2.4. COGNITIVE CAPACITY AND LEARNING
the relation between subjects’ cognitive capacity and their cognitive hierarchy in a scaling-up environment, rather than being limited to the analysis of two-or three-person games. In addition to this distinction, our paper and Gill and Prowse (2012) are also distinguished from Georganas et al. (2010) and Branas-Garza et al. (2012a) in that we both have multiple rounds of the experiment, instead of a one-shot game. This design serves the purpose of studying learning well, and is an important subject to which we now turn.8
2.4 Cognitive Capacity and Learning
2.4.1 Cognitive Capacity, Cognitive Hierarchy and Learn-ing
Learning is normally not assumed in level-k reasoning, as if the model can best apply or only apply to the situation when the environment presented to the subject is novel. Once agents have experiences and have learned, their level of reasoning becomes irrelevant since they may all be updated. This concern implicitly assumes that learning can eliminate the initially observed heterogeneity in subjects’ levels of reasoning, and therefore, subjects’ cogni-tive capacity should be irrelevant when learning takes effect. For example, the main feature found in repeated BCG is that behavior converges to equilib-rium over time. If all subjects converge to choose equilibequilib-rium zero, there will be no behavior heterogeneity. In this case, we should not expect a cognitive hierarchy revealed and the effect of cognitive capacity. This “convergence hypothesis” has been well supported by Schnusenberg and Gallo (2011) in their repeated beauty contest game, while their analysis did not bring in the element of cognitive hierarchy. This results implies that learning dominated, or even stronger, that learning is independent of cognitive capacity in the repeated environment.
However, there are also experimental studies showing that learning is not independent of cognitive capacity. By that, subjects’ learning capability or dynamics can be affected by their cognitive capacity. For example, Casari, Ham and Kagel(2007) showed that subjects with lower composite SAT/ACT scores were more likely to suffer from a winner’s curse. Even if they are
re-8Branas-Garza et al.(2012a) have only one round for each of their six different versions of the beauty contest game, but the use of this multiple-version design may still allow subjects to learn from their experiences such as the learning observed in the multiple-stage game. Nevertheless, the learning dynamics, i.e., how learning can happen by carrying over the experience from one version of the game to other subsequent versions, is not the focus of their analysis.
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2.4. COGNITIVE CAPACITY AND LEARNING
cruited as experienced subjects participating in similar experiments, they can still not avoid the “curse”. This implies that the effect of cognitive capacity persists even after learning is taken into account. Gill and Prowse (2012) also found that the performance gap caused by cognitive capacity will not decay away simply due to learning. In fact, their study shows that cogni-tive capacity may posicogni-tively affect learning in the way that subjects with higher cognitive capacity may learn more actively than subjects with lower cognitive capacity and hence, in the end, their performance gap will become even more significant than the initial time.9 In addition, they also applied a level-k learning model to understand the behavioral mechanism that leads to these differences. Gill and Prowse (2012) found that high cognitive abil-ity subjects follow level-k choice type that are significantly higher than low ability subjects, either in own-matched group or in cross-matched group.
In our case, the issue is then whether subjects of different working memory capacity may actually learn in the way that their initial levels of reasoning, absolutely or relatively, remains unchanged, or, alternatively, in the way that the initial gaps among subjects can be filled. In other words, our study examines, to what extent, the level-k reasoning holds for not just a one-shot game, but may in effect work also in repeated games. More generally, this study makes a fundamental inquiry into the effect of cognitive capacity on cognitive hierarchy in contexts with experiences and without experiences.
2.4.2 Cognitive Capacity and EWA Learning
Learning behavior in BCG has been widely studied by applying various learn-ing models (Nagel,1995;Stahl,1996;Duffy and Nagel,1997;Stahl,1998;Ho, Camerer, and Weigelt, 1998; Camerer and Ho, 1999). Among them, a gen-eralized reinforcement learning model, said, experience-weighted attraction (EWA) learning was proposed to combine reinforcement and belief learning, two seemingly unrelated mechanisms, and to include them as special cases.
EWA learning was created by Camerer and Ho (1998, 1999) to model the learning behavior in several repeated games. BCG is one of them.
The formulation of EWA learning model is described as follows. In gen-eral, the choice probabilities of strategies are determined by the logit transfor-mation of their attractions, which reflect initial propensity and are updated according to payoff experience. There are two variables, attractions Aji(t) and an experience weight N(t) in EWA model. Both variables are updated
9Gill and Prowse(2012) found that, while in the first five rounds there is no significant difference in earnings between high and low cognitive ability subjects, their earnings dif-ference becomes bigger in the last five rounds, and the difdif-ference is statistically significant.
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2.4. COGNITIVE CAPACITY AND LEARNING
according to the experience after each period. The first variable N(t) begin with an initial value N(0) and is updated according to
N(t) = ρN(t − 1) + 1, t ≥ 1
The parameter ρ is a discount factor that captures decay in the strength of prior beliefs. The second variable Aji(t) denote player i’s attraction of strategy j. It starts with some prior values Aji(0) and is updated according to t. In general, the attraction Ai(t) is the running total of past attractions, which are constituted by a depreciated experienced-weighted past attraction Aji(t) plus the payoff yielded from period t. The key component of this up-dating rule is the weighted payoff term [δ + (1 − δ) · I(sji, si(t))] · πi(sji, s−i(t)), which captures two basic principles of learning. First, the attractions of cho-sen strategies si(t) are updated by actual payoff, which means that successful strategies are given more reinforcement and are more likely to be repeated subsequently. This is so called the ’law of effect’ (Thorndike, 1911; Herrn-stein, 1970) in the literature of learning by behavioral psychologist. Second, the attractions of unchosen strategies are updated by forgone and hypothet-ical payoff with a weight δ (0 ≤ δ ≤ 1). Camerer and Ho (1999) introduced this effect and called it the law of simulated effect and renamed the former one as the law of actual effect. In this setting, both chosen and unchosen strategies are ’reinforced’ by the payoff that strategy either yielded or would have yielded.
EWA model can be reduced to choice reinforcement or belief-based mod-els under different parameters configurations. In other words, reinforce-ment learning and belief learning are special cases of EWA learning. Define κ = (φ − ρ)/φ. The family of choice reinforcement model corresponds to δ = 0. With further restriction of parameter κ, the EWA learning can be ei-ther cumulative choice reinforcement when κ = 1, or averaged-reinforcement when κ = 0. The family of belief learning model corresponds to δ = 1.
Similarly, further restricting some parameters derive specific belief learning models such as Cournot best-response dynamics when φ = κ = 0, standard fictitious play when κ = 0 and φ = 1, weighted fictitious play when κ = 0.