Blackjack as an optimisation problem
Posted 3 February 2007 at 15:55
In the middle of the last century blackjack stopped being written about as a game of judgement and started being written about as a decision problem with a computable answer. The shift is worth studying for its own sake, because it shows what happens to a game when someone finally writes its rules down in a form a machine can evaluate.
The problem has a clean statement. A player faces a known own total and one visible card on the other side, and must choose one action from a small set. Each action leads to a new state or to a resolution. The value of a state is the average result over the distribution of the remaining cards, given that every later choice is also made to maximise that average. That is a recursion, and recursions of that shape can be evaluated exhaustively if the state space is small enough.
The state space here is small enough, barely. The visible card takes ten values, the own total takes a few dozen depending on how soft totals and pairs are counted, and the actions number four or five under most rule sets. The result is a grid of a few hundred cells, each holding the action with the highest average value. Published as a chart, the grid is what most people mean when they say basic strategy.
Two properties of that grid are often lost. First, it is specific to a rule set: change the number of decks, the drawing rule on soft totals, or which resolutions are permitted after splitting, and cells change. A chart quoted without its rule set is an incomplete statement. Second, the grid maximises an average that is still negative under ordinary rules. It is an optimisation, not a reversal.
That second property is the reason this entry is filed under mathematics rather than anything else. The interest of the problem is that a game with an apparent element of judgement turned out to have a closed decision table sitting inside it, computable in an afternoon once the recursion was written correctly. Whether the resulting average is agreeable is a separate question, and not one arithmetic answers.
Posted in Table games | Permalink