The wisdom of crowds has been recognized as an effective decision making mechanism by aggregating information from different individuals to derive an overall decision. However, in this information aggregation process, individuals may be influenced by various factors and provide biased claims (or individual level decisions), especially when such claims are related to their economic benefits. In this research, we investigate crowd's claims in binary decisions under explicit constant influence and aim to understand their real but hidden belief (distribution) on the decision. Particularly, we take fixed odds betting on binary events as a representative scenario in this study. We model the relationship between event probability and crowds' belief distribution as a linear combination of Beta distributions. Taking a Maximization Likelihood Estimation (MLE) paradigm, we estimate the parameters of this distribution based on observed crowds' bets. In this process, we model individual betting decisions under the influence of odds using prospect theory. We apply the framework on a real world dataset of Olympic Games outcome betting. After identifying betting participants' hidden belief distribution, we also found that crowds' belief tend to tilt to the high probability side of the event (if there is no outside influence), which partially explains why the wisdom of crowds can make decision marking easier. We believe this paper contributes to the literature of crowd intelligence and can help generating more accurate digestions of the wisdom of crowds.
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