Rock-paper-scissors (RPS) is a fair game. Since each choice has an equal strength and weakness–rock is beaten by paper, paper is cut by scissors, and scissors are smashed by rock–the correct way to play is to randomize equally over all three choices. Neither player can gain an edge, and if the game is played for money, both people can expect to break even.

What happens when you modify the game just slightly?

Consider modified RPS where a player gets $1 if the win is from playing rock or scissors, but the player gets $2 if the win is from paper. (Conversely, a player has to pay $1 if the loss is from scissors losing to rock or paper losing to scissors; and the player has to pay $2 if the loss is from rock losing to paper).

Here is a comparison of standard RPS and modified RPS:

The only differences in modified RPS are that winning with paper gives double the payout and losing with rock to paper gives double the loss.

The question is, what’s the correct strategy in modified RPS? Do you end up playing paper more or less than in standard RPS?

Unless you’ve studied game theory, chances are you will quickly arrive at the wrong answer, as explained after the jump.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Most people get it wrong

Many people reason very simplistically. They think, “Well, I can win twice as much money if I win with paper. So clearly I should play paper a lot more now and I should play rock and scissors less.”

Can you spot the error in this reasoning?

The problem is it’s a very self-centered line of reasoning. You cannot base your strategy on what you alone want to have happen. You need to consider that you have an opponent who is also looking to profit.

If you start playing paper a lot, then your opponent can exploit the pattern by playing scissors. But if your opponent might counter with scissors, you might react by playing rock. Then your opponent happily plays paper, and so on.

The point is you can’t make a decision based solely on what you want to happen. You have to base your choice on what you think your opponent will do.

Game theory is the tool that can solve these problems with a mathematical answer of the best strategy.

While playing paper more is the natural instinct, it turns out to be completely the wrong thing to do.

Solving the game

I’ve explained the technical process to solve for games like this here and also here.

There is also an excellent video on how to solve this particular game. William Spaniel offers a step-by-step explanation of how to solve the game. I refer you to these videos.

Part 1: intro to modified RPS

Part 2: solving modified RPS

For my purposes, I am happy to plug this game into a zero-sum game solver.

The correct answer is that each person should play:

rock with probability 0.25

paper with probability 0.25

scissors with probability 0.5

The result is a bit counter-intuitive. Even though paper has a larger reward, the correct strategy is to play it LESS than in standard RPS.

Why is paper played less? The reason is the effect of the rule change on the other strategies. Since winning with paper is more attractive, losing with rock to paper is also less attractive. Rock turns out to be a relatively bad choice to play: it only wins $1 against scissors but it loses $2 to paper. Since rock is undesirable, that also makes playing paper less desirable since paper only wins against rock. Both rock and paper still get played, but ultimately it is best to play scissors most of the time.

But most people fail to recognize the complete effect of the rule change. I came across a state science fair project from 2006 where the student was surprised to see how badly people play modified RPS:

I thought that people would more frequently play the choices worth fewer points to defeat their opponent’s strategy or strategies. This was based on my understanding of game theory and my experience with games. My results did not support my hypothesis. People seemed to jump automatically to the strategies worth more points. They focused on the more obvious, seemingly more powerful, tried and true strategies rather than attempting to play mind games with their opponents.

Modified RPS is a simple example that demonstrates our instinctual strategic ineptitude and why game theory is important for finding correct solutions.