The Four Horsemen and the First Complete Strategy
Before computers could simulate millions of blackjack hands, four U.S. Army engineers solved the game with desk calculators. Roger Baldwin, Wilbert Cantey, Herbert Maisel, and James McDermott — later nicknamed the Four Horsemen — published the first mathematically correct basic strategy in 1956 in the Journal of the American Statistical Association. They did not run a Monte Carlo simulation. They worked through the decision tree by hand, using only adding machines to grind out the expected value of every possible action for every player total against every dealer upcard.
The distinction matters because the two methods answer different questions. Monte Carlo simulation approximates outcomes by playing a game many times with random inputs and observing what happens. Combinatorial analysis, which is what produced basic strategy, enumerates every possible state of the game and computes the exact expected value of each decision from that state. One samples; the other solves.
What Combinatorial Analysis Computes
A blackjack hand is a finite decision problem. The player sees their own two cards and one dealer upcard. From that state, the player can stand, hit, double, or split. Each action leads to a distribution of possible outcomes. The analyst's job is to compute the expected value of each action under the specific rules of the table, then choose the action with the highest expected value — or the least negative one, since blackjack is a negative-expectation game for the player who does not count cards.
The Four Horsemen worked through every combination. A player total of 16 against a dealer 10 is one state. A player total of 16 composed of three cards against a dealer 10 is another state, because the composition changes the probabilities of drawing each remaining card. The full decision space includes every possible player hand composition, every dealer upcard, and every applicable rule: whether the dealer stands on soft 17, whether doubling after splitting is allowed, how many decks are in play. Each rule change alters the expected values slightly, which is why basic strategy charts differ across rule sets.
The arithmetic is exhaustive rather than sampled. For a single-deck game, the number of distinct player states is large but finite. The analysts computed the probability of each resulting total after a hit, the dealer's probability of reaching each final total, and the payoff for each outcome. Summing the probability-weighted payoffs gives the expected value of the action. Repeating that calculation for every state and every action yields the complete strategy.
Where Monte Carlo Simulation Fits In
Monte Carlo simulation arrived later and serves a different purpose. Instead of solving every state exactly, a Monte Carlo approach deals millions of random hands according to the rules and records the results. The average outcome across those hands approximates the true expected value, with the approximation improving as the number of simulated hands grows. The method is powerful for games or situations where the state space is too large for exact enumeration, or where the analyst wants to measure variance, bankroll swings, or the effect of a betting pattern over time.
For blackjack basic strategy, Monte Carlo is unnecessary. The game's decision tree is small enough that exact computation was possible even with 1950s tools. The Four Horsemen's paper did not need randomness; it needed patience. Later researchers, including Edward Thorp, extended the work with more powerful computers and published strategy refinements, but the foundation was combinatorial, not simulated.
| Aspect | Combinatorial analysis | Monte Carlo simulation |
|---|---|---|
| Method | Enumerates every possible state and computes exact expected values | Samples many random outcomes and averages them |
| Result | Exact, given the rule set | Approximate, improving with sample size |
| Use in blackjack | Produced basic strategy charts | Used for variance studies, bankroll modelling, testing betting systems |
| Computational cost | High but finite; feasible by hand for four engineers | High; requires many iterations for precision |
| Sensitivity to rule changes | Requires recomputation for each rule set | Requires new simulation runs for each rule set |
How the Expected Value of a Single Decision Works
Consider a player holding a hard 16 against a dealer 10. The player must choose between standing and hitting. The analyst computes the expected value of standing: the probability that the dealer busts, the probability the dealer finishes with 17 through 21, and the payoff for each case. Then the analyst computes the expected value of hitting: for each possible card drawn, the probability of that card, the new player total, and the resulting expected value from that new state, accounting for the possibility of busting. The action with the higher expected value — which for hard 16 against a 10 is hitting, though it remains a losing hand overall — goes into the strategy chart.
The same logic applies to doubling and splitting. Doubling restricts the player to one additional card, so its expected value is the expected value of the resulting two-card total multiplied by two, since the wager doubles. Splitting separates the pair into two hands and requires a second wager, so the analyst computes the expected value of each resulting hand independently and sums them. Every entry in a basic strategy chart is the output of this kind of calculation, not a rule of thumb.
Why the Chart Is Rule-Specific
A basic strategy chart is only correct for the rule set it was computed under. The dealer standing on soft 17 versus hitting soft 17 changes several decisions. The number of decks changes the probabilities of drawing specific cards, because removing cards from a single deck alters the composition of the remaining deck more than removing cards from a six-deck shoe. Whether doubling after splitting is permitted changes the expected value of splitting pairs. Whether surrender is offered adds a third action to consider in some states, and the analyst must compute its expected value as well.
This is why a player who copies a chart from one source and applies it to a game with different rules may be making systematically wrong decisions. The error is usually small in expected-value terms, but it is real. The correct procedure is to find a chart computed for the specific rule set in play at the table where the player sits.
What Basic Strategy Does and Does Not Do
Basic strategy minimises the expected loss per hand under the given rules. It does not turn blackjack into a positive-expectation game. The house edge remains, because the player acts first and busts lose immediately even if the dealer also busts. Basic strategy reduces the house edge to its mathematical minimum for a player who is not tracking cards, but the edge is still negative. The player's expected result per hand is a small loss, and the expected result over many hands is that loss multiplied by the number of hands played.
Decisions per hour amplify this. A player at a table with other players might see 60 to 80 hands per hour. A player alone at a fast table might see 200 or more. Each hand carries the same expected loss, so the total expected loss scales with the number of hands. Basic strategy does not change the sign of that expectation; it changes its size.
Monte Carlo and the Illusion of Betting Systems
Monte Carlo simulation is useful for demonstrating why progression systems fail. A player who doubles their bet after each loss — the Martingale — will win back previous losses plus one unit whenever a win eventually occurs, but the bet size required grows exponentially, and table limits cap the progression. A Monte Carlo simulation of a Martingale player shows that the system produces many small wins punctuated by rare catastrophic losses. The expected value of the system is unchanged from flat betting: the house edge applies to every unit wagered, regardless of how the wager sizes are arranged.
Simulation makes the variance visible. A player using a progression system experiences a different distribution of outcomes than a flat bettor, but the mean of that distribution is the same negative number. The system redistributes when the losses occur; it does not remove them. A Monte Carlo run with enough iterations will show the average result converging toward the same expected loss per unit wagered as flat betting.
Reading a Strategy Chart as a Decision Table
A basic strategy chart is a lookup table. The rows are player hand totals, sometimes split by hard totals, soft totals, and pairs. The columns are dealer upcards. Each cell names an action: hit, stand, double, split, or surrender where offered. The chart assumes the player has no information about the remaining cards beyond what is visible. When a card counter has additional information, the optimal action can change, but that is a different exercise built on top of basic strategy, not a replacement for it.
The chart also assumes the player's objective is to minimise expected loss per hand. A player with a different objective — maximising the chance of a large win in a short session, for example — might make different decisions, but those decisions carry a higher expected loss. Basic strategy is the answer to a specific mathematical question: which action maximises expected value in each state under the given rules.
FAQ
Did the Four Horsemen use a computer to invent basic strategy?
No. Baldwin, Cantey, Maisel, and McDermott used desk calculators to perform the combinatorial calculations by hand. They published their results in 1956, before electronic computers were widely available for this kind of work. Their method was exact enumeration of the game's decision states, not simulation.
Can Monte Carlo simulation produce a basic strategy chart on its own?
In principle, yes, but it would be inefficient and less precise. A Monte Carlo approach would need to simulate each decision state many times to estimate the expected value of each action, and the result would carry sampling error. Exact combinatorial analysis gives the true expected value for each state, which is why it remains the standard method for generating strategy charts.
Why do different basic strategy charts disagree with each other?
They disagree because they were computed under different rule sets. The number of decks, whether the dealer hits or stands on soft 17, whether doubling after splitting is allowed, and whether surrender is offered all change the expected values of certain decisions. A chart is only correct for the rules it was computed for, so a player should match the chart to the table.
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