Betting and fairness
Betting and fairness are closely intertwined concepts that explore the principles of probability and expected value in various gambling contexts. The notion of a "fair bet" arises when the cost of a bet equals its expected payout based on possible outcomes. This idea was notably advanced by mathematicians Blaise Pascal and Pierre de Fermat in the 17th century, who sought to fairly divide winnings in games of chance with incomplete information. In modern society, betting occurs not only in casinos and sporting events but also in lotteries and financial markets, where the fairness of bets can vary significantly.
Lotteries, for instance, are often deemed unfair because a portion of the revenue is retained by the state, leading to a scenario where players are unlikely to see a return on their investment. Similarly, casino games typically feature a "house edge," ensuring the establishment profits over time, with rare exceptions like blackjack potentially offering players advantageous betting opportunities. Furthermore, subjective probabilities play a crucial role in betting, as individual perspectives on the likelihood of an event can influence the perceived fairness of a bet. This complexity is further exemplified in stock trading, where personal beliefs about a stock's value drive buying and selling decisions, akin to placing bets on future performance. Understanding these dynamics provides insight into the broader implications of betting practices and their fairness.
Betting and fairness
Summary: Mathematics is used to analyze betting and probabilities for games of chance and for investing in the stock market.
A pivotal moment in the early development of probability occurred in 1654, as the French mathematicians Blaise Pascal and Pierre de Fermat exchanged a series of letters. Pascal and Fermat were wrestling with questions involving the fair payoff for a gambler who is forced to quit in the middle of a game. In modern language, they were calculating the “expected value” of the game’s payoff (the average payoff under the various possible outcomes, weighted according to the likelihood of those outcomes). A bet is said to be “fair” if the price of placing it is equal to the expected value of the payoff. Betting plays an integral part in our modern society. People place bets in casinos and at sporting events, as well as by buying lottery tickets. They are also placing bets when purchasing insurance or investing in the stock market. Some of these bets are fair, some are unfair, and some cannot be objectively categorized.
The primary problem that Pascal and Fermat solved (each employing a different method) can be used to illustrate some important ideas on fairness. In the problem, two gamblers are playing a game in which a coin is repeatedly tossed. The game is interrupted at a point where 2 more heads are required for Player A to win and 3 more tails are required for Player B to win (whichever occurs first). How should the potential winnings be divided at this stage of the game?
Fermat solved the problem by observing that at most 4 tosses remain in order to identify the winner, and that there are 16 equally likely ways in which 4 tosses could occur:
HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, and TTTT.
In 11 of these possibilities (the first 11 items on the list), Player A would win, because 2 heads occur before 3 tails; in the other 5 possibilities, Player B would win, because 3 tails occur first. Therefore, Fermat reasoned that Player A should receive 11/16 of the winnings, and Player B should receive 5/16 of the winnings. In modern language, Player A would win the game with probability 11/16 and Player B would win with probability 5/16; Fermat was calculating the “expected value” of the winnings for each player.
Suppose that up to this point in their game, neither Player A nor B has paid any money for the opportunity to play, but that they are now required to pay a total of $1, altogether, and that this dollar will constitute the winnings. Fermat’s solution to the previous problem allows for a fair method of dividing the payment: Player A should pay 11/16 of the dollar and Player B should pay 5/16, so that the payments match the expected winnings. In other words, if the game is being played for a $1 payoff, then the price for a fair bet is 11/16 of a dollar for Player A and 5/16 of a dollar for Player B.
Lotteries and Casinos
State-run lotteries are unfair to the player who purchases a ticket, because some of the revenue goes to the state and is not redistributed to the winner(s). Of course, even if all of the ticket revenue were paid to the winner(s)—so that the bets were fair—a lottery would be unfavorable to almost every player. Nonetheless, lotteries attract large numbers of players because people are willing to pay a small amount for the minuscule chance of winning a fortune.
A similar motivation attracts bettors to casinos, where almost all games are unfair. This casino advantage is known as the “house edge.” On average, the house edge at a casino is 2% to 3%, which means that for each dollar that is bet, the house makes a profit of 2 or 3 cents. Over thousands of bets, this adds up to a significant profit. Some games, like slot machines, can have a house edge of up to 15%. Typically (in roulette, slot machines, and craps, for instance), the odds for each bet are slightly in favor of the house. Blackjack is a rare example of a casino game in which a player might be able to place bets that are better than fair from the player’s perspective. In blackjack, two initial cards are dealt to each player as well as to the dealer. Certain strict rules dictate whether additional cards are dealt to the dealer, while each player has the choice of whether to receive additional cards. The objective of each player is to hold a total card value closer to 21 than the dealer holds, without going over. Each player knows which cards he or she holds, as well as some of the cards held by the dealer and other players, since some cards are dealt face up. An adept player can also keep track of cards that have been used in previous games following the last shuffle—though casinos often dissuade such card counting by combining several decks and shuffling regularly. By using this information, it is possible for a player to calculate the probability of drawing a particular card and, therefore, the expected value of the payoffs under the options of either receiving an additional card or not; often, one of these expected values is greater than the amount of the bet.
Subjective Probabilities
Early in the twentieth century, mathematicians realized the need to define probability in a rigorous way, if it were to be a formal part of mathematics. In problems involving tossing fair dice or coins, or counting card hands, it was obvious what should constitute the probabilities of the various occurrences, but in many other situations it was unclear. Usually people thought of probabilities as idealized frequencies: if a fair coin is tossed many times, for example, then the fraction of tosses which land heads should be approximately 1/2; so a fair bet for a $1 payoff on heads should cost $0.50. But there is not an obvious analogy for two boxers, for example, about to fight a match. Also, probability was becoming an increasingly important tool for the physical sciences, and mathematical theorems were required. As such, an axiomatic system was necessary. The Russian mathematician A. N. Kolmogorov and the Italian philosopher and mathematician Bruno de Finetti independently provided such a framework in the 1930s. Although different in appearance, their definitions are equivalent in most situations.
De Finetti’s concept of a probability stems from gambling: the probability of an event is the price for a $1 payoff bet on that event. These prices may be assigned in whatever way one wants (hence the label “subjective probabilities”), provided certain consistency conditions are met. For example, suppose even money is coming into a betting house on two teams preparing to play a baseball game. This indicates that the bettors collectively value the two teams as equally likely to win the game. Ignoring the house fees, the price for a $1 payoff bet on either team is $0.50, because after the game, the entire pool of money will be redistributed to those who bet on the winning team.
Suppose, however, a particular bettor favors the home team, believing that team to have a 3/4 probability of winning the game. Then this bettor would price a $1 payoff bet on that team at $0.75; for this bettor, the $0.50 price generated by the betting pool is a bargain. From this bettor’s perspective, a bet on the home team is better than fair: the price for a $1 payoff bet is $0.50, but the expected value of the payoff is $0.75. Such situations occur beyond sporting events, perhaps most prominently in the stock market. The fact that individuals’ valuations often differ from those of the collective public is the driving force behind the trading of stocks. Individuals buy stocks that they believe to be undervalued and sell stocks that they believe to be overvalued. Because they are predicting the future performance of these stocks, they are essentially placing bets that they believe to be better than fair. In 1956, John Larry Kelly, Jr., a physicist who worked at Bell Labs, formulated and described the Kelly criterion. This algorithm for determining an optimal series of investments (or bets) is based on probability and economic utility theory, which tries to mathematically quantify satisfaction. In recent years, the Kelly criterion has been incorporated into many mainstream investment theories and betting strategies.
Bibliography
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Devlin, Keith. The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern. New York: Basic Books, 2008.
Epstein, Richard A. The Theory of Gambling and Statistical Logic. 2nd ed. San Diego, CA: Academic Press, 2009.
Packel, Edward. The Mathematics of Games and Gambling. 2nd Ed. Washington, DC: The Mathematical Association of America, 2006.
Von Plato, Jan. Creating Modern Probability. New York: Cambridge University Press, 1994.