The WMC task was always conducted after finishing beauty contest experiment and administrated in the order of DSpan, SSTM, MU, SentSpan, OpsSpan for all subjects. It took about 90 minutes to finish all 5 tests. NT$200 was paid to those subjects who completed all five sets. The score for each test was calculated and then normalized by the mean and standard deviation of the scores derived from our Experimental Subject Database (ESD), which included 740 subjects completing the same task. Then a single measure of working memory capacity was derived by averaging these five normalized scores. We then briefly describe the stimuli, design and procedure of all five tests.. 立. 政治大. ‧ 國. 學. ‧. Dspan This task was to recall a set of digits in reverse order. Following a fixation cross presented for 1 sec, a set of 4 to 8 digits were displayed one by one, for 1 sec each. After that, subjects were required to enter this set of digits in reverse order without time constraints. There were 15 trials total, 3 at each set size.. io. sit. y. Nat. n. al. er. SSTM The subjects were required to memorize the location of a set of dots in a 10×10 grid. This task started with a fixation cross for 1 sec and the grid was shown. There were 2 to 6 solid dots appeared, one by one, in individual cells, for 900 msec each. The interstimulus interval was 100 msec. The subjects were instructed to remember the spatial relation of the dots instead of absolute position of each dot. After presenting all of the dots, the subjects were asked to replicate the pattern of dots. There were 30 trials, 6 at each set size.. Ch. engchi. i n U. v. MU This task was to encode a set of digits, each presented sequentially in a set of frames, and then to update these digits by arithmetic operations. On each trial, the subjects are presented with 3 to 5 frames containing to-beremembered digits in each. Each trial was initialized by a keypress and then the initial digits were displayed one by one, for 1 sec each. After that, 2 to. 25.

(27) 3.3. COGNITIVE HIERARCHY 6 arithmetic operations, such as “+3” or “-1”, were displayed in individual frames one by one for 1.3 sec each and followed by 250-msec blank interval. Subjects were required to apply these operations to the digits that they currently remembered in that particular frame and to update the content with the result. There were 15 trials in total. SentSpan On each trial, an alternating sequence of Chinese sentence and to-be-remembered consonants were presented. The subjects had to judge the meaningfulness of sentences and to remember the following consonants for later serial recall. The sentences were composed of 17 Chinese characters. For example, a meaningful sentence might be I went out without taking any money, but fortunately I ran into an old friend who helped me out. Replacing fortunately with unfortunately obtained the meaningless counterpart of this sentence. Following a fixation cross presented for 1.5 sec, subjects saw the first sentence appeared on the screen. It disappeared either when subjects made a response or after the maximal response time of 5 sec had elapsed. The subjects were instructed to use the “/” and “z” key to make Yes, this is correct and No, this not correct responses, respectively. After the judgement of sentence, a consonant was presented for 1 sec. After the consonant disappeared, next sentence appeared. List length, defined as the number of sentences and letters needed to be judged and remembered, ranged from 4 to 8. There were 15 trials, 3 trials per list length.. 政治大. 立. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. OpsSpan This task was almost the same as the SentSpan task except that the subjects had to judge the correctness of arithmetic equations (e.g., 3+2 = 5). A minor difference is that the maximum response time to equation was set to 3 sec due to the simplicity of this processing task.. 3.3. i n U. C. v. h e n g c hofi Cognitive Hierarchy Characterization. A simple way to identify subjects’ reasoning levels-k, known as the Cournot myopic best response algorithm, was proposed by Nagel (1995). Denote player i’s guess in period t by gi (t) and the number of players in session j by nj . For t = 1, player i is classified exactly as   k0 (level 0), if gi (1) = 50,     k1 (level 1), if gi (1) = 50 × p ≈ 33.33, (3.1)  k2 (level 2), if gi (1) = 50 × p2 ≈ 22.22,     k3 (level 3), if gi (1) = 50 × p3 ≈ 14.81. 26.

(28) 3.3. COGNITIVE HIERARCHY (a) Define exact levels ki k3. 2 2 2 3 50 ⋅ ( 3 ) 50 ⋅ ( 3 ). 0. k0. k1. k2. 2 50 ⋅ ( 3 ). 100. 50. . . (b) Define adjacent intervals d >3. d =3 d =2 k2. k1. 2 2.5 50 ⋅ ( 3 ). 2 1 .5 50 ⋅ ( 3 ). k3 0. 2 3.5 50 ⋅ ( 3 ). d =1. d <0. d =0. 2 0.5 50 ⋅ ( 3 ). 50 =. 100. k0. . Figure 3.1: Level classification procedure for guesses in period 1. 治政大 (a). Although subjects may These critical values are shown in Fig. 3.1, panel not choose these 立 critical values, making a guess closer to any one of them ‧. ‧ 國. 學. could be roughly considered as belonging to the same level. Therefore, we divide [0, 100] into several adjacent intervals corresponding to different levels (see Fig. 3.1, panel (b)). For t > 1, the subjects are given information about the previous target number. It is plausible that they make the best response to the behavior in the previous period by assuming others to be the same. To sum up, formally, subject i will be classified as level d in period t, and denoted by di (t), if3 ( m(t − 1)pd+0.5 , m(t − 1) , if d = 0, (3.2) gi (t) ∈ d+0.5 d−0.5 m(t − 1)p , m(t − 1)p , if d 6= 0.. n. er. io. sit. y. Nat. al. where. Ch. e(n50.g c h i. m(t − 1) =. 1 nj. P nj. i=1. i n U. v. if t = 1,. gi (t − 1), if t > 1.. Nagel (1995) found that d = 0, 1, 2 and 3 can identify approximately 80% or more guesses. The subjects with guesses larger than the upper limit of 3. In Equation (3.2), the upper limit of the level d = 0 is bounded from the right side by m(t − 1), instead of m(t − 1)p−0.5 . About this asymmetry, Nagel (1995) indicates that the results, for the first period, would not change if a symmetric bound were to be taken instead (ibid, p.1317). For the later periods, it is pointed out that the chosen numbers tend to be below the mean of the previous period (ibid, p.1320). To make our results comparable with those of Nagel (1995) (Section 4.2), the same asymmetric bound is taken in our analysis.. 27.

(29) 3.4. EWA LEARNING MODEL the level zero (d = 0) are grouped into “d < 0”. Similarly, the subjects with guesses smaller than the lower limit of the level 3 (d = 3) are grouped into “d > 3”. The subjects, finally, are categorized into 6 classes: d < 0, d = 0, 1, 2, 3 and d > 3. In addition, the median reasoning level for each subject, denoted by di , is obtained by simply transforming these 6 ordinal classes to 6 numbers, from 0 to 5, and calculating the median of each subject’s 10 corresponding levels.. 3.4 3.4.1. EWA Learning Model The Model. We will apply different versions of EWA learning model to structurally characterize the learning behaviors in repeated BCG.. 立. 政治大 Camerer and Ho (1999) provide the first attempt to. ‧. ‧ 國. 學. Camerer’s EWA calibrate the EWA parameters with the data in repeated BCG. They proposed some necessary details of design, in addition to the general structure described in Section 2.4.2, in order to implement the model, in particular, in the context of BCG. The strategy space for the subjects is in the interval [0, 100] with only integers allowed. These 101 strategies should be endowed with some initial attractions Aj (0) and being reinforced over time. If we treat all Aj (0) as the model parameters, we have to calibrate all of these 101 attractions. By simply assuming that initial attractions are equal in ten-number intervals [0, 9], [10, 19], ... [90, 100], we can alleviate the problem of using too many degrees of freedom. Therefore, the first design detail is that 101 initial attractions will be reduced to only 10, denoted by A1 (0), A2 (0), ..., A10 (0). We will call it as Assumption A1 later on. Second, we also assume that subjects know the winning number w = arg mingi [|gi − τ |] (Assumption A2) and they also neglect the effect of their own guesses on the target number (Assumption A3). In fact, A2 is not true in both our and Camerer and Ho (1999)’s designs of experiment, because we only show the target number to the subjects and the winning number is unknown, unless they happen to be the winner of that particular round. Denote the distance between the winning number and the target number as d = |τ − w|, and also denote the prize for each round as nπ where n is the number of winners in a particular round. By the assumption A2 and A3, we can easily define the reinforcement intervals without introducing additional parameters. In particular, all subjects reinforce numbers in the intervals (τ − d, τ + d) by δnπ. The winners reinforce what they chose, which is one of the boundary number, either τ − d. n. er. io. sit. y. Nat. al. Ch. engchi. 28. i n U. v.

(30) 3.4. EWA LEARNING MODEL or τ + d, by π, and reinforce the other boundary number by δnπ. The losers reinforce both boundary numbers τ − d and τ + d by (δnπ)/(n + 1). The assumptions A1 and A2 were later removed and replaced in Camerer, Ho and Chong (2002). First, instead of treating initial attractionsI Aj (0) as parameters and simultaneously calibrating them with other EWA parameters, Camerer, Ho and Chong (2002) empirically obtained Aj (0) from the choices data in first period. Formally, we recover initial attractions by the following equations j. eλ·A (0) P10 λ·Ak (0) = f j , j = 1, ..., 10. k=1 e. where f j represent the observed probability of strategy j in the first period. We have 11 unknown variables, including 10 Aj (0) plus a λ, but only 10 conditions. The number of parameters are too many to be identified. Therefore, we fix the strategy j with the lowest f j to have Aj (0) = 0. Following Camerer, Ho and Chong (2002), we also remove the unrealistic assumption of A2 and introduce an additional parameter d to describe how losers compute forgone payoffs. We assume they reinforce numbers in the interval [τ − δnπ , τ + δnπ ]. The amount of reinforcement is δnπ at the target number, d d which is the maximum. Departing from the target number, the amount of reinforcement decrease at a rate of d, which is a parameter to be estimated. The forgone payoff for the losers to be reinforced will be of a triangular form(see the panel (b) of Figure 3.2). Camerer, Ho and Chong (2002) also assign this parameter d to the winners and assume that they reinforce the numbers in the interval [τ − e, τ − e − δnπ ] and [τ + e, τ + e + δnπ ] with similar d d triangle form of reinforcement amount, if there is only one winner. Note that we didn’t incorporate the parameter d to the winners for the reason of simplicity. We estimated both versions of EWA model described above.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. EWA rule learning In both versions of EWA model in Camerer and Ho (1999) and Camerer, Ho and Chong (2002), subjects are assumed to be able to initialize 101 strategies attractions, maintain all of them in memory and then update over time. Considering the limited capacity of memory and attention in human brain, it is unlikely for subjects to learn in the way as Camerer’s EWA. One plausible approximation is to assume that subjects apply only few rules during the experiments. By the descriptive power of level-k model, revealed in repeated BCG (Duffy and Nagel, 1997), we assume that subjects apply and reinforce level-k rules instead of 101 numbers. According to the level classification in Section 3.3, the strategy space be29.

(31) 3.4. EWA LEARNING MODEL. 立. 政治大. ‧. ‧ 國. 學. Figure 3.2: Reinforcement and the corresponding interval for winner and loser in beauty contest experiments. er. io. sit. y. Nat. comes sj ∈ {d < 0, d = 0, d = 1, d = 2, d = 3, d > 3}. In this setting, subjects are only required to initialize, maintain and update 6 attractions Aji (t), where j ∈ {1, 2, 3, 4, 5, 6}. We define the payoff function as follows, ( 100 if sji = target-d, πi (sji , s−i (t)) = 0 if sji 6= target-d.. al. n. v i n where target-d denotesCthe level in which the target number located. Besides U h i e h n target-d, all other levels will not g bec reinforced. The amount of reinforcement 4. for target-d will be 100 if it is chosen and will be 100δ if it is not chosen. 4. Remember that, in our definition, a level is an interval including several numbers.. 30.

(32) 3.4. EWA LEARNING MODEL. Table 3.1: Summary of Estimated Learning Models Parameters Strategy. N(0), φ, ρ, δ, λ. Aj (0). Model I. Number. Estimated. Estimated. Model II. Number. Estimated. Initialized. Model III. Number. Estimated. Initialized. Model IV. Level-k rule. Estimated. Estimated. Model V. Level-k rule. Estimated. Initialized. d. Estimated. 政治大. Model I is referred to Camerer and Ho (1999). Model II and Model III are referred to Camerer, Ho and Chong (2002). 立. ‧. ‧ 國. 學. Summary of estimated learning models Table 3.1 summarize the models which were estimated in this study. Model I, II, III tried to replicate EWA learning proposed by Camerer and Ho (1999) and Camerer, Ho and Chong (2002). The common feature of these Camerer’s EWA learning is that subjects try to learn the attractiveness of all guessing numbers. On the contrary, Model IV and V assumed that subjects try to learn the attractiveness of levelk rules. We also tried some perturbations and modifications, including that initializing initial attractions Aj (0) by observed probability of strategy in the first period and introducing additional parameter d to replace the unrealistic assumption about the knowledge of winning number.. n. a l Strategy Estimation Ch. engchi. er. io. sit. y. Nat. 3.4.2. i n U. v. Attractions are transformed to the choice probability by the logit function which is given by j. Pij (t. eλ·Ai (t−1) + 1) = Pmi λ·Ak (t−1) i k=1 e. where mi denotes the number of choices, and mi = 101 in Camerer’s EWA and mi = 6 in our EWA rule learning. Define player i’s initial attractions i as a vector Ai (0) ≡ (A1i (0), A2i (0), ....Am i (0)). We assume a representative agent and calibrate his learning behavior, so Ai (0) = A(0) ∀i. The number of subjects is denoted by N and the total sample size, 10 · N is denoted by M. Then the log-likelihood function LL(A(0), N(0), φ, ρ, δ, λ), is 31.

(33) 3.4. EWA LEARNING MODEL. LL(A(0), N(0), φ, ρ, δ, λ) =. 10 X N X. ln. t=1 i=1. =. 10 X N X. mi X. I(sji , si (t)) · Pij (t). j=1. ln. t=1 i=1. mi X. j. I(sji , si (t)). j=1. We also imposed some restrictions of parameters, 0 ≤ Aj (0) ≤ 1000,∀j φ > 0, λ > 0, 0 ≤ ρ ≤ 1, 0 ≤ δ ≤ 1, 1 0 ≤ N(0) ≤ 1−ρ. eλ·Ai (t−1) · Pmi λ·Ak (t−1) i k=1 e. !. 政治大. 學. ‧ 國. 立. !. ‧. The parameters are first estimated by the choices data of all 108 subjects in all ten rounds. In this way, our results can be compared with the original work done by Camerer and Ho (1999) and a modified version by Camerer, Ho and Chong (2002). We then separated our data into two groups, high and low WMC, and obtained two sets of parameter estimates, to see how cognitive capacity affect learning behaviors. We tried several numerical nonlinear global optimization methods in Mathematica, including simulated annealing, Nelder-Mead, random search and differential evolution, to maximize the likelihood function. When a specific method derived superior results, we further explored some options of this method, such as number of search points, number of random seeds and post process for local search, to avoid reporting local optima.. n. er. io. sit. y. Nat. al. Ch. engchi. 32. i n U. v.

(34) Chapter 4 Results. 政治大. We conducted 6 experimental sessions with 108 subjects. In what follows, we will first present the results of the effects of cognitive capacity on guessing performance (Section 4.1), i.e., the lower left edge of Figure 1.1. We then move to the lower right edge of Figure 1.1 and present the effects of cognitive capacity on cognitive hierarchy (Section 4.2). In both cases, the effects of cognitive capacity are not just studied in a static manner, but also in a dynamic fashion by taking learning into account. This can be illustrated by our examination of not just the reasoning level distribution conditional on different WMCs (Section 4.2.1), but also the evolution of this distribution over time (Sections 4.2.2 and 4.2.3). Finally, we shall come to the upper edge of Figure 1.1 and present our findings on the connection between guessing performance and the cognitive hierarchy (Section 4.3).. 立. ‧. ‧ 國. 學. er. io. sit. y. Nat. Cognitive Perfora l Capacity andi Guessing v n mances C h e U i. n. 4.1. ngch. Do subjects with higher working memory capacity (WMC) perform better than subjects with lower WMC? To answer this question, we have to decide a performance measure. In this context, the subjects’ performance can be measured by their behavior (guessing accuracy) or the consequence of their behavior (payoffs). The two measures will be equally good if the latter is a strict monotone transformation of the former. However, by the usual payoff design of the beauty contest experiment, this is not the case; we, therefore, consider the guessing accuracy to be a more direct and informative measure for performance than the payoffs. To measure subject i’s guessing accuracy in period t, we consider the 33.

(35) 4.1. WMC AND GUESSING PERFORMANCE. Table 4.1: Correlation coefficients between WMC scores and guessing differences in the beauty contest experiment Kendall’s τb. Spearman’s ρ. Period. Estimates. p-value. Estimates. p-value. 1. −0.2368∗∗. 0.0003. −0.3415∗∗. 0.0003. 2. −0.1600. 0.0144. −0.2339. 0.0148. 3. −0.0957. 0.1441. −0.1349. 4. −0.0040. 5. −0.1601. 0.1639 0.7854 0.0297 0.1691. 0.0041. 0.9512. −0.0004. 0.9967. −0.0401. 0.5444. −0.0617. 0.5255. −0.0797. 0.2285. −0.1119. −0.1366. 0.0537. −0.2057. −0.2016∗∗. 0.0020. −0.2944∗∗. 0.0020. −0.1606∗. 0.0138. −0.2321∗. 0.0158. −0.1433∗. 0.0279. −0.2059∗. 0.0325. 4-10. 0.0591. 5-10. −0.0982. 6-10. −0.0627. 7-10. −0.03. 8-10. −0.0776. 9-10. −0.1663. 10. 2-10. io. 3-10. Nat. 1-10. al. −0.1231. Ch. ∗. 0.1322 eng chi. ‧. 9. ∗. er. 8. ‧ 國. −0.1333. 學. 0.1871. y. ∗. sit. 7. ∗. 治 −0.0265 政 0.9513 大 0.0156 −0.2094. ∗. n. 6. 立 −0.0875. ∗. −0.1762 iv n U −0.1503. 0.2488 0.0327. 0.0681 0.1206. 0.3365. −0.0913. 0.3475. 0.646. −0.0447. 0.6458. 0.2353. −0.1098. 0.0113. −0.2450. 0.258 ∗. 0.0106. and ∗∗ denote the significance of estimates at the 5 percent and 1 percent levels, respectively. ∗. 34.

(36) 4.1. WMC AND GUESSING PERFORMANCE absolute error: εi (t) = |gi (t) − target(t)|, where target(t) = 23 m(t). With this performance measure, we are curious to know whether there is any correlation existing between the cognitive capacity (WMC) and the performance measure. To find out, we run the rank correlation between WMC and εi (t) for each t (t = 1, 2, ..., 10). Table 4.1 gives the Kendall rank coefficient (the 2nd column) and the Spearman rank coefficient (the 4th column). The table is presented in two parts: the upper panel shows the coefficient for each single period, whereas the lower panel shows the coefficient over the consecutive periods from the specified beginning period to the final period. Therefore, the first row of the lower panel, “1-10”, gives the aggregate result, i.e., the rank coefficient by taking into account all subjects over all periods. Table 4.1 shows that this result is negative for both the Kendall rank coefficient (τb ) and Spearman rank coefficient (ρ), and the minus sign is statistically significant (columns 3 and 5). This result suggests that, as a whole, subjects with higher WMC perform better, and hence supports a positive effect of cognitive capacity. However, if we break down this aggregate result into individual periods (the upper panel of Table 4.1), negative correlations are also observed in all periods, while only a few of them are statistically significant.1 A closer look indicates that these few results occur mainly in the initial periods, say, periods 1 and 2 (the upper panel) and in those windows which begin with these initial periods, such as windows “1-10”, “2-10” and “3-10” (the lower panel). This suggests that the WMC effect in the initial periods substantially influences the overall significance. Then, as time goes on, subjects may learn with experience, and their learning may help to narrow the original gap in cognitive capacity. However, the question is: will the effect of learning completely annihilate the effect of cognitive capacity, and will the subjects with lower WMC eventually catch up in their performance compared to the subjects with higher WMC? Would the effect of cognitive capacity persistently matter being conditional on the effect of learning, or will the effect of learning eventually dominate and make up for the deficit in talents? This question is a little intriguing because it is difficult to know in general what is the minimum time required for the learning to work, and all experiments must inevitably be based on a prechosen limited duration to avoid human fatigue. However, even though only ten iterations are run for each experiment, we believe that, in the context of BCG, it will be reasonably long enough to examine the above catching-up or convergence hypothesis.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. 1. Ch. engchi. i n U. v. The only exception is period 7 (τb ), but it is also insignificant.. 35.

(37) 4.1. WMC AND GUESSING PERFORMANCE. . Figure 4.1: Guessing Errors between the High WMC groups and the Low WMC groups. 立. 政治大. ‧. ‧ 國. 學. Subjects are divided into the high WMC groups and Low WMC groups using different thresholds. The six curves shown above are the mean guessing errors of the six groups, the bottom one-fourth (WMC < P25 ), the bottom one-third (WMC < P33 ), the second half (WMC < mean), the top one-fourth (WMC < P75 ), the top one-third (WMC > P67 ), and the first half (WMC > mean). The numbers inside the brackets are the number of subjects belonging to the respective group.. n. al. er. io. sit. y. Nat. To better trace the effect of learning, a more apparent way to present the result is to draw the guessing error over time and over groups with different WMCs. This is done in Figure 4.1. Since there are 108 subjects possibly with 108 possible learning curves, drawing all of them in one figure will not help us see anything. We, therefore, present the learning curves in groups. We first divide the subjects into two groups, one with higher WMCs, and one with lower WMCs. We consider three different thresholds to group them; they are the mean, one third and one fourth. Based on these three thresholds, subjects or subsets of subjects are divided into two groups, the bottom (the bottom one-half, the bottom one-third, and the bottom one-fourth) and the top (the top one-half, the top one-third, and the top one-fourth). With a little abuse of the notations, we can denote these groups by percentiles; hence, P25 (the 25th percentile) denotes the bottom one-fourth, P33 the bottom one-third, P67 the top one-third, and P75 the top one-fourth.2 We then figure out the mean guessing error of each group in each period,. Ch. engchi. 2. i n U. v. All the thresholds or percentiles are computed based on our Experimental Subject Database (ESD) which includes the WMC scores of 740 subjects.. 36.