3.1.1. Definition of Case-Based Attraction

CBL is defined by Equation (2). *CBA<sup>j</sup> i* (*t*) is the case-based attraction of agent *i* to strategy *j* at time *t*; as discussed above, an attraction corresponds to the probability of selecting a strategy *j*. Here we present the equation and discuss each component in turn:

$$\text{CBA}\_{i}^{j}(t) = A\_{0}^{j} + \sum\_{m = \max(t - M, 0)}^{t - 1} I(\text{s}\_{i}^{j}, \text{s}\_{i}(m)) \cdot S(\text{x}\_{t}, \text{x}\_{m}) \cdot [\pi(\text{s}\_{i}(m)) - H] \tag{2}$$

The first term, *A<sup>j</sup>* <sup>0</sup>, is a taste parameter for strategy *j*. On the first instance of play, the second term is zero (we will explain below), so *A<sup>j</sup>* <sup>0</sup> also equals the initial attraction to strategy *j*. On the first instance of play, there are no prior cases to inform the experimenter of the subject's preferences, so it might be natural to assume that the agent ought to be indifferent among all actions, which would suggest that agents ought to choose all actions with equal probability in the first round. This does not appear to be the case in the data, hence the inclusion of this taste parameter (if initial actions are selected with equal probabilities, then these taste parameters will be estimated to be equal).

Now, let us consider the second term:

$$\sum\_{i=-\max(t-M,0)}^{t-1} I(s\_{i\prime}^j s\_i(m)) \cdot S(\mathbf{x}\_{t\prime} \mathbf{x}\_m) \cdot [\pi(s\_i(m)) - H]$$

The variable *M* the (maximum) length of memory considered by the agent. The first case considered by the agent is listed as *m* = max(*t* − *M*, 0). This has a straight-forward interpretation: either considered memory begins at period 0, which is the beginning, or, if *t* > *M* (and, therefore, *t* − *M* > 0), then only the last *M* periods are considered in memory. For example, if *M* = 3, then every utility calculation only considers the last three periods. If all experiences are included in memory then *M* is equal to ∞. We test the importance of the choice of *M* in the Section 6.

*I*(*s j i* ,*si*(*m*)) is an indicator function that maps cases in memory to the appropriate attraction for the strategy chosen: that is, when the strategy chosen, *si*(*m*), is equal to strategy *s j i* , then this function equals one and it contributes to the attraction for strategy *s j i* . Otherwise, this function is zero and it does not contribute.

*S*(*xt*, *xm*) is the similarity function, which translates the elements of the problem into relevance: the greater the similarity value, the more relevant problem *xm* is to problem *xt* to the decision-maker.

[*π*(*si*(*m*)) − *H*] is the payoff in memory, net the aspiration level, so results that exceed aspirations are positive and results that fall short of aspirations are negative.
