
THEORETICAL MODEL (1A)
For my research, I use a Bayesian approach, which provides the flexibility to keep the prior odds separate from the conditional likelihood. This makes it possible to measure how much weight the coaches are giving new information compared to the original information when making decisions, which in turn allows me to directly test for the representativeness heuristic.
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THEORETICAL MODEL (1B)
This is Bayes' rule in odds form.
On the left hand side of the equation we have what is called the posterior, which in this circumstance is the ingame odds of team C winning the game given the fourth down decision that the coach made.
The first term on the right hand side is called the prior, which is the odds of team C winning the game given the gamestate, while the second term is called the conditional likelihood, which in this case is the inverse conditional odds of team C making that decision given that they won the game.
In other words these two components are the original information and the new information.
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EMPIRICAL MODEL
In this frame, we have the equation for testing the hypothesis. In this equation BetaOne represents the parameter estimates for prior information while BetaTwo represents the parameter estimates for the new information. To test for representativeness we will compare those two numbers.
As you can see down here, the null hypothesis is that coaches are equally weighting the two components and are therefore not guilty of the representativeness heuristic. The first alternative hypothesis is that they are underweighting new information, while the second alternative hypothesis is that they are overweighting new information, which indicates the presence of the representativeness heuristic in their decisionmaking process.
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PRIOR (1A)
If this frame looks similar it's because we are again utilizing Bayes' rule, but this time it is with the posterior as the PREDECISION odds of team C winning the game given the gamestate, which is shown on the lefthand side of the equation.
On the righthand side we again have the prior and conditional likelihood. In this case, the first term is literally the prior (pregame) odds of team C winning the game while the second term is the odds of team C being in that gamestate given that they ended up winning the game.
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PRIOR (1B)
The next step is to calculate these two components. Down here you can see that to estimate the first component (the prior) we are using a logistic regression to predict game outcomes from closing point spreads.
For the second component we use a multinomial logistic regression to predict the probability of the team being in that situation given that they won the game. In this case XPrime represents the matrix of variables that make up the gamestate: the score margin, time remaining in the game, timeouts remaining, field position with respect to the offensive team, current down, yards to go to gain a new set of downs, and an indicator variable for possession with respect to the home team, as well as the bookmakers' over/under data for the game.
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CONDITIONAL LIKELIHOOD
This frame shows the conditional likelihood component of the theoretical model. On the left hand side of the equation we have the probability of that decision being made in game G at time T, given that team C ended up winning the game.
On the righthand side we have another multinomial logisitic regression where XPrime again respresents the matrix of variables that make up the gamestate. The one difference you will notice here is that this time it does not include the current down, and that is because this equation is looking only at fourth down plays, while the previous was looking at all plays, regardless of down.
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