8/14/2019 Ec 1723 Pset 4

1/2

3. a) A winner-takes-all contract that costs $p and pays off $1 if and only if a specificevent occurs will elucidate theprobability of the event occurring, assuming risk-neutrality.

An index contract that pays according to the value of a certain number that variescontinuously should cost the mean value of this number.

A spread bet costs a fixed amount and pays a predetermined amount if the bettorwins but $0 if the bettor loses. The bettors choose the cutoff y* at which they are willingto make the bet. For example, if the contract costs $1 and the payoff is $2 in the goodcase and $0 in the bad case, then y* will be the median value of the bet.

b) Favorite-long shot bias is a betting phenomena in which very unlikely bets (longshots) are overpriced and the most likely bets (favorites) are underpriced. This occursbecause bettors are bad at predicting the likelihood of an event with a small probabilityand thus tend to overestimate.

If markets showed equal bias, then the bid and ask prices on Tradesports wouldcorrespond with estimated prices from actual S&P options. However, in such

comparisons, Wolfers and Zitzewitz see that bettors on Tradesport slightly overvalueunlikely bets and undervalue most likely bets compared to estimates from actualDecember S&P options. They interpret it as the inability of investors to correctly valuesmall bets, as well as a certain irrationality of bettors who trade according to their desiresunder certain situations.

c) Saturday, Oct 27 - 4:02PM

P(nominated) P(elected) Implied

P(elected | nominated)

Giuliani 42.5 17.6 0.414117647

Romney 26.2 8 0.305343511Thompson 11.4 5.1 0.447368421

According to Bayes formula,P(elected | nominated) = P(nominated | elected) * P(elected) / P(nominated)However, since we know P(nominated | elected) = 1, this simplifies toP(elected | nominated) = P(elected) / P(nominated)

d) Not necessarily, because the results from part (c) do not imply causality, simplycorrelation. Thus, the results reflect the popular sentiment on the chances of eachcandidate winning given their performance on the primary. For example, an underdog in

the primary may have a higher probability in part (c) of winning; this may reflect anunderstanding that if the candidate is so effective in winning over the public in theprimary as to win the primary, then that candidate would fare much better in the actualelection than the current favorite. However, making the underdog the best-fundedcandidate will help them bypass the selection process when they are not actually asqualified.

8/14/2019 Ec 1723 Pset 4

2/2

4. a) The probability that the Red Sox will win can be proxied by the price of eachlot, or the security that pays $100 in the case that the Red Sox win the World Series.The probability increases slowly over the last few months, spikes to about 40% in earlyOctober, fluctuates, and rises to about 68% the day before the first game is played.Surprisingly, it remains at this level through the first two games and jumps ~12% the day

after the second Red Sox win and keeps jumping up, to 87% the next day and 96% on theday the Red Sox win the World Series.

b) If information is revealed only during games, then price jumps should happenonly after games and not before.

c) The implied probability of the Red Sox winning the World Series would updateafter each game because after the result of each game is revealed, the event that the RedSox win that game is no longer a random variable; thus, the probability that the Red Soxwill win the World Series is the probability that, going forward from that point, the RedSox will win the requisite number of games out of the total games remaining.

For example, after the first game is played, the probability will either go up if theRed Sox win, or go down if the Red Sox lose. At the next game, the same thing willhappen, until after the last game, the probability of winning is 1 or 0 depending onwhether they won or lost. The exact probabilities can be calculated; I will not do so onthis problem set since its a pretty time-consuming mechanical calculation.

As for whether the data from Tradesports are consistent with this model, it is hardto say. Off the bat, if we use the data as exact proxies for the probabilities that the RedSox will win at each stage, then no, because the probabilities should go up after eachgame and not before the series begins.

Its probably a decent approximation of what actually happens if we assume thatno-one knows what that probability is; its an underlying probability that drives the gamebut we may not discover it even when the game is over. For example, there may be amore or less fixed probability that the Red Sox will win each game. However, eachbidder has a different assessment of what this probability is. Before the World Seriesbegins, there is a lot of speculation and variance of beliefs about what this underlyingprobability is. After the first game, the variance of beliefs narrows but the expectation ofthe underlying probability also changed; the two effects balance to keep the investorsassessment of the probability that the Red Sox win the same. After the second game,there is a great jump in prices because people believe that two wins in a row implies amuch smaller range of possible underlying probabilities. Consistent with the theory, thevaluations of winning rise until the series is finally won by the Red Sox.