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which we take to be a normally distributed random variable z with standard deviation σ . The spread s$_{0}$ is the estimated mean of z according to the oddsmakers but an individual may have a different estimate s . The subjective probability of favorite game victory is given by the cumulative probability that z > 0: ρ ( s ) = 1 $_{2}$erfc $^{(}$− s/ ( σ √ 2) ) where erfc( x ) = 2 √ π ∫ ∞ x e − v $^{2}$dv . The function ρ ( s ) is concave for s > 0. That NFL fans have an intuitive understanding of this concave relationship — that the probability of game victory flattens out as the expected margin of victory increases — has been observed empirically (Simmons and Nelson 2006:422). 

A similar expression holds for the probability of the favorite winning against the spread: p ( s ) = 1 $_{2}$erfc $^{(}$− ( s − s$_{0}$ ) / ( σ √ 2) $^{)}$. If s > s$_{0}$ , then p > 0 . 5 and one should bet on the favorite. One can then solve for s as a function of p , which in turn can be written as a function of w using Equation 5. Substituting for s in ρ ( s ) above then yields: 

As for p ( w ), ρ ( w ) is concave for w > 0 as displayed in Figure 2. 

<img src='content_image/312427.jpg'>

We use ρ ( w ) in our simulations assuming identical risk aversion α and standard deviation σ . Rather than using α directly, we define the more intuitive parameter p$_{max}$ which corresponds to the subjective probability p ( w$_{max}$ ) for which an individual will make the maximum wager, w$_{max}$ = $7; α can then be calculated via α = (2 p$_{max}$ − 1) / (8 w$_{max}$p$_{max}$ (1 − p$_{max}$ )). While p$_{max}$ is a free parameter, we fix σ = 12 . 8, which is the empirical standard deviation for the margin of victory in NFL games (Sapra 2008). The derivation of ρ ( w ) represents a first-principles calculation that relates the rhetorical issue to contextually meaningful parameters and hence represents a more stringent test than using an arbitrary S-shaped function. It also emphasizes that even if both favorite and underdog wagers are arrived at by a rational procedure, 

