Ec 1723 Pset 4

3. a) A winner-takes-all contract that costs $p and pays off $1 if and only if a specific event occurs will elucidate the probability of the event occurring, assuming riskneutrality. An index contract that pays according to the value of a certain number that varies continuously should cost the mean value of this number. A spread bet costs a fixed amount and pays a predetermined amount if the bettor wins but $0 if the bettor loses. The bettors choose the cutoff y* at which they are willing to make the bet. For example, if the contract costs $1 and the payoff is $2 in the good case 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 (long shots) are overpriced and the most likely bets (favorites) are underpriced. This occurs because bettors are bad at predicting the likelihood of an event with a small probability and thus tend to overestimate. If markets showed equal bias, then the bid and ask prices on Tradesports would correspond with estimated prices from actual S&P options. However, in such comparisons, Wolfers and Zitzewitz see that bettors on Tradesport slightly overvalue unlikely bets and undervalue most likely bets compared to estimates from actual December S&P options. They interpret it as the inability of investors to correctly value small bets, as well as a certain irrationality of bettors who trade according to their desires under certain situations. c)

Saturday, Oct 27 - 4:02PM P(nominated)

Giuliani Romney Thompson

42.5 26.2 11.4

P(elected)

Implied P(elected | nominated) 17.6 0.414117647 8 0.305343511 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 to P(elected | nominated) = P(elected) / P(nominated) d) Not necessarily, because the results from part (c) do not imply causality, simply correlation. Thus, the results reflect the popular sentiment on the chances of each candidate 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 an understanding that if the candidate is so effective in winning over the public in the primary as to win the primary, then that candidate would fare much better in the actual election than the current favorite. However, making the underdog the best-funded candidate will help them bypass the selection process when they are not actually as qualified.

4. a) The probability that the Red Sox will win can be proxied by the price of each “lot,” 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 early October, 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 the day the Red Sox win the World Series. b) If information is revealed only during games, then price jumps should happen only after games and not before. c) The implied probability of the Red Sox winning the World Series would update after each game because after the result of each game is revealed, the event that the Red Sox win that game is no longer a random variable; thus, the probability that the Red Sox will win the World Series is the probability that, going forward from that point, the Red Sox 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 the Red Sox win, or go down if the Red Sox lose. At the next game, the same thing will happen, until after the last game, the probability of winning is 1 or 0 depending on whether they won or lost. The exact probabilities can be calculated; I will not do so on this problem set since it’s a pretty time-consuming mechanical calculation. As for whether the data from Tradesports are consistent with this model, it is hard to say. Off the bat, if we use the data as exact proxies for the probabilities that the Red Sox will win at each stage, then no, because the probabilities should go up after each game and not before the series begins. It’s probably a decent approximation of what actually happens if we assume that no-one knows what that probability is; it’s an underlying probability that drives the game but we may not discover it even when the game is over. For example, there may be a more or less fixed probability that the Red Sox will win each game. However, each bidder has a different assessment of what this probability is. Before the World Series begins, there is a lot of speculation and variance of beliefs about what this underlying probability is. After the first game, the variance of beliefs narrows but the expectation of the underlying probability also changed; the two effects balance to keep the investors’ assessment 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 a much smaller range of possible underlying probabilities. Consistent with the theory, the valuations of winning rise until the series is finally won by the Red Sox.