Expected Payoffs and Fair Entry Prices

What Makes an Entry Price Fair?

A game can offer a large prize and still be a poor bet if the chance of receiving it is small. To assess the price of entry, we need to account for every possible outcome, its probability, and the cash received or paid in that outcome. Expected value provides a precise benchmark: the fair entry price is the price that makes the expected net payoff equal to zero.

This is a mathematical definition of fairness, not a promise about what happens in one play. A player who pays a fair price might win more than the fee, or lose the entire fee. The definition concerns the probability-weighted average across the model's possible outcomes.

Definition: A fair entry price is an entry fee at which the expected net payoff to the player is zero. If $R$ is the cash payout received from the game, and the player pays a fixed entry fee $c$ regardless of the outcome, then the net payoff is $N=R-c$. When the payout distribution does not depend on the fee, the fair price is $c^*=E[R]$.

Here, “payout” means the cash the player receives from the game, not the profit after subtracting the entry fee. Keeping those quantities separate prevents a common pricing error. If a game awards $12 and the player paid $3 to enter, the net payoff on that outcome is $9, not $12.

The Expected-Net-Payoff Equation

Suppose a game has a finite set of possible outcomes. Let $r_i$ be the cash payout in outcome $i$, and let $p_i$ be the probability of that outcome. The probabilities satisfy $p_i\geq 0$ and $\sum_i p_i=1$. Assume the entry fee $c$ is paid once, regardless of the outcome, and the listed payouts are the total cash received from the game. Then the net payoff in outcome $i$ is $r_i-c$.

The definition of expected value and the fact that the outcome probabilities sum to 1 give the expected net payoff:

E[N]=\sum_i p_i(r_i-c) =\sum_i p_i r_i-c\sum_i p_i =E[R]-c.

The fair price makes this quantity zero. Thus $E[R]-c^*=0$, so $c^*=E[R]$. In words, when the payout distribution is fixed and the fee is paid in every outcome, the fair entry price is the expected cash payout. This calculation uses the linearity of expectation: subtracting a fixed fee subtracts that same amount from the expected payoff.

Model assumptions: The outcome probabilities describe the game being priced, the listed cash payouts apply to those outcomes, and the fee is paid once regardless of the result. This basic formula also treats all amounts as comparable cash amounts, with no additional costs or timing adjustments. If the payout itself depends on the entry fee, write the net payoff in terms of that fee and solve $E[N(c)]=0$ instead.

Worked Example: A Prize for Landing on Red

A hypothetical spinner lands on red with probability $1/4$, in which case the player receives $12. Otherwise, the player receives $0. What entry fee makes the expected net payoff zero?

Check the model. The two probabilities are $1/4$ and $3/4$, which are nonnegative and sum to 1. The $12 payout is received only on red, and the fee is paid either way.

Let $R$ be the payout. Its expected value is

E[R]=\left(\frac14\right)(\$12)+\left(\frac34\right)(\$0) =\$3+\$0=\$3.

Therefore, the fair entry price is $c^*=\$3$. At this price, the net payoff is $9 if the spinner lands on red and $-\$3$ otherwise. Its expected value is

E[N]=\left(\frac14\right)(\$9)+\left(\frac34\right)(-\$3) =\$2.25-\$2.25=\$0.

The calculation does not say that a player paying $3 will break even. The player either finishes $9 ahead or $3 behind in this single play. It says that the probability-weighted average net payoff, under the stated model, is zero.

Finding the Price from Different Game Descriptions

A probability table is often the clearest way to track payouts and calculate their expected value. Each row should give an outcome, its probability, and the cash payout for that outcome. Payouts of zero must be included: an outcome in which the player receives nothing still contributes to the probability model.

Worked Example: An Award Based on a Die Roll

In a hypothetical game, a fair six-sided die is rolled. The player receives $8 if the result is 1, 2, or 3, and receives nothing otherwise. What is the fair entry price?

Check the model. A fair die gives each face probability $1/6$. Three of the six faces produce the $8 payout, so that payout has probability $3/6=1/2$; the $0 payout also has probability $1/2$. The probabilities sum to 1, and the entry fee is paid for either payout.

Outcome	Probability	Cash payout
Roll is 1, 2, or 3	$1/2$	$8
Roll is 4, 5, or 6	$1/2$	$0

The expected payout, and therefore the fair entry price, is

E[R]=\left(\frac12\right)(\$8)+\left(\frac12\right)(\$0) =\$4+\$0=\$4.

At a $4 fee, the net payoff is $4 on a roll of 1, 2, or 3 and $-\$4$ on a roll of 4, 5, or 6. The expected net payoff is $(1/2)(\$4)+(1/2)(-\$4)=\$0$. If the fee were $5 instead, the net payoff would be $3 on the first three faces and $-\$5$ on the other three, so its expected value would be $(1/2)(\$3)+(1/2)(-\$5)=-\$1$. That fee is not fair by the zero-expected-net-payoff definition.

Sometimes a game description mentions a prize but leaves several possible award amounts. The expected payout must include every possible amount and its probability, not just the most likely award or the largest advertised award.

Worked Example: Several Possible Prize Amounts

A hypothetical ticket pays $0 with probability $0.5$, $5 with probability $0.3$, and $25 with probability $0.2$. The player pays the same entry fee regardless of the result. Find the fair price and check the expected net payoff at that price.

Check the model. All probabilities are between 0 and 1, and $0.5+0.3+0.2=1$. The payouts are stated as cash received, and none depends on the fee. Therefore, the fixed-payout formula applies.

E[R]=(0.5)(\$0)+(0.3)(\$5)+(0.2)(\$25) =\$0+\$1.50+\$5.00=\$6.50.

The fair entry price is $6.50. At this price, the net payoffs corresponding to the three payouts are $-\$6.50$, $-\$1.50$, and $18.50. Checking directly,

E[N]=(0.5)(-\$6.50)+(0.3)(-\$1.50)+(0.2)(\$18.50) =-\$3.25-\$0.45+\$3.70=\$0.

A price of $6.50 is not a prediction that a particular ticket will return $6.50. It is the probability-weighted average payout under this hypothetical probability model.

When the Payout Includes the Entry Fee

The simple formula $c^*=E[R]$ applies when the cash payout is specified independently of the entry fee. A different calculation is needed when a winning outcome returns the fee as well as awarding a prize. In that case, the cash received can depend on $c$, so the expected net payoff must be written explicitly as a function of the fee.

Worked Example: A Prize Plus a Returned Fee

Suppose a hypothetical ticket costs $c$ dollars. With probability $0.1$, it returns the entry fee and adds a $10 prize; with probability $0.9$, the player receives nothing. Find the fair entry price.

Check the model. The probabilities are $0.1$ and $0.9$, which sum to 1. In the winning outcome the cash payout is $c+\$10$, not just $10; in the losing outcome the payout is $0. The player pays $c$ in either case.

The net payoff is $10 if the ticket wins, because the returned fee cancels the fee paid. If it loses, the net payoff is $-c$. Consequently,

E[N(c)]=(0.1)(\$10)+(0.9)(-c).

Set the expected net payoff equal to zero and solve:

0=\$1-0.9c \quad\Longrightarrow\quad c^*=\frac{\$1}{0.9}=\$1.111\ldots.

To check, at $c^*=\$1.111\ldots$, the winning net payoff is $10 and the losing net payoff is $-\$1.111\ldots$. Thus

E[N(c^*)]=(0.1)(\$10)+(0.9)(-\$1.111\ldots) =\$1-\$1=\$0.

The fair fee is about $1.11. It is not $1.00: that number is the expected $10 bonus, but a losing player also loses the fee, while a winning player gets the fee back. The fee therefore affects the cash flows differently across the outcomes.

What Fairness Does—and Does Not—Say

A fair price is a benchmark based on expected net payoff. If a fixed-payout game costs $c$, its expected net payoff is $E[R]-c$. A fee below the fair price gives a positive expected net payoff under the model; a fee above it gives a negative expected net payoff. Neither statement guarantees a gain or loss on a particular play. It compares expected values, not certain outcomes.

The word “fair” here also does not mean that every player will consider the game worthwhile. Two games with the same expected payout can have very different possible outcomes. Expected value alone does not describe how spread out or unpredictable those outcomes are; those questions require additional summaries. Nor does a zero expected net payoff establish that the probabilities or payouts in the model are correct. The answer is only as reliable as the stated game rules and probabilities.

The price calculation also does not include unlisted costs. If playing requires a separate fee, or if a prize is subject to a cost that varies by outcome, include that cash flow in the net payoff before setting its expectation to zero. The general method is to list what the player receives and pays in each outcome, then solve for the fee that makes the expected net payoff zero.

Common Mistake: Pricing from the Prize Alone

A common mistake is to call the advertised prize the fair entry price, without weighting it by the chance of receiving it. For example, “the prize is $12, so a $12 ticket is fair” ignores the probability of winning and the possibility of receiving nothing. In the spinner example, the expected payout was $(1/4)(\$12)+(3/4)(\$0)=\$3$, not $12.

The correction is to identify each possible cash payout, multiply it by its probability, and add the results. Then subtract the entry fee in the net-payoff calculation. Also check whether a stated payout includes a returned fee: if it does, the net payoff may depend on the fee itself, and the equation $E[N(c)]=0$ must be solved directly.

Key takeaway: Fairness in this setting means zero expected net payoff under a specified probability model. With fixed payouts and a fee paid in every outcome, the fair entry price equals the expected cash payout. It does not guarantee that an individual player breaks even.

Check Your Understanding

For each question, distinguish cash payout from net payoff and state the assumptions needed for the calculation.

A game pays $20 with probability $0.1$ and $0 otherwise. The fee is paid regardless of the result. What is the fair entry price?
Why is a fair entry price not a guarantee that a player will break even on one play?
A ticket pays $4 with probability $0.5$ and $10 with probability $0.5$. Find the fair price if the fee does not affect either payout.
A game returns the fee and awards an additional prize when the player wins. Why might the fair fee not equal the expected value of the additional prize alone?
A game costs $7 to enter and has an expected payout of $6. What is its expected net payoff, and is the fee fair by the zero-expected-net-payoff definition?