What the house edge actually measures
Posted 20 February 2007 at 11:30
The house edge is quoted constantly and defined rarely. Stated carefully, it is the expected loss per unit staked on a single resolved wager, expressed as a proportion. It is an average, it is taken over the distribution of outcomes of that one wager, and it says nothing about any particular sequence of play. Every misunderstanding of the quantity comes from dropping one of those three clauses.
Because it is an average per unit staked, it scales with total amount staked and not with time, seats, or the number of people at a table. Two hours of slow play at a large stake and six hours of fast play at a small stake can carry identical expected results if the totals staked match. The quantity is indifferent to everything except the sum of the stakes and the wagers they were placed on.
Because it is taken over a distribution, it describes the centre of a spread and not the spread itself. Two wagers can carry the same edge and behave completely differently: an even money wager resolves close to its average quickly, while a wager paying several hundred to one sits far from its average for very long stretches. Anyone reading only the edge is reading only one of the two numbers that describe a wager.
And because it applies to a single resolved wager, it composes by addition across wagers rather than by any more interesting rule. Placing two wagers with different edges gives an expected result that is the stake-weighted sum of the two. There is no arrangement of wagers within a game that cancels the edges of its parts, because the sum of negative terms is negative regardless of the order the terms are written in.
That last point is worth stating plainly because it is where a great deal of published material goes astray. Combining wagers changes the shape of the distribution, sometimes dramatically. It does not change the sign of the average. The edge is a property of each wager offered, fixed by its payout table, and it is the one number in the subject that stays still while everything else moves.
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