{"id":170,"date":"2015-06-13T03:14:24","date_gmt":"2015-06-13T03:14:24","guid":{"rendered":"http:\/\/commoncents.blogwyrm.com\/?p=170"},"modified":"2023-04-01T09:11:30","modified_gmt":"2023-04-01T13:11:30","slug":"bookies-how-to-set-odds","status":"publish","type":"post","link":"https:\/\/commoncents.blogwyrm.com\/?p=170","title":{"rendered":"Bookies & How to Set Odds"},"content":{"rendered":"

In honor of American Pharoah winning the first Triple Crown in 37 years, this week\u2019s column will concern itself with that fine art of subtle economics \u2013 making book.\u00a0 For those you aren\u2019t familiar with the terminology, to make book is the phrasing used to describe how a bookie balances the risks associated with backing a gambling opportunity. The idea behind making book is to guarantee a positive expected income for the bookie no matter what happens in the event.<\/p>\n

The treatment here is patterned after \u2018The Parable of the Bookmaker\u2019 found in the book \u2018Financial Calculus: an introduction to derivative pricing\u2019, by Baxter and Rennie and the very readable discussion of \u2018The Art of Bookmaking\u2019<\/a> Matt Elliott.<\/p>\n

Both treatments consider, for simplicity, the case with a sporting event with two possible outcomes.\u00a0 I\u2019ll stick with the horse racing theme used by Baxter and Rennie.<\/p>\n

Suppose that we are having a runoff between the great Triple Crown winners of yesteryear and American Pharoah.\u00a0 I think that, without a doubt, Secretariat is the greatest of the past winners; an opinion that is shared in sporting circles<\/a>.\u00a0 Now let\u2019s imagine that we are pitting Secretariat against American Pharoah as our sporting event of the millennium.<\/p>\n

As bookies, we want to accept wagers from bettors who are both favor Secretariat and those who favor American Pharoah, but we want to do so in such a way that regardless which horse actually wins, we can meet our payouts and still take home a tidy profit.\u00a0 How do we do this?<\/p>\n

First, let\u2019s start by examining the probability that each horse will win the race.\u00a0 Since this is a fantasy race, we can run the race as often as we like, and suppose that in doing so we find that Secretariat is 3 times more likely to win than American Pharoah.<\/p>\n

Now we have to set the odds.\u00a0 The actual language and notation associated with quoting the odds seems to differ from country to country and culture to culture so I am going to give a set of definitions that maximize the overlap with all cases.\u00a0 First define the stake<\/em><\/strong> as the amount of money that the bettor (or punter as it is sometimes known as) places on the outcome.\u00a0 A winning bet pays the bettor back his original stake plus an additional amount of money called the return<\/em><\/strong>, since it represents the return on his investment.\u00a0 Thus the successful bettor walks away from the track with the sum of these two, which is called the payout<\/em><\/strong>.\u00a0 In symbols, if st<\/em><\/strong> is the stake and r<\/em><\/strong> is the return then p = s + r<\/em><\/strong> is the payout.\u00a0 The unsuccessful bettor simply walks away.<\/p>\n

Now if we set our odds consistent with the probabilities found in our fantasy running, we would set the odds as follows:<\/p>\n

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