B.O. Vaandrager (11931175)
University of Amsterdam
MSc Business Economics
Dr Z.E. Öztürk
Machiavelli and Keynes:
‘Does Priming of a Theory of Mind Improve
Strategic Reasoning in Keynesian Guessing
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Executive Summary
Extending from the findings in Coricelli and Nagel (2009) that the medial prefrontal cortex is essentially the “neural signature of high-level strategic reasoning”, this thesis studies the possibility of improving strategic competence by priming players with a Theory of Mind (an ability mediated by the mPFC) prior to competing in the Number Guessing Game. Capitalising on experimental data, this these concludes that an entry game task may instigate a deeper level of strategic reasoning and results in superior performance compared to a control group. Furthermore, a correlation was found between the degree of priming attained by the entry game task and a measure of Strategic IQ. A second treatment condition involving a ToM story task, on the contrary, did not produce any significant results. Outcomes suggest that entry game-like tasks may benefit strategic reasoning of economic agents in competitive interaction. In addition, this study contributes to our understanding of the mechanisms governing strategic thought and social cognition.
Statement of Originality
This document is written by Student Bastiaan Ole Vaandrager who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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Table of Contents
I.
Introduction
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II.
Related Literature
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Evolutionary Foundations of Social Cognition
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Neural Correlates of Strategic Reasoning
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The Number Guessing Game and Level-k Thinking
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III. Methodology
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Experimental Design and Procedures
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Formulation of Hypotheses
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IV.
Results
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H1 – Depths of Strategic Reasoning
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H2 – Optimal Choice and Strategic IQ
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H3 – Degree of Priming and Performance
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H4 – Adaptive Behaviour of Strategic Types
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V.
Conclusion
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VI.
Appendices
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I.
Introduction
“It is double pleasure to deceive the deceiver.” – Niccolò Machiavelli
It may be argued that some primate species – and mankind in particular – are much cleverer than they need be. It contradicts the basic principles of evolution to find animals with capacities that far exceed the needs for everyday survival and reproductive success. A growing body of evolutionary psychology and neuroscience literature suggests our intelligence (and that of non-human primates) is, ultimately, an adaption to the complexities that arise from living in large social environments. Group living implicates having to constantly compete with others for shelter, resources and sexual partners, thereby exerting evolutionary pressures on cognitive development (Humphrey, 1976; Whiten, 2000; Brüne, 2005). The term Machiavellian
Intelligence – a reference to teachings of Renaissance philosopher Niccolò Machiavelli –
appropriately illustrates the idea that our intelligence is primarily the result of social competition: the need to, essentially, outsmart others in socially complex environments (De Waal, 1982; Byrne and Whiten, 1989).
If increased intelligence is, indeed, the adaptive solution to adequately dealing with socially complex situations, then exactly what cognitive abilities have we developed? A number of traits have been identified as being selective group living, including competent visual discrimination (‘who am I dealing with?’) and efficient memory (‘what is my history with this person?’) (Byrne, 1997). Imperative to sustaining cooperative behaviour is, also, the ability to successfully anticipate the intentions of other group members, for example: is the person I am dealing with more likely to cooperate or defect? Theory of Mind refers to the cognitive machinery to infer the mental states of others and understand the consequences of one’s own action to others. Theory of Mind, or mentalising, is deemed indispensable to fruitful social human interaction as it allows us to integrate the (predicted) actions of other individuals into our own decision making. Likewise, the success of economic agents often depends on accurately inferring the opposing agent’s state of mind – think of stock markets, wage negotiations, auctions, etc. It thus logically follows that Theory of Mind is at the very heart of economic decision making (Singer and Fehr, 2005; Hampton et al., 2008).
Games of strategic interaction provide a testbed to study our economic behaviour in a controlled and restricted circumstance. Strategic games typically demand a Theory of Mind: players need to implicitly reason about the (expected) behaviour of others and adjust their own strategy accordingly. A game that unquestionably involves a mentalising ability is the Number
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Guessing Game. In this game, inspired by Keynes’ beauty contest to describe stock market
investment – hence “Keynesian Guessing Game” – all participants pick a number between 0 and 100. The winner is the one whose choice is closest to 2/3 of the group average; versions of the game in which the average is multiplied by a half or a number larger than one are also not unusual. The game-theoretic solution of the Number Guessing Game is for all participants to choose 0, found after iterated elimination of weakly dominated strategies. In practice, however, people appear to behave far from fully rational; the median pick is generally in the region of 25 to 35. The game’s structure and objective permits studying the degree to which players incorporate the behaviour of others into their own reasoning and, subsequently, describe the observed behaviour according to a model of Cognitive Hierarchy or level-k thinking (Nagel, 1995; Camerer et al., 2004).
Neuroimaging techniques, such as functional MRI, have opened the gates to extensively investigate the neural underpinnings of social cognition and decision-making in strategic games. In this regard, Coricelli and Nagel (2009) studied the neural correlates of strategic reasoning in the Number Guessing Game. The authors distinguished high- from level-level strategizing subjects based on the “depth” of reasoning their choices reflect; “high” in this case corresponds with at least two steps of iterated reasoning. Coricelli and Nagel found high-level reasoning players to have significantly higher activations of the medial prefrontal cortex (mPFC), an area of the brain implicated in Theory of Mind. Low-level reasoning players, on the other hand, were shown to have increased activity in the rostral anterior cingulate cortex (rACC), a region associated with self-referential, first-order thinking (figure 1). Similarly, a strong correlation was observed between mPFC activity and a measure of “Strategic IQ”. Coricelli and Nagel (2009), therefore, argue that the “mPFC implements more strategic thinking
Fig. 1 (From Coricelli and Nagel, 2009) Subjects showing a high level of reasoning, on the right, show stronger activations of the mPFC. Whereas low-level reasoning players were found to have higher activity in the rACC, an area implicated in self-referential (first-order) thinking.
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about other players’ thoughts and behaviour”, suggesting that the mPFC’s functioning is essentially the “neural signature of high-level strategic reasoning.”
The evidence regarding the role of the mPFC in strategic reasoning raises the question whether we can instigate a “deeper”, enhanced level of thinking in competitive interaction by
a priori activating the mPFC. In other words, can we augment subjects’ depth of reasoning by
engaging them in tasks known to activate the mPFC before playing strategic interaction games, such as the Number Guessing Game. For obvious reasons, this thesis cannot resort to neuroscientific instruments, such as Transcranial Magnetic Stimulation, to directly activate the medial prefrontal cortex, nor can it conclude with certainty the tasks activated the mPFC specifically. We can, however, draw from neuroscientific literature and let subjects perform tasks that were shown to stimulate the Theory of Mind network of the brain (including the mPFC) and, instead, “prime” subjects with a Theory of Mind prior to playing the first round of the guessing game. This thesis, therefore, given the means available studies the research question it believes to be closest to the original question:
“Does priming of a Theory of Mind improve strategic reasoning in Keynesian Guessing Games?”
In addition to the numerous implications relevant to society and economic decision-making that can be thought of, the importance of this research question extends to addressing the causal relationship between the medial prefrontal cortex and strategic intelligence. The work by Coricelli and Nagel (2009) shows a correlation; the claims they make regarding the role of the mPFC and high-level reasoning, however, mostly rely on reverse inference. This study aims to contribute to the convergence of evidence on the causal mechanisms governing our strategic thought processes. Can priming of a ToM encourage people to think more strategically? Machiavelli meets Keynes: can we induce people to behave more strategic in a game inspired by the most influential economist of the 20th century.
To address the main research question and test the various hypotheses derived from existing literature, this study capitalises on empirical data obtained from a series of experiments. The “guess 2/3 of the average” game was played with 7 different groups of high-school students in The Netherlands. All participants belonged to same type of education and cohort (5th year of pre-university education), thus permitting a between-subject comparison. Two treatment conditions were compared to a baseline level produced by a control group. To prime subjects, in the treatment conditions subjects were either asked to perform a Theory of Mind story
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exercise (TOMS) or engage in an entry game (EG) prior the playing the first round of the Number Guessing Game. Statistical analyses suggest that an entry game task may, indeed, improve strategic reasoning: behaviour of players in the entry game was more often found to reflect higher-order strategic sophistication, and their choices were, on average, closer to the optimal choice, resulting in superior performance compared to the control group. The ToM story, on the contrary, was not found to significantly affect strategic ability. In addition, we found some evidence for a positive correlation between the degree of priming by the entry game and performance in the subsequent game. Finally, we conclude that any differences between high- and low-level reasoning players, regardless of treatment, ceases to exist over time. In later stages of the game, behavioural differences become virtually negligible.
This thesis maintains the following structure: first, the literature relevant to the topic is discussed, followed by the methodology section in which the experimental design and procedures are presented, as well as a rationale for the hypotheses addressed in this study. The subsequent result section is divided by the various hypotheses that were developed and include a representation of the outcomes of the experiment, statistical tests and a discussion of potential confounds. Finally, in the conclusion, the main research question of this thesis is answered and some limitations and suggestions for further research are discussed. The last few pages include a list of appendices and references.
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II.
Related Literature
Evolutionary Foundations of Social Cognition
In order to understand the mind, we must understand its origins. We are, after all, the product of our genes and genes are the product of evolution. In this, the mind is no exception. One way to shed light on the evolutionary foundations of the mind is to study the behaviour of our closest relatives. The Machiavellian intelligence hypothesis – derived from, amongst others, extensive observations of chimpanzee communities – posits that we thank our cognitive capacities mainly to the need to think ahead and outsmart other group members in the face of competition for food, shelter and reproductive success. Hence, the analogy between primate behaviour and the doctrines of the Italian philosopher and political theorist; a recurrent theme in Machiavelli’s work is the justification of the use of deceit and other morally dubious tactics in the pursuit of political power. While there are apparent advantages to cooperation and group living (Axelrod and Hamilton, 1981), as primate societies grew more and more complex, social competition eventually became the foremost biological problem, for which increased intelligence is an adaptive solution (Byrne, 1997). The concept of “Machiavellian intelligence” is given further support by anatomical evidence: there appears to be a correlation across primate species between the relative volume of their neocortex (responsible for higher-order cognition) and the size of the social group they are part of (Dunbar, 2003).
If harmonious group living, on the one hand, increases the likelihood of survival (Trivers, 1971; Axelrod and Hamilton, 1981), while, on the other hand, it leads to competition within the social group, our intelligence must have evolved such that it competently manages both forces. With respect to the former, evidence from behavioural game theory suggests that we, indeed, have a specialised mechanism to detect defection or deceit. Evolution allegedly favours abilities that help sustain cooperation and minimise the number of free-riders (Tooby and Cosmides, 1992). A subsequent study by Sugiyama et al. (2002) found a superior ability to detect social deceit across a wide variety of cultures (including non-literate tribes in South-America), supporting the notion of a universal and innate presence for cheater-detection mechanisms in humans. It can be argued, however, that both forces, in the end, come down to the same skill: the ability to understand the intentions of others and choose action adaptively to the benefit of oneself and the group. In that matter, our unique capacity to “mind-read” and infer what might be going on in the minds of others, a Theory of Mind, has likely evolved to facilitate cheater-detection mechanisms and Machiavellian social skill (Brüne, 2015).
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Critique on the Machiavellian intelligence hypothesis comes primarily from two directions. The first being the air of paradox this view (admittedly) presents: we wish to preserve cooperation and reap the benefits of harmonious group living, while at the same time being “hard-wired” to behave Machiavellian. This critique is parried in a convincing manner by Orbell et al. (2004). In a computerised model of natural selection, the authors demonstrate how a combination of well-developed Machiavellian intelligence and simultaneously a highly cooperative disposition may become the “most adaptive configuration” of the brain over time. The second criticism relates to the fact that many insects form incredibly complex societal structures, despite having very limited cognitive capacities.
Theory of Mind, strategic reasoning and its neural underpinnings
If Theory of Mind is the term we use to describe the cognitive machinery to infer other people’s intentions and to understand the consequences of one’s own action to others, what neural foundations then facilitate these mental representations? Thanks to modern neuroimaging techniques, we can now draw on converging literature regarding the brain regions implicated in Theory of Mind. Subjects in these studies are typically asked to perform tasks which require a Theory of Mind, such as a mentalising stories. Consistently, brain scans revealed three brain areas to be activated while performing these tasks: the medial prefrontal cortex (mPFC), temporal poles (TPJ), and posterior superior temporal sulcus (pSTS) (Gallagher et al., 2000; Rilling et al., 2004; Carrington and Bailey, 2009). In addition, a majority of neuroimaging studies, too, found evidence for involvement of the rostral anterior cingulate cortex (rACC) in mentalising (Brunet et al., 2001; Vogeley et al.,
2001). Together, these four regions comprise, what is commonly referred to as, the Theory of
Mind network of the brain (figure 2).
Hampton et al. (2008) subsequently developed a computational model describing the “dissociable contributions” of the various components of the ToM-network. In this study, subjects played a competitive game in which the “employers” could choose to “inspect” or “not inspect” and the subordinates could either “work”
Fig. 2 (From Yang et al., 2015) Visual representation of the Theory of Mind-network of the brain. Neuroimaging data reveals mentalising processes are largely governed by the frontal lobe, including the medial prefrontal cortex.
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or “shirk”. By comparing a number of learning models, the authors first demonstrate that, indeed, mentalising governs decision-making in games of strategic social interaction. They conclude that an “influence learning” model gave the best representation of the behaviour observed from the players; in this model players base their decisions on assumptions of how the opposing player will respond to their own choices in previous rounds. The model, therefore, unambiguously involves a Theory of Mind. Second, the neuroimaging data accompanying the behavioural results showed that subjects who assumed greater influence over the competing player’s behaviour (suggesting more strategic thinking) had stronger activations of the mPFC. Finally, the authors conclude that, within their computational framework, the mPFC and pSTS fulfil very different functions in human strategizing. This last result can be summarised as follows: expectations are likely formed in the mPFC, by that guiding decisions in strategic interaction; the pSTS, contrarily, plays a role in updating a player’s choice after observing the actual outcome relative to expectations. The pivotal role of the mPFC in successful mentalising is highlighted in several other works. In games of rock-paper-scissors against either another human player or a computer, for instance, the human vs. human condition was found to be associated with higher levels of mPFC activity compared to the human vs. computer condition (Gallagher et al., 2002). The importance of the prefrontal cortex in social cognition – a Theory of Mind naturally falls under the broader umbrella of social cognition – is further endorsed by neuroimaging data of subjects with autistic spectrum disorders (Baron-Cohen et al., 1994; Happé et al., 1996), lesion studies of focal brain damage to the mPFC (Koenings and Tranel, 2007) and studies of patients suffering from a degenerative brain disease (McNamara et al., 2007).
In spite of the evidence that humans generally develop a Theory of Mind from the age of four (Baron-Cohen et al., 1994), we often find people to be heterogeneous in terms of their strategic sophistication. While some people reason no further than their own frontiers (first order), others account for the thinking of others into their own thinking (second order), or even the fact that others will likewise incorporate the thinking of others (third order), and so forth. What determines how deep we think? An obvious explanation might be due to cognitive limitations (Devetag and Warglien, 2003). Another explanation relates to overconfidence: people might – arrogantly – belief that the competing players fail to reason as far as they do (Camerer and Lovallo, 1999). A study by Coricelli and Nagel (2009) addresses the neural correlates of strategic heterogeneity and, more specifically, distinguishes between low- and high-level reasoning players in terms of their brain activity. Neuroimaging data was gathered
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from subjects playing the Number Guessing Game (“guess 2/3 of the average”), a game inspired by Keynes’ original beauty contest analogy. A more detailed, game-theoretic description of the game is provided in the next paragraph of this literature review. The authors found that playing the Number Guessing Game, as expected, activated the brain’s aforementioned ToM-network. Within this network, however, only activity in the mPFC distinguished high-level reasoning subjects from low-level reasoning subjects. In this case, a subject was categorised as “high-level” when his or her choices reflected a “depth” of strategic reasoning of second order or higher. Brain activity of low-reasoning players, on the contrary, was characterised by activation of the rostral anterior cingulate cortex (rACC); an area commonly associated with self-referential (first-order) thinking in social circumstances. Interestingly, Coricelli and Nagel (2009), too, present evidence that indicates a relatively strong correlation between the observed degree of mPFC activation and subjects’ Strategic IQ (measured by the average distance from the winning number). This result suggests that a higher degree of mPFC activity leads to more successful outcomes in social interaction (figure 3). The authors state that the findings are novelty in the Theory of Mind literature, “providing evidence for the fundamental role of the mPFC in successful mentalising.” The conclusions of this study need to be interpreted with some caution however. For one thing, the
results rely on a process of reverse
inference, in which behaviour or psychological processes are deduced from a pattern of brain activity. In this case, higher-order strategic reasoning is observed in combination with superior mPFC activation. The two are, therefore, assumed to be directly related, but conclusive statements regarding causality may, for that matter, be questioned. A second critique pertains to the generalisability of results obtained from the Guessing Game – but does this not hold for any game-theoretic evidence?
Figure 3 – mPFC Activation and Performance
Fig. 3 (From Yang et al., 2015) Visual representation of the Theory of Mind-network of the brain. Neuroimaging data reveals mentalising processes are largely governed by the frontal lobe, including the medial prefrontal cortex.
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The Number Guessing Game and Cognitive Hierarchy Theory
“Professional investment may be likened to those beauty contests in newspaper competitions, in which the competitors have to pick out the 6 prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole. It is not a case of choosing those which are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree – to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth, and higher degrees.” - J.M. Keynes (1936, p.156)
Although there being some debate on the original inventor of the Number Guessing Game or
(p-)Beauty Contest Game, it is widely agreed that the game is inspired by the metaphor Keynes
uses to depict stock market investments (Nagel, 1995; Bosch-Domenech et al., 2002; Bühren and Frank, 2012). Countless adaptions of the game can be thought of, but the version of the guessing game this thesis considers, takes the following form: all players choose a number from the closed interval [0, 100]. The winner is the player whose choice is closest to the mean choice of all players multiplied by a factor p (meant * p), where p is a positive parameter and known
to all those participating. For 0 < p ≤ 1, there is a unique Nash equilibrium of the game: all players choose the lowest possible number (0). Zero is also the only possible strategy that survives from iterated elimination of weakly dominated strategies under common knowledge of rationality. Under the assumption of full rationality of all players, game theory for that matter predicts all players to always choose zero. Empirical evidence from various types of experiments, however, consistently reveals a vastly different pattern. For p = 2/3, the mean choice in one-shot (or first-round) interaction is typically between 22 and 35 (Bosch-Domenech et al., 2002); even in experiments with groups of grandmaster chess players yield relatively high mean and median outcomes (Frank and Bühren, 2012).
The structure and objective of the game (provided that p<1) permit the use of models of bounded rationality to help explain the observed out-of-equilibrium behaviour in most – if not all – experiments (Nagel, 1995). This analysis is in fact not much different from what Keynes refers to as the “first, second, and third degree”. In behavioural game theory literature, Cognitive
Hierarchy has become the predominant model to describe empirical behaviour and thought
processes in strategic games. The Cognitive Hierarchy model hinges on two key components: (1) players using iterative decision rules using k steps of thinking, and (2) the relative frequency
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of “step-k thinkers”. Cognitive Hierarchy Theory (CHT) assumes that a step-k thinker believes to be the most strategically sophisticated players, i.e. he/she believes all other opponents to be distributed from level 0 to k-1. The distribution of other players is assumed to follow a normalised Poisson distribution (Camerer et al., 2004). Following CHT, players can then be categorised according to the “depth” of their strategic reasoning. Applying the model to the Number Guessing Game (p=2/3) yields the following implications:
• Level 0 are players with no strategic sophistication whatsoever; behaviour is characterised by random decisions with an average of 50. Choices of 50 (or higher are) therefore associated with level 0 (L0) thinking.
• Level 1 players choose a best reply to random behaviour (L0); numbers close to 33 (2/3 x 50) are thus thought to reflect L1 thinking.
• Level 2 players assume their opponents to be a combination of L0 (50) and L1 (33), but not L2 or higher. In its simplest form, level-k players best respond to the belief that all other players are k – 1 types. In this case, 22 is the “hallmark” of L2 reasoning.
Note that, 50 is generally considered to be the reference point from which the iterative steps of reasoning depart. An obvious drawback of this approach is the incidence with which random behaviour is wrongfully interpreted as higher-order reasoning. Suppose a player at random chooses, say 22, he will then be categorised as level 2, when in fact he is a L0-player. Regarding this matter, however, a meta-analysis of empirical data shows that relative frequencies of choices are typically concentrated – or elevated at least – around 22 and 33 (see
figure 4), suggesting that people, indeed, follow the predicted iterative steps of reasoning in
Keynesian Beauty Contest games (Bosch-Domenech et al., 2002). Erroneous interpretation of random behaviour nonetheless remains a point of concern. Other issues with CH- or Level-k models include the question what number aptly serves as the reference point (i.e. level 0) for further iterative reasoning and the distribution of types – does the relative frequency of types follow a Poisson distribution or do all players simply assume their opponents to be of k – 1 strategic sophistication? Someone who has employed 2 steps of iterative thinking (L2) may, theoretically, choose 33, because he believes all other players to be L0. Another critique concerns the model’s limited predictive powers (Dean, 2015). Furthermore, Georganas et al. (2013) found no correlation for estimated types from one game to another.
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Learning models and Adaptive Behaviour
In laboratory of classroom experiments with subjects playing the Number Guessing Game, commonly multiple rounds of the game are played. Although the subsequent rounds are in principal unrelated, this design promotes numerous further analyses, including the effects of learning (how does the game unfold over time) and the sort of updating rules subjects employ from one round to another. In multi-round designs (p=2/3) mean and median choices appear to move towards the equilibrium relatively quick. While median first round choices vary between 28 and 33; median choices in the fourth and final round are typically below 10. Naturally, the “rate of decrease” (measured by the relative decrease from the first to last round) is decreasing in p; that is, stronger rates of decrease are observed if p=1/2 (Nagel, 1995).
An important question when capturing behaviour in the Guessing Game with a model of k-level thinking, concerns whether subjects’ depth of strategic reasoning increases over time. In other words, are the so-called “types” fixed? Camerer et al. (2004) and Nagel (1995) conclude that if the mean of the previous round is treated as Level 0, i.e. the reference point from which iterated thinking departs, level k reasoning does not improve significantly from the first to the fourth round. Yet, when “naïve” players have no influence, k was found to increase in later stages of the game (Duffy and Nagel, 1997).
The data multi-round designs of the Keynesian beauty contest games generate, a variety of learning models may be applicable to formally describe players’ adaptive behaviour – e.g. naïve, adaptive or Bayesian learning models. Instead of testing these competing theories, Rosemarie Nagel (1995) in her seminal work on Number Guessing Games describes the
Figure 4 – Relative Frequencies of Choices
Fig. 4 (From Bosch-Domenech et al., 2002) Aggregated data from various experiments (p=2/3) showing the relative frequencies of choices in one-shot (or first round) interaction. Higher relative frequencies around 22 (L2) and 33 (L1) suggests players follow iterative steps of reasoning, departing from the reference point 50.
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observed behaviour in subsequent stages of the game by a qualitative learning-direction theory. According to Nagel, learning-direction theory (LDT) might help explain “why the modal frequency of depth of reasoning does not increase”. The (simplified) version of LDT she proposes prescribes a process in which players compare their own adjustment factor, i.e. the factor by which they updated their choice from one round to the next, to the ex-post optimal factor. This approach presupposes that if players finds out that his or her adjustment factor was in retrospect too high (low), the players should then decrease (increase) their rates towards the optimum. The study shows that roughly 73 percent of the observed behaviour corresponds to the proposed learning-direction theory. Additionally, Nagel (1995) rejects the null hypothesis that experience is irrelevant at the 1-percent level. Her original work, however, fails to distinguish between different archetypes of players with respect to their adaptive behaviour. One may ask whether strategically more sophisticated players – as categorised by CH – use different learning mechanisms.
From an assessment of various learning mechanisms and their neural correlates, Griessinger and Coricelli (2015) suggest that a cognitive hierarchy of strategic learning mechanisms may, in fact, exist. L0 strategic thinking, characterised by mere striatal activity in the brain, may be associated with reinforcement learning (RL). First-order (L1) reasoning, on the contrary, is shown to correspond with rACC activity. At the same time, Zhu et al. (2012) found rACC activity to mediate the learning of a lower degree of strategic sophistication. L1 players are therefore thought to update according to fictitious play learning – a model of first-order strategic sophistication. In the study by Hampton et al. (2008) discussed in the literature section, the authors show that mPFC activity – the “neural signature” of higher-order strategic reasoning – also implements the capacity to dynamically integrate the adaptive behaviour of others into one’s own learning process. Hampton et al. refer to this type of learning mechanism as influence
learning (IL) as it accounts for the influence one’s own behaviour may have on the behaviour
of competing players. The link between influence learning and higher-order strategic thinking (L2) follows unmistakably.
Recent advancements in literature on strategic and social cognition
In view of the emergence of neuroeconomics over the last decade or so, this thesis believes it is important to briefly address some of the more recent developments in literature related to this the topics of this study. Following the result that higher mPFC was found to be associated with more strategic thinking in the NGG, Nagel et al. (2015) studied whether results are replicated in other games of strategic interaction. The authors conclude that in Entry Games, too, mPFC
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activity correlates with strategic sophistication and payoffs. Another avenue of research has focused on cognitive measures to gain a better understanding of strategic heterogeneity. Bühren and Frank (2012), for instance, studied whether training of certain cognitive domains carry over to other interactive games (e.g. from the chess board to the Guessing Game). Their results mostly reject the notion of “cognitive abilities transfer”: chess players generally behave highly comparable to the average performance. Furthermore, the depth of strategic reasoning appears to be increasing in cognitive ability. More specifically, less impulsive subjects who have higher scores on the Cognitive Reflection test typically choose lower numbers (Brañas-Garza et al., 2012).
Over the last decade, neuroscientific studies using transcranial magnetic stimulation (TMS) have received more and more attention. TMS is, simply put, a technique in which a magnetic field generator (“coil”) is placed near the scalp of the subject. In this way, the coil essentially stimulates the neurons in the region of interest. In consideration of this thesis’ central research the, namely studying the effects of activating (or priming) a Theory of Mind on strategic reasoning, a brief review of related TMS literature may not be ignored. Most relevant are those studies that have focused specifically on the social interaction and the role of the brain’s ToM-network. Santiesteban et al. (2012), for instance, showed that stimulation of the right Temporoparietal Junction (TPJ), one of the areas implicated in Theory of Mind, may enhance social cognition. Other TMS studies have researched the extent to which affective (associated with empathy) and cognitive mentalising rely on different neural structures (Costa et al., 2008; Kalbe et al., 2010). Lev-Ran et al. (2012) employed rTMS methods to study the role of the (ventro)medial prefrontal cortex in Theory of Mind learning and showed that “knocking-out” the vmPFC significantly impairs learning abilities in social interaction. Krause et al. (2012), contrary to their own expectations, found that stimulation of the dorsomedial PFC did not influence performance in a cognitive ToM task – a result that challenges the implicated pivotal role of the mPFC in successful mentalising. The authors, however, assert that some caution regarding their findings is advised: reducing activity in one region of the ToM-network may not sufficiently disrupt its functioning. This last result, nonetheless, contributes to the relevance of this study. As far as this thesis is concerned, no such brain stimulation techniques of any kind have been used to manipulate behaviour in Keynesian Beauty Contest-like games.
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III. Methodology
Experimental Design
As mentioned in the introduction, to study the effects of medial prefrontal activation on strategic reasoning in the Number Guessing Game, ideally one would resort to methods like transcranial
magnetic stimulation, which allow for a direct, more causal analysis of any relationship that
might exist between mPFC activation and strategic sophistication. Given the obvious limitations that graduate research incurs, the original research question was adapted to permit studying the closest possible research question in a (classroom) experimental setting.
“Does priming subjects with a Theory of Mind improve strategic reasoning in Keynesian Guessing Games?”
This is derived from the idea that the mPFC is implicated in the brain’s Theory of Mind network – the areas jointly responsible for facilitating people’s capacity to mentalise. For that reason, by “priming” subjects with a Theory of Mind by committing them to a task that instigates reasoning about others’ thinking, we may likewise activate the medial prefrontal cortex and stimulate higher-order strategic reasoning.
Given the lack of existing empirical data required to appropriately answer the main research question of this thesis and test the various hypotheses (formulated in the next subsection), the following experimental setup was devised. Four rounds of the Number
Guessing Game (p = 2/3, interval [0,100]) were played with groups of high-school students.
The groups as a whole belonged to one of three treatments: a control condition serving as a baseline and two treatment conditions in which subjects received a task aimed at activating the brain’s mentalising capacity. The main objective of this design is to study any cross-treatment behavioural differences (focusing on strategic sophistication) that may arise from priming players with a Theory of Mind prior to proceeding to the first round of the Guessing Game. The rationale for adopting a between-group design – rather than applying treatment variation within a group of subjects – lies primarily in the possibility to derive any treatment-related differences that may exist from the way the game in its entirety unfolds (e.g. mean and median choices,
rate of decrease, etc.). A second, more practical argument relates to the organisational
complexities that follow from having multiple treatment conditions and, for that matter, multiple procedures in the same session.
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Baseline Treatment (Control)
The control condition serves the purpose of providing a baseline level against which observations obtained from the treatment conditions are compared and, thus, derive potential treatment effects. Subjects in control sessions of the experiments played 4 rounds of an “ordinary” version of the Number Guessing Game, with p = 2/3 and a choice interval of [0,100]. Theory of Mind Story Treatment (TOMS)
Subjects in the TOMS treatment received a mentalising story right before playing the Number Guessing Game – the game itself was no different compared to the control condition. The idea of using a story to prime subjects with a Theory of Mind is derived from neuroimaging evidence presented by Gallagher et al. (2000). In this study subjects were asked to read two kinds of short stories and answer a question related to the content of texts. One text requires the subject to, essentially, infer the thinking of others to answer the question correctly (ToM), the other does not necessarily involve a Theory of Mind (non-Tom). Compared to the non-ToM story, significant brain activations were observed in the medial prefrontal cortex for the ToM task, indicating that mentalising is largely mediated by the mPFC. This result suggests that using a mentalising story, alike the one used by Gallagher et al. (2000), is an appropriate way of activating the brain’s Theory of Mind network and prime subjects with a mentalising “state of mind”. The text and corresponding question (see Appendix B) were an exact copy of the one used by Gallagher et al. (2000). In addition – following neuroscientific practices – a second question asked subjects to rate the degree to which they felt the task triggered them to “think about the thinking of others”. This is not only informative of the relevance of the task, but also provides a cautious measure of the “level” of Theory of Mind priming and, ergo, the degree of mPFC activation – a big leap admittedly.
Entry Game Treatment (EG)
In line with the TOMS treatment, the EG treatment had the goal of priming subjects with a Theory of Mind prior to commencing the Number Guessing Game. The EG treatment addresses two potential pitfalls: (1) the TOMS task does not require subjects two reason about the thinking of competing players per se, and (2) the TOMS task itself cannot be paired with financial incentives – players might thus not be intrinsically motivated to take the task serious. In the EG condition, subjects were asked to play an Entry Game of the following sort (see Appendix C): subjects were given the choice between a certain payoff (€3) and a conditional, uncertain payoff (€10 provided that at most 10 other players also choose this option). Clearly, a well-considered
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decision in this situation requires one to reason about the expected actions of competing players. The concept of using an Entry Game is borrowed from a neuroimaging study by Nagel et al. (2017). The authors find that (high-level) reasoning in an Entry Game situation may be associated with stronger (dorso)medial PFC activation compared to Stag-Hunt games. At the of the experiment, one player would be randomly selected for reward and paid according to his/her choice. Analogous to the TOMS treatment, an additional question was included regarding the “degree of trigger”.
A critical concern relates to the question whether a ‘mentalising state of mind’ can be
primed. That is, if a subject performs a task that explicitly instigates thinking about others, such
as the entry game or Theory of Mind story, to what extent – if at all – does this way of thinking persist in the Number Guessing Game? The effects of priming (or anchoring) players with certain concepts on behaviour in economic games have been studied quite extensively (Shariff and Norenzayan, 2007; Al-Ubaydi et al., 2013; Cohn et al., 2015). Alas, very little relevant literature is available on the effects of priming subjects with a specific “way of thinking”. Perhaps, the irony of the expression “don’t think about a pink elephant” answers the question sufficiently. In any case, the timing between the “priming task” and the first round of the Number Guessing Game is naturally crucially important.
The multi-round design of the Guessing Game adopted in all treatments supports cross-treatment analyses of learning effects (how do games unfold over time?) and, more importantly, study subjects’ adaptive behaviour (what type of mechanisms do players engage to update their choices from one round to another?). Following common practice in experimental economics literature (Holt, 2006), participants were financially incentivised to perform well in the Number Guessing Game. After the last stage of the experiment, a randomly selected round was used for pay-out. The winner of the stochastically determined round received 10 euro’s cash at the end of the experiment. This of approach of, randomly selecting one of the four rounds for pay-out, should motivate players in each round of the game. After consulting with some of the teachers, the amount (€10,-) was assumed to sufficiently incentivise students of the age group and economic situation at issue.
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Experimental Procedures
Experimental procedures were the same across treatments, of course with the exception that in the treatment conditions, subjects received an additional task before proceeding to the actual Number Guessing Game. Every session of the experiment involved a different subject pool. All subjects in a given group belonged to one of three treatment conditions, implying that all players in a given group received the same treatment. All sessions were conducted in a classroom-like setting; tables were separated to minimise contact with other players. Materials (instructions, participant numbers, answering forms, etc.) were placed on the tables before participants entered the room. As soon as participants arrived, they were kindly requested to turn off their phones or any other device they may have and remain fully silent until the end of the experiment. Instructions (see Appendix A) were then read out loud; any questions that participants had were answered in private. Subsequently, in the EG and TOMS treatments subjects were asked to turn over answering form “Q” (see Appendices B and C), representing the mentalising task that characterises the respective treatments. Note that, subjects in EG and TOMS treatments had no prior knowledge whatsoever on the type of task. Based on a number of trial runs, a time limit of 3 minutes to answer the accompanying questions was imposed in either treatment. Upon signalling time had elapsed, we would then immediately proceed to the first round of the Number Guessing Game in order to, essentially, “preserve” a Theory of Mind (if any) primed in subjects as much as possible. For self-explaining reasons, stage “Q” was skipped in the control treatment.
In all treatments, procedures for the Number Guessing Game took the following form:
i. Bell rang to indicate subjects may turn over answering form of the respective round. ii. 60 seconds to write down estimate of “2/3 of the group average”. To minimise
spill-overs, subjects were asked to fold over answering form immediately after finishing. iii. Bell rang again to signal time had elapsed; experimenter collected forms and
processed answers in a spreadsheet.
iv. For every round, the average and 2/3 * average was reported on the blackboard. Note that the answering forms contained the subjects’ unique participant number as well as a brief remark of the objective of the game (see Appendix D). Procedures in all four rounds were exactly the same. After the final stage, a 6-sided die was thrown to randomly select the round used for pay-out. The winner’s participant number was announced and the reward was then handed over in cash. In case of multiple winners, the amount (€10) was split equally amongst
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the winners. In the EG treatments, a random number between 1 and n was generated corresponding to the numbers of the participants. Subsequently, the randomly selected participant was paid according to his or her choices in the Entry Game and, given the nature of the game, the choice of the competing players.
Formulation of hypotheses
Based on a review of related literature from experimental economics and neuroscience, the following hypotheses have been developed in order to address the main research question of this thesis – does priming subjects with a Theory of Mind improve strategic reasoning in Keynesian Guessing Games? The following section presents an overview of the formulated hypotheses and a brief discussion of the underlying justification. Note that the formulation of hypotheses does not distinguish between treatment conditions TOMS and EG. Given the content of the two tasks, dissimilar results may nonetheless be expected.
Hypothesis 1:
“Subjects in treatment groups, primed with a Theory of Mind, have higher depths of strategic reasoning according to a model of Level-k thinking.”
Hypothesis 1 is derived mainly from the findings in Coricelli and Nagel (2009), i.e. higher-order thinking is associated with mPFC activation. By (successfully) priming subjects with a Theory of Mind, and in that way install a mode of thinking beyond oneself – presumptively through activating the mPFC – this thesis therefore expects treated subjects to exhibit behaviour that may be associated with a higher “depth” of reasoning. In this case, according to a model level-k thinking, a higher relative frequency of subjects is assumed to reason strategically of level 2 or further. In the Number Guessing Game of the kind adopted in this study, higher-level thinking should be reflected by (1) choices being concentrated around 22, if L2, and (2) lower first round mean and median choices.
Hypothesis 2:
“Subjects in treatment groups are, on average, closer to the optimal choice.”
Strategic competence is not only reflected in the number of iterative steps players employs in their decision process. Ultimately, the most strategic players are the ones that are closest to the winning number in each round, indicating an accurate ability to adapt one’s own actions to the predicted behaviour of competing players. Thus, in case the treatments (EG and TOMS), as predicted, sort a positive effect on strategic sophistication in the treated subjects, this should be
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reflected in lower mean deviations from the winning number. Coricelli and Nagel (2009) capture this in a measure they refer to as “Strategic IQ”, which is a player’s average squared deviation from the winning number of the various rounds. Since the “priming effect” likely becomes irrelevant after the first round of the Guessing Game, alternatively the squared distance in the first round may be indicative of enhanced strategic competence.
Hypothesis 3:
“There exists a positive correlation between the degree to which the tasks triggered a Theory of Mind and performance.”
Following the result that the degree of mPFC activation strongly correlates with Strategic IQ (Coricelli and Nagel, 2009), this thesis likewise expects to find a positive correlation between Strategic IQ and a self-reported degree of trigger, i.e. how strongly subjects felt the task encouraged them to consider the thinking of others. Given the lack of accompanying neuroimaging data, the degree of trigger, like a proxy, serves as an alternative for mPFC activity – a somewhat far-fetched association admittedly.
Hypothesis 4:
“Adaptive behaviour of higher-level reasoning subjects from one round to the next involves a Theory of Mind.”
The final hypothesis is obtained from the conclusions presented by Hampton et al. (2008) regarding the type of learnings mechanisms high-level strategic players apply. The authors found that the adaptive behaviour of L2-thinking subjects (albeit in a different game) may be associated with influence learning algorithms. Influence learning unquestionably requires players to capitalize on their Theory of Mind abilities. In similar vein, this thesis will test whether the updating rules of strategically more sophisticated players fit with a learning mechanism that involves a Theory of Mind.
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IV.
Results
The results section first considers a general discussion of the experiment in general, the various sessions and key statistics. The remainder of the results section is structured according to the four hypotheses that were developed in the methodology section (p. 21-22).
Data
This study considers experimental data obtained from 7 sessions. Across all sessions, all participants were high-school students enrolled in the same level of education, year and subjects (Dutch: “5vwo economie”). In this manner, homogeneity among treatment groups was maintained as much as possible. To minimise the risk of spill-over effects between students from the same school, each session was conducted at a different school in the Utrecht/Amsterdam region (Appendix E). Students generally seemed to enjoy participating in the experiment and during none of the sessions any noteworthy misconduct was encountered. Not a single participant said to be familiar with the game.
Table 1 – Summary Statistics
Treatment Condition
Total Control EG TOMS
No. of Sessions 7 2 3 2 N 147 42 57 48 Round 1 Mean (SD) Choice 35.3 (14.8) 38.3 (15.5) 32.1 (11.0) 36.6 (17.6) Median Choice 35 37 34 36 D11 343 372 233 448 Round 2-4
Round 4 mean (median) 9.5 (7) 7.9 (7) 6.0 (4) 15.1 (12.5) Rate of Decrease2 0.73 (0.80) 0.79 (0.81) 0.81 (0.88) 0.59 (0.64)
Strategic IQ3 162 150 97 248
Degree of Trigger4 3.9 n/a 4.0 3.8
Table 1 reports the summary statistics of the empirical data obtained from the experiment. The first column shows the aggregated results; the last three columns are separated by treatment conditions. 1D1 measures the average squared distance from the winning number (X1-M1)² in the first round. 2Rate of Decrease indicates how quickly mean (median) choices decrease from round 1 to round 4. 3Strategic IQ is subjects’ mean squared error from optimal choice (M
t) over 4 rounds: (Xt – Mt) / 4 4Degree of Trigger corresponds with a self-reported “degree of trigger” (Likert scale from 1 to 5).
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Raw outcomes nevertheless suggest that only very few participants either did not take the experiment serious or did not comprehend the game’s objective: only 5 choices above 66 were observed. Summary statistics of the results are provided in table 1. A total of 147 subjects took part in the experiment – due to the number of sessions and varying group sizes (between 16 and 28), it proved impossible to distribute the number of participants perfectly even across treatments. As can be seen in the last row of table 1, the self-reported degree of trigger – a measure of how strongly subjects felt the task encouraged “thinking about others’ thinking” – was high in both the Entry Game condition (4.0) as well as the ToM-story condition (3.8). In fact, 79 percent of subjects in the EG treatment and 66 percent in the TOMS treatment rated the respective tasks 4 or higher on a Likert scale from 1 to 5. This suggests that both tasks, as desired, stimulated the majority of players to engage their Theory of Mind abilities. This thesis shall henceforth regard a rating of 4 or 5 as indicative for successfully priming players with a Theory of Mind. Due to the non-normal distribution of degree of trigger (Shapiro-Wilk test; W=0.953; p<0.01), a non-parametric Mann-Whitney U test was employed to compare median values. The two tasks are not statistically significantly different in the extent to which they triggered subjects to consider the thinking of others – based on a self-reported valuation (Mann-Whitney two-tailed test; U=1159; p=0.180). Despite the evidence from neuroscientific literature that either task is associated with the Theory of Mind-network of the brain (Gallagher et al., 2002; Nagel et al., 2017), we may not simply assume that these areas (and the mPFC in particular) were de facto activated in primed subjects. Since we lack neuroimaging data, caution is advised with regard to any of such claims. The diverging contents of the two tasks, too, should not be overlooked; regardless of the implicated similarity in terms of their neural correlates.
Table 2 presents an overview of the results of all 7 sessions in which the Number
Guessing Game was played, including the outcomes per round and several other key statistics. Median values, too, are included as mean values may be heavily influenced by very high and seemingly random choices. In all sessions (except for #2), outcomes were very comparable to other classroom or laboratory settings with p=2/3 (Nagel, 1995; Bosch-Domenech et al., 2002). Similar conclusions may be drawn with respect to the rate of decrease, which measures how quickly the game’s mean or median value falls from the first to the final round (Nagel, 1995). Yet very little can be said about the rate of decrease in relation to the various treatment conditions; a comparison across treatments suggests a rather stochastic pattern when sessions are considered individually.
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Hypothesis 1
“Subjects in treatment groups, primed with a Theory of Mind, have higher depths of strategic reasoning according to a model of Level-k thinking.”
To examine the assumed enhancing effects of the treatment on strategic reasoning, the behaviour of players in the first round of the Guessing Game is presumably the best indicator of one’s level of strategic sophistication. Not only is the first-round of the game equivalent to a one-shot interaction (no learning effects, reference-dependent behaviour, etc.), in case any priming effect does arise from the two treatment conditions (EG and TOMS), it is most likely to pertain in the first round. To briefly recapitulate on the idea: the mPFC has been described as the “neural signature of strategic thinking” reflecting a higher depth of strategic reasoning (Coricelli and Nagel, 2009). Since the mPFC is also found to play a key function in successful mentalising, we belief that subjects primed with a Theory of Mind will behave in strategically more sophisticated manner relative to subjects in the baseline treatment. Following the thought that more strategic players employ (at least) 2 steps in their process of iterated reasoning – and thus choose lower numbers in the Number Guessing Game – the most obvious indicative statistic is the mean of first-round choices (X1). A comparison of mean first-round values across
treatments is presented in table 3. Note that, a Shapiro-Wilk test for normality holds that in all treatments, the null hypothesis that choices in the first round are normally distributed cannot be rejected.
Choices in the EG treatment are, on average, significantly lower than in the control condition. As expected, a stronger result is observed when the sample is limited to players with a self-reported degree of trigger of at least 4. Average values in the TOMS condition, on the other hand, are not statistically significantly different compared to the baseline – also when the sample is limited to primed subjects (table 3 – column 6). These results may be interpreted as
Table 3 – First-Round Choices Per Treatment Condition
Control EG TOMS (Trigger≥4) EG (Trigger≥4) TOMS
Mean (SD) X1 38.3 (15.5) 32.1** (10.9) 35.5 (15.9) 31.5** (9.8) 33.5 (16.0) p n/a 0.022 0.404 0.016 0.197 N 42 57 47 45 32 Table 3 shows the mean values of first-round choices across treatment. The last two columns represent samples restricted to subjects with a (self-reported) degree of trigger of 4 or higher, indicating successful priming. The table reports outcomes from a normal two-sided t-test to compare mean values relative to the control condition. ***p<0.01; **p<0.05;*p<0.10.
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a first, careful indication that (1) the entry game treatment instigates more strategic thinking in subjects and (2) the Theory of Mind story sorts a weaker (if any) effect.
As mentioned previously, this thesis adopts a model of Level-k thinking to categorise players’ depth of strategic reasoning. According to level-k thinking, a transition from low- to higher-order strategic thinking (i.e. from L1 to L2), should be mirrored by first-round choices concentrated around 22 (L2) or even 15 (L3). For instance, under the assumption that 50 (L0) serves as the reference point, 22 is the number that is found after two steps of iterated reasoning. Since players’ strategies do not necessarily correspond to the exact theoretical iteration steps (50, 33, 22, and so forth), specific “intervals” belonging to each kth iteration step were determined inspired by Nagel (1995). Neighbourhood intervals are intervals around the kth iteration step: 50*(2/3)k+1/4 to 50*(2/3)k-1/4 rounded to the nearest integer. For example, the lower bound for L1 is: 50*(2/3)1+1/4 = 30, while the upper bound is determined by 50*(2/3)1-1/4 = 37. Interim intervals are the intervals between two neighbourhood intervals. The value of (¼) follows from the geometric means of the successive intervals. Interim intervals are on a logarithmic scale approximately as large as the neighbourhood intervals if rounding effects are ignored. Note that for L0, the upper limit is bounded by 50. In this way, first-round choices are categorised according to the intervals that were created. Xi,1 = 36, for example, is now associated
with the L1 interval and therefore reflects first-order strategic sophistication. The relative frequencies of first-round choice intervals are reported in table 4. Likewise, a visual portrayal of relative frequencies per treatment is provided in figure 5.
Table 4 – Relative Frequencies of First-Round Choices
Interval All Control TOMS EG (Trigger≥4)EG 1
TOMS (Trigger≥4)1 0-12 0.04 0.05 0.06 0.02 - 0.09 13-16 0.03 0.02 0.06 0.02 0.02 0.06 17-19 0.03 0.02 0.06 0.04 0.04 0.06 (L2) 20-25 0.18 0.10 0.10 0.32 0.33 0.06 26-29 0.12 0.12 0.19 0.07 0.07 0.22 (L1) 30-37 0.18 0.24 0.08 0.23 0.27 0.9 38-44 0.18 0.21 0.21 0.14 0.13 0.13 (L0) 45-50 0.07 0.02 0.08 0.11 0.09 0.09 51-100 0.15 0.21 0.19 0.07 0.04 0.19 N 147 42 48 57 45 32 Table 4 reports the relative frequencies of first-round choices by the intervals they are associated with. Rows shaded in grey reflect the critical first- and second-order choice intervals.
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A closer look at table 4 and figure 5 clearly suggests that substantially more subjects in the Entry Game condition chose numbers that may be associated with a level of strategic sophistication of 2. In the control group, only 10 percent of subjects seem to employ this depth of thinking, while 32 percent of observations in the EG treatment belong to the L2 interval. If the sample is restricted to primed subjects only (i.e. characterised by a degree of trigger of 4 or higher), the relative frequency of L2 choices is not affected. In the TOMS condition, however, the relative frequency of L2 choices is no different from the control condition. The relative number of picks between 20-25 actually falls when the sample is limited to subjects with a high self-reported degree of trigger. These findings concur with the result obtained earlier that mean guesses are significantly lower in the entry game treatment relative to the control condition, but not in the TOMS treatment. To statistically test if the distributions of L1 and L2 players in the EG and control groups are dependent, we employed a 2x2 chi-square test (given the ordinal nature of the data). At conventional levels of significance, we may reject the null hypothesis that the frequency of L1 and L2 players is independent from the treatment (χ2=6.98; p=0.008).
To further investigate the treatment effect on player’s depth of strategic reasoning – rather than simply test independence – we ran a probit estimation to verify if the coefficient on EG treatment has the expected positive sign. The dependent variable is a dummy whether the observation reflects L2-thinking (1 if 20≤X1≤25; 0 if otherwise), the independent variable is
dummy indicating successful priming from the EG task (1 if EG and trigger≥4; 0 for Control). The sample is restricted to EG and control observations only. Additionally, in a second specification we limit ourselves to the L2 up until L0 intervals of first-round choices [20,44],
L2 L1 L0 0,05 0,10 0,15 0,20 0,25 0,30 0,35 ≤12 13-16 17-19 20-25 26-29 30-37 38-44 45-50 51-100 R el ati ve Fre que nc y Choice Interval Control EG TOMS
Figure 5 – Relative Frequencies of First-Round Choices
Fig. 5 Relative frequencies of first-round choice intervals and associated level k, separated by treatment. Note that samples in this figure are not restricted to trigger ≥ 4.
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such that “random” choices – and the exceptional cases of L(k≥2) – are omitted from analysis. In both regressions, the estimated coefficient on EG has the expected positive sign and is significant at the 5-percent level (table 5). This result concurs with the suggestion that the likelihood of attaining a second-order depth of strategic reasoning increases with the entry game treatment (if trigger≥4). We will refrain from marginally interpreting probit estimations.
A remark of caution regarding the last result is the following: as can be seen from table
4, the relative frequency of L2 choices does not change when restricting the sample to primed
subjects (column 5 to 6). This observation can be interpreted in two – if not more – ways. First, players with a low self-reported degree of trigger were nonetheless (subconsciously) influenced by the entry game task, resulting in more strategic behaviour. Second, the sample of subjects in the EG treatment is positively biased, which would entail that the effect of the entry game task on strategic thinking is overestimated. The latter argument is of course more troublesome. In favour of the first explanation, however, we can say that the lion’s share of subjects in the entry game treatment gave the impression to work on the task zealously. All in all, we can accept the first hypothesis partly. Evidence suggests that players (successfully) primed with a Theory of Mind after having received an entry game task are significantly more likely to engage a second step of iterated reasoning. Subjects in the TOMS treatment, however, appear to behave no more strategically; both the mean value of first-round choices, as well as the relative frequency of L2 players, are not significantly differently different from the control group. Categorisation and behaviour of strategic types in later periods of the game is addressed under Hypothesis 4.
Table 5 – Probit Estimation of EG Effect on L2-Thinking L2 (1) (2) EG 0.878*** (0.266) 0.857** (0.361) Z 2.66 2.38 N 87 64
Table 5 shows outcomes of a Probit estimation with a binary dependent variable (L2=1 if 20≤X1≤25) and a binary independent (EG=1 if treatment was EG and trigger≥4). In (2) observations are restricted to the range 20– 40; (1) considers the full range of first-round choices.
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Hypothesis 2
“Subjects in treatment groups are, on average, closer to the optimal choice.”
One of the niceties of the Number Guessing Game holds that playing the Nash equilibrium (0) or applying the most steps of iterated reasoning does not necessarily mean one is certain to win the game. Ultimately, the player with the highest strategic competence is the one whose choices are consistently closest to the winning number, suggesting an accurate ability to predict the actions of competing players and adjust one’s own strategy accordingly. For this reason, a mere account of the depth of reasoning (L1, L2, L3, etc.) seems an unjust measure to infer players’ strategic ability. To obtain a more relevant, additional indicator of strategic performance, for each player his or her squared distance from the winning number (i.e. the optimal choice) in all four rounds was computed. Analogously to Coricelli and Nagel (2009), we shall refer to this measure as Strategic IQ:
• Strategic IQi = Σ(Xit – Mit) / 4
• Average squared deviation from the winning numbers.
One can argue that the use of the word “IQ” is perhaps somewhat misleading and pretentious, but it serves the purpose of referring to a player’s performance and quality of strategic reasoning. Note that, in contrary to the way IQ is conventionally measured, in this case a lower score actually indicates better performance. Akin to the previous analyses in H1, any treatment effects that may follow from the entry game or ToM story are arguably only relevant to, or most obvious in, first-round choices. Behaviour in round 2-4 is likely overshadowed by, amongst other factors, an improved understanding of the game and/or outcomes in previous rounds. This idea pertains to the observed stochasticity of the rate of decrease across treatments – the games appear to unfold in a rather unique ways independent of the treatment condition. We, therefore, likewise consider the squared deviation in the first round (D1i). Whether D1 is an appropriate
statistic is debatable. In a group with many L0 players, for instance, a strategically sophisticated player choosing a relatively low number will be disadvantaged. Across our sessions, however, the mean of the optimal choice in the first round (M1) was 23.4. For that matter, D1 scores for
L2 players (20≤X1≤25) will be substantially lower than those of low-level reasoning players.
An overview of the mean and median values of Strategic IQ and D1 per treatment are reported in table 6. Median values are included as mean values are evidently strongly affected by (irrationally) high choices – especially given the quadratic nature of the computations.
