We have discussed the rationality behind voting behavior, and I think that the pB + [D] - C > 0 model could also be used to create a “rational gambler hypothesis” with sports bettors. Each individual bet can be seen as a “vote,” with p representing the probability of it being successful, B being the potential payout, and C representing the wagered amount as well as the time and effort spent to place the bet. Similar to voting, pB will virtually always be less than C because sportsbooks both set accurate lines and offer odds that are advantageous to the house. If we assume that lines favor the house and gamblers know this, then the “D” value, the added utility of watching a game with a bet placed on it, is the reason that people gamble on sports.
If we attempt to analyze the model to get the optimal bet size (similar to Peltzmann’s work), then the first step is to take the derivatives of (pB - C) and (D) with respect to bet size. (pB - C)’ is negative as the size of the bet increases due to the disadvantageous odds, and there is also a significant "fixed cost" associated with placing any bet. D’ is positive until some satiation point where the bet becomes stressful, and D’’ is negative before this due to the law of diminishing returns.
A normative implication of this model is that gamblers should place bets that are big enough to be worth their time but not too large that they start to become stressful. These moderately-sized wagers could benefit both the bettor and the house, who knew that gambling could be pareto-efficient?
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