A short story illustrating how game theory can resolve the well‐known paradox of the Unexpected Exam.
The Unexpected Deal
Henry Hawkins the history teacher made an unexpected announcement to his high‐school class one Thursday. “Some morning next week,” he said, “I will give you a history test. For all you know, it may be on Monday, Tuesday, Wednesday, Thursday or Friday. But it will be unexpected. I won’t tell you which morning it will be on until the day of the test.”
“Aha!” trumpeted William the whiz kid. “The Unexpected Exam. Alias the Unexpected Tiger.” The class groaned. “Alias the Unexpected Blackout, the Unexpected Egg, the Unexpected Hanging, the Unexpected Spade. What do you mean by unexpected: merely that we won’t know for certain, or that we won’t be able to predict with more than 50% confidence?”
“You won’t be able to predict with more than 50% confidence” answered Henry (with more than 50% confidence).
“And when you say it will be unexpected, is that merely what you believe, or is it a guarantee?”
“A guarantee,” replied Henry, unwisely. “A one‐hundred‐percent, cast‐iron guarantee”.
“Then you won’t mind a bet on it, even if the odds are long, say a hundred to one?” asked William. The class were more interested now. William was well known for his crazy bets. Crazy, except that he seemed to have an uncanny instinct in the matter, and any of his classmates would have known better than to wager with him. “I bet you £10, at odds of ninety‐nine to one, that when I arrive at school on the morning of the test, I will be expecting it. Here is my £10 now. Take it. And if, when I arrive at school on the morning of the test, I am expecting it, then you give me £1,000, which includes my £10 stake.”
“Not so fast,” said Henry, a little more wisely this time. “I wasn’t born yesterday. If I give you the test on, say, Tuesday, and you tell me you were expecting it, then I’m not exactly going to believe you, am I?”
At this point Mr Martin the Mathematics man made an unexpected entrance. He was fascinated by this mental duel. “I am willing to act as referee,” he volunteered. “Mr Hawkins, you can tell me tomorrow, Friday, on what day you intend to give the test, or how you will decide on what day to give it, and why you think William will be unable to predict it. And William, each morning when you arrive at school, if you expect the test on that day then you tell me so, and tell me why you think you can rationally predict it with more than 50% confidence. I’ll judge whether your reasoning is sound. What is to happen if the test is cancelled?”
“If I were to cancel it purely to avoid paying up, that would be cheating,” volunteered Henry.
“Very noble. So I propose that if Mr Hawkins does not give the test during the week, whether the circumstances are within or beyond his control, then he must give William whatever sum, from zero to £1,000, I find to be fair and reasonable in the circumstances. Mr Hawkins, William, if the bet is accepted will you trust me and will you both accept my decision as final?”
There was a brief pause before Henry and William said “Yes” simultaneously.
“Done,” said Henry, as he pocketed the tenner. He had got himself into this, and he did not intend to lose face now. Meanwhile, for his part, William smiled inwardly at this unexpected bargain.
Back home, Henry was confident. “Obviously,” he reasoned, “I can’t give the test on Friday, or William will expect it. So William knows it won’t be on Friday. If I leave it until Thursday, then William will expect it, knowing that it can’t be on Friday. So I can’t have it on Thursday. And William is clever, so he knows that too.” Henry was no logician: he was blind to the continuation of that line of reasoning that appears to rule out Wednesday, Tuesday and Monday in turn, and to the logical minefield that ensues. But as he worked his way back through the days of the week it seemed to him that the earlier in the week he gave the test, the safer was his money. “So,” he concluded, “I will give the test on Monday. There is no way that William can predict that with more than 50% confidence.”
Suddenly he had doubts. “William is clever. He may anticipate my reasoning. If I tell Mr Martin that I have chosen Monday for the test because William can’t rationally predict it so early in the week, and William tells Mr Martin that he expects the test on Monday for that very reason, then Mr Martin will surely adjudicate in his favour.”
“Ok, I won’t risk that. I’ll give the test on Tuesday.
“Except that William is not just clever, he is very clever. William may well anticipate that reasoning too.
Slowly it dawned on Henry that whatever day he chooses and for whatever reason, William may outwit him, correctly guessing not only the day, but also the reason. And whatever day he chooses and for whatever reason, if William gives the same reasoning to Mr Martin then suddenly Henry is £1,000 poorer. From Henry’s side, it didn’t now seem such a clever bet after all.
Henry needed advice, and fortunately he was on friendly terms with Professor Banach‐Tarski of the Mathematics Department of the local university. The two met over a coffee, and Henry explained his unexpected problem.
“You seem to be looking for a strategy that this young whippersnapper won’t guess,” suggested the Professor. “And you’re not confident of winning that psychological battle?”
“Not at all confident,” confessed Henry. “I never was any good at poker, and this is too much like poker for my liking.”
“Then you must randomise,” explained the Professor. “What you have here is a classic piece of game theory. This is a two‐person zero‐sum game. You’re quite right about the earlier days of the week being safer. But you mustn’t choose a day heuristically, or you risk being outwitted. You must adopt what game theorists call a mixed strategy. You must choose Monday with a probability of one half, Tuesday with a probability of one quarter, Wednesday with a probability of one eighth, Thursday and Friday each with a probability of one sixteenth. The easiest way to achieve that is to toss a coin each day. On Monday if the coin comes up heads, give the test on Monday. Otherwise, on Tuesday if the coin comes up heads, give the test on Tuesday. And so on through the week. If you reach Thursday you must pray that the coin comes up heads. If it comes up tails, give the test on Friday.”
“And lose £1,000!” quivered Henry. “Surely if I reach Thursday I should give the test on Thursday: leaving it until Friday is admitting defeat.”
“No,” insisted the Professor. “You must hold your nerve. Remember you have agreed to declare your strategy to the referee. The referee will want to witness the coin tossing and you can hardly refuse. If you abandon your randomising on Thursday, then you risk William saying to the referee that he expected you to do so. Stick to the mixed strategy. Declare your strategy in advance to William in the presence of the referee. Then on each day from Monday to Thursday, if the class are still waiting for the test William knows there is exactly a 50% chance that it will take place on that day, and if he says he expects it on that day with more than 50% confidence then his expectation is irrational and will be recognised as irrational by the referee.”
“So I have a one‐in‐…” – his mental wheels ground slowly – “a one‐in‐sixteen chance of losing my £1,000?”
“That’s right. But it’s the best you can do. You accepted a bet of £10 for this? You were unwise. A fair bet would have been one sixteenth of £1,000, or £62.50.”
“Thank you.” And a resigned Henry wandered home. On the following morning, Friday, and in a very subdued manner, he declared his strategy to Mr Martin the referee, who nodded with understanding. He could not face telling William.
William arrived bright and early at the school on Monday, and made an unexpected encounter with Henry who met him at the gate. “About this bet, William. I’m not as confident as I was. If I give you, say, £40 now, can we call the bet off?”
“Sixty‐two pounds fifty,” shot back William.
They smiled, they shook hands, and the unexpected deal was done.