--- title: Betting odds #description: author: Issa Rice creation-date: 2015-01-27 last-major-revision-date: 2015-01-27 language: English # accepts "notes", "draft", "in progress", or "mostly finished" status: notes # accepts "certain", "highly likely", "likely", "possible", "unlikely", "highly unlikely", "remote", "impossible", "log", "emotional", or "fiction" belief: possible # accepts "CC0", "CC-BY", or "CC-BY-SA" license: CC-BY tags: untagged aliases: odds math: yes --- Betting someone with an odds of $x$ to $y$ and a stake of $\$n$ means that if you lose, you pay the $\$n$ stake. If you win, you keep the stake, and, in addition, win $\$\frac{x}{y}\cdot n$. # Example We take an example from Noah Smith's "[Bets do not (necessarily) reveal beliefs](http://noahpinionblog.blogspot.com/2013/05/bets-do-not-necessarily-reveal-beliefs.html)". First, DeLong gives Smith 50-to-1 odds that inflation would go over 5%. Let the stakes for Smith be $\alpha$. This means that if inflation goes over 5%, Smith wins, and gets $50\alpha$. If inflation stays under 5%, Smith loses, and loses the $\alpha$ from the stakes. Next, Smith gives Chovanec 25-to-1 odds that inflation would stay under 5%. This means that if inflation goes over 5%, Smith loses, and must pay $25\alpha$. On the other hand, if inflation stays under 5%, Smith wins, and wins the stakes of Chovanec, namely $\alpha$. (Perhaps it's easier to see this looking from Chovanec's view: ) This means that, overall, if inflation goes over 5%, then Smith gets: $50\alpha - 25\alpha = 25\alpha$, or "25 pizza dinner equivalents", since $\alpha$ was the cost of a pizza dinner. On the other hand, if inflation stays under 5%, then Smith gets: $-\alpha + \alpha = 0$, or breaks even. # Example 2 Alex Tabarrok in "[A Bet is a Tax on Bullshit](http://marginalrevolution.com/marginalrevolution/2012/11/a-bet-is-a-tax-on-bullshit.html)" gives the example of Nate Silver betting on the outcome of the presidential election. Tabarrok says: > A properly structured bet is the most credible guarantor of rigorous > disinterest. In order to prove his point, Silver is not required to take > the Obama side of the bet! At the odds implied by his model (currently > between 3 and 4 to 1) Silver should be willing to take either side of a > modest bet. Indeed, we could hold a coin toss, heads Silver takes the > Obama side, tails he takes Romney. > > In fact, the NYTimes should *require* that Silver, and other pundits, > [bet their beliefs](http://hanson.gmu.edu/futarchy.pdf). Furthermore, to > remove any possibility of manipulation, the NYTimes should escrow a > portion of Silver’s salary in a *blind trust bet*. In other words, the > NYTimes should bet a portion of Silver’s salary, at the odds implied by > Silver’s model, randomly choosing which side of the bet to take, only > revealing to Silver the bet and its outcome after the election is over. > A blind trust bet creates incentives for Silver to be disinterested in > the outcome but *very* interested in the accuracy of the forecast. Suppose Silver thinks Obama will win with odd 3-to-1, and suppose he's willing to stake $\alpha$. Now, we can make a tree diagram of all the possibilities: ``` /\ Obama / \ Romney .5 / \ .5 / \ /------/ \------\ / \ O wins /\R wins O wins /\ R wins ``` So the expected value is: $$ \frac{1}{2} \left( \frac{x}{x+y} \cdot \frac{x}{y} \cdot \alpha \\ - \frac{y}{x+y} \cdot \alpha \\ - \frac{x}{x+y} \cdot \alpha \\ + \right) $$ links: - - - - - - -