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    Payoff Information
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<style>
table, th, td {
    border: 1px solid black;
    border-collapse: collapse;
    text-align: center;
    vertical-align: middle;
}
</style>

<p>
    Your chances to make up to $9 extra in this section are best if you report your actual belief that the Yellow box is selected and your actual belief about Participant 2's beliefs on the same event.
</p>
<p>
    For each possible sample, your report determines the chances of getting $1.50.
    This applies both to your own estimate and your estimate of the other person’s estimate.
    You will be making decisions in three independent tasks, and therefore you can make up to $3 x 3 = $9.
</p>
<p>
    A detailed explanation of how you increase your chances of making up to $9 extra by reporting your true beliefs will be shown when you click the 'View Payoff Details' button.
    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";
  }
</script>


<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.
        An independent Box and sample will be selected for <strong>Participant 2</strong>, who chooses the strategy <strong>q</strong>.
        Remember you also have to report your guess about <strong>q</strong>, and let's call it <strong>r</strong>.
        These two guesses you make (<strong>p</strong> and <strong>r</strong>) determine the probability of making up to $3 in each task:
    </p>

    <ol>
        <li>
        <p>
            Guess on the box selected by the computer: <strong>p</strong> (your report divided by 100)
        </p><p>
        For every possible sample, indicate a probability <strong>p</strong> between 0 and 1 that you believe the Yellow box was selected.
            If the computer selects the Yellow Box, you will make $1.50 with probability (1 - (<strong>1-p</strong>)<sup>2</sup>).
            If the computer selects the Green Box, you 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 you think that the Yellow box was selected for sure, choosing <strong>p=1</strong> gives your 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 your chances of winning.
                For example, if you 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 you 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 you think that the Green box was selected for sure, choosing <strong>p=0</strong> gives your best chance of winning;
                the probability of winning would be 1 - (<strong>0</strong>)<sup>2</sup> = 1, or 100%.
                If you 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 you 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 you believe that the Yellow box was selected with a 50% chance, choosing <strong>p=0.5</strong> gives your 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 your chances.
                For example, if you 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 you 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 your belief that Yellow Box was selected  will decrease your chances of winning.
        </p>
        </li>
        <li>
        <p>
        Guess on Participant 2’s guess: <strong>r</strong> (your report divided by 100)
        </p><p>
        <strong>Participant 2</strong> will also bet on
            the probability that a Yellow box was selected for them by reporting the number <strong>q</strong>.
            You will indicate a probability <strong>r</strong> between 0 and 1 that is your guess about <strong>q</strong>.

        You will make $1.5 with probability:
            <center>
            (1- (<strong>q</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>q</strong>.
            Participant 2 will have the same payment scheme as you do in your first guess above.

            Here are some examples of how the payment works:
        <ul>
            <li>
                If you think that Participant 2 guessed the Yellow box was selected for sure(<strong>q=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 Participant 2 guessed the Yellow box was selected with a 50% probability (<strong>q=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 Participant 2's strategy will decrease your chances of winning.
        </p>
        </li>
    </ol>

    <p>
These bets were designed such that
    reporting your beliefs truthfully
    maximizes your chances of getting up to $9 in this section.
    Be as accurate as possible,
    and carefully consider the probability of the Yellow Box selected given each sample,
        and what <strong>Participant 2</strong> would choose.
</p>


<p>

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</p>

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