# Intro to Econ: Second Lecture – Arbitrage with Sports Bets

Let me turn to another area in which the absence of arbitrage – due to people preferring more money over less – implies severe restrictions: sports betting. Consider a set of potentially fictional betting odds for three football (soccer) games, given in the following table.

$\begin{tabular}{c|ccc} & Game 1 & Game 2 & Game 3 \\ \hline A & 1,1 & 4,75 & 1,9 \\ x & 11 & 3,6 & 4,2 \\ B & 21 & 1,78 & 5 \\ \end{tabular}$

How should you read this table? Each column represents one football game between some teams A and B. For each football game you can bet on the event that team A (the home team) wins the game, that the game ends in a draw – coded as x – and that team B (the away team) wins. The number in a cell represents the betting odds for the respective event. For instance the number 1,1 represents the betting odds on the home team (team A) winning in the first of the three games. What does this number mean? If you place 1 Euro on team A winning and team A actually ends up winning you receive 1 Euro and 10 cents back. So in the event of team A winning you gain 10 Euro-cents. If the game ends in a draw or in team B winning you get nothing back, so you simply lose your Euro.

Take a good look at this table of betting odds and think about whether you find these odds plausible or implausible. You may want to use a calculator to aid your thinking. Take your time, this is not easy.

Are you done thinking? It turns out that two of these games are real games and one is a fake game, and that in fact the betting odds for the fake game are impossible in a world without arbitrage opportunities.

The first two games took place in the evening after the first time I gave this lecture, on Wednesday the 18th of October 2017. The first was Bayern Munich against Celtic Glasgow with odds of 1,1 on Bayern, 11 on a draw, and 21 on Celtic (I believe Bayern did win that night). The second was Benfica Lisbon against Manchester United with odds of 4,75 on Benfica, 3,6 on a draw, and 1,78 on Man U (I believe Man U did win that night). These are fine odds as we shall see below.

But let me first turn to the last game, which was not a real game. What is wrong with these odds of 1,9, 4,2, and 5? Well, with these odds one could do arbitrage. What you could do is to place Euro amounts on all three bets proportionally to the reciprocal of the odds. The reciprocal of the betting odds is what I would call the event’s “implicit probability”. The implicit probabilities are then 1/1,9 = 0,5263 for team A winning, 1/4,2 = 0,2381 on a draw, and 1/5 = 0,2 on team B winning. Suppose you take a target 100 Euros and place these on the three bets proportionally to the implicit probabilities. So you would place 52,63 Euros on team A winning, 23,81 on a draw, and 20 on team B winning. In total you would have placed 52,63+23,81+20=96,44 Euros on the three bets. Note that some of the 100 Euros remain in your wallet. Now what can you win with this betting scheme? Well, only three things can happen. Either team A wins, or there is a draw, or team B wins. How much money will you receive in these three events? If team A wins you get 1,9 Euros for every Euro placed on team A. As you have placed 52,63 Euros on team A you get 52,63*1,9 = 100 Euros back if team A wins. If there is a draw you get 4,2 Euros for every Euro placed on a draw. As you have placed 23,81 Euros on a draw you get 23,81*4,2 = 100 Euros back if there is a draw. If, finally, team B wins, you get 5 Euros for every Euro placed on team B. As you have placed 20 Euros on team B you get 20*5=100 euros back if team B wins. So no matter what happens in this game you always get 100 Euros back. But you have only placed 96,44 Euros. So you win 3,56 Euros in every possible case! You have made arbitrage! If you think 3,56 Euros is a miserly sum for all the trouble then multiply all your bets with 1000 and you win a sure 3560 Euros without risk.

Now note that you cannot do this with the two real games. In both cases the sum of the implicit probabilities exceeds one and, because of that, there are no arbitrage opportunities. You can try to find arbitrage opportunities in sports (or other) betting odds, but I doubt that you will find any.

There is a nice joke about economics that has to do with the topic of arbitrage. Here it is. Two people walk along the street, one of them is an economist. While they are walking they spot a 100 dollar bill on the pavement. The non-economist starts to scoop down to pick up the 100 dollar bill. The economist says: “Don’t bother! This can’t be a real 100 dollar bill. If it were a real 100 dollar bill, someone would already have picked it up.” My takeaway from this is twofold. If you do see a 100 dollar bill on the street, of course, do pick it up. But I wouldn’t start walking miles and miles of streets in the hope to find many 100 dollar bills waiting for me to pick them up.

Here is the Video (in German):