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Best of Alan Krigman

Gaming Guru

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Why Maximum Expected Win Isn't the Only Betting Criterion

12 May 1997

Conventional casino canon claims the best bets maximize expected win. Similarly in business, the best options ostensibly maximize expected profits, stock prices, executive pay, or other monetary value. Maximum expectation is an objective criterion which can be readily analyzed using run-of-the-mill college math.

Here's how it works in gambling. Which is better - $10 to win $10 on the red at single-zero roulette or $10 to win $18 on the four at craps? In each case, expected value is the probability of success times the win minus the probability of failure times the loss. For roulette bets on red this is 48.6 percent of the $10 win minus 51.4 percent of the $10 loss, or ?$0.28. For craps bets on four this is 33.3 percent of the $18 win minus 66.7 percent of the $10 loss, or -$0.67. The roulette wager has the higher expected value so it's the better choice based on this criterion.


The shortcoming of maximum expectation is its tacit assumption that the same decision is made repeatedly. After "enough" trials, actual results closely match statistical predictions so monetary returns are indeed maximized. But few casino patrons make enough bets in their lifetimes for their bankrolls to converge on the expected values. And, vital business decisions are typically one-of-a-kind. In these and other practical situations, mathematical expectation often doesn't reliably anticipate performance.

Other factors, often highly subjective in nature, then enter the decision equation. Here are two non-casino illustrations.

1) A telemarketer calls a company, selling $150 worth of office supplies for $50. The firm can accept or reject the offer. The purchasing manager knows it's a gamble and thinks there's a 50 percent chance of getting a $100 bonus and a 50 percent chance of losing $50 to a scam artist. The expected value of placing an order is 50 percent of $100 minus 50 percent of $50, or $25. Hanging up on the telemarketer has an expected value of zero. Based on higher expectation, ordering is the better choice.

2) A pharmaceutical firm can buy a gene-splicing patent for $50 million. The directors know it's a gamble. They think the patent has a 60 percent chance of being worth $150 million and a 40 percent chance of proving useless. The expected value of the purchase is 60 percent of the $100 million gain minus 40 percent of the $50 million loss, or $40 million. Not acting has zero expected value. Based on higher expectation, it's better to buy the patent.


In the first scenario, the purchasing manager might gamble on the higher-expectation option - the purchase - because the company can easily absorb $50 loss if the proposition is a scam. A homeowner facing a similar offer for garden tools, though, might be unwilling to accept the same monetary risk.

In the second instance, the pharmaceutical company might decline the patent despite this choice having much lower expected value, because losing $50 million would force a bankruptcy. A bigger firm might be willing to take the risk and make the acquisition.

These examples show that maximum expected value isn't always the sole or even the strongest criterion for rational action when decisions are not repeated sufficiently frequently for statistics to reliably predict performance. Attitude about risk, level of assets, emotional response to triumph and defeat, losses which can be comfortably sustained, and winnings to which a person aspires are also valid inputs to the decision process.

So gambling strategies which don't maximize expectations aren't necessarily "wrong." Solid citizens sometimes make the best decisions - for themselves - by flouting the experts with such "sucker bets" as high-edge longshots, insurance at blackjack, single coins played in slot machines, or hedges at craps. The immortal Sumner A Ingmark, whose poetry may not maximize expectations, either, put it this way:

If every decision, though made with precision,
Completely lacks vision, it's prone to derision.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.