Calculating the Lowest Possible House Edge
By Colin McCrae
Gambling on the blockchain has been one of the early success stories of decentralized smart contracts on the blockchain. The immutable and provably fair properties of smart contracts make the blockchain a natural home for unstoppable code which can facilitate fixed odds betting games.
Smart contracts on the blockchain are developing and advancing at a breathtaking speed. Much of the reason for this is that in order for these contracts to be provably fair, the developers must release their source code to the public. This also allows other developers to catch up to the current state-of-the-art and indeed improve and easily re-deploy smart contracts.
It is not possible to offer fixed odds bets with no house edge, without risking ruin, unless the contract has access to an infinite bankroll. An infinite bankroll could indeed be achieved by pricing bets and payouts in a native token which can be minted as required by the smart contract. However, this means that the supply of tokens is not fixed, leading to an inherently unpredictable token supply, adding to price volatility, and reducing token value over time through inflation.
Therefore, any fixed odds betting platform that uses valuable native tokens can only have access to a finite bankroll, and hence must charge an edge to ensure there is no risk of ruin for their bankroll. This has started a ‘Race to The Lowest House Edge’, where every new platform must charge a lower house edge, otherwise bettors and game operators will simply use a cheaper competing smart contract.
It is EdgeFund’s goal to accept the widest range of fixed odds bets at the lowest mathematically possible house edge, returning profits to token holders. This post describes how this minimum house edge can be found using the Kelly Criterion.
The Kelly Criterion
The Kelly Criterion is a formula used to determine the optimal size of a series of bets where the bettor has a positive edge (i.e. the odds are favorable). It was originally derived by John Kelly, Jr. in 1956 during his time at Bell Labs. For a given bet, where the probability of winning and payout odds are both known, the formula returns what the bet should be as a fraction of the bettor’s current bankroll to optimize long term profit. The bet size that satisfies this formula is known as the Kelly Bet, and sizing bets like this is known as the Kelly Strategy.
The equation below is the simplest form of the Kelly Criterion.
f* — bet as a fraction of the current bankroll (bet size / bankroll)
b — decimal payout odds received on the bet (b>1)
p — probability of winning (0<p≤1)
The formula provides several interesting results. If the odds are fair for the probability of winning, the Kelly Criterion states you should not bet at all. For example, evens payout (b = 2) on a coin flip (p = 0.5) would cause f* to be zero. This demonstrates that no matter how big your bankroll or how small your bet, you cannot make a Kelly Bet without there being a positive edge. A smart contract following this rule therefore must have a positive edge to be able to offer any bets.
It can be demonstrated that in simple fixed odds gambling scenarios, the Kelly Bet will increase bankroll faster than any other strategy in the long-term. The downside of the Kelly Bet is short-term volatility that can often be severe.
If avoidance of short-term volatility is important, then some long-term gains can be sacrificed to reduce short-term volatility by employing a ‘Fractional Kelly’ system. In Fractional Kelly, the bets are sized as a proportion of the Kelly Bet (e.g. 50% of the Kelly Bet is known as ‘Half Kelly’).
The equation below incorporates a factor, K, which is the fraction of the Kelly bet to use (0<K≤1).
House Edge
The Kelly Criterion formula is useful for sizing bets when a bettor has a positive edge. However, in a normal fixed odds betting scenario, it is the house which has the positive edge. Therefore, the Kelly Criterion can be used by casinos when deciding if their bankroll is large enough to accept certain bets. This is especially relevant for small game operators, who may not have a large bankroll. Casinos normally provide games which have a fixed house edge for each type of bet. For example, the house edge on all bets in American Roulette is 5.26%.
For a unit bet, the edge can be calculated from the probability of winning (p) and payout odds (b) as shown below. It can be thought of as simply the probability of winning (p) multiplied by the sum won (b-1), plus the probability of losing (1-p) multiplied by the sum lost (-1).
The term ‘pb-1’ in the Kelly Criterion formula can now be replaced by ‘Edge’. Rather than comparing the probability of winning to the payout odds, we now have a single term ‘Edge’ by which to describe how favorable a bet is compared to its payout odds.
EdgeFund’s core philosophy is that its smart contract will accept any bet at any odds, assuming it can meet the Kelly Criterion. If the bet can meet the Kelly Criterion, EdgeFund will take the bet at the requested odds, and it will use the Kelly Criterion to offer the lowest edge. If we re-arrange the above formula for Edge, we have a formula to calculate the minimum acceptable edge for a given bet and payout odds.
We now have a method for calculating the minimum possible edge that can be offered, whilst protecting our bankroll from risk of ruin. The only inputs required for the formula are the requested bet size relative to the current bankroll (f*), the requested payout odds (b), and our fraction Kelly factor (K).
Winning the Race to The Lowest House Edge
The method described here will be used by EdgeFund to create a smart contract offering fixed odds at the lowest mathematically possible house edge. The only way to compete with this game theory optimum strategy would be to offer zero edge by minting new tokens to pay winning bets — although as described earlier this will diminish the token’s value. Any other smart contract offering zero edge bets will be risking ruin, and will not survive long-term.
Using this method, EdgeFund can win the ‘Race to The Lowest House Edge’.
EdgeFund will allow start-up game operators to start their businesses with no bankroll and immediately offer bets with large potential payouts. Every bet will make them a profit at zero risk.
A positive feedback loop is created whereby as use increases, the value of EdgeFund tokens will increase, which allows for larger bets to be accepted, and for the house edge to decrease even further. EdgeFund can quickly become the cheapest way for even established game operators to offer fixed odds games.
We see the combinations of these network effects being incredibly powerful and leading to EdgeFund being the de facto decentralized fixed odds platform. Once the bankroll has sufficient value, it will be near impossible for any other decentralized fixed odds platform to provide cheaper odds or a larger bet range.
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I am currently working on EdgeFund, an open-source platform which offers a decentralized shared bankroll on the Blockchain. To learn more about EdgeFund, please visit our website. Join our Telegram group to chat to the team and follow us on Twitter!
