**Appendix A. Proofs**

**Proposition A1.** *Let N(t)=N(t+1) and H* = 0*, A*<sup>0</sup> = 0 *for all choices, only time is in the definition of the problem for CBL, and the similarity function is simple inverse weighted exponential given in Equation (10). Then there exists φ* > 0 *such that RL attraction RLA with parameter φ and the case-based attraction implied by*

*Games* **2020**, *11*, 38

*the similarity function S are equal, and therefore the attractions decay at the same rate and φ and w are related in Equation* (A1)*.*

$$\phi = S(\mathbf{x}\_t, \mathbf{x}\_{t-1}) = \frac{1}{w} \tag{A1}$$

For Proposition 1 we provide the following proof. When considering attractions from RL and CBL with the following simplifications: *N*(*t*) = *N*(*t* + 1), *H* = 0, *A*<sup>0</sup> = 0, and only time is in definition of the problem for CBL. Under the condition that the similarity function is an inverse exponential of the difference in the time index, RL attraction degrades at the rate *φ* while CBL degrades at the rate defined by the following similarity function,

$$
\phi = \frac{1}{w^{\left| t - t - 1 \right|}}.\tag{A2}
$$

Then, for all past attractions more generally, *φ<sup>k</sup>* = <sup>1</sup> *wt* <sup>|</sup>*k*<sup>|</sup> . As past attractions are discounted in RL they get discounted each time period by *φ* so to is the weight in CBL by an equivalent adjustment in the distance between time periods. This rate is held constant across time in both models since *N*(*t*) = *N*(*t* + 1).

The typical similarity and distance functions used in the literature do not have equivalence between the similarity and recency in RL, and therefore this may be seen as a special case of the relationship between RL and CBL. We do not use these specialized forms to estimate against the data, but rather use them to demonstrate the simple similarities and differences in how CBL and RL are constructed.
