In a game where there is no hidden information and no luck, there will always exist a series of moves for one player that will lead to a win or a draw. Tic-Tac-Toe (Noughts and Crosses) is a trivial example – to guarantee a win or draw, the first player must always play in the centre square – while more complex examples like Chess or Go are presently impossible for computers to map out.
However, this is not true for most of the games we play.
In Rock-Paper-Scissors, unlike Tic-Tac-Toe, there is no single series of moves that you should always play in response to what your opponent does. If you always play Rock, your opponent.
The core concept of Rock-Paper-Scissors – that A beats B, B beats C, but C beats A – is a fundamental idea in games. Surprisingly, the earliest trace of the game is in writings from the Chinese Ming dynasty, in the 1600s. Compare this with dice and playing boards, which go back more than 5000 years!
From China, the game spread to Japan and the rest of Asia, and then to North America and Europe as migration carried people to those continents. The symbols and their names have changed in various cultures, but the current Rock, Paper and Scissors actually match what was originally used in China.
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One option in playing a series of Rock-Paper-Scissors games is to try to guess what your opponent is going to do and craft a strategy around that. For example, if your opponent always plays Rock, or plays it more than a third of the time, you switch to playing Paper more often.
This is called an ‘exploitation strategy’, because you’re trying to take advantage of suboptimal play by your opponent. The danger with exploitation strategies is that they are open to counter-exploitation. If you switch to playing mostly Paper, your opponent may switch to playing Scissors more often. So, you switch to Rock, and around and around we go.
However, is there a way to get off this exploitation strategy merry-go-round?
Mathematicians studying game theory have proven that, for any game, there is a mix of strategies called the ‘optimal strategy’. If you play the optimal strategy, it doesn’t matter what your opponent does – you will end up with a consistent result.
Now, for Rock-Paper-Scissors, the optimal strategy is simply to play Rock, Paper and Scissors each one-third of the time. But you can’t play them in sequence or in a particular order. This is known as a ‘mixed strategy’, where you play a suite of moves in different proportions, selected randomly.
There are several interesting properties of using this strategy in Rock-Paper-Scissors, and mixed strategies in general. The first is that it does not guarantee that you will win any single game. However, if you play many, many games, there is no strategy your opponent can use that will cause you to lose over the long haul.
Next, the outcome of this strategy is invariant – it does not change, regardless of what your opponent does. It factors out what your opponent does. For example, let’s say your opponent plays Rock 100 per cent of the time. Now, against this strategy you will win one-third of the time, when you select Paper, lose one-third of the time, and draw the other third. In fact, against any strategy your opponent picks, you will have an equal number of wins and losses.
Of course, if your opponent is playing a suboptimal strategy, you may be tempted to change your strategy to try to exploit it – switching to more Paper if your opponent is playing more Rock, as we discussed earlier. But then you are opening yourself up to your opponent taking advantage of your play, and thereby ending up with a worse result.
So, the optimal strategy is the safest strategy to take, in that it guarantees you a certain result. It is not necessarily a strategy that helps you to win more than your opponents, or even as often as them.
If all the players are playing optimal strategies, then any player who does something different gets a worse result. This is called a Nash Equilibrium, developed in 1950 by John Nash, the mathematician who was the subject of the 2001 film A Beautiful Mind and who first explored this area of game theory.
Finally, there is the ‘Bart Simpson strategy’:
Lisa: Look, there’s only one reasonable way to settle this, Bart – Rock-Paper-Scissors.
Lisa (thinking): Poor predictable Bart. Always takes Rock.
Bart (thinking): Good old Rock. Nothing beats that!
How to Win at Rock-Paper-Scissors
A 2014 study from Zhejiang University titled ‘Social Cycling and Conditional Responses in the Rock-Paper-Scissors Game’ (described in the media as ‘How to Win at Rock-Paper- Scissors’),1 had 360 students play 300 games each of Rock- Paper-Scissors. The students earned points based on how they did: zero points if they lost a game, one point if they tied, and a set amount if they won. The students were broken up into five groups, each assigned a different amount if they won – ranging from 1.1 (only slightly better than a tie), up to 100 (way better than a tie).
So, did the players play an optimal strategy – randomly choosing each play one-third of the time? The researchers first looked at the total distribution of Rock, Paper and Scissors. They found that each was played very close to one-third of the time, the Nash Equilibrium.
They next looked at the relationship between what participants played in one game, and in the next. And this is where things got interesting.
The researchers defined moving from Rock to Paper to Scissors as moving in the ‘positive’ direction (from losing to winning options), and moving from Rock to Scissors to Paper as the ‘negative’ direction (from winning to losing options).
To play an optimal strategy means that regardless of what happens, if players play Rock in the first round, 33 per cent should stay with Rock in the second, 33 per cent should switch to Scissors, and 33 per cent should switch to Paper.
However, the study found that if players won their match, they tended to stay with the previous choice. Moving in the negative direction was the second most popular option. If they won with Rock, 50 per cent stayed with Rock, 30 per cent switched to Scissors, and only 20 per cent switched to Paper.
If players lost their match, they predominantly moved in the negative direction. If they lost with Rock, 40 per cent switched to Scissors, while 33 per cent stayed with Rock, and 27 per cent switched to Paper.
And if players tied, they were slightly more likely to either pick the same choice again (staying with Rock about 38% of the time) or move in the positive direction (switching to Paper). Moving in the negative direction (switching to Scissors) was least likely in this instance.
The authors do not speculate on why students exhibited this behaviour. Were they trying to out-think their opponents? Or was this just an unconscious bias?
Well, in almost all cases, players were least likely to shift to the move that would lose if their opponent played the same thing again. I think that this has to do with something called ‘loss aversion’ – if your opponent stays the same and you lose, you will regret it more than if you changed and lost. (More on this in Chapter 6.) And this subtly pushed the players’ choices.
Okay, so players in this game weren’t perfect random-number generators. But can you exploit these results?
The authors constructed an optimal strategy based on what they had observed, which leads to an increase of wins from the standard 33 per cent to 36 per cent. So, the effect is real, and can be exploited, but even a sophisticated analysis only increases your score slightly. Let’s say you get zero points for a loss, one for a tie, and two for a win. Exploiting this effect would increase your average score from 1 to 1.1. This may be enough if you’ve got a lot of money on the line, but I’m not sure that it deserves the hyperbolic title ‘How to Win at Rock-Paper-Scissors’.
However, there is another way to lift your overall win percentage.
Rock, Paper, Scissors, Choice
In another study conducted on Rock-Paper-Scissors, researchers at the University College London paired participants up to play a series of Rock-Paper-Scissors games.
In some cases, both participants were blindfolded. In others, one person was blindfolded and the other person was not. In either case, they still did the old one-two-shoot method of timing their hand-position selection.
A prize was given to the player who won overall, so there was an incentive to win.
When both players were blindfolded, the number of draws was exactly equal to what you would expect – one-third of the matches were draws, to within half a per cent. But in the blind–sighted pairings, the number of draws was above what you would expect – close to 40 per cent of the matches were draws, more than a 10 per cent increase over the expected values.
What’s going on?
There is a psychological imperative to imitate what we see others do – monkey see, monkey do, if you will. This is hardwired pretty far down in our brains and can be difficult to override.
In order to imitate what the blindfolded player is doing, the sighted player must slightly delay putting their hand out. (In fact, when experimenters took out the games where the sighted player put their hand out first, or basically at the same time, the percentage of draws really shot up.)
However, this must be subconscious. First, there is very little time to react and adjust. And second, if the players really are trying to cheat by adjusting to what they see, they are doing a bad job of it. If you see that your opponent is throwing out Rock, you should quickly switch to Paper, not Rock. So, this is reflective of the brain’s imitative circuits, not a higher-level function.
Interestingly too, the second-highest choices were not optimal either. For example, when the blind player played Scissors, the sighted player also played Scissors 38 per cent of the time, rather than the expected 33 per cent. But the second most commonly played symbol in this case wasn’t Rock – which is what you would expect if there was conscious decision-making to beat Scissors. It was Paper, which was played 33 per cent of the time. Rock, the best choice against Scissors, was only played 29 per cent of the time.
The moral of the story is that people don’t – or maybe even can’t – cheat at Rock-Paper-Scissors. The blindfolded players actually won more than the sighted players. So, next time you play, you might try closing your eyes as an advanced strategy!
Biofeedback: Count on Your Heart to Trust Your Gut
There is a famous experiment called the Iowa Gambling Task. In the Iowa Gambling Task, players get a starting stake of money and are presented with four decks of cards. Each turn, they pick a card from any of the decks and it tells them how much they gain or lose.
The trick is that the decks are biased, but the players don’t know that. Some are very high-risk, some are low-risk, some are high-reward, and so on.
Most players take about 50 cards before they start to draw only from the decks that give them the best results. And it often takes another 30 draws before they realise they are doing it and can logically explain their actions.
Recently, experimenters used the Iowa Gambling Task to gain a deeper appreciation of how we learn about the world. The experiment was twofold. First, the experimenters measured the body’s classic stress reactions during the game – things like pulse rate and sweatiness of palms. It turned out that after only ten card draws, the stress reactions increased when drawing from a negative deck. The unconscious had already started to associate certain decks with bad results, even though this information had not entered the conscious brain yet.
The experimenter also asked the players to count their heartbeats while they played. Those who were better at that task also switched to the good decks sooner, and did better at the card game overall. The researchers hypothesised that those players who were more in touch with feedback from their bodies noticed the stress signals sooner and were able to bring subconscious information up to the conscious brain faster.
So, while you shouldn’t trust your body’s imitative circuits to win you a blind–sighted Rock-Paper-Scissors tournament, it could mean the difference between winning and losing in other games. As gamers, we should learn when to listen to our bodies and gut reactions. If you have a bad feeling about something, there may be a reason for it. But our brains are also supremely capable of seeing patterns where there aren’t any.
The above chapter is an excerpt from GameTek, a book about the big questions of life through games by Geoffrey Engelstein. Engelstein is an adjunct professor of Board Game Design at the NYU Game Center and an award-winning table-top game designer.
He has degrees in Physics and Electrical Engineering from the Massachusetts Institute of Technology, and is currently the president of Mars International, a design engineering firm.
Since 2007 he has been a contributor to the leading table-top game podcast Dice Tower, presenting ‘GameTek’, a series on the math, science, and psychology of games. He also hosts Ludology, another weekly podcast on board games.