Noisy Introspection in the 11–20 Game

Abstract

Previous experiments based on the 11–20 game have produced evidence for the level‐k model with observed levels of strategic thinking consistently ranging from 0 to 3. Our baseline treatment uses the 11–20 game and replicates previous results. We apply four models of strategic thinking to the baseline‐treatment data and use these to predict behaviour and beliefs in five other treatments that employ games with a very similar structure. The best predictive performance is achieved by models that incorporate ‘common knowledge of noise’. A model of noisy introspection, which does so, predicts behaviour remarkably well.

Cognitive Sophistication and Deliberation Times

“…Finally, our work sheds light on the recent literature exploring the limits of models of iterative thinking, as the experiment of Goeree et al (2016) mentioned above. It has been pointed out that strategic sophistication, as captured by level-k models, might be heavily dependent on the situation at hand.…”

“…Alaoui and Penta (2016b) used the 11-20 game to test their model of endogenous depth of reasoning. Goeree et al (2016) introduced a graphical version of the 11-20 game that allows to vary the payoff structure without affecting the underlying best-reply structure of the game. We now describe a generalized version of this graphical 11-20 game.…”

“…In the baseline version (BASE) the amounts are increasing from the left box to the rightmost box, containing the highest amount of 20. BASE corresponds to the original version of Arad and Rubinstein (2012) and to the baseline version of Goeree et al (2016). In BASE 5 Note that the best reply to an opponent choosing box 9 is to choose box 0, hence for k > 9 the best-reply structure cycles repeatedly from 0 to 9.…”

“…We test these predictions in an experiment employing two different games commonly used to study iterative thinking: the beauty contest game (or guessing game; Nagel, 1995), which is the workhorse in that literature, and several variants of the 11-20 money request game, recently introduced by Arad and Rubinstein (2012), in the graphical version of Goeree et al (2016). These variants all share the same best-reply structure, but the payoff structures are manipulated in order to change the salience of certain features which might encourage iterative thinking.…”

“…These variants all share the same best-reply structure, but the payoff structures are manipulated in order to change the salience of certain features which might encourage iterative thinking. The reason why we use the latter game is that in some of the variants considered in Goeree et al (2016), reconciling observed behavior with a model of iterative thinking would require inordinately high levels of sophistication, compared to those usually observed in the literature. This provides a natural setting where deliberation times can discriminate whether observed behavior actually corresponds to different levels of cognitive effort.…”