Contests with entry fees: theory and evidence

Abstract

We provide some theory and experimental evidence on contests with entry fees. In our setup, players must simultaneously decide whether or not to pay a fee to enter a contest and the amount they wish to bid should they choose to enter the contest. In a general n-bidder game, we show that the addition of contest entry fees increases the contest designer’s expected revenue and that there is a unique revenue maximizing entry fee. In an experimental test of this theory we vary both the entry fee and the number of bidders. We find over-bidding for all entry fees and bidder group sizes, n. We also find under-participation in the contest for low entry fees and over-participation for higher entry fees. In the case of 3 bidders, the revenue maximizing entry fee for the contest designer is found to be significantly greater than the theoretically optimal entry fee. We offer some possible explanations for these departures from theoretical predictions.

Appendix

1.1 A Proof of Proposition 1

Proof

Consider the following differentiable function

$$\begin{aligned} f\left( x\right) =\sum _{i=0}^{n-1}\frac{1}{\left( i+1\right) ^{2}} C_{i}^{n-1}\left( x\right) ^{i}\left( 1-x\right) ^{n-i-1}. \end{aligned}$$

It is straightforward to check that

$$\begin{aligned} f\left( 0\right) =1 \end{aligned}$$

and

$$\begin{aligned} f\left( 1\right) =\frac{1}{n^{2}}. \end{aligned}$$

Note that

$$\begin{aligned} f^{\prime }\left( x\right) =\sum _{i=1}^{n-1}\left[ \frac{i^{2}-\left( i+1\right) ^{2}}{i\left( i+1\right) ^{2}}\right] \frac{\left( n-1\right) !}{ i!\left( n-i-1\right) !}\left( x\right) ^{i-1}\left( 1-x\right) ^{n-i-1}<0 \text { for }x\in \left( 0,1\right) \text {.} \end{aligned}$$

Therefore, function $f\left( x\right)$ is monotonically decreasing on the interval $\left[ 0,1\right]$ and the range of function f is $\left[ \frac{ 1}{n^{2}},1\right]$. Hence, equation ( 4) has a unique solution $p^{*}\in \left[ 0,1\right]$ for any $\frac{c}{V} \in \left[ \frac{1}{n^{2}},1 \right] .$ $\square$

1.2 B Experimental instructions

Here we present the instructions for the pairs treatment. The instructions for the triples treatment are similar.

Welcome to this experiment in the economics of decision-making. You are guaranteed $7 for showing up and completing this study. These instructions explain how you can earn additional earnings from the choices you make. Please silence any mobile devices and refrain from talking with others for the duration of this study. If you have any questions, please raise your hand.

Today’s study involves 6 rounds of decision-making and the completion of a questionnaire.

1.3 Decisions

Prior to the start of each of the 6 decision rounds, you will be randomly matched with one other participant in the room. Thus, the participant you are matched with will likely change from one round to the next. In each round you will be randomly assigned the role of “participant 1” or “participant 2”. This labeling helps in identifying each person’s choices in the round but otherwise it makes no difference. You will not know the identity of the other participant you are matched with in each decision round – “your match” – nor will they know your identity even after the study is over.

For each decision round, you and your match for that round have to simultaneously make one or two decisions.

The first decision is whether you want to enter a contest with the other participant. The contest always yields a prize of 100 points to the winner and 0 to the loser. In order to enter the contest, you have to pay an entry fee in points which will be shown on your decision screen. The entry fee will be the same for you and your match.

Let us denote the fee to enter decision contest number $k=1,2,...,6$ by $f_k$ points. The actual entry fee will differ from round to round so pay careful attention to the fee in each decision round. A fee of 0 points means there is no entry fee, but in that case you still have to choose to pay that fee to enter the contest.

Prior to deciding whether you want to enter contest k, both you and your match for that contest will each be given $120 + f_k$ total points.

If you choose “Don’t Enter” then you keep and earn the $120 + f_k$ points you were given for decision round k.

If you choose “Pay the Fee and Enter the Contest,” then you give up $f_k$ points and you have to decide how many of your remaining 120 points you want to bid toward winning contest round k.

Specifically, on the first screen for each round you will see this information:

The prize to the winner of the contest is: 100.0 points

The fee for entering the contest this round is: $\mathbf {f_k}$ points

You are given 120.0 points plus the fee of $\mathbf {f_k}$ points for a total of $\mathbf {120.0+f_k}$ points this round.

Do you want to pay the fee and enter the contest?

Below this you click on either the “Don’t Enter” button or the “Pay the Fee and Enter” button. Then click the Next button to confirm your decision. You can change your mind anytime prior to clicking the Next button.

If you choose to Pay the Fee and Enter the contest then you give up $f_k$ points and on the next screen, you make a second decision: how many of your remaining 120 points you want to bid toward winning the contest you have entered. You make this second decision by moving a slider on your screen between 0 and 120 or by entering the number of points you want to bid between 0 and 120 in an input box. Once you have made your bid, click the Next button. You can change your mind anytime prior to clicking the Next button.

If you choose Don’t Enter then you will see a “Please Wait” screen.

In either case, you will NOT know when making your own decisions whether your matched participant has chosen to Pay the Fee and Enter the Contest or has chosen Don’t Enter. You also don’t know the bid that your match makes if they do choose to enter the contest until after the round is over.

1.4 Decision outcomes

There are several possible outcomes for each decision round (contest):

• 1. Both you and the other participant chose to pay the fee and enter the contest. In this case you each give up the entry fee of $f_k$ points. Your probability of winning the contest is calculated as:

  $$\begin{aligned} \hbox {Your Probability of Winning} = \frac{\hbox {Your Bid}}{\hbox {Your Bid} + \hbox {Other Participant's Bid}} \end{aligned}$$

  The other participant’s probability of winning is calculated in the same manner and is equal to 1-your probability of winning.³ Using these two probabilities, the computer program determines the winner in a manner such that the participant with the higher (lower) probability of winning is more (less) likely, though not certain to win the contest. For example, suppose in a round that you are participant 1 and based on the points bid, your probability of winning is .60 (60%) and your match (participant 2) has a probability of winning equal to $1-.60 = .40$ (40%). In this case the computer program draws a number randomly from the interval [1, 100]. If the number drawn is 60 or less, than you are declared the contest winner, while if the number drawn is greater than 60, then the other player is declared the contest winner. If you are the winner, then your payoff in points for the round is

  $$\begin{aligned} 120 - \hbox {Your bid} + 100 \end{aligned}$$

  If you are not the winner, then your payoff in points for the round is:

  $$\begin{aligned} 120 - \hbox {Your bid} \end{aligned}$$

  These payoff consequences are symmetric for the other participant in the contest.

• 2. You enter the contest but the other participant does not enter. In this case, you automatically win the contest with any bid that you make but, of course, you don’t know in advance whether the other participant entered the contest or not. Your payoff in points for the round in this case is:

  $$\begin{aligned} 120 - \hbox {Your bid} + 100 \end{aligned}$$

  The other participant earns $120 +f_k$ points for the round where $f_k$ is the contest entry fee.

• 3. You do not enter the contest and the other participant does. In this case the other participant automatically wins the contest with any bid. Your payoff in points for the round in this case is:

  $$\begin{aligned} 120 + f_k \end{aligned}$$

  where $f_k$ is again the contest entry fee in points. The other participant’s payoff in points for the round in this case is 120 - Other Participant’s Bid + 100.

• 4. You and the other participant both Don’t Enter the contest. In this case, there is no winner of the contest. The points earned by both you and the other participant for the round in this case are:

  $$\begin{aligned} 120 + f_k \end{aligned}$$

  where $f_k$ is again the contest entry fee in points.

1.5 Feedback

At the end of each round, you learn what the other participant chose to do and the outcome of the round. If one or both of you chose to enter the contest, then you will learn what was bid by each participant (1 and 2) and your probability and/or the other participant’s probability of winning. You will learn who (if anyone) won the prize of 100 points for that round. Finally, you will see your total earnings in points for the round which you can write down. When you have viewed this information click the Next button.

1.6 Earnings

Following completion of all 6 rounds, the computer program will choose one of the six decision-rounds randomly. All six rounds have an equal chance of being chosen. Your points from the one chosen round will be converted into dollars at the exchange rate of 1 point = 10 cents ($0.10 USD).

1.7 Questionnaire

To finish the study, you must complete an online questionnaire. Following completion of the questionnaire, you will be awarded your earnings from the experiment plus your $7 show-up payment on a final screen that also shows your unique subject ID number. Please leave this screen open for verification purposes.

1.8 Questions?

Now is the time for questions. If you have a question, please raise your hand.

1.9 Comprehension quiz

The following questions are intended to check your understanding of the instructions. Please answer all parts of all 6 questions. If you make a mistake you will be asked to re-do your answer until you get it right.

• 1. Circle One: True or False: I will be matched with the same other participant in all 6 rounds.

• 2. Suppose in round k, the contest entry fee, $f_k =30$ points.

  • a. How many points will you earn for the round if you do not enter the contest?

  • b. If you do enter the contest, how many points can you bid toward winning the prize?

  • c. Suppose you enter the contest, you bid 20 points but you do not win the prize. What are your earnings in points for the round?

  • d. Suppose you enter the contest, you bid 20 points and you do win the prize. What are your earnings in points for the round?

• 3. Suppose in round k, the contest entry fee $f_k =80$ points.

  • a. How many points will you earn for the round if you do not enter the contest?

  • b. If you do enter the contest, how many points can you bid toward winning the prize?

  • c. Suppose you enter the contest, you bid 50 points but you do not win the prize. What are your earnings in points for the round?

  • d. Suppose you enter the contest, you bid 50 points and you do win the prize. What are your earnings in points for the round?

• 4. Circle One: True or False: Paying the fee and entering a contest will always result in more points earned in a round than choosing not to enter the contest.

• 5. Circle the correct answer. If both participants enter a contest and both make positive bids then:

  • a. Whoever bids the most is guaranteed to win the prize.

  • b. Whoever bids the most has a greater chance of winning but either participant can win.

  • c. The probability that each participant wins the prize can never be the same.

• 6. Circle One: True or False: After playing 6 decision rounds, one round will be randomly chosen and the points you earned in that round will be converted into money earnings a the rate of 1 point = $0.10 (10 cents).

1.10 C Screenshots

See Figs. 8, 9, 10, 11 and 12.

Fig. 8

First entry choice screen in round

Fig. 9

Screen with bid slider if entered contest

Fig. 10

Contest results screen, participant 1

Fig. 11

Contest results screen, participant 2

Fig. 12

Final payoff screen

1.11 D CRT and demographic questions

1.12 CRT questions

The CRT questions we used differ from Frederick (2005) (which are already well known) and are taken from Toplak et al. (2014)

• 1. The ages of Anna and Barbara add up to 30 years. Anna is 20 years older than Barbara. How old is Barbara? [Correct Answer 5; Intuitive Wrong Answer 10]

• 2. If it takes 2 nurses 2 minutes to check 2 patients, how many minutes does it take 40 nurses to check 40 patients? [Correct Answer 2; Intuitive Wrong Answer 40]

• 3. On a loaf of bread, there is a patch of mold. Every day, the patch doubles in size. If it takes 24 days for the patch to cover the entire loaf of bread, how many days would it take for the patch to cover half of the loaf of bread? [Correct Answer 23; Intuitive Wrong Answer 12]

• 4. If John can drink one barrel of water in 6 days, and Mary can drink one barrel of water in 12 days, how many days would it take them to drink one barrel of water together? [Correct Answer 4; Intuitive Wrong Answer 8]

1.13 Demographic questions

• 1. What is your age?

• 2. What is your gender? Choices: Male, Female, Non-binary

• 3. What is your university major?

• 4. What is your grade point average (GPA)?

• 5. In general, how willing are you to take risks? Please use a scale from 0 to 10, where 0 means you are “completely unwilling to take risks” and a 10 means you are “very willing to take risks.”

1.14 E Additional tables

See Tables 8, 9 and 10.

Table 8 Random effects regression analysis of contest participation and bids, all data, comparison of specifications involving different cut-off values for CRT scores and risk tolerance

Table 9 Random effects regression analysis of contest participation and bids, $n=2$ treatment comparison of specifications involving different cut-off values for CRT scores and risk tolerance

Table 10 Random effects regression analysis of contest participation and bids, $n=3$ treatment comparison of specifications involving different cut-off values for CRT scores and risk tolerance