The Expected Value of the Game

What motivates people to gamble?

One might answer that money is the motivation--- it is plausible that a gambler is motivated by the expected monetary gain or the expected value of the game.

However, the expected values of casino games are always to the advantage of the house. Otherwise, how can casinos pay off all kinds of expenses and still earn profits.

There must be something beyond expected value that motivates the player to participate in gambling.

The conventional analysis of gambling is based on the expected values of games. The expected value of a game is defined as the sum of the outcomes multiplied by their relevant probabilities.

Moreover, the expected value of a game is always negative for the player and positive for the casino house. The absolute values of the two are exactly the same.

Therefore, what the player loses equals what the house wins. If the expected value of a game for the player is 0, then the game is 'fair'. It has 'true odds' against the player.

But fair game would earn zero revenue for the casino, so the casino cannot afford to provide players with fair games.

If casino games were all fair, the casino would be out of business.

To earn revenue for the casino, games must be 'unfair', to the advantage of the house.

The 'unfairness' of casino games is well-known to players. The players, however, knowingly play the 'unfair' games.

Evidently, the expected value of a game does not explain why people gamble.

Early in the eighteenth century, the motivation of gambling puzzled many scholars who tried to use expected value to explain, but failed.

A typical example is the 'St. Petersburg Paradox', and experimental game first designed by mathematician Nikollaus Bernoulli of Switzerland in the eighteenth century.

Unlike today's casino games, the St. Petersburg experiment has odds to the advantage of the player or a positive expected game value for the player.

To play the game, the player continues to toss the coin until the head side lands upward on the ground and the game is over.

The player is awarded one ducat (Swiss currency in the eighteenth century) if he or she has heads at the first toss, two ducats if heads appear at the second toss, four if at the third, eight at the fourth, and so forth.

Each additional toss will double the amount of the prize. To maximize the prize, the player wants to see heads appear as late as possible. The purpose of the experiment was to determine the fair price at which the player was willing to play the game.