*3.1. Case-Based Learning*

We bring a formulation of case-based decision theory as introduced by Gilboa and Schmeidler [1] into the "attraction" notation discussed above, ultimately ending up with a case-based attraction *CBA<sup>j</sup> i* (*t*) for each strategy *sj*.

The primitives of Case-Based Decision Theory are: a finite set of actions A with typical element *a*, a finite set of problems P with typical element *p*, and a set of results R with typical element *r*. The set of acts is of course the same as the set of actions or strategies as one would find in a typical game theoretic set-up. The set of problems can be thought of a set of *circumstances* that the agent might face: or, more precisely, a vector of relevant information about the present circumstances surrounding the choice that the agent faces, such as current weather, time of day, or presence of others. The results are simply the prizes or outcomes that result from the choice.

A problem/action/result triplet (*p*, *a*,*r*) is called a *case* and can be thought of as a complete learning experience. The set of cases is C = P×A×R. Each agent is endowed with a set M⊆ C, which is called the memory of that agent. Typically, the memory represents those cases that the agent has directly experienced (which is how it is used here) but the memory could be populated with cases from another source, such another agent or a public information source.

Each agent is also endowed with a similarity function *<sup>s</sup>* : P×P → <sup>R</sup>+, which represents how similar two problems are in the mind of the agent. The agent also has a utility function *<sup>u</sup>* : R → <sup>R</sup> and a reference level of utility *H*, which is called the aspiration value. An aspiration value is a kind of reference point. It is the desired level of utility for the agent; when the agent achieves her aspiration value of utility, she is satisfied with that choice and is not moved to seek alternatives.

When an agent is presented with problem *p*, the agent constructs the case-based utility for each available action *<sup>a</sup>* ∈ A and selects an action with the highest CBU.<sup>6</sup> CBU is constructed from memory in Equation (1).

<sup>5</sup> We discuss the functional form of the probability distribution in Section 3.6.

<sup>6</sup> As discussed in the introduction of this section, our implementation uses attractions, so choice is not deterministic, but rather stochastic with the probability of choosing an action increasing in the CBU.

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

$$\text{CBI}(p, a) = \sum\_{(q, a, r) \in \mathcal{M}(a)} s(p, q) \left[ u(r) - H \right] \tag{1}$$

where M(*a*) is defined as the subset of memory in which act *a* was chosen; that is M(*a*) = {(*q*, *a*,*r*)|(*q*, *a*,*r*) ∈ M}. (Following Gilboa and Schmeidler [1], if M(*a*) is the empty set—that is, if act *a* does not appear anywhere in memory—then *CBU*(*p*, *a*) is assumed to equal zero.)

The interpretation of case-based utility is that, to form a forecast of the value of choosing act *a*, the agent recalls all those cases in which she chose action *a*. That typical corresponding case is called (*q*, *a*,*r*). The value associated with that case is the similarity *s*(*p*, *q*) between that case's problem *q* and her current problem *p*, times the utility value of the result of that decision, minus the aspiration value *H*. Subsequently, her total forecast is the sum of those values across the entire available memory.

Now, let us bring the theory of case-based learning to an empirical strategy for estimating case-based learning in these experiments. Note that the experiments studied here—2 × 2 games with information about one's history—provide an environment for testing the theory of case-based learning, because the information vectors P presented to subjects is well-understood and controlled by the experimenter. (Outside the lab, more and stronger assumptions may be required to define P).
