Mathematical Considerations For Optimising Association Football Entries

In this post I would like to lay out some mathematical foundations for finding the best value on ternary moneyline events, notably association football, which has win lose and draw outcomes. Unlike American sports, where the rare occasion a draw happens it can be absorbed into the +0.5 and -0.5 spreads in assessing your options, the draw has a very significant probability of happening in football to the extent it opens up double chance and draw no bet markets (in which the draw can be significant enough to render both pick'ems unprofitable at the same time, a phenomenon not observed in American sports). This leads directly on to part of the challenge in football betting: the draw can nuke evenly poised spreads since spreads require an edge on one team over the other, so moneylines have to considered to get a full picture of potential value.

Whereas finding an edge on American sports primarily lies in identifying a mispricing in one team which is anticorrelated to a mispricing on the other team, a mispricing on a football moneyline is only correlated with an aggregate mispricing on the other two options. If you identify an outcome as overpriced, it is not immediately clear which of the other two options are underpriced. You may decide one team is overpriced and bet on the other only to find the draw was where the value lay all along. Since my approach is predicated on expected value analysis, I will consider what the implications of a mispricing in one outcome are for the other two moneyline options to optimise value when betting on football.

Suppose we have a game between team A and team B with win probabilities a, d and b (for team A to win, draw and team B to win). Suppose we have odds with break even probabilities a', d' and b'. It immediately follows that a + d + b = 1 since all outcomes are considered and a' + d' + b' = 1 + r, where r is the overround. Suppose we see an edge on team A and decide to back them. This immediately implies a > a'. We can now calculate the expected value of team A to win, and it's given by x = a/a' - 1. We can now consider the expected value of betting on their opponent not to win. Incorporating the draw and taking into account the overround, we have an expected value of (a + d) / (a' + d') - 1. In order for no on the other team to maintain the same edge, you would need a / a' to be equal to (a + d) / (a' + d'), which is the case if a / a' = d / d', which is to say, the draw is underpriced by the same factor as the win. In practice, these edges are different and the edge on the no lies in between the edge on the win and the edge on the draw.

What are the implications of this? Generally, when a clear favourite is overvalued, the underdog is undervalued to win and the draw is undervalued too. In this case, taking no on the favourite can reduce the variance on choosing between the underdog to win and the draw. Where the practical edge comes in is that it is generally easier to sell no on the favourite than it is to buy yes on the underdog and especially easier than buying yes on the draw. That's why when backing a clear underdog, I prefer to take their opponent not to win rather than the choosing between win and draw. On the other hand, when two teams that are relatively evenly matched, one team being overvalued does not necessarily propagate evenly into the other two options; it can very easily be the case that both teams are overvalued, hence value on the draw. In that scenario, you can get good value by selling an equal number of contracts on both sides, which is the basis for the synthetic draw strategy.

In the same way that two teams can be simultaneously overvalued implying an edge on the draw, it can also be the case that two teams are undervalued and that the draw is overvalued. This can be the case in more volatile games between two teams that are good at attacking but not good at holding on to leads. Sometimes the premium on a draw is a little too fat and there can be an edge in selling it. As a matter of liquidity, this is not always so reliable as there is far less interest in trading draws than there is in trading teams moneyline. In that scenario you can still buy both teams individually and there may be enough people looking to sell them that you can cover for a winner, even if you don't have a lean on who it will be. Either way, if a team is undervalued, it's best to try and buy them outright rather than selling the other team.

These options are more easily accessed if you on tradeable betting exchanges such as Polymarket or Kalshi as you can easily buy and sell event contracts on various outcomes rather than only having the bookmaker options available. You can still play these options indirectly via the spread markets if the teams are not evenly poised, through the double chance markets or a combination of win and draw moneyline to form a synthetic +0.5. These generally are not priced the same way so you can optimise your entry by comparing the odds and expected returns of these options if you don't have access to Polymarket or Kalshi.