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  1. 1. The measurement and neural foundations of strategic IQ 1. Summary of Work under Previous Grant In [2] and [3], we extend our prior work on “experience-weighted attraction” (EWA) learning ([1]) to include sophistication and strategic teaching in repeated games. Sophisticated players understand that others are either learning or sophisticated like themselves; sophisticated teachers account for the way their current-round actions influence learners’ future-round behavior. Including teaching also allows an empirical learning-based approach to reputation formation which predicts better than a quantal-response extension of the standard Bayesian-Nash approach. The adaptive EWA learning model has several key parameters. The “self-tuning EWA” model ([4]) replaces these parameters with functions of subject’s earlier experience in the games. The self- tuning EWA model has only one (response sensitivity) parameter to be estimated, but it is as accurate as the parametric EWA and improves the subjects’ payoff the most if they were to use it to forecast others’ behaviors. In [5], we adapt the EWA model to predict consumer choice using a panel-level data from sixteen product categories and 133,492 purchases. The central feature of the model is that buying one product leads to partial reinforcement of “nearby” products with similar features (brand names, package sizes, flavors, etc.) The model has about half the number of parameters of the leading models of consumer choice but forecasts substantially more accurately out-of-sample. Because the model assumes people learn about the value of product features, it can also be used to forecast test- market results from products which combine existing features in a new way. In [6], we develop a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games (for learning models). The CH approach assumes that players use different numbers of reasoning steps; higher-level thinkers best-respond to what lower-level thinkers do. The CH model fits and predicts behavior better than Nash equilibrium in 60 experimental samples of matrix games, mixed-equilibrium games and entry games. It provides an empirical alternative to Nash equilibrium. [7] is a short published report describing this model. [8] is a published version of a Nobel Symposium summarizing all our previous work on the CH “thinking model”, the EWA learning model, and the teaching version of the EWA model. [1] Camerer and Ho, “Experience Weighted Attraction Learning in Normal Form Games,” Econometrica, 67 (1999), 827-873. [2] Camerer, Ho and Chong, “Sophisticated EWA Learning and Strategic Teaching in Repeated Games,” Journal of Economic Theory, 104 (2002), 137-188. [3] Chong, Ho, and Camerer, ``Strategic Teaching and Equilbrium in Repeated Entry and Trust Games” (under revision for Games and Economic Behavior), 2003. [4] Ho, Camerer and Chong, “Economics of Learning Models: A Self-tuning Theory of Learning in Games,” 2004. [5] Ho and Chong, “A Parsimonious Model of SKU Choice,” Journal of Marketing Research, Vol. XL (Aug 2003), 351-365. [6] Camerer, Ho and Chong, “A Cognitive Hierarchy Theory of One-shot Games,” Quarterly Journal of Economics, August 2004. [7] Camerer, Ho and Chong, “Thinking, learning and teaching in games,” American Economic Review, 93 (2003), 182-186. 1
  2. 2. [8] Camerer, Ho and Chong, ``Behavioral Game Theory: Thinking, learning and teaching in games," in Essays in Honor of Werner Guth (Steffen Huck, Ed.) (in press). 2. Introduction We propose to create a system to measure how well different people reason in social (game- theoretic) situations, where the choices of other people (or organizations, including companies and nation-states) affect their own outcomes. Choices unfold over time in most of these situations, so the proposal will help us understand social dynamics with an ultimate goal of enhancing human performance. Our work is practical because the measure of “strategic IQ” we create will be used to calibrate how well people with different skills think strategically. It also has scientific value because these data, along with measures of neural activity from lesion patients, fMRI and eye- tracking, can also be used to understand the neural foundations of strategic thinking. The work is interdisciplinary, bringing together strands of mathematical game theory, psychology, and neurobiology. A strategic IQ measure also has practical value for enhancing human performance by helping people understand how their strategic thinking can be improved, and helping organizations understand who is good at strategic thinking. Our research starts with game theory. Game theory has proved enormously useful in economics, is spreading rapidly to other disciplines, and can potentially unify diverse disciplines by providing a unifying language. Game theory provides a way to define strategic situations mathematically—viz., in terms of players, their information and strategies, an order of moves, and how players evaluate consequences which result from all the strategy choices. Most analyses of how people actually behave in games use some concept of equilibrium. In equilibrium, players are not surprised at what others do because they have already figured it out. While the concept of equilibrium is useful as an idealized model, hundreds of experiments with many groups of people (including subjects who are highly financially-motivated and trained in game theory) show that actual behaviors are often inconsistent with the equilibrium assumption (Camerer, 2003). The fact that people do not always make equilibrium choices implies that there may be reliable, measurable differences in the ability of people to think strategically, which we call strategic IQ.1 Strategic IQ measures a person's ability to guess accurately what others will do in situations where two or more parties do not have prior contractual agreements (agreeing on a contract may be the strategic situation of interest), each party can choose several possible courses of action, and each individual payoff depends on his or her actions, the actions of the other parties, and on chance events. Examples include war, international trade negotiation, rivalry between firms to gain market share, and child custody fights between divorced couples. A person's strategic IQ is her total normalized payoff when playing with others in a series of carefully designed competitive situations. Note that IQ is dynamic-- a person's IQ can increase or decrease as a result of others changing their choices, and can be improved by sharpening a person's understanding of typical behavior. A premise of the engineering applications of our approach is that people differ in their IQ-- this variation enables us to understand the neural foundations of strategic thinking-- and that it is possible to improve IQ by training. We propose to measure strategic IQ by how much people earn in a battery of games, when playing against the previous choices of a population of specified players. Measures of strategic IQ's will be compared to three benchmarks-- equilibrium choices from game theory analyses; a ``clairvoyant'' best guess about what others do (based on the data we collect), which by definition 1 Indeed, if everybody plays the equilibrium strategy, then everybody has the same strategic IQ. 2
  3. 3. an upper bound on strategic IQ; and a ``cognitive hierarchy'' model developed in previously- supported NSF research. As noted, an important use of strategic IQ is basic scientific research. To this end, we will give the battery of games to five special populations: (i) Undergraduates who are superbly skilled in mathematics (screened groups of Berkeley and Caltech students); (ii) undergraduate and graduate students specially trained in game theory2; (iii) chess players recruited at tournaments (since their numerical chess-skill ratings can be compared to their strategic IQ’s); (iv) highly- experienced business managers (available through executive education activities at Berkeley where PI Ho teaches); and (v) patients with specialized brain lesions (available through the widely-used Iowa patient population which PI Adolphs has used extensively). Other interesting subject pools will be sampled as opportunities arise. Evidence about what games the lesion patients (group v) play poorly, in conjunction with fMRI brain imaging, will help diagnose which brain circuitry is used in various types of strategic thinking. This part of the project also contributes to basic neuroscience by providing a new set of tasks which can help refine our understanding of the functions of higher-order cognition in prefrontal cortex, which is not thoroughly understood (e.g., Wood and Grafman, 2003). Another use of a strategic IQ measure is practical engineering: To help people evaluate and refine their own skills, to aid in personnel selection (e.g., screening corporate strategists, diplomats, or military personnel), and to show people where their strategic thinking can be improved. Before we continue, it is important to note that strategizing is not merely outfoxing opponents in “zero-sum” games, where one player loses if another player wins. Strategic thinking generally refers to guessing correctly what others are likely to do. Therefore, strategic thinking can often be mutually beneficial, in games where more strategic thinking by both players can help them achieve joint gains. So raising one person’s strategic IQ may help others. Also, the term “IQ'” just denotes a numerical measure of how well different people perform in a specific test. We hope to avoid debates about whether there is such a thing as general intelligence, or whether such tests are inherently biased or used for socially harmful purposes. The strategic IQ measures simply starts to put an important skill on a scientific basis, which may also have some practical use. Our approach follows Gardner (1983) and others who distinguish “multiple intelligences” (cf. Salovey, Mayer and Caruso, 2002 on “emotional intelligence”). Gardner's seven types of intelligence include logic/mathematical reasoning and interpersonal intelligence. Strategic IQ is a combination of these two subtypes. 3. Proposed Activities The proposed research has several basic steps (which are detailed further below): 1. Strategic IQ measurement: First we will develop a battery of games and a database of how various groups play those games under conditions which are typical in experimental economics-- namely, the game is described abstractly and subjects earn payoffs which depend on their choices and the choices of others whom they are paired with. 2 Earlier research on the economic value of models (Camerer, Ho and Chong, 2004) showed that players could often earn more by following Nash equilibrium advice. This fact suggests that students trained in game theory will raise their strategic IQ (since earning more when playing others and strategic IQ are linearly related). However, in some games equilibrium choices are bad choices against a typical population. So it is conceivable that students trained in game theory will have lower strategic IQ’s than some other untrained group with better strategic intuition. 3
  4. 4. 2. Creating theoretical benchmarks: It is useful to compare human performance on these games with two theoretic benchmarks-- the game-theoretic advice from equilibrium models, and a simple descriptive model based on “cognitive hierarchy” (CH) models of naturally-limited cognition (extending earlier work by PI Camerer and Ho). Doing this comparison requires extending the CH models to dynamic “extensive form” games, which is a basic scientific contribution. 3. Studying special groups: Experiments with lesion patients from the Iowa database will show which neural regions are important in the various components of strategic thinking identified in (1). Other experiments with specialized populations, including experts and others will tell us more about likely regions used in different kinds of strategic thinking. 4. Imaging neural activity: fMRI imaging and tracking of eye movements with typical subjects will supplement what we learn from (3) about neural regions active in strategic thinking. Imaging will be done at Caltech's Broad Imaging Center where PI Camerer has been active (e.g., Tomlin et al, 2004) and there is good access to scanner time. The fMRI scanner goggles worn by subjects also include tiny cameras so the eye movements of subjects can be measured while they are in the scanner (see Camerer et al, 1994, for an early application of eye-tracking to game theory). 4. Strategic IQ measurement To illustrate the skeleton of a strategic IQ measure, we will discuss five classes of games which we hypothesize to tap different dimensions of strategic IQ. We illustrate each class with only one exemplar game, but in practice we will use many games with similar strategic properties (some of which are enumerated in footnotes annotating the section that describes each exemplar game). Standard psychometric methods like factor analysis will be used to see which components of strategic IQ congeal statistically. That is, each game will be treated as a separate test item, and we will evaluate statistically which test items correlate into distinct factors. Of course, the items may cluster into categories which are different than the five we hypothesize. The simple psychometrics is important because most recent mathematical models in behavioral game theory assume that players have distinct emotional or cognitive “types” which will lead to correlated behavior across games, but little is known about how reliable types are across a wide range of games. Psychometric testing of which components of game performance are correlated will reveal whether there are separate dimensions of strategic thinking-- for example, whether anticipating how others will react emotionally to outcomes which give different payoffs to different players is correlated with planning ahead in dynamic games. We will also compare strategic IQ measures to a measures of general intelligence from a short-form of the Wechsler scale (Satz and Mogel,1962) and a measure of emotional intelligence (MSCEIT, e.g. Lopes et al, in press). Analyses with lesion patients who have localized brain damage, and expert subgroups (e.g., the undergraduate chess players), and fMRI measures of brain activity on typical normal control subjects, enable us to potentially link performance on different item-clusters to distinct neural regions which underly components of strategic thinking. This analysis may also provide new ways to categorize brain function which might interest neuroscientists. All these games have been well-studied experimentally (see Camerer, 2003, for a review). The experimental data from many previous studies are what permit the construction of an IQ measure, since that number measures how well new players do against population averages from previous experiments. 4
  5. 5. Table 1 below summarizes the five strategic principles, the exemplar games, hypothesized psychological processes underlying each principle, and candidate brain regions in which the psychological processes take place which will be explored with lesion patients, experts, and fMRI and eye-tracking. Assume there are J dimensions altogether and each dimension j has K(j) test questions (i.e., games). If N people take the test, each participant i's strategic IQ(i) is determined as follows: Each individual will have N-1 possible matches in each of the competitive situations. The total payoffs for player i will be Πi = Σj=1J Σk=1K(j) Σm=1N-1 πi(j,k,m), where π i(j,k,m) is player i’s payoff from test question k in dimension j, in each match m with one of the N-1 other players. The payoffs will be scaled so each item receives equal weight. Strategic IQ(i) is then normalized by subtracting the mean and dividing by the variance. Each section below describes one component of strategic thinking, an exemplar game which illustrates the strategic thinking component, some hypothesized psychological processes, and tentative candidate brain regions that can be explored as neural loci of the psychological process. Note that very little is known about the brain circuitry that creates these processes— a major contribution of our proposed research is to learn more-- so the hypothesized brain processes are tentative and the work will be exploratory. Figure 1 shows a sagittal view of the brain with regions of interest marked. a. Strategic reasoning: The central feature of strategic thinking is the ability to forecast what other players will do purely by reasoning about their likely choices (and about the reasoning of others, and others’ reasoning about reasoning, etc.) An example which distinguishes pure strategic thinking from emotional forecasting and dynamic planning ahead (which are discussed separately below) is the “p-beauty contest game” (e.g., Ho et al, 1998, named after a passage in John Maynard Keynes about how the stock market is like a beauty contest). In this game each of the players chooses a number in the interval xi ∈ [0,100]. The player who is closest to p times the average (with p<1) wins a fixed payoff. The Nash equilibrium in this game is the number x everyone would pick which is closest to the average when everyone picks x; the result is zero. In 24 different subject pools with p=2/3, the average number picked ranges from 20 to 35. These choices suggest that people are only doing 1 to 3 steps of strategic thinking on average (Camerer, Ho, and Chong, 2004). The availability of a very large amount of data on these games enables us to construct an IQ item which gives a percentage chance that a person would win the game, if she played in a group sampled from previous data. The percentage chance of winning times the winning payoff gives a numerical score.3 Reasoning in this game requires people to use working memory to store iterations of reasoning (e.g. “If I think the average will be 50, I should choose 33; but if people think like I do they will pick 33 so I should choose 22…”). It also presumably requires “theory of mind” (ToM, e.g. Baron-Cohen 1995), the capacity to form beliefs about what other minds know, as well as the ability to iterate theory-of-mind beliefs. 3 Other games which measure strategic reasoning are those in which players delete strategies which are dominated iteratively (see Camerer, chapter 5). A deeper feature of strategic thinking is the realization that what other players know-- in game theory terms, their ``private information''-- may affect how they behave, and making probabilistic inferences from what players do about what they know. This is important in “signaling games” and in auctions for objects of unknown common value, in which players must infer what the bids of other bidders tell them about the guesses those other bidders have about the object’s value. 5
  6. 6. Table 1: Strategic principles, exemplar games, cognitive processes, and candidate brain regions Strategic principle Exemplar game Cognitive process Brain regions to explore Strategic reasoning Beauty contest Working memory, VM (pilot), DLPFC ToM Emotional anticipation Trust ToM (emotions), VM, insula, cingulate social emotion (Sanfey et al 2003) Strategic foresight Shrinking-pie Planning BA 10, PFC bargaining Coordination Pure matching ToM, social meta- VM, BA10 knowledge Learning Iterated beauty contest Reinforcement, regret, VM (pilot), Hippocampus forgetting, novelty- detection b. Emotional anticipation: In the beauty contest game the payoff is fixed (in game theory jargon, the game is “constant-sum”) so there is little scope for emotional reaction to inequality in payoffs. In most games, however, the distribution of payoffs (and their total) depends on the choices peple make. A wide variety of data suggest people dislike unequal payoffs and will often sacrifice their own payoffs to reduce inequality (e.g., Fehr and Schmidt 1999), to harm a player who has treated them badly or help a player who has behaved nicely (e.g. Rabin, 1993). Therefore, a separate feature of strategic thinking is the ability to forecast how others will behave when emotions influence their reaction to unequal outcomes. A well-studied exemplar4 game is the “trust game” (Camerer and Weigelt, 1988). In the simplest version of this game (Berg et al, 1995) one player starts with $10 which she can partly invest, or keep. The amount she invests is tripled—representing a productive return on investment —and given to a second player, the “trustee”. The trustee can repay as much as she wants to the first player, or keep as much as she wants. Presumably trustees repay money because they feel a sense of altruism toward the first player, or a sense of reciprocal moral obligation since the first player took a risk to enlarge the available “pie” for both players. Therefore, the first player must anticipate the emotional reaction of the trustee—the trustee’s sense of altruism or reciprocity. Many studies show that players invest about half of their $10 on average (although the investments are widely dispersed) and, on average, trustees repay about $5 so the first player just breaks even. Investing wisely requires theory of mind as well as anticipation of social emotion. Studies of autistics (who are thought to have poor ToM) show that about a third do not anticipate emotional reactions of others (Hill and Sally, 2003). An fMRI study by Sanfey et al (2003) of the related 4 Other games we will study to measure emotional anticipation include “ultimatum” bargaining, in which one player makes a take-it-or-leave-it offer to another player (see Camerer, 2003, chapter 2). In ultimatum games players often reject low offers, presumably because they would prefer to get nothing than to accept an unequal share. Another emotionally-charged game is the well-known prisoner’s dilemma (PD), in which players can “defect” and earn more money, but if both players “cooperate” then both earn more. (The trust game is like a PD in which the players’ moves are sequential rather than simultaneous). The economic analogue of the PD is “public goods contribution” in which players can keep tokens for private gain, or invest them in a way that benefits everybody. In an economic analogue of the trust game, “gift exchange”, firms prepay a wage to workers, who can decide how much effort to choose. Effort is costly to workers and valuable to firms, so firms must anticipate how much moral obligation a high or low wage offer will instill in workers. 6
  7. 7. “ultimatum” game shows that when the second player is deciding what to do, there is activity in prefrontal cortex, insula (a region which is active in discomfort like disgusting odors and pain), and cingulate cortex (a “conflict resolution” region which presumably weighs the desire to earn more money with emotional reactions to inequality). These studies provide candidate regions for emotional anticipation in strategic thinking we will explore in fMRI and with lesion patients. c. Strategic foresight: The two games discussed so far are played simultaneously (the beauty contest game) or only require one step of planning (the trust game). Another component of strategic IQ is strategic foresight in games with many steps (in psychological terms, “planning”). Many studies suggest that players do not plan ahead more than a couple of steps. An exemplar game is alternating-offer “shrinking pie bargaining”5, a workhorse example widely used in economics and political science. In a three-stage example studied experimentally, one player (P1) makes an offer of a division of $5 between herself and a second player, P2. If P2 accepts the offer they earn the proposed amounts and the game ends. However, if P2 rejects the offer then the available money “shrinks”, say to $2.50, and P2 has a chance to make an offer of how to divide the $2.50. If P1 rejects P2’s offer the pie shrinks further, say to $1.25, and P1 makes a final offer. If P2 rejects that offer, the game ends and neither player earns anything. The “subgame perfect” equilibrium of this game, assuming players have no emotional reaction to unequal outcomes (i.e., they don’t care how much the other player gets), is for P1 to offer $1.25. In experiments, however, offers are around $2.10; offers below $1.80 are rejected about half the time. Direct measurement of looking ahead (using a computer analogue to eye- tracking) shows that about 10-20% of the time, players do not even look at the third stage to see how much would be available if the first two offers were rejected (Camerer et al, 1993). These data suggests that strategic foresight is limited to only a couple of steps, due to constraints on working memory or a “truncation heuristic” which leads players to ignore steps far in the future (as in chess- playing programs that look ahead only a few steps because looking very far ahead is computationally difficult). Temporal planning requires players to have ToM beliefs (perhaps in Brodmann area 10; e.g. McCabe et al, 2001) and use prefrontal cortex and working memory. d. Coordination In the games above there is a unique equilibrium prediction. However, in many games of economic interest, there are multiple equilibria. In these games, the behavioral challenge for players is figuring out which equilibrium is likely to occur. This requires a kind of "social common sense" or understanding or norms of social convention, or (as Schelling, 1960, put it), a sense of which outcomes are "focal" or "psychologically prominent". 5 Another game which requires strategic foresight is the PD when it is repeated. For example, if the PD is played 10 times, and players know that, then players often cooperate for several periods until the end draws near, when one player typically defects. Maximizing payoffs requires emotional anticipation, as well as planning ahead to guess accurately when another player will defect. Many other repeated games can be used to test for strategic foresight as well. 7
  8. 8. An exemplar game6 to study coordination is “pure matching”. In a pure matching game, players simultaneously choose objects from a set, and earn a fixed reward if their choice matches the choice of another player. For example, in Mehta, Starmer and Sugden’s (1990) experiments players are asked to name a date of the year, a flower, a number, and a male name (you can test this component of your strategic IQ by thinking about what you would pick; the "answers" are below7). Pure matching requires ToM and a sense of what the conventional or well-known choice is, which requires a kind of “social meta-knowledge”—i.e., what people know people know. For example, in Mehta et al’s experiments, when people were simply asked to name a favorite day of the year, the 88 subjects chose 75 different dates—presumably their birthdates. But when they were trying to match choice of others, those with healthy ToM realize that others are not likely to know and choose their birthdates, so most chose focal dates like December 25, rather than their birthdates. Of course, the answer can also depend heavily on the group you are playing with, so social metaknowledge about what is culturally known and shared is an important component of strategic IQ. There is little guidance from neuroscience about where this sort of processing of shared knowledge is likely to occur, so fMRI studies of these games will be exploratory and may provoke some new thinking in neuroscience. e. Learning from experience: An important component of strategic IQ is learning from experience, and anticipating how other players will learn from experience (i.e. understanding of human social dynamics). The "EWA" model pursued in earlier NSF-supported work (e.g., Camerer and Ho, 1999) is a benchmark for learning which is psychologically rich. The EWA model combines two kinds of learning mechanisms. One kind is direct reinforcement of chosen strategies according to their payoffs (which may be neurally instantiated by midbrain dopaminergic neurons, or parietal neurons; e.g., Schultz, ? and Glimcher, 2003). Another kind of learning is based on counterfactual reasoning, or "regret", about how much higher the payoff would have been if another strategy was chosen (which probably requires cognition in frontal cortex.8 The EWA model combines these two types of learning, putting a weight of 1 on the strength of reinforcement and δ on the counterfactual-learning mechanism (with 0< δ <1). The model also assumes that past reinforcements are weighted by a decay rate φ between zero and one (a lower φ corresponds to decaying the past more heavily). 6 Many other coordination games are described in Camerer (2003), chapter 7. An example is the “battle of the sexes”: Two players choose either 1 or 3; if the two numbers add up to 4 then each person receives their number, and if the numbers don’t add up to 4 they get nothing. This game pits the desire to coordinate on some pair of numbers which add to 4, with the private desire to get 3 rather than 1. Another interesting game is “stag hunt”: Both players choose H or L. L pays 5 for sure, and H pays X (say, X=10) only if the other player picks H as well, and zero otherwise. This game pits the desire to coordinate on H, so both players can earn more, with the desire to avoid social risk in case the other player chooses L rather than H. 7 In their experiments (conducted with UK students in the late 1980’s), the most common choices were December 25 (44%), rose (67%), the number 1 (40%), and the name John (50%). 8 Interestingly, "fictitious play" learning, based on updating beliefs of what other players are likely to do, is equivalent to a special parametric restriction of the EWA model in which there is only regret-driven learning, and no pure reinforcement (i.e., all strategies are reinforced equally, whether they were chosen or not, or δ=1). 8
  9. 9. Exemplar games to study learning9 from experience can be obtained from simply repeating any of the games described above. Here we shall use the beauty contest game, choosing a number from 0 to 100 and trying to get closest to 2/3 of the average, with feedback after each round. The thin line in Figure 2 shows a time series of choices by a single control subject with temporal-lobe damage. (The control subject’s learning path is similar to those usually seen in other subject populations.) The numbers this player picked fall steadily toward the Nash equilibrium of zero. Statistically, these paths are fit reasonably well by the EWA model with a δ=.78, and φ=.36 (Ho, Camerer, and Chong, 2004). In an extension of the EWA model, Ho et al (2004) allow the parameters themselves to be functions of experience rather than fixed throughout the learning process. For example, the decay rate on the past φ can be interpreted as forgetting, or as a “self-tuning” response to the detection of novelty (which often activates the hippocampus). For example if other players suddenly switch their behavior, a self-tuning learner will realize that old history may be a poor guide to action and will deliberately decay previous reinforcements. Note that such a learner is not “forgetting” per se (she may easily recall previous outcomes), but is instead suppressing old history on purpose. The EWA model suggests that learning involves psychological processes of reinforcement of received payoffs, regret-driven counterfactual reasoning about whether other strategies would have given higher payoffs (with strength δ), and forgetting and novelty-detection which affect φ. These processes may occur in VM, possibly in emotional regions which register regret, and in the hippocampus (which detects novelty and also creates long-term memories). 4. Creating theoretical benchmarks Measures of strategic IQ can be compared to three benchmarks. The first theoretical benchmark is the Nash equilibrium player. In each match of a test question, we can compute the payoff for a Nash player. Adding these up gives the IQ of an artificial equilibrium player. A key interest of measure is where the Nash player falls in the distribution of payoffs and whether it is above or below the average human strategic IQ (i.e. a normalized IQ score of 100). This is an indirect way to measure whether equilibrium models give good advice (by raising the strategic IQ of typical players; cf. "economic value" in Camerer, Ho and Chong (2004)). The second benchmark is a player who clairvoyantly guesses the actual distribution of choices by the other players and chooses the best response to that distribution. This benchmark is an upper bound because strategic IQ is defined as the payoff earned by playing against the actual distribution-- this is the maximum achievable strategic IQ, which is what the clairvoyant benchmark player earns. The third theoretical benchmark is an artificially-intelligent player who uses a "cognitive hierarchy" model to predict others' actions and best-respond to that forecast (e.g., Camerer, Ho, and Chong, 2004). The CH model assumes that there are frequencies f(k) of players who use k-steps of reasoning in an iterative fashion. 0-step players choose all strategies with equal probability. One- step players believe they are facing 0-step players and best-respond to that belief. Two-step players think they are playing a mixture of zero- and one-step players (with mixture frequencies f(0)/ (f(0)+f(1)) and f(1)/(f(0)+f(1))) and best-respond to their belief, and so on. (In general, k-step 9 Repeating virtually any game provides a way to study learning and many such games have been studied experimentally (see Camerer 2003, chapter 6). However, it is important to choose games in which the “repeated game equilibria” (which presume strategic foresight, so players anticipate the effects of current choices on future actions of other players) are the same as the equilibria in one-shot games. (For example, in a repeated PD it is an equilibrium to cooperate until another person defects, but this is not an equilibrium in a single one-shot PD.) In practice, this can also be achieved by having players randomly rematched in each period with people who do not know the previous history. 9
  10. 10. players think they are facing a mixture of players using k-1 or fewer steps of reasoning.) For tractability, we assumed that f(k) follows a Poisson distribution f(k)=eτ τk/k! which has only one parameter, τ, which represents the average number of thinking steps (and also the variance). This simple model is easy to compute once τ is specified. In earlier work it was applied to about 120 different games and found that it can usually explain where Nash equilibrium predicts poorly and where equilibrium predicts surprisingly accurately. An artificial CH player uses the CH model to form beliefs and chooses a strategy which has the highest expected payoff given those beliefs. However, the CH model is only designed to apply to normal-form matrix games (in which players are assumed to choose at the same time). Of course, many games unfold dynamically over time; these "extensive-form" games are often represented in the form of a tree with nodes that follow in temporal order. Extending the CH model to dynamic games is a challenge we will address in this research as well, in order to create an all-purpose CH benchmark that can be applied to any game. One way to apply CH model to extensive-form games is to combine it with backward induction principle and proceed recursively. That is, 0-step players randomize at all decision nodes. Then 1-step players optimize starting from future information sets and subgames and working backward, and so forth for higher step players. But the CH model is designed to capture limited cognition. It seems contradictory to assume that players are constrained in their strategic thinking and yet are able to do backward induction. One idea is to link the number of steps of strategic thinking that players do about other players to the number of steps ahead that they plan. This can be accomplished by a nested-logit procedure which creates limits on forward-lookingness in a simple way. It includes backward induction as a special case. The nested logit was developed to model stochastic choice in situations where there are natural hierarchies (nesting) of choice sets or features of choices. This procedure posits that players replace a future subgame by its “inclusive value” in order to decide on a current move. At the end of their planning horizon, they insert an arithmetic average of the possible future payoffs at the nodes which are the last ones considered, and use backward induction principles on the truncated tree. Preliminary estimates of this model on one game indicate that it fits dynamic data reasonably well. 5. Studying special groups: Expertise, and deficits in lesion patients We plan to administer the battery of social situations to five special populations: (i) Undergraduates who are superbly skilled in mathematics (e.g., high SAT quantitative score): Since strategic IQ combines logic/mathematical reasoning and interpersonal intelligence, this group of subjects allows us to isolate strategic IQ from logic/mathematical reasoning intelligence. Our theory predicts that this group of subjects will not necessarily score well in the strategic IQ test. (ii) Undergraduate and graduate students specially trained in game theory: To the extent that these subjects are more likely to behave like “equilibrium” players, their performance is likely to be close to the equilibrium benchmark. They will not earn above average payoffs if they fail to understand that others may not be as rational as they are. (iii) Chess players recruited at tournaments: Skilled chess players are good at planning many steps ahead. So we would expect them to do well in games along the planning dimension. We plan to correlate their numerical chess-skill ratings with their scores along the planning dimension to determine where the two scores are indeed correlated. 10
  11. 11. (iv) Highly-experienced business managers: Experienced business managers who must consider how their competitors will react in choosing their managerial actions may be more skilled in guessing what others are likely to do in social situations. It will be interesting to determine whether this group of subjects will do better than the average population in the strategic IQ test. (v) For higher-order cognitive tasks, it is well-established that several distinct brain regions typically participate in a "neural circuit" to produce behavior. Patients with lesions are invaluable for learning which parts of the neural circuitry are necessary for the circuit to function properly, just as clipping one crucial wire in an electric circuit tells which wires are necessary for the circuit to function. Our research will use PI Adolphs's experience with the Iowa lesion patient database to correlate deficits in different types of strategic thinking with lesion locations. There are many subgroups that could be studied but we have chosen two which have a large, homogeneous sample of patients, who are he most likely to tell us something about localization of strategic thinking regions (describe below). b. Hippocampal damage and learning One interesting group is patients with memory damage restricted to the hippocampus (N=10 or so) (without concomitant damage to the nearby amygdala, a small walnut-shaped region important in fear and learning). The hippocampal patients usually have damage due to anoxia (lack of oxygen to the brain, e.g., following carbon monoxide poisoning). They have a very selective problem: their anterograde declarative memory is impaired, but their procedural memory and emotional memories are intact. (Their ability to form new episodic and semantic memories is impaired, like a buggy software program that “saves” your file every minute, but erases it by mistake rather than saving and “remembering” it.) The hippocampal-damage patients will not remember your name, or a list of numbers to memorize, but can learn and remember how to drive a car, and whether people and situations caused them emotional reactions (even if their declarative memory for those people and situations is absent). The film "Memento" dramatized patients of this sort. Whether hippocampal patients can learn in games is important because mathematical models of learning in games generally assume that people store memories of the quality of previous strategies in some numerical form, which is related statistically to the chance of choosing those strategies in the future (Camerer and Ho, 1999) and appears to be encoded, in some games, in the firing rates of neurons in the parietal lobe (e.g., Glimcher, 2003) and orbitofrontal cortex (Barraclouch et al, in press). However, these models do not commit to any interpretation of where or how these memories and updated numerical ratings are stored in the brain. The hippocampal patients enable us to learn about an intriguing question that would not have occurred to mathematical modellers-- viz., are memories about what happened previously in dynamic games explicit, declarative memories (which hippocampal patients cannot form), or procedural memories about physical acts, or emotional memories about whether a person harmed or helped you (which the hippocampal patients can form)? Put plainly: Is being betrayed by somebody in a repeated trust game, for example, more like a fact about a person (like their name), or more like an emotional memory so specially associated with a person that even a hippocampal-damage patient won't forget? If people with hippocampal damage (and intact procedural and emotional memory) "remember" an experience of being repaid a small amount in a repeated trust game, and 11
  12. 12. invest less in future periods, then their memory is clearly emotional rather than declarative, which is a huge clue about the nature of the memory and, hence, about how models of learning should be structured. Furthermore, the nature of learned memories about previous histories in games may depend on whether the histories are emotionally charged-- as previous experiences in dilemma and trust games are likely to be-- or more declarative and cognitive. c. Ventromedial prefrontal cortex Patients with ventromedial (VM) frontal lobe damage are an important source of insight, because the frontal lobes are unique to humans and only a few primate groups, and are thought to be crucial for higher-order thinking and especially planning. Patients with damage in the ventromedial prefrontal cortex are selectively impaired in their ability to plan ahead, so their behavior is a useful way to measure the possible neural circuitry of the planning ahead component of strategic IQ, and perhaps other dimensions (see Bechara et al, 1998, for details on typical VM patient profiles). These patients are useful to study because their lesions are fairly homogenous and localized, and a lot is known about their skills on a wide range of tasks and psychometric measures. Figure 2 below shows pilot data in the beauty contest game repeated six times, from four VM patients, and one control (with temporal-lobe damage) who were given a battery of games and risky choices tasks over the last four months in Iowa under the supervision of PI Adolphs. The four VM patients (three males, one female) range in age from 51-60 and range in measured IQ from 97-147. The control patient is a 49-year old female with unknown IQ. The samples are small, and obviously not random, but that is typical in lesion-patient studies (which is why triangulating results with fMRI and other measures is important). The control patient’s choices, shown by the thin line, are quite close to those in typical normal-subject samples (see Camerer et al, 2004)— she started 33 (one step of strategic thinking) and converged downward steadily toward over time the equilibrium prediction of 0. The two dotted lines show the mean, plus and minus one standard error, of the four VM patient choices. The VM patients’ number choices are significantly above the control in the first two periods. This is a suggestion, albeit with a limited sample, that VM is an important region in the first and fifth 12
  13. 13. components of strategic thinking listed in Table 1-- strategic reasoning and Figure 2: VM patients (n=4) & control, (2/3)-beauty contest game 60 50 40 mean+1 SE number 30 mean-1 SE control 20 10 0 1 2 3 4 5 6 period learning over time (at least the first period of learning). Their slower learning raises the possibility that VM patients do not produce enough regret or counterfactual thinking to learn as rapidly from feedback other than their own payoffs, compared to normal subjects. These data show that is possible to study strategic thinking using lesion patients, and are illustrative of what might be learned in larger samples of lesion patients doing a wider range of games 7. Measuring neural activity At the same time, we can also use brain imaging (fMRI) and eye tracking to establish neural circuitry in normally-functioning adults (as well as fMRI imaging of lesion patients, which is a rather new combination of tools). Part of our research is to do fMRI scans of normal subjects playing a battery of strategic IQ tests. Caltech has a new (c. 2003) imaging center with a 3T Siemens scanner dedicated to research, which is relatively underutilized and (currently) priced very cheaply. PI Camerer has successfully collaborated in a "hyperscan" consortium with neuroscientists at Baylor, Emory, and Princeton, using subjects in two scanners making simultaneous, web-linked decisions (e.g., Tomlin et al, 2004). In our scanner setup, subjects wear a pair of goggles which show a computer screen. The corner of the goggles contain a tiny eye-tracking camera which records the movements of the subjects' eyes (by contrasting the whites of the eyes with the center). Eye tracking is ideally suited to studying strategic thinking, because the payoffs of players from combinations of strategies are usually shown in distinct cells of a matrix, and it is easy to relate rules for strategic thinking to the order in which different matrix cells are "looked up" visually. 10 Eye-tracking has proved useful in 10 For example, if players think others are completely random, and choose strategies which maximize expected payoffs assuming others are random, then those players don't need to look up the payoffs of the other players. If players do look at payoffs of others, it follows that they are either thinking more strategically, or perhaps expressing 13
  14. 14. measuring the number of steps of strategic thinking in studies by Costa-Gomes, Crawford and Broseta (2001) and Costa-Gomes and Crawford (2003), building on a mouse-driven analog first used to study game theory by Camerer et al (1994), Johnson, Camerer et al (2002) and Camerer and Johnson (in press). section 8. Research plan and expected project significance Year 1: Extend CH model to dynamic games to enhance CH as a benchmark for strategic IQ. Conduct experiments with many games that tap the five strategic dimensions. Psychometric factor analysis to see whether results cohere into hypothesized factors or some other configuration, and whether strategic thinking is dissociable from general intelligence (short-form Wechsler scale) and emotional intelligence (MSCEIT). Preliminary fMRI and eye-tracking work on the CH steps-of- thinking model to establish neural circuitry of higher-order strategic thinking. Year 2: Web-based programming of a strategic IQ test that can be widely taken, capturing data from a wide range of demographic and educational groups. Administer the strategic IQ measure to specialized lesion patient populations. PI Adolphs budgets studies with 50 patients (LIST HERE) Year 3: Continuing studies with lesion patients. Collect and assemble web-based data from strategic IQ test. Dissemination and publication of findings. Expected Project Significance: This project will provide several significant scientific contributions. 1. We will extend the CH model to dynamic games (to provide a strategic IQ benchmark). This is novel because we know of no boundedly-rational model of behavior in dynamic games that has been fit to data. 2. We will provide new data on behavior of lesion patients, and expert populations, in a wide battery of games. 3. We will provide fMRI and other (e.g. eye-tracking) neurally-relevant evidence of behavior in a large new class of games. Little is known about the neural underpinning in a broad class of games. 4. Studying games will provoke some new questions in neuroscience about higher-order social cognition in dynamic situations. 5. When it is appropriate to do so, we will exploit the new capacity to “hyperscan” more than one brain at the same time. These simultaneous scans may be important for defining concepts of dyadic activation which can be relevant for understanding social outcomes. For example, the rejection of an offer is basically a dyadic behavior (it is a reflection of both the offer, and perhaps the offerer’s mistaken guess about what offer would be rejected, and the behavior of the responder). Understanding this behavior by scanning only one brain is like understanding a tennis match by watching only one player’s strokes. social preferences about whether they are earning less or more money than other players (a necessary part of emotional forecasting). Furthermore, the regret-driven component of EWA learning requires players who chose one row of a matrix, for example, to look at how much other row choices would have earned, so we can use eye-tracking as a measure of how much counterfactual thinking is being used as part of learning. 14
  15. 15. 6. We will develop a database of games, and actual choices, that permit widespread use of the strategic IQ measure for personnel evaluation and training. 11 Because the system does not require subjects to interact with others over time (except for the learning component), it is also amenable to being put on a website, which also permits gathering very large samples (albeit nonrandom ones). 9. Conclusion 11 PI Ho’s former student reported that her tech company uses the beauty contest game as a kind of mini-IQ test of general intelligence to evaluate prospective employees. 15
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