Betting odds

Created: 2015-01-27
Status: notes; belief: possible

Example
Example 2

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”.

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” 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. 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: \[ \frac{1}{2} \left( \frac{x}{x+y} \cdot \frac{x}{y} \cdot \alpha \\ - \frac{y}{x+y} \cdot \alpha \\ - \frac{x}{x+y} \cdot \alpha \\ + \right) \]

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Betting odds

Contents

Example

Example 2