# A Little Game Theory

What is the ultimate strategy for winning at rock-paper-scissors?  According to three physicists in China, the answer does not lie in having absolutely no strategy and ensure that your choice of weapon is completely random, unlike previously thought.  If that strategy seemed obvious, perhaps you haven’t played the game enough to delude yourself into thinking that this might be a winning strategy…

If you’ve ever played Rock-Paper-Scissors, you might have wondered about the strategy that is the most likely to beat your opponent.  Actually, game theorists have long puzzled over this and other similar games in the hope of finding the ultimate approach.

# The Nash Equilibrium

The best strategy is to choose a weapon at random.  This is known as a mixed strategy Nash equilibrium in which every player chooses the three actions with equal probability in each round.

Nash equilibrium is an equilibrium for a game in which

Neither player has an incentive to unilaterally move away from its choice in the pair of strategies.  If both move at the same time, they could both benefit, but neither wants to do that alone.

Such equilibria may not always exist for a given game.  But if the game is played probabilistically, or played with ‘mixed strategies’, then it is guaranteed that there will always be a Nash Equilibrium.

The game goes like this.  Rock crushes Scissors.  Scissors cuts Paper.  Paper covers Rock.

Over the long run, the random strategy makes it equally likely that you will win, tie, or lose against your opponent.  At least, that’s how the game is usually played.  And so far, small-scale experiments have demonstrated that it is indeed the strategy that tends to evolve out of the game.

Or so game theorists had thought.

# The Experimental Setup

Zhijian Wang et al. carried out 5 sets of experimental sessions at different days during December 2010 to March 2014, with each set consisting of 12 individual experimental sessions.  Each set of experimental sessions required 72 Rock-Paper-Scissors players, distributed uniformly at random by a computer program, into 12 groups of 6.

In total, 360 undergraduate and graduate students from different disciplines of Zhejiang University, China, were  recruited to participate in the game.

The players then sited separately in a classroom, facing a computer screen.  They were not allowed to communicate with each other during the whole experimental session.  Written instructions were handed out to each player and the rules of the experiment were also explained by an experimental instructor.

The rules of the experimental session are as follows:

1. Each player plays the Rock-Paper-Scissors game repeatedly with the same other five players for a total number of 300 rounds.
2. Each player earns virtual points during the experimental session according to the payoff matrix shown in the written instruction.  These virtual points are then exchanged into RMB as a reward to the player, plus an additional 5 RMB as show-up fee. (The exchange rate between virtual point and RMB is the same for all the 72 players of these 12 experimental sessions.  Its actual value is informed to the players.)
3. In each game round, the six players of each group are randomly matched by a computer program to form three pairs, and each player plays the RPS game only with the assigned pair opponent.
4. Each player has at most 40 seconds in one game round to make a choice among the three candidate actions “Rock”, “Paper” and “Scissors”.  If this time runs out, the player has to make a choice immediately (the experimental instructor will loudly urge these players to do so).  After a choice has been made it can not be changed.

Before the start of the actual experimental session, the player were asked to answer four questions to ensure they fully understood the rules of the experimental session.

These four questions are:

1. If you choose “Rock” and your opponent chooses “Scissors”, how many virtual points will you earn?
2. If you choose “Rock” and your opponent chooses also “Rock”, how many virtual points will you earn?
3. If you choose “Scissors” and your opponent chooses “Rock”, how many virtual points will you earn?
4. Do you know that at each game round you will play with a randomly chosen opponent from your group (yes/no)?

# Social Cycling and Conditional Response

On average, the RPS players in all the groups chose each action (rock, paper or scissors) about a third of the time, which is expected if their choices were random.

On closer inspection, the players’ behaviour reveals something else.

### The players who win tend to stick with the same action, while those who lose appear to switch to the next action in a clockwise motion (where R → P → S → R is the chosen clockwise direction).

In evolutionary game theory, this is a conditional response.

According to Zhijian et al., the collective cyclic motions of the game cannot be understood in terms of Nash equilibrium, but they are successfully explained by the empirical data-inspired conditional response mechanism.  As it has never been observed before in Rock-Paper-Scissors experiments, the team speculated that it could be due to the smaller-scale of past experiments.

From the point of view of psychology, a “win-stay, lose-shift” strategy is entirely legitimate because people usually stick with a winning strategy.

The researchers plan to investigate this aspect in greater details:

“Whether the conditional response is a basic “built-in” decision-making mechanism of the human brain, or just a consequence of more fundamental neural mechanisms is a challenging question for future studies.”

# How You Can Win at Rock-Paper-Scissors

The discovery has practical implications.  If players invariably use a predictable strategy in Rock-Paper-Scissors, this weakness can be exploited by one of the players.

Theoretical calculations revealed that this new strategy may offer higher payoffs to individual players, compared to the mixed strategy Nash equilibrium already discussed.

Worth bearing in mind next time you bet a round of drinks at your local pub…

It would seem, however, that when you know someone well enough, there is a 75-80% probability of any Rock-Paper-Scissors games that you play with that person to end up in a tie.  Fortunately, you can always add a slight variation to the game that reduces that probability.

If you ever get bored, try the following…

# Rock-Paper-Scissors-Lizard-Spock

The game was invented by software engineer Sam Kass, with his wife, technical writer Karen Bryla.

Famously, it goes like this.

Scissors cuts Paper.
Paper covers Rock.
Rock crushes Lizard.
Lizard poisons Spock.
Spock smashes Scissors.
Scissors decapitates Lizard.
Lizard eats Paper.
Paper disproves Spock.
Spock vaporizes Rock.

And as it always has

Rock crushes Scissors.

Still bored, why not try this?

# Rock-Paper-Scissors-Lizard-Spock-Spiderman-Batman-Wizard-Glock

It begins much the same.

Scissors cuts Paper.
Paper covers Rock.
Rock crushes Lizard.
Lizard poisons Spock.

But then, unexpectedly…

Spock zaps Wizard.
Wizard stuns Batman.
Batman scares Spiderman.
Spiderman disarms Glock.
Glock breaks Rock.
Rock interrupts Wizard.
Wizard burns Paper.
Paper disproves Spock.
Spock befuddles Spiderman.
Lizard confuses Batman.
Spiderman defeats Lizard.
Lizard confuses Batman (presumably because it looks like Killer Croc).
Batman dismantles scissors.
Scissors cut Wizard.
Wizard transforms Lizard.
Lizard eats Paper.
Paper jams Glock.

It gets nasty.

Glock kills Batman’s mum.
Batman explodes Rock.
Rock crushes Scissors.
Scissors decapitate Lizard.
Lizard is too small for Glock.
Glock shoots Spock.
Spock vaporizes Rock.
Rock knocks out Spiderman.
Spiderman rips Paper.
Paper delays Batman.
Batman hangs Spock.
Spock smashes Scissors.
Scissors cut Spiderman.
Spiderman annoys Wizard.
Wizard melts Glock.
Glock dents Scissors.

If you’re STILL bored, there is no hope. You’re probably a nerd…

In which case, welcome to the club!  🙂