**2. Applying Case-Based Learning to Experiments**

First, we compare the case-based approach to traditional expected utility. The expected utility framework requires that the set of possible states is known to the decision-maker and that the decision-maker has a belief distribution over this set of states. Case-based decision theory replaces the state space and its corresponding belief distribution with a "problem" space—a space of possible circumstances that the decision-maker might encounter—and a similarity function defined over pairs of problems (circumstances). One limitation of the expected utility approach is that it is not well-defined for the decision-maker to encounter a truly "new" state, which is, a state the decision-maker had never thought of before (it could be modeled as a state that occurs with probability zero, but then Bayesian updating would leave it at probability zero). The case-based approach overcomes this difficulty: the decision-maker can naturally encounter a "new" problem or circumstance, and need only be able to judge how similar that problem is to other problems the decision-maker has encountered: no ex-ante determination is required.2 The problem space is also, arguably, more intuitive for many practical decision-making problems than the corresponding state space. For example, consider the problem of hiring a new assistant professor, where one's payoff includes the success of this candidate, fit with the current department, willingness of the candidate to stay, etc. Describing each candidate as a vector of characteristics that can be judged more or less similar is fairly intuitive, while constructing

<sup>2</sup> It is worth noting that there is a mapping between expected utility and case-based decision theory [25], which implies that in a formal sense replacing the state space with the problem space is not 'easier,' if one requires that the decision-maker must *ex-ante* judge the similarity between all possible pairs of problems.

a corresponding state space—possible maps of candidates to payoff-relevant variables—may not be. Reasoning by analogy, through similarity, can also make complex decisions more manageable. Moreover, the similarity between vectors of characteristics provides a specific means of extrapolating learning about one candidate to other candidates; the assumption of prior distributions, and updating those distributions, provides less guidance about how that extrapolation should be done.

In our setting multiple games are played in a laboratory in which players interact with each other to determine outcomes. The state space in this setting is large. The broadest interpretation of the appropriate state space is: the set of all possible maps from all possible histories of play with all opponents to all future play. While this is quite general, learning (that is, extrapolation from past events to future ones) requires the specification of a well-informed prior; if literally any path of play is possible given history, and if one had a diffuse (i.e., "uninformed") prior over that set, then any future path is equally likely after every possible path of play. Alternatively, the state space might assume a limited set of possible player types or strategies; in that case, the state space would be all possible mappings of player types/strategies to players. While this provides more structure to learning, it requires that the (correct) set of possible player types is known.

On the other hand, defining an information vector about history is less open-ended. There are natural things to include in such a vector: the identity (if known) of the player encountered; the past play of opponents, a time when each action/play occurred; and, perhaps other features, such as social distance [26] or even personality traits [27]. This implies a kind of learning/extrapolation in which the behavior of player *A* is considered to be more relevant to predictions of *A*'s future behavior than is the behavior of some other player *B*; that if two players behave in a similar way in the past; that learning about one player is useful for predicting the play of another; and, that more recent events are more important than ones far in the past. These implications for learning naturally arise from a similarity function that considers vectors closer in Euclidean distance to be more similar (as we do here).<sup>3</sup> Interestingly, others have adopted the concept of similarity as a basis of choice in cognitive choice models [28].4

Note that this kind of extrapolation can be constructed in a setting with priors over a state space, under particular joint assumptions over the prior over the state space and the state space itself. Case-based decision theory can be thought of as a particular set of testable joint assumptions that may or may not be true for predicting human behavior.
