Expected Utility and Lotteries

Numerous lotteries and gambles give the participant a likelihood of a potential gain or loss at a certain cost. The ability to predict the outcomes and expected utility from each lottery can enable the participant to have a comparative advantage over the rest, increasing the chances of a win. If a lottery offers n distinct prizes, and the probability of winning the prize is xi, all the values of xi add up to one. In such a scenario, for a unique lottery (Y) with unique prizes y1, y2…yn with unique probabilities x1, x2…xn, then the expected value of the lottery is

E(Y) = y1x1 +y2x2 + y3x3+ …+ynxn.

            The expected value is the weighted sum of the potential outcomes. The weights used are the respective probabilities. The unique prizes from the lottery or gamble provide different levels of utility to the participant and I can determine the expected utility from each type of a lottery.

A practical example is the daily number for instance:

Assume you purchase a one-dollar ticket with a three-digit number. Each number is chosen uniquely and can be equal to 0-9.

The results of the digits can be any number between 000 and 999, which gives 1000 outcomes with an equal likelihood.

Any potential win will give me a total of $500 minus the cost of the ticket, which is $1. The loss is $5. 

The expected utility from this type of a bet is

= (1/1000)*($500-$1) + (999/1000)*(-$1)

= -0.5

            This reveals a low utility and participating in that bet has minimal chances of a potential win. This shows that I would go for a different bet with a higher expected utility with potential higher costs. If there exists different types of potential bets and lotteries, then a table of the expected utilities and prices can be drawn to arrive at the most suitable value of the expected utility, which maximizes the chances of a win.  This will also minimize the losses I incur from the bets that I do not involve the element of prediction.