Cognitive ability and the effect of strategic uncertainty

Another Nash equilibrium, (L, l), involves a weakly dominant strategy l by Player B. The existence of a clear-cut theoretical benchmark distinguishes this game from another well-known \(2 \times 2\) coordination game, the stag hunt, in which each Nash equilibrium is supported by a certain solution concept—either payoff dominance or risk dominance.

Other studies, somewhat less related to ours, used robots that did not follow equilibrium strategies as a way to control for subjects’ beliefs about the behavior of their opponents. For instance, Ivanov et al. (2010) used robots to replicate past behaviors of their subjects, and Embrey et al. (2015) and Costa-Gomes and Crawford (2006) used robots to make some players follow the predetermined distribution of boundedly rational behaviors.

Raven’s progressive matrix test (often called Raven’s test) is a picture based, non-verbal measure of fluid intelligence, that is “the capacity to think logically, analyze and solve novel problems, independent of background knowledge” (Mullainathan and Shafir 2013, p. 48). It is widely used by, e.g., psychologists, educators and the military (Raven 2000). It consists of a series of tasks to be solved within a fixed amount of time (for instance, we use a series of 16 tasks to be solved in 10 min). In each task, a subject should pick a single element (among 8 options) that best fits a set of 8 pictures. These pictures are put into a certain logical order and presented in a \(3 \times 3\) table with a blank space in the bottom right corner. The level of difficulty increases from one question to the other. See Raven (2008) for an overview.

Dohmen et al. (2010) reported similar correlations between cognitive ability (measured with a verbal and a non-verbal task related to the Wechsler Adult Intelligence Scale) and risk and time preferences in a representative sample of the representative German population.

The “Race to 5 (or 10 or 15)” game is a two player sequential move game in which two players, moving alternatively, can put either 1, 2 or 3 stones in a common hat which is empty at the beginning. The player who puts the 5th (or 10th or 15th, respectively) stone in the hat wins. The first mover has a clear advantage in this game, and one can derive the winning strategy by a backward induction. The difficulty of deriving the winning strategy increases with the number of target stones.

Baron-Cohen et al. (2001) developed the “Reading the Mind in the Eyes” test (RMET) to measure one’s theory of mind—the capacity to infer the internal emotional states of others. RMET consists of a series of photos of the area of the face involving the eyes. Subjects are asked to choose one of the four words that best describes what the person in the photo is thinking or feeling. Ibanez et al. (2013) found that people with high scores in Raven’s test also perform better in RMET. In an experimental investigation of the Level-k model, Georganas et al. (2015) found a positive correlation between the score in RMET test and the propensity to adapt Level-1 reasoning.

Repetition allows to assess the extent to which inefficient behavior is sensitive to learning. For this sake, we use an indefinitely repeated game with one-round compensation rule1 as an attempt to homogenize incentives across rounds, and allow for an accumulation of experience from a series of uniform one-shot interactions. Kamecke (1997) shows that our perfect stranger, round-robin procedure is optimal for this purpose since it maximizes the number of rounds for a given number of players and the one-shot nature of each interaction between subjects.

In the present study, we focus on the determinants on Player As’ behavior, considering Player Bs’ solely as a source of strategic uncertainty. Jacquemet and Zylbersztejn (2013) offer a systematic analysis of the patterns of Player Bs’ decisions in this game.

Some studies have also documented the effect of the saliency of monetary incentives in coordination games. See, for example, Battalio et al. (2001) for symmetric \(2 \times 2\) games, and Goeree and Holt (2005); Devetag and Ortmann 2010) for n-player minimum and median effort games.

We decided to implement the administrative questionnaire at the beginning of the experiment to reduce the noise in answers and to avoid an accumulation of post-experimental surveys. As correctly stressed by a referee, this might raise concerns about anonymity in subjects’ decision-making. However, this part of the design is identical in all session and thus should not affect our main results that are based on the between-treatment differences.

The unexpected behavior initially observed for matrices BG1 and EG2 led us to complement our design with matrices BG2 and EG1, hence the delay between the two sets of experiments. To assure an in-depth exploration of players’ behavior, these complementary sessions also included Raven’s test.

In one EG2 Robot treatment session, we had 18 subjects instead of 20, so for the Robot treatment there were 40 subjects for BG1, BG2 and EG1, and 38 subjects for EG2. For the Human treatment sessions, we had 60 subjects (half of whom are Player As) for each of the four games.

All sessions took place at the Laboratoire d’Economie Experimentale de Paris (LEEP) at Paris School of Economics. Subjects were recruited via an online registration system based on Orsee (Greiner 2004) and the experiment was computerized through software developed under Regate (Zeiliger 2000) and z-Tree (Fischbacher 2007).

As will be described below, Raven’s test was included in half of our experimental sessions and was carried out as a post-experimental task. For this post-experimental task, 15 additional minutes were needed beyond the usual duration of the sessions (around 45 min, including the time to read the instructions, answer the questionnaires, play 10 rounds of the experimental game and be paid for participation).

BG1: \(p = .402\); BG2: \(p = .385\); EG1: \(p = .557\); EG2: \(p = .002\).

We test the difference in proportion of a given outcome between two experimental conditions by carrying out a two-sided bootstrap proportion test that accounts for within-subject correlation—i.e., the fact that the same individual takes 10 decisions. The procedure consists of bootstrapping subjects and their corresponding decisions over all ten rounds instead of bootstrapping decisions as independent observations (see, e.g.,  Jacquemet et al. 2013, for a detailed description of the procedure). In Round 1, data are independent and thus allow us to analyze behavior with a standard bootstrap proportion test. Frequencies in Round 1 are 23.3 % in BG1 and 50.0 % in BG2 (\(p=.027\)), and 30.0 % in EG1 and 50.0 % in EG2 (\(p=.091\)).

The analogous frequencies in Round 1 for Player Bs are 80.0 % in BG1 and 83.3 % in EG1 (\(p=.731\)), and 76.7 % in BG2 and 86.7 % in EG2 (\(p=.232\)); for Player As, they are: 23.3 % in BG1 and 30.0 % in EG1 (\(p=.583\)), and 50.0 % in BG2 and 50.0 % in EG2 (\(p=.889\)).

Our first-order stochastic dominance test is based on a bootstrap version of the univariate Kolmogorov–Smirnov (KS) test which allows for ties (see, e.g., Abadie 2002; Sekhon 2011).

These shifts are significant according to bootstrap proportion tests at the 1 % threshold in BG1 (\(p=.001\)) and in EG1 (\(p=.005\)), at the 5 % threshold in EG2 (\(p=.033\)) and are not statistically significant in BG2 (\(p=.305\))

First-order stochastic dominance (FOD) in BG2 is induced by a sharp increase in the proportion of subjects playing R in all 10 rounds (from 30.0 % in the Humans treatment to 65.5 % in the robot treatment). This is enough to induce a statistically significant FOD. In EG2, the magnitude of this increase is small: from 40 % of subjects playing R in all 10 rounds in the humans treatment to 57.9 % in the robot treatment.

The data from BG2 and EG1 are pooled to focus on the overall effect of removing strategic uncertainty and guarantee sufficient sample sizes in each category.

To ensure a sufficient sample size in each ability group, we pooled the outcomes from both games in each treatment. The tests are performed using two-sided bootstrap K-S tests. The p values of the two-by-two comparisons are: \(p=.288\) for the low-ability group vs. the medium-ability group, \(p=.599\) for the low vs. the high and \(p=.695\) for the medium vs. the high.

Given the group sizes, the bootstrap test is based on re-sampling subjects and the number of times they choose decision R. To account for asymmetry in the empirical distribution, we computed an equal-tail bootstrap p value. See Davidson and MacKinnon (2006) for further details on this procedure.