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Do not leave games to chance
Research

Do not leave games to chance

A hint: If you want to win a game of rock-paper-scissors, try to think like a self-driving car… How does that work?

7 minutes
24 November 2020

The stakes of a game of rock-paper-scissors can be surprisingly high. So it is better to know how to turn it to your own advantage…

The head of a Japanese electronics giant put part of his valuable collection of impressionist art up for auction, but could not decide between two big auction houses. He decided that a game of rock-paper-scissors would resolve the matter. One auction house wrote ‘paper’ on a piece of paper, without any particular strategy. The other researched the psychology of the game and came to the conclusion that ‘scissors’ is the most promising opening move against a new opponent. And won.

Mathematical engineer Giovanni Samaey tells us the anecdote to illustrate that purported games of chance are not always a question of luck. Or at least not if humans – in this case literally – have a hand in them. A Chinese scientist discovered that after winning a round of rock-paper-scissors, his students tended to repeat the winning symbol. Those who lost, on the other hand, tended to move to a different symbol in the series.

Patterns on the playground

Giovanni Samaey recommends trying to identify patterns in series yourself. For example, you might note that your opponent – either consciously or not – always alternates: stone-paper-scissors-stone-paper-scissors … Or that he always repeats a winning move, like in the Chinese research. Or perhaps just does the opposite.

As soon as you recognize the patterns, you can start exploiting them.

“You have to play several times to discover this,” Samaey says. “But as soon as you recognize the patterns, you can start exploiting them. I recently played rock-paper-scissors against my eleven-year-old daughter. During the first games, I still had to identify her strategy. She had a 3-0 lead when I thought I saw a pattern. Eventually, I won 5-4. It was more difficult than I expected, with a lot of draws. She often plays the game on the playground and had already discovered that you can find patterns: If this person does the same thing twice, they will do something else the third time.”

In other words, you try to make a prediction on the basis of what you know from past experience and the present situation. That is also how artificial intelligence works, Samaey says. “Take a self-driving car, for example. If it has to decide whether a pedestrian will cross the road or not, it looks at its memory. ‘I will slow down just in case, because in ninety percent of all the comparable situations I know, the pedestrian crossed.’ That is the same principle as: ‘My opponent will probably choose ‘scissors’ because he did that in almost all the previous sequences that were similar to this one.’”

Help from a clock

What if you realize that your opponent has also figured out your strategy and is winning more games than you? When that happens, it is better to leave the choice between paper, rock and scissors up to chance, to confuse your opponent. But that is easier said than done: people are not able to choose completely at random, even if they try to.

This is also evident from experiments in which, for example, a teacher asks his pupils to toss a coin 200 times at home. They must note down whether it was heads or tails each time, ‘h’ or ‘t’. When the teacher corrects the homework, he will immediately see who really did the task. Pupils who simply write down a series of h’s or t’s without actually tossing the coin will rarely write the same letter three times in a row, because it does not look random enough. But actually, in series of 200, it is more likely than not to occur several times.

But back to our game: how can we let chance decide? Giovanni Samaey has a tip: “You can glance at your watch. If the second hand is between 12 and 4, choose ‘paper’; if it is between 4 and 8, choose ‘paper’; and if it is in the last third of the clockface, opt for ‘scissors’. If you want to make a genuinely random choice, then you must use these kinds of external aids.”

Stone in the middle

Mathematicians take little pleasure in games that depend entirely on chance. Think of the goose game, for example, in which dice determine your progress and thus also whether you win or lose. It is more fun to unleash some maths on the game rather than playing it, Samaey says. For example, there is a 23% chance of a draw if there are two players: one ends up in the well and the other in prison. Or you can calculate each player’s chances of winning, if you know the positions of each player. But human choices or strategies play no role in the goose game. “Terrible game,” Samaey concludes.

It is more fun to unleash some maths on the goose game rather than playing it.

He says he has the same opinion of Connect 4. This is not surprising because that is an example of a so-called ‘solved game’. The player makes choices, but for each situation, you can calculate what to do to win. And you don’t even have to do it yourself because there are apps that will tell you. The player who starts can always win if he drops his first disc into the middle hole. “As soon as I knew that, I was no longer interested,” Samaey says. “That is typical of mathematicians: if it has been demonstrated that a solution exists, we sometimes don’t even care to find it (laughs).”

Chess up a tree

Chess is a different matter entirely. By comparison: a traditional Connect 4 rack of six by seven holes can be filled with discs 4,531,985,219,092 different ways, the number of possible chess games is estimated to be approximately 10120. “That is far more than the number of atoms in the universe,” Samaey says. “From each position, the number of possible moves is enormous, and the multitude of situations that result from it in the following moves is gigantic. If you want to think five moves ahead, you have to map out a whole tree of possible situations in your head. Computer programmes that analyse these things have to make decisions at lightning speed: which branches of the tree can be ignored because they are no longer interesting?”

And chess is not even the toughest nut for a computer to crack: “The boardgame Go has even more possible branches. But a computer programme has now also been developed – AlphaGo – that can beat a human.”

If you want to think five moves ahead in a game of chess, you have to map out a whole tree of possible situations in your head.
If you want to think five moves ahead in a game of chess, you have to map out a whole tree of possible situations in your head.

Daring plays a role

A good game has to be balanced, Samaey says. A beginner has to have a chance, but you also want players to get better the more they play. Are mathematicians not already at an advantage thanks to their strategic insight? “To win, you often have to take a risk that either pays off or leads to a heavy loss. If you play in an excessively calculated way, with a meticulous strategy, you can avoid those risks. That is why I seldom end up first or last, but usually come second or something.”

In everyday life, we often apply elements from game theory without realizing it. “Imagine, your company organizes an event with a whole series of tasks that have to be divided. You have to think about it carefully: ‘Do I accept the first task, or do I wait until something comes along that I really like?’ The risk is that you ultimately end up with a task that is worse than the first, but also the possibility that you don’t have to do anything at all. These are situations that can be analysed using game theory. But of course it is healthier simply to consult with your colleagues without turning everything into a tactical game.”

This article is partly based on the chapter ‘Van ganzenbord tot Machiavelli: wanneer speltheorie ernst wordt’ from the book X-factor, 20 verhalen over de onzichtbare kracht van wiskunde by Giovanni Samaey and Joos Vandewalle.