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    Payoff Information
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<style>
table, th, td {
    border: 1px solid black;
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<p>
    The chances to make up to $4.50 extra in this section are best if the actual belief that the Yellow box is selected
    and the actual belief about the other participant's beliefs on the same event are reported.
</p>
<p>
    For each possible sample, your report determines the chances of getting $1.50.
    This applies both to the belief and the estimate of the other person’s belief of the Yellow Box being selected.
    Each participant will be making decisions in three independent tasks, and therefore you can make up to $1.50 x 3 = $4.5.
</p>
<p>
    A detailed explanation of how you increase your chances of making up to $4.50 extra by reporting your true beliefs will be shown when you click the 'View Payoff Details' button.
    These payoffs were designed such that
    reporting your beliefs truthfully
    maximizes your chances of getting more money.
    Feel free to read the details of the payment method.
</p>


<center>
<!-- payoffs box button -->
<button type="button"
        onclick="detailsNext()"
        style="width: 200px; height: 40px;
            background-color: darkgrey;
            border-color:black">
    View Payoff Details
</button>
</center>

<script>
  function detailsNext(){
    if (document.getElementById("displaytable").style.display === "none")
        document.getElementById("displaytable").style.display="block";
    else
        document.getElementById("displaytable").style.display="none";
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<div id="displaytable" style="display:none">
<h3>Payoff Calculations Detailed</h3>
    <p>
        In every task, the computer will select a Box, generate a sample of 4 marbles, and implement your guess about the probability of the Yellow Box selected (<strong>p</strong>) to calculate your payoffs.
        You will be randomly paired with another participant in this session.
        An independent Box and sample will be selected for <strong>one participant</strong>, who chooses the strategy <strong>q</strong>.
        Remember <strong>another participant</strong> also have to report their guess about <strong>q</strong>, and let's call it <strong>r</strong>.
        These two guesses (<strong>p</strong> and <strong>r</strong>) determine the probability of making up to $1.50 in each task for each participant:
    </p>
    <p>
        Next we describe the payoffs for each participant:
    </p>

    <ol>
        <li>
        <p>
            (For <strong>Participant 2</strong>) Guess on the box selected by the computer: <strong>p</strong> (your report divided by 100)
        </p><p>
        For every possible sample, <strong>Participant 2</strong> will indicate a probability <strong>p</strong> between 0 and 1 that <strong>Participant 2</strong> believe the Yellow box was selected.
            If the computer selects the Yellow Box, <strong>Participant 2</strong> will make $1.50 with probability (1 - (<strong>1-p</strong>)<sup>2</sup>).
            If the computer selects the Green Box, <strong>Participant 2</strong> will make $1.50 with probability (1- <strong>p</strong><sup>2</sup>). This is also shown below:
        </p><p>
        <center>
            <table>
<thead>
  <tr>
    <th>Selected Box</th>
    <th>Probability of winning</th>
  </tr>
</thead>
<tbody>
  <tr>
    <td>Yellow Box</td>
    <td>1 - <strong>(1-p)</strong><sup>2</sup></td>
  </tr>
  <tr>
    <td>Green Box</td>
    <td>1- <strong>p</strong><sup>2</sup></td>
  </tr>
</tbody>
</table>
        </center>
        </p><p>
            Here are some examples of how the payment works:
        <ul>
            <li>
                If <strong>Participant 2</strong> think that the Yellow box was selected for sure, choosing <strong>p=1</strong> gives the best chance of making $1.50.
                This happens because the probability of winning would be 1 - (1-<strong>1</strong>)<sup>2</sup> = 1, or 100%.
                Choosing any other number will decrease <strong>Participant 2</strong>'s chances of winning.
                For example, if <strong>Participant 2</strong> chose <strong>p = 0.5</strong>, the probability of winning would be 1 - (1-<strong>0.5</strong>)<sup>2</sup> = 1 – (0.5)<sup>2</sup> = 1 - 0.25 = 0.75, or 75%;
                if <strong>Participant 2</strong> chose <strong>p = 0</strong>, the probability of winning would be 1 - (1-<strong>0</strong>)<sup>2</sup> = 1 – 1 = 0, or 0%.
            </li>
            <li>
                Similarly, if <strong>Participant 2</strong> thinks that the Green box was selected for sure, choosing <strong>p=0</strong> gives the best chance of winning;
                the probability of winning would be 1 - (<strong>0</strong>)<sup>2</sup> = 1, or 100%.
                If <strong>Participant 2</strong> chose <strong>p = 0.5</strong>, the probability of winning would be 1 - (<strong>0.5</strong>)<sup>2</sup> = 1 – 0.25 = 0.75, or 75%;
                if <strong>Participant 2</strong> chose <strong>p = 1</strong>, the probability of winning would be 1 - (<strong>1</strong>)<sup>2</sup> = 1 – 1 = 0, or 0%.
            </li>
            <li>
                If <strong>Participant 2</strong> believes that the Yellow box was selected with a 50% chance, choosing <strong>p=0.5</strong> gives the best chance of making $1.50;
                the chance of winning would be the average between the probabilities of winning given each box: 0.50*(1 - (1-<strong>0.5</strong>)<sup>2</sup>) + 0.50*(1- (<strong>0.5</strong>)<sup>2</sup>) = 0.50*(1 - 0.5<sup>2</sup>) + 0.50*(1 - 0.5<sup>2</sup>) = 0.50*0.75 + 0.50*0.75 = 0.75, or 75%.
                Choosing any other number will decrease <strong>Participant 2</strong>'s chances.
                For example, if <strong>Participant 2</strong> chose <strong>p = 1</strong>, the probability of winning would be: 0.50*(1 - (1-<strong>1</strong>)<sup>2</sup>) + 0.50*(1- (<strong>1</strong>)<sup>2</sup>) = 0.50*(1) + 0.50*(0) = 0.50, or 50%.
                Similarly, if <strong>Participant 2</strong> chose <strong>p = 0</strong>, the probability of winning would be 0.50*(0) + 0.50*(1) = 0.5, or 50%.

            </li>
        </ul>
            Any number different from <strong>Participant 2</strong>'s belief that Yellow Box was selected  will decrease your chances of winning.
        </p>
        </li>
        <li>
        <p>
        (For <strong>you</strong>) Guess on Another Participants’s guess: <strong>r</strong> (your report divided by 100)
        </p><p>
        <strong>Participant 2</strong> will bet on
            the probability that a Yellow box was selected for them by reporting the number <strong>p</strong>.
            You will indicate a probability <strong>r</strong> between 0 and 1 that is your guess about <strong>p</strong>.

        You will make $1.5 with probability:
            <center>
            (1- (<strong>p</strong>-<strong>r</strong>)<sup>2</sup>)
            </center>
        </p><p>
            You would wish to choose <strong>r</strong> to be as close as possible to <strong>p</strong> (<strong>Participant 2</strong>'s guess on the box selected by the computer).
            The other participant will have the same payment scheme as explained above.

            Here are some examples of how the payment works:
        <ul>
            <li>
                If you think that the other participant guessed the Yellow box was selected for sure(<strong>p=1</strong>), choosing <strong>r=1</strong> maximizes your chances of making $1.50.
                The probability of winning would be 1 - (1-<strong>1</strong>)<sup>2</sup> = 1, or 100%.
                Choosing any other number will decrease your chances of winning.
                For example, if you chose <strong>r = 0.5</strong>, the probability of winning would be 1 - (1-<strong>0.5</strong>)<sup>2</sup> = 1 - 0.5<sup>2</sup> = 0.75, or 75%;
                if you chose <strong>r = 0</strong>, the probability of winning would be 1 – (1-<strong>0</strong>)<sup>2</sup> = 0, or 0%.
            </li>
            <li>
                If you think that the other participant guessed the Yellow box was selected with a 50% probability (<strong>p=0.5</strong>), choosing <strong>r=0.5</strong> maximizes your chances of making $1.50.
                The probability of winning would be 1 - (0.5- <strong>0.5</strong> )<sup>2</sup> = 1 – 0 = 1, or 100%.
                Choosing any other number will decrease your chances.
                For example, if you chose <strong>r = 1</strong>, the probability of winning would be (1 - (0.5-<strong>1</strong>)<sup>2</sup>) = 1 - 0.25 = 0.75, or 75%;
                if you chose <strong>r = 0</strong>, the probability of winning would be (1 - (0.5-0)<sup>2</sup>) = 1 – 0.25 = 75%.
            </li>
        </ul>
            Any number different from your belief about the other participant's strategy will decrease your chances of winning.
        </p>
        </li>
    </ol>

    <p>
These payoffs were designed such that
    reporting your beliefs truthfully
    maximizes your chances of getting up to $4.50 in this section.
    Be as accurate as possible,
    and carefully consider the probability of the Yellow Box selected given each sample,
        and what <strong>the other participant</strong> would choose.
</p>


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