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The ISL guessing game

The ISL Guessing Game is an optional activity to all MOP students who haven't seen the IMO shortlist. It's a chance to try to guess the shortlist numbers for problems on the tests. The best guesser gets bragging rights and maybe a prize at the talent show.

Overview

Task description

For each problem, you submit a \(10\)-tuple \((a_1, a_2, \dots, a_9, a_0)\) of nonnegative integers such that \(p_i := \frac{a_i}{a_1 + a_2 + \dots + a_0}\) represents the probability you think the problem is numbered \(i\), with \(p_0\) denoting the probability you think the problem is not from the shortlist. For technical reasons, we require \(0 \le a_i < 10^6\).

The scoring function is the KL loss function.1 For each \(i\), you then score \(s_i := 1 + \log p\) where \(p \in \{p_i, 1-p_i\}\) is the probability that you assigned to the truth and \(\log\) is natural logarithm. This number may be negative. If \(p=0\), we let \(s_i = -15\) rather than \(s_i = -\infty\).

Thus, you may earn up to \(10\) points per problem. Your overall score for the event is the sum across all problems across all tests.

Example

Suppose you saw a geometry problem and you think it's 45% likely to be G2, 20% likely to be G3, and 35% likely to not be from the shortlist. Then you could submit the \(10\)-tuple \((0, 45, 20, 0, 0, 0, 0, 0, 0, 35)\).

If it turns out the problem is indeed G2, then you earn

\[ \begin{aligned} s_1 = s_4 = s_5 = s_6 = s_7 = s_8 = s_9 &= 1 + \log(1) = 1 \\ s_2 &= 1 + \log(0.45) \approx 0.201 \\ s_3 &= 1 + \log(0.80) \approx 0.776 \\ s_0 &= 1 + \log(0.65) \approx 0.569 \end{aligned} \]

So your total score for the problem is \(10 + \log(0.45) + \log(0.8) + \log(0.65) \approx 8.548\).

Detailed rules

  1. We'll run for MOP tests 1-3 and mock IMO; in the former case only the problems V, W, X, Y, Z are considered. We'll announce which problems correspond to which codes after each test.

    You can submit for any problem, even if it didn't appear on the test you took. So you'll want to look at your friends' tests to inform your strategy.

  2. Shortlist problems are often edited for MOP tests, and multiple versions (easier, harder, or simply different than the original) could appear. This does not affect the scoring, e.g. "nerfed A3" still counts as A3 for scoring purposes.

  3. You should submit via the link on Forms. The link is the same for every round.

  4. Each submission must be a space-separated list of \(10\) nonnegative integers. For instance, the earlier example tuple should be encoded as:

    45 20 0 0 0 0 0 0 35

  5. You need to submit between before test review to count. If you submit multiple times, only the first submission is counted.

  6. Blank submissions, or submissions with \(a_i=0\) for all \(i\), are treated as uniform submissions with \(p_i = \frac{1}{10}\) for all \(i\).

  7. The choice of subject (A/C/G/N) doesn't affect scoring, and is intended to be hidden until after the submission deadline. This is part of the game (e.g. if you aren't sure whether a problem is A or N, this adds some uncertainty).

  8. Obviously, you shouldn't discuss your guesses with anyone who already knows what was on the IMO shortlist like staff or international guests. You may discuss your guesses with other participants in the guessing game, but ultimately each person makes their own submission.

Strategy and advice

  • In general, be wary of placing any guesses under 5%, and super wary of anything less than 1%. The penalty for being confidently wrong is enormous.

  • Keep item 2 of the rules above in mind! In general don't be overconfident.

  • Again: don't be overconfident. I can't stress this enough.

  • Seriously, don't be overconfident.

  1. KL loss function has the property that the expected value of your score is maximized if you honestly report your beliefs. It's up to you whether expected value is the metric you want to optimize; for example, if you only care about winning 1st place, you may prefer to guess more aggressively. ↩