Rock, paper, and scissors are one of the classic games of our childhood. The game that let us decide on things that otherwise we couldn’t. In 1950 John Nash suggested that any game with a finite number of players and a finite number of options like in the case of Rock, paper, and scissors, a mix of strategies always exists where no player can do better by changing the strategy of only one player. These stable strategies have a theory behind it called “Nash equilibria” which revolutionized the field of game theory. Thus, changing the way we studied and analyzed things like political treaties, network traffic, etc. It earned Nash his Nobel prize in 1994.
Rock-Paper-Scissors falls in the only one mixed Nash equilibrium. Let’s model the game such that there are two players, Player A and Player B. They are playing the game over and over again. In each round, the winner earns a point whereas the loser loses a point, and tie tallies zero.
Now, let us assume that Player A adopts the silly strategy of always choosing rock. After a few rounds of winning, losing and tying, the strategy is noticed by Player B. Hence, playing the counter-strategy of always playing paper. Now, Player A will never win. Similarly, if Player B always plays paper, Player A will start playing scissors. This will lead to Player A always winning. This will lead to a continuous cycle of rock, then paper, and then scissor. Thus, leading to no equilibrium. For this reason, there exists no pure Nash equilibrium for this game as no player would choose one choice for the entire game as it is predictable.
Probability and Nash Equilibrium
Now, let RA, PA, and SA be the probability of player A playing rock, paper, and scissors respectively. Similarly, RB, PB, and SB be the probability of Player B playing rock, paper, and scissors respectively. Since none of these probabilities will be 1. Hence, the Expected Values for Player A is:
EV(RA)=0*RB +(-1) *PB +(1) * SB
EV(PA)= (1) *RB +(0) *PB +(-1) * SB
EV(SA)= (-1) * RB+(1) *PB +(0) * SB
Also, RA+ PA+SA=1.
Using these equations, we reach the Nash Equilibrium of Rock, Paper, and Scissors for Player A to be:
RA = PA= SA= 1/3
Similarly, for Player B to be:
RB =PB = SB = 1/3
In the game of rock, paper, and scissors the players play multiple rounds, and according to Nash equilibrium, it is best to stick with it for rock, paper, and scissors but is it really the best?
In an experiment where multiple people played the game with one another, it was seen that the Nash equilibrium theory came into play. It was noticed that winners were those players who stuck to their strategy and losers were those who kept changing their strategy. It is called “conditional response” in game theory. In fact, the conditional strategy proved to be more reliable for winning than the Nash equilibrium by 10 per cent.
Thus, one should be more cautious when using Nash equilibrium as we found the equilibrium it might not be the best strategy. So, even in a simple game like rock, paper, and scissors one should be careful before using the Nash equilibrium. The invisible hand of John Nash may guide some games, but others may resist its hold and trap the players in a never-ending competition for gains that are just out of reach.
Reference: Cornell University
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