# Arbitrage Betting Case Study

Consider a football match, there are three possible outcomes, either of the two teams winning or the match ending in a draw. Below in figure 1 are example odds given from three different bookmakers covering all three outcomes of a certain match. For each of the bookmakers odds, no errors have been made, so there are no opportunities for an arbitrage bet if you bet on all outcomes with

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Bookmaker 1 earns a percentage profit of $6.96\%$ using these odds. Similarly, with bookmaker 2 and bookmaker 3 their percentage profit is $3.78\%$ and $6.24\%$ respectively. To see whether there is an arbitrage bet within the odds given in the table consider placing bets with more than one bookmaker covering all the possible outcomes of the match. Logically, take the largest odds for each outcome and see whether their inverses add up to a value less than $1$. In this case the largest odds for outcome 1 is with bookmaker 1, the largest for outcome 2 is with bookmaker 2 and the largest for outcome 3 is with bookmaker 3. Taking the inverses of these odds:vspace{12pt}

egin{equation} otag 2.8^{-1} + 2.9^{-1} + 3.6^{-1} = 0.9797 end{equation}vspace{12pt} This shows, even though each of the bookmakers odds individually ensure a profit for themselves, placing bets with all three bookmakers using the odds shown above, a profit can be made as the value above is less than $1$. The last thing required to perform an arbitrage bet is to calculate how much to bet on each outcome to guarantee a return profit. Betting a total stake of pounds100 over the three outcomes, not just placing a third of it on each outcome. The odds are divided in relation to the values of the odds. vspace{12pt}

egin{defn} hfill \

egin{itemize}

item $b_1 =$ bet on outcome 1 (Team A Win) item $b_2 =$

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There is no obligation saying you must bet on draws, in theory, one could bet on any of the three outcomes as long as the odds being offered are above the 2.618 threshold. In the 2014/2015 English premier league the rate of a draw was 24.5 percent, almost 1 every 4 matches. Strictly going by this figure the Fibonacci Betting System will pay out approximately every 4 games. That is not to say that it will not go much past 4 matches. The main drawback for this progressive betting system is if one were to be unfortunate and not have a success for a large number of matches then the stake sizes increase exponentially. The 21st number of the Fibonacci sequence and therefore the size of the 21st bet would be pounds10,946, which is very large considering the first bet was just pounds1. One could be unlucky enough to have a losing streak like this and not have the capital available to continue the betting sequence, therefore, losing all of the money bet before totalling pounds17,710. There is an up side of the stake sizes increasing, it also returns greater profits. The 21st bet with odds of 3.5 for a draw and ending successful would return pounds38,311, a total profit of