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In talking about betting for value and the relative lack of value in non-fixed odds bets, we’ve been fairly vague about what exactly gives a bet value. We’ve used this term as shorthand for expected value, and in this article we are going to break down how it’s defined, how it’s calculated and why it’s important.
What is Expected Value?
Expected value, simply put, is the average value of any randomly distributed variable. In any system where some outcome is randomly determined, and different outcomes can be assigned different numerical values, expected value represents the average outcome.
Formally, it is defined as:
For some random variable X, where x1 ,…xk are the possible outcomes and p1 ,…pk are the respective probabilities of each of those outcomes occurring.
This is a bit mathy, so let’s break it down in simpler terms, starting with a very simple example: you’re playing a game where you flip a coin. If the coin comes up heads, you win $2. If it comes up tails, you lose $1. Thus, the two possible outcomes are +2 and -1, and the probabilities for them are 50% and 50% (assuming, of course, that this is a fair coin). Let’s write it out:
E[X] = x1p1 + x2p2 = (2 * 0.5) + (-1 * 0.5) = 0.5
You’ll notice that this is equal to the mean (sum of the outcomes divided by the number of outcomes); (2 – 1) / 2 = 0.5. That’s because expected value is really just a generalization of the mean, and specifically of the weighted mean. Different outcomes can have different probabilities of occurring, and it just so happens that for a fair coin, both outcomes have the same 50% probability.
So what does it mean to have an expected value of 0.5 in this game? On any individual flip, you can either win $2 or lose $1, so you’re never winning $0.50. But what if you could play this game more than once? If you flip the coin ten times, you’ll most likely get five heads and five tails. In that situation, you’ll have won $10 and lost $5, for a total gain of $5. Divide that by our ten games, and you’ve averaged $0.50 per game.
You’re not guaranteed to get five heads and five tails, and so any set of ten flips may have a different outcome. However, the more times you can repeat the coin flip, the closer to a 50/50 split of heads and tails you’ll approach. This is called the law of large numbers, and we’ll cover it in detail in a future article. For now, think of expected value as the average outcome if things align exactly with their probabilities.
Expected value is commonly shortened to “EV,” so we’ll use that as shorthand going forward.
What is “good” value?
In talking about value previously, we’ve also thrown around ideas such as “the reward isn’t worth the risk.” That was also just an approximation of EV, and now we can quantitatively define what it means for a reward to be worth it.
First, though, let’s build up some intuition. Would you want to play the above game? Your answer should almost certainly be yes. Sure, sometimes you play and lose a dollar, but in the long run you should win more money than you lose, which makes it worth it. What if the results were inverted? On heads you lose $2 and on tails you win $1. We’ll leave the calculation as an exercise for the reader, but the EV of this game is -0.5. Would you want to play with these payouts? You might get lucky a few times, but in the long run you’re losing $0.50 every time you play, so it should be a fairly obvious no.
That’s all there really is to it:
If a game has positive EV, you should play it.
If a game has negative EV, you should not play it.
Expected Value and betting
So far, we’ve talked about arbitrary games, but you can probably see where this is going for something like a League of Legends match.
Team A is playing Team B, with payouts of 1.50 and 3.00 respectively. As we had brought up, this is actually offering you two games: you can bet on Team A (and win or lose that bet), and you can bet on Team B (and win or lose that bet). So, we’ve got two potential games and want to see if either of them is worth playing (i.e. if either bet is worth taking). Let’s calculate some EV:
To simplify, let’s assume we’re always going to bet $100. For team A, our two outcomes are a payout of $150 or a loss of $100. For Team B, the outcomes are a $300 payout or $100 loss.
When we’re talking about EV, what we actually care about is our profit. A game is only worth playing if it has positive expected profit, not positive expected payout. So, for Team A, we have outcomes of +$50 and -$100, and for Team B +$200 and -$100. Let’s summarize quickly:
|Bet on||Team A win||Team B win|
We’re missing one final piece: the probabilities of each outcome. In future articles, we’ll talk about how you can derive probabilities from the payouts and why it’s important to have your own independent views of those probabilities, but for now let’s say we’ve got a model that says Team A has a 60% chance to win and thus Team B has a 40% chance.
EV(Team A) = (profit if Team A wins) * (probability Team A wins) +
(profit if Team B wins) * (probability Team B wins)
EV(Team A) = (50 * 0.6) + (-100 * 0.4)
EV(Team A) = -10
EV(Team B) = (profit if Team A wins) * (probability Team A wins) +
(profit if Team B wins) * (probability Team B wins)
EV(Team B) = (-100 * 0.6) + (200 * 0.4)
EV(Team B) = 20
Based on expected value, betting on Team A isn’t worth it, but betting on Team B is.
It’s worth noting that the amount you bet doesn’t really matter. Since your bet size scales both your potential profit and potential loss, the magnitude of EV will change, but the directionality (i.e. whether it’s positive or negative) will not. We generally quote the EV of a $1 bet because it’s easy to scale from there. If a $1 bet has an EV of 0.5, a $10 bet will have an EV of 5, and so on.
Don’t Get Hung Up on Outcomes
Let’s say you bet on Team B but they ended up losing. That stings a bit, but did you make the wrong choice? The field of statistics would say no.
If something has positive EV, that’s by no means a guarantee of profit. In this case, we knew going in that we had only a 40% chance to make money. The point of EV is to tell you what happens on average.
If you could replay this same match many more times, then it would start paying off. Obviously you can’t actually replay the same match in reality, but you can sort of approximate it by placing bets on many matches, and over the long run should be profitable. In short, you can’t expect to win every bet, but as long as your bets all have positive EV, they are worth making.
What if there was another match happening at the same time and you calculated that betting on Team C to win against Team D had an EV of, say, 60? It seems like Team C is the better deal, so shouldn’t you only bet on them and not on Team B? There are a few conflicting forces here, as it turns out.
On the one hand, higher EV is better, so betting on Team C is more appealing. On the other hand, we’ve mentioned a few times now how important it is to place lots of bets to smooth out what happens and get closer to the average, so you’d probably want to bet at least something on both matches. Finally, EV isn’t the whole picture. Two bets can have the same EV but vastly different odds of paying out (if you’re betting on the underdog in one and the favorite in the other), and you’d want to take that into account somehow.
The short answer is all of these factors are important. There are some formulas for how much you should be betting given all of these factors, and the end result is you should bet on every match where you can find positive EV, but scale your bets. The long answer is we’ll cover exactly how to do this in a future article. In the meantime, our bet suggestions incorporate this, which is why you’ll see us suggest different bet sizes on different matches.
Vanya is the founder of alacrity.gg, where he leverages seven years of quantitative finance experience to build esports prediction models. He'll happily debate you on anything from math to video games to the impending subjugation of the human race by artificial intelligence.