We develop a novel graphical paradigm of a strict-dominance-solvable game to study the developmental trajectory of steps of reasoning between 8 years old and adulthood.
Most participants play the equilibrium action either always or only when they have a dominant strategy. Although age is a determinant of equilibrium choice, some very young participants display an innate ability to play at equilibrium. Finally, the proportion of equilibrium play increases statistically-significantly until 5th grade and stabilizes afterward, suggesting that the contribution of age to equilibrium play vanishes early in life.
…To minimize these concerns, we propose a novel graphical interface in which subjects possess 3 objects with 3 attributes each: a shape, a color, and a letter. Their goal is to select an object with a certain characteristic, which depends on the object selected by another player in the game. This is true for all but one player, who must simply match a feature of a specific single object. This player’s decision constitutes the starting point of the iteration process, and the problem of the other players can be iteratively solved by successive elimination, with a maximum of 3 steps of reasoning.
To analyze the developmental trajectory of behavior in our paradigm, we recruited 3 populations. The first experiment involves a population of children and adolescents (8–18 years old) recruited at a single private school in Los Angeles [Lycée International de Los Angeles (LILA), a French-English bilingual private school in Los Angeles], as well as a control young-adult population from the University of Southern California (USC). This experiment tests the effect of age on strategic sophistication. In the second experiment, we recruited younger children (5–8 years old) from that same school and we implemented a simpler version of the same game. This experiment is designed to assess whether the skills detected in children older than 8 years old are already developing before that age. Last, we recruited a third population of middle schoolers (11–14 years old) from a single public school also in Los Angeles. This experiment aims to inform us on the potential impact of school characteristics and student demographics on sophistication.
Our analysis yields 3 main results. First, the vast majority of participants either play always at equilibrium or they play at equilibrium only when they have a dominant strategy. There are few random players, and virtually no one exhibits an “intermediate” level of reasoning (ie. plays at equilibrium when it requires 2 steps of reasoning but not when it requires 3 steps). This is in sharp contrast with the existing adult literature that emphasizes large heterogeneity in levels of reasoning and abundance of intermediate types (Costa- et al 2001; et al 2002; Costa-2006; Brañas- et al 2011; et al 2014; 2015; 2016; et al 2018). Second, although age is an important determinant of equilibrium thinking, there is an ability component that is either innate or acquired at a very young age. Furthermore, the evolution over the entire window of observation is not as steep as one might expect. Indeed, the proportion of individuals who consistently play at equilibrium is statistically-significantly above 0 at 8 years old (24%) and statistically-significantly below 1 at 17 years old (59%). Third and related, the change in equilibrium play is not constant. Choice improves statistically-significantly between 3rd and 5th grade and stabilizes afterward. In other words, the contribution of age to equilibrium behavior vanishes relatively early in life (12–13 years old).
Our data reveal important predictors of performance. We find that female participants and subjects with a self-reported preference for science subjects perform statistically-significantly better. Finally, differences across schools and across tracks within schools are also associated with differences in sophistication. In particular, we find that students enrolled in different programs or in different GPA-based tracks within programs exhibit different levels of sophistication. [IQ/intelligence was not measured.] Overall, even though the main pattern of behavior (namely, the absence of an intermediate level of reasoning) is replicated in all populations, the distribution of sophistication is modulated by individual and group characteristics.
…We designed a simple paradigm in which subjects were matched in groups of 3 and assigned a Role as player 1, player 2, or player 3, from now on referred to as Role 1, Role 2, and Role 3. Each player in the group had 3 objects, and each object had 3 attributes: a shape (square, triangle, or circle), a color (red, blue, or yellow), and a letter (A, B, or C). Players had to simultaneously select one object. Role 1 would obtain points if the object he chose matched a given attribute of the object chosen by Role 2. Similarly, Role 2 would obtain points if the object he chose matched a given attribute of the object chosen by Role 3. Finally, Role 3 would obtain points if the object he chose matched a given attribute of an extra object. The attributes to be matched were different for different Roles and specified by the experimenter. Accordingly, in each game any number of participants could obtain points. All options and objectives of players were common knowledge and displayed on the computer screen. Figure 1 provides a screenshot of the game as seen by Role 2. The game can be easily solved with an inductive argument starting from Role 3. In the example of Figure 1, Role 3 has to match the shape of the outside object, so he obtains the points if he chooses the red square C. Conditional on that choice, Role 2 obtains points if he chooses the red triangle B, and, again conditional on that choice, Role 1 obtains points if he chooses the yellow circle B (the original software uses easily distinguishable colors).
…Figure 3 reports the proportion of subjects by grade [strategy type] who are classified under each type, from most sophisticated (bottom) to least sophisticated (top). In strong support of Hypothesis 1, our theoretical model provides a very solid behavioral template. Indeed, the choice of 76% of LILA students and 97% of USC students can be accounted for by one of the 4 types described in §II.B. The proportion of subjects who do not fit in one of these types (O) decreases with age, although it is statistically smaller only for 10th graders. In other words, the level-k behavioral theory that has proved successful in explaining nonequilibrium behavior of adults performs well also with children and adolescents.
…Hypothesis 2 is not supported by the data. Subjects either recognize only a dominant strategy or always play at equilibrium. Also, some very young players display an innate ability to play always at equilibrium while some young adults are unable to perform 2 steps of dominance.
…Hypothesis 3 is weakly supported by the data. Equilibrium performance increases with age very statistically-significantly during elementary school but it stabilizes in 6th grade.
…While equilibrium performance increases with age, there is also a substantial innate component: some of our youngest participants play perfectly from the first trial whereas some of our oldest participants do not go beyond one step of reasoning. Even though there is some evidence of learning, repeated exposure is ineffective at bringing participants to play Nash. Finally, performance increases statistically-significantly 8–12 years of age and stabilizes afterward, suggesting that most of what is needed to solve dominance-solvable games is acquired by the end of elementary school. Interestingly, most students acquire complex mathematical skills during adolescence. Our observations suggest that this extra knowledge does not translate into better strategic decision making.
…Adolescents are particularly exposed to situations in which strategic sophistication is crucial to avoid wrong decisions. Examples include engaging in risky activities, such as accepting drugs from peers or engaging in unprotected sex. Also, with the development of the internet, naive users are often preyed upon, asked to provide personal information, or tricked into making harmful decisions. Information deliberately intended to deceive young minds also circulates through social media. Making correct decisions in such environments requires understanding the intentions of others and anticipating the consequences of following their advice or opinions. More generally, children and adolescents are gradually discovering the dangers hiding behind social interactions and need to come equipped to detect them, assess them, and navigate around them.
We conjecture that failures in these abilities are closely related to underdeveloped logical abilities, and we predict that the level of sophistication of an individual detected through a simple task matches their behavior in social settings.