See the pdf file for a full compiled version.
Learning through experimentation is typically characterized as a multi-armed bandit problem, where the agents face the decision of choosing among a set of actions, whose consequences might be contingent on the state of the world. In the simplest case with two actions, we refer to the two-armed bandit problem that involves one safe arm with a known expected payoff and a second risky arm with an unknown expected payoff that may be higher or lower than the safe payoff. The problem here is typically portrayed as a trade off between exploiting what we already know (and is safe) and exploring the condition of something that might be possibly better than the safe course of action. As opposed to other types of learning environments, where typically the cost of information is exogenous, in bandit problems the cost of learning is simply the opportunity cost of not exploiting the returns of the safe action. So, one faces the decision of to what degree they should experiment and when to stop experimenting. Typically, the strategies are defined as mappings from the belief space to the allocation of a resource among the actions, where it's assumed that the agent makes her decision only depending on her current subjective belief of the world. The strategies are then chosen in order to maximize total expected payoffs.
Historically, the environment usually involved only a single-agent acting on her own and only observing her own actions and their consequences, whereas in the modern literature the strategic interactions in a social learning environment, allowing agents to learn from each others' actions is the focal point of the analysis. With the recent developments, notoriously intractable nature of the bandit problems are understood to a greater degree. Mainly, we talk about two effects that evolves from the strategic consideration in a social learning environment: the free-riding effect and the encouragement effect. The first is a natural result of the two main properties of the model: ability to learn from each others' actions and the assumption that the experimentation procedure and the state of the world is identical among the agents. This conjunction leads agents to shy away from costly experimentation as long as they believe that there are other people, who are undertaking a sufficient amount of experimentation and (depending on the model) are going to share their experimentation results fully and truthfully with them. Believing this, agents have no incentive to undertake costly experimentation, when they can just as well continue to use the safe arm and be indirectly benefiting from the experimentation at the same time. The latter effect can also be thought of as a result of free-riding. Given that incentives to experiment are problematic and as nobody wants to be the guinea pig of the group, experimentation process can slow down undesirably, even though it's beneficial for everyone. Hence, knowing that a good experimentation outcome today might lead someone to join experimentation tomorrow one has then the incentive to be the leader of the group to motivate social welfare. This effect is only sustained if the good news are inconclusive i.e. when there is still something to learn after a good outcome. Nevertheless, typically the free-riding effect is still persistent enough to cause inefficient levels of experimentation. Although, with recent developments, we see that efficiency is in fact achievable under different equilibrium considerations.
As a technical simplification, the literature typically treats the problem in a setting where the state of the world is considered to be binary: good or bad. This naturally leads to the conclusion that prior to conducting experimentation, agents have to know exactly what two outcomes they might be facing. In an artificial game where rules are communicated in advance (like a gambling machine) this assumption might be non-restrictive. Although, as especially in the economics literature the setup is thought of as an abstraction of real-world experimentation like drug trials or mineral discoveries; thinking of the state space as higher dimensional (potentially continuous) should allow for an environment that is more suitable for real-world justifications, which may also lead to a more elastic learning environment and higher levels of efficiency.
A potential benefit of achieving tractable analysis under a continuous state space might be that, then, the state of the world can be thought of as the resulting outcome of a completely different non-binary process. For example, the goodness of the arm might be time-dependent. This might relate to concepts like time-dependent preferences or external shocks that change the underlying characteristic of payoffs. In the case of drug trials, the binary setup only allows for whether the drug is more effective than a benchmark level or not. In a non-binary state, how effective the drug is can also be characterized in order to potentially compare with other opportunities like simultaneous experimentation. Further, the effectiveness of the drug might be dependent on time-dependent complications with other stimulants and hence the signal that experimentation produces might also be biased depending on at which point in time the experimentation is undertaken. With time dependent tastes, expected benefit of experimentation (and it's relation to the opportunity cost) can be variable at different points in time, which might lead to time variable cutoff beliefs that agents use to decide when to experiment. Such a model might be used to explain periodic risk sensitives depending on factors like self-confidence and liquidity.