Negative marking and guessing: the expected value of an answer

What the rules say, and what they do not

The scoring rules are Bocconi's: $+1$ for a right answer, $0$ for a blank, $-0.2$ for a wrong answer, and $-0.33$ for a wrong answer on a critical thinking question with only three options. Bocconi gives no advice on whether to guess. Everything below is our analysis of those rules, using one tool from probability: expected value.

Expected value

The expected value of a decision is the average number of points it would earn if you could repeat it many times:

$$E = P(\text{right}) \times (+1) + P(\text{wrong}) \times (\text{penalty}).$$

A blank always earns $0$, so an answer is worth giving, on average, whenever its expected value is above $0$.

A blind guess on a four-option question

If you pick one of four options completely at random, $P(\text{right}) = \frac{1}{4}$ and $P(\text{wrong}) = \frac{3}{4}$:

$$E = 0.25(1) + 0.75(-0.2) = 0.25 - 0.15 = +0.1.$$

The penalty does not cancel the reward. One right answer gains as much as five wrong answers lose ($5 \times 0.2 = 1$), but a blind guess is wrong only three times out of four, not five times out of six. On average, a blind guess adds a tenth of a point.

A blind guess on a three-option critical thinking question

With three options, $P(\text{right}) = \frac{1}{3}$ and $P(\text{wrong}) = \frac{2}{3}$, and the penalty is $-0.33$:

$$E = \tfrac{1}{3}(1) + \tfrac{2}{3}(-0.33) = 0.3333\ldots - 0.22 \approx +0.113.$$

The larger penalty is matched by the better odds of a three-way guess, and the expected value stays slightly positive.

Elimination raises the value

Each option you can rule out with confidence raises $P(\text{right})$, and the expected value climbs quickly:

Situation	Chance of being right	Expected points
Four options, none ruled out	$\frac{1}{4}$	$+0.10$
Four options, one ruled out	$\frac{1}{3}$	$+0.20$
Four options, two ruled out	$\frac{1}{2}$	$+0.40$
Three options, none ruled out	$\frac{1}{3}$	about $+0.113$
Three options, one ruled out	$\frac{1}{2}$	$+0.335$

For example, with one of four options ruled out: $\frac{1}{3}(1) + \frac{2}{3}(-0.2) = \frac{1}{3} - \frac{0.4}{3} = \frac{0.6}{3} = 0.2$.

The break-even probability

Write $p$ for your chance of being right. On a four-option question the expected value is $p - 0.2(1 - p) = 1.2p - 0.2$, which is positive when

$$p > \frac{0.2}{1.2} = \frac{1}{6} \approx 0.167.$$

On a three-option question it is $p - 0.33(1 - p) = 1.33p - 0.33$, positive when $p > \frac{0.33}{1.33} \approx 0.248$.

A blind guess ($\frac{1}{4}$ or $\frac{1}{3}$) is above both thresholds. So, as far as expected value goes, answering beats leaving blank unless your chance of being right is below the threshold. That can happen: if a question is built so that a tempting wrong option attracts you, and you are confident in that wrong reasoning, your real chance can be below a random guess. Expected value assumes your probability is honest.

Expected value is an average, not a promise

On one question, a guess earns either $+1$ or a penalty, never $+0.1$. The average shows up only across many questions. Guess blindly on five four-option questions and the expected total is $5 \times 0.1 = +0.5$, but the actual result can be anywhere from $-1$ (all wrong) to $+5$ (all right), and the chance that all five are wrong is $\left(\frac{3}{4}\right)^5 \approx 0.237$, almost one in four.

So the honest summary is: guessing raises your average total, slightly when blind and clearly after elimination, and it also widens the spread of results. Whether you accept that spread is your decision; Bocconi does not tell you either way.

Why this matters more on this test

You cannot come back to a screen. A question left blank "to return to" is a blank for good, so the decision to answer or not has to be made while the screen is open, with the numbers above in mind. A few seconds spent ruling out one option moves a four-option guess from $+0.1$ to $+0.2$, and a second ruled-out option doubles it again.