Gambling on Sporting Events: Some people gamble on sporting events strictly to make money while others care directly about which teams win quite apart from whether or not they gambled on the game.
A. Consider your consumption level this weekend and suppose that you have $1,000 available. On Friday night, Duke is playing UNC in an NCAA basketball tournament, and you have the opportunity to beaten amount $ X < $1,000 on the game. If you bet $X, you will only have ($1,000-$X) if you lose the bet, but you will have ($1,000+$X) if you win. We would say in this case that you are being given even odds (since your winnings if you win are as big as your losses if you lose). Suppose that you believe that each team has a 50% chance of winning (and that, if a game is tied, it goes into overtime until the tie is broken.)
(a) First, suppose you don't care about sports and only care which team wins to the extent to which it increases your consumption. I offer you the opportunity to place a bet of X = $500 on either team. Will you take the bet?
(b) Suppose you got a little inebriated and wake up in the middle of the game to find that you did place the $500 bet on Duke. You notice the game is tied — and you ask me if you can get out of the bet. How much would you be willing to pay to get out?(c) Suppose that, just as you come to realize that alcohol had made you place the bet, Duke scores a series of points and you now think that the probability of Duke winning is δ > 0.5. Might you choose to stay in the bet even if I give you a chance to get out for free?
(d) Suppose you were actually risk-lover. If you could choose to place any bet (that you can afford), how much would you be ton the game (assuming you again think each team is equally likely to win)?
(e) Illustrate your answer to (a) and (c) again, but this time in a graph with xD on the horizontal and xUNC on the vertical (with the two axes representing consumption in the “state” where Duke wins (on the horizontal) and where UNC wins (on the vertical).) (Hint: The “budget constraint” in the picture does not change as you go from re-answering (a) to re-answering (c))
(f) Suppose that you love Duke and hate UNC. When Duke wins, everything tastes better— and if UNC wins, there is little you want to do other than lie in bed. Might you now enter my betting pool (prior to the start of the game) even if you are generally risk averse and not at all drunk? Illustrate your answer.
(g) True or False: Gambling by risk averse individuals can arise if the gambler has a different probability estimate of each outcome occurring than the “house”. Alternatively, it can also arise from state-dependent tastes.
(h) True or False: If you are offered a bet with even odds and you believe that the odds are different, you should take the bet.
B. Consider again the types of bets described in part A, and suppose the function u(x) = xa allows us to represent your indifference curves over gambles using an expected utility function.
(a) Suppose α = 0.5. What is your expected utility of betting$ 500 on one of the teams, and how does this compare to your utility of not gambling?
(b) Consider the scenario in A (b). How much would you be willing to pay to get out of the bet?
(c) Consider the scenario in A(c). For what values of δ will you choose to stay in the bet?
(d) Suppose α = 2. How much will you bet?
(e) Consider what you were asked to do in A (e). Can you show how the MRS changes as δ changes? (Hint: Express the expected utility function in terms of xUNC and xD and derive the MRS.) For what value of δ is the $500 bet on Duke the optimal bet to place?
(f) Suppose that uD(x) = αx0.5 and uUNC (x) = (1−α) x0.5 are two functions that allow us to represent your tastes over bets like this using an expected utility function. What is the equation for the MRS in your indifference map assuming that you think the probability of each team winning is in fact 0.5? For what value of α will the $500 bet be the optimal bet.