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Comparing Game Strategies with Expected Value

Discrete Mathematics · Chapter 7: Probability in Games, Gambling, and Strategic DecisionSection 7.6 · lesson 45 of 8711 min · free with a free account

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In this lesson

One number to compare dice, cards, poker and lottery bets on equal footing.

  • Use house edge as a common yardstick
  • Choose among alternatives with a payoff table
  • Rank strategies across 4 different games
  • Add variance as a second lens on risk

Transcript

Everything said in the video, section by section (1,512 words). Read along, or search it.

Probability in Games, Gambling, and Strategic Decision

Comparing Game Strategies with Expected Value

This section closes chapter seven by comparing strategies across all the games in it. No new probability calculations are needed. Every probability here was established earlier. What's new is the comparison itself. Expected value as a common yardstick, a framework for making decisions with it, and what it leaves out.

The Opening Problem

1 Number for 4 Different Games

Imagine a gaming site that displays a small figure next to every game. Expected loss per one hundred dollars wagered. A newcomer might wonder how one number could compare a dice game, a card game, a hand of poker, and a lottery draw. They have completely different rules, payouts, and bet sizes. The answer is that the number isn't really about the rules. It's the expected value of a one dollar wager, restated as a percentage. Once every game's outcomes and payouts go through the same formula, the result means the same thing, whether the game uses two dice, fifty-two cards, or a numbered ball machine.

7.6.1 A Common Yardstick

Normalize, Then Compare

Expected value, from section six point two, is each outcome times its probability, summed. It's the long-run average per repetition. The formula doesn't change from game to game. The obstacle to comparison is that raw dollar amounts depend on how much is wagered. A one dollar dice bet and a one hundred dollar blackjack hand could have proportionally identical odds, but wildly different dollar expected values. So normalize. The house edge is minus the expected value per one dollar wagered, as a percentage. A house edge of one point four one percent means losing, on average, one point four one cents of every dollar, over the long run. There's no new probability in it. Just a normalization, and a sign convention.

7.6.2 A Decision Framework

Alternatives, Outcomes, Payoffs

Comparing two bets works pairwise. With several, a framework helps. Decision theory organizes the choice into three pieces. A decision alternative is a choice actually available, like which bet to place. An outcome is a possible result once chance resolves, like win, lose, or push. And a payoff is the dollar consequence of an alternative meeting an outcome. Laying all of that out together is called a payoff table. Then the expected value is computed for each alternative. The expected-value criterion picks the alternative with the highest expected value. Someone who always follows it is called risk-neutral. They're indifferent between two options with equal expected value, however differently those options might swing. Two other rules are worth naming for contrast, since neither uses probabilities at all. Maximin picks the best worst case, a rule for someone strongly risk-averse. Maximax picks the best best case, a rule for a risk-seeker.

Table 7.6.1

3 Dice Bets, 1 Payoff Table

Here are three one dollar craps bets as alternatives in one payoff table. The pass line wins with probability about zero point four nine three, and pays even money. Its expected value is about minus one point four one cents. Don't pass wins about zero point four seven nine, loses about zero point four nine three, and pushes, a tie, about zero point zero two eight of the time. Its expected value is about minus one point three six cents. And any seven, a one-roll bet paying four to one on a seven, has expected value minus one sixth, about minus sixteen point seven cents. By the expected-value criterion, don't pass is preferred to the pass line, and both are strongly preferred to any seven.

Blackjack

Insurance Feels Like a Hedge

The same machinery works across completely different games. House edge doesn't care what produced a probability, only what it is. In blackjack, insurance is a side bet offered when the dealer shows an ace. It pays two to one if the dealer's hidden card makes blackjack, and loses otherwise. In the simplified single-deck baseline, sixteen of the fifty-one unseen cards are ten-value. So the expected value is sixteen fifty-firsts times two, minus thirty-five fifty-firsts. That's minus one seventeenth, about minus five point nine cents. A house edge of five point eight eight percent. Insurance feels like a hedge rather than a bet. But its house edge is much worse than either even-money dice bet. Here the criterion corrects an intuition, instead of confirming it.

Poker - A Hypothetical Bonus Bet

A Big Top Payout, a Bigger Edge

Now poker. Some casinos offer optional bonus bets that pay by the rank of a hand. Here's a purely hypothetical one, using the hand probabilities from section seven point three. For each one dollar staked, it returns five hundred dollars on a royal flush, fifty on a straight flush, and twenty on four of a kind. It returns seven dollars on a full house, five on a flush, and two on two pair. Multiply each return by its probability. Two pair contributes the most, about nine and a half cents. Everything together comes to about twelve cents. So the expected value is about twelve cents, minus the one dollar staked. That's about minus eighty-eight cents, a house edge of about eighty-eight percent.

Why It Fails

The Common Hands Pay Nothing

Why is it so bad? Every hand that isn't listed loses the stake, and that's about ninety-five percent of hands. One pair and high card alone are over ninety-two percent of all hands, and they pay nothing here. So no top-end payout can rescue the bet. The eye-catching five hundred dollar top prize is exactly what makes it easy to misjudge. A large maximum payout and a large house edge aren't in tension. They routinely coexist.

Lottery - A 6/49 Ticket

Jackpot Only, and Still Losing

Finally, a lottery. A standard six-from-forty-nine lottery has thirteen million, nine hundred eighty-three thousand, eight hundred sixteen combinations, as in section five point eight. Consider a two dollar ticket, against a twenty million dollar jackpot, ignoring smaller prizes, taxes, and split jackpots. The expected value is twenty million over that count, about one dollar forty-three, minus the two dollar price. That's about minus fifty-seven cents, a house edge of about twenty-eight and a half percent. Setting the expected value to zero, the jackpot would need to reach about twenty-eight million dollars just to break even on the jackpot alone. And since taxes and split jackpots only lower the value, the true break-even point is higher still.

Figure 7.6.2

6 Strategies, 1 Scale

Here are all six strategies on the single house-edge scale. Don't pass, one point three six percent. Pass line, one point four one. Blackjack insurance, five point eight eight. Any seven, sixteen point six seven. The lottery ticket, twenty-eight point four nine. And the hypothetical poker bonus bet, eighty-seven point eight seven. Every bar is the same quantity, negative expected value per dollar wagered, even though the games are structurally unrelated.

7.6.4 A Second Lens

Variance: How Hard Outcomes Swing

Every comparison so far ranked alternatives by expected value alone. But expected value only describes the long-run average. It says nothing about how far a single outcome might swing above or below it. That missing dimension is variance, introduced here. With mu as the expected value, the variance is each outcome's squared distance from mu, times its probability, summed. Its square root, the standard deviation, sigma, restates that spread in the same units as the bet itself. Dollars, rather than squared dollars.

Worked Example

Identical EV, Very Different Risk

Here are two simplified one dollar bets, deliberately built to be exactly fair, with expected value zero, so variance is the only difference. Bet A wins or loses one dollar, each with probability one half. Bet B wins ninety-nine dollars with probability one percent, and loses one dollar otherwise. Both have expected value zero. But bet A's variance is one, so its standard deviation is one dollar. Bet B's variance is zero point zero one times ninety-nine squared, plus zero point nine nine. That's ninety-nine, so its standard deviation is about nine dollars ninety-five. Bet B swings nearly ten times harder, around the identical average. The expected-value criterion treats them as the same. A risk-averse player would generally prefer bet A. Neither preference is mathematically wrong. Expected value and variance simply answer different questions, and a full picture needs both.

Bankroll and Ruin

When the Long Run Never Arrives

This matters beyond a constructed example. A bettor with a limited bankroll can be driven to ruin by a high-variance strategy long before the long-run average has any chance to assert itself. Even when that strategy's expected value is equal, or better. House edge describes where a game's average outcome sits. A volatility measure describes how roughly a player might get there. That's why a transparent platform might reasonably show both, side by side.

Takeaways

EV Compares, Variance Warns

To summarize. House edge, expected value per dollar wagered as a percentage, lets dice, blackjack, poker, and lottery bets be ranked on one scale. A payoff table and the expected-value criterion organize a choice among alternatives. Maximin and maximax offer payoff-only alternatives, for risk-averse or risk-seeking decision-makers. A large top payout doesn't imply a small house edge. And two strategies can share an expected value while differing enormously in variance. Here's something to think about. A player has exactly twenty dollars, and must choose once. Either bet A, repeated many times over an evening, with expected value zero. Or a single wager with expected value plus two dollars, but a fifteen percent chance of losing all twenty at once. The second has the better expected value. Is it still the rational choice for this player? And what role does the size of the total bankroll play? Is there a size at which your answer would change?

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FAQ

Questions students ask us

About Comparing Game Strategies with Expected Value

What is house edge, and how does it let you compare different games?
House edge is the negative of a bet's expected value per one dollar wagered, written as a percentage. Because it rescales every game to the same unit, a dice bet, a blackjack side bet, and a lottery ticket can be ranked directly. For example, a house edge of 1.41\% means losing about 1.41 cents per dollar, on average, over the long run.
How do you use a payoff table to choose between betting options?
A payoff table lists each decision alternative, each possible outcome, and the dollar payoff when they meet. Compute the expected value of each alternative, then pick the highest, which is the expected-value criterion. For one dollar craps bets, don't pass (\approx -0.0136) edges out pass line (\approx -0.0141), and both beat any seven (-1/6).
What is the expected value of blackjack insurance?
In the simplified single-deck baseline, insurance has expected value -1/17 per dollar, about -5.9 cents. Sixteen of the 51 unseen cards are ten-value and the bet pays 2 to 1, so EV = \frac{16}{51}(2) - \frac{35}{51} = -\frac{1}{17}. That is a 5.88% house edge, much worse than the roughly 1.4% on even-money dice bets, even though insurance feels like a hedge.
Why isn't expected value enough to compare two strategies, and what does variance add?
Expected value gives only the long-run average, while variance, \sigma^2 = \sum (x-\mu)^2 p, measures how hard outcomes swing around it. Take two fair one dollar bets. Bet A wins or loses 1 dollar with probability 1/2, so \sigma = 1. Bet B wins 99 dollars with probability 0.01 and loses 1 otherwise, so \sigma^2 = 99 and \sigma \approx 9.95.

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