### Rock Paper Scissors

Remember playing Rocks, Paper, Scissors back when you couldn’t choose
“who gets the last piece of pie”? Where you good, or did you not use the right
strategy? Yeah that’s right, strategy! Before analyzing your opponent everyone
has some sort of strategy. Due to the complexity of the game there is room for
multiple strategies for a player to follow. But which one is the best? In my
version of the day old game of Rocks, Paper, Scissors I can display the potential
outcomes for many potential strategies.
“Score one for the Skipper!”
I logically assigned an arbitrary score to each tool. Each tool’s score was
assigned in coordination to its value of destruction. These psychological presets
grant a rock a score of 5 points. A rock is easily the most destructive of the
tools. Scissors would be granted 3 points for being more destructive than paper
but less destructive than a rock. Consequently paper is assigned a 2 point value.

“Can I have directions to. . .”
Rocks, Paper, Scissors is a zero sum game always resulting in a collective
total of zero. The two players play simultaneously through three games. The
player with the higher score at the end of the third game wins. Rock beats
scissors, scissors beats paper, and paper beats the rock. The victor is granted
the points of the dominant tool. i.e. Player 1 uses a scissors and Player 2 uses a
rock. Player 2 has just won 5 points for beating a scissors with a rock. The two
players have two more games to play until they can decide on who is the winner.

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This game, because of the three tools, can be broken down into a 3×3
game matrix. Display 1 is an example of what the matrix would appear to Player
1. The matrix for Player 2 would look very similar despite the fact that all of the
numbers in the matrix will be opposite that of what is in Player 1 matrix. The
matrix provides no dominance for either player meaning there is no saddle point.
The game provides for several mixed strategies that can get very complicated.
It also provides for simple strategies that players may decide on using.

“Ya’ll play rational now, ya hear!?”
A rational player plays to win with the highest score. There are only three
moves that a player can make. The first move by either player can in no way be
an irrational move because there is no dominance in the matrix. Similarly the
second move is strictly a strategic move done so by analyzing the character of
your opponent, and can in no way be an irrational move. The final move,
however, is a key move. By analyzing the rationality of some the first two moves
a player can assume what their opponents final move will be. Here is an
example; The first two games play as such. Both use paper for the first move.
The second move Player 1 chooses paper (P) and Player 2 chooses scissors (S).
The score is now (-3,3) in Player 2 favor. The third move is where rationality is
important. Player 1 is down by 3 points, consequently it would be irrational to
use paper because he/she could only win a total of 2 points. So Player 1 only
has 2 choices, either rock or scissors. If he/she chooses rock the possibility of
winning with a total of (2,-2) is appealing. Scissors would equal out the score to
(0,0) resulting in a tie. From a reverse perspective, Player 2 can acknowledge
Player 1 situation and use paper for his/her final tool. This would result in a
score of (-5,5). In the sense of this “reverse perspective” the two player can
never truly discover their opponents final move.

“Why do you have to be so irrational?”
Rationality is a rule of my version of Rocks, Paper, Scissors. A player
must never pick an irrational move. This is important because some strategy
revolves around cornering their opponent into an irrational situation. Some
games are won by the second move. Take for instance a hypothetical game
where Player 1 has chosen to play a simple strategy of PPP. The first set results
in Player 1 choosing paper and Player 2 choosing paper as well. The score is
now (0,0). The second set is Player 1 paper and Player 2 scissors. The score is
now (-3,3). Now Player 2 has cornered Player 1 into choosing either a rock tool
or a scissors tool. This is because a rock is the only last tool that can win the
game for Player 1 (2,-2). Scissors is the only tool that can tie Player 1 and
Player 2 (0,0). See Player 1 can’t choose paper for the last move because there
is no way that they could win. Player 2 aware of Player 1’s situation knows that
rock would put them even further into the lead. Player 2 knows that Player 1
can’t choose paper, the only tool to beats rock, because it is irrational. This
hypothetical game was won by Player 2 in the second move. Some strategy is
completely reliant on this rule of rationality.

“It is in the play that I shall catch
Due to the complexity of this game I will only give examples of two
possible strategies for either player. The first strategy is what I call a simple
strategy. This means that the player will choose the same tool all three times.
Display 2 is a chart that lists all possible outcomes for a simple strategy. As you
can see the best simple strategy appears to be rock. The chart plays through 20
hypothetical games all of which Player 1 uses the same tool. When totaling the
scores for each game, it shows that an all rock simple strategy results in 63
points. Of course this strategy doesn’t always appear to work, for example, look
at hypothetical games 1, 2, 14, 17, and 18. Another strategy is the expectation –
equalizing mixed strategy that is explained in chapter 2 of our text book. This
game does not have a sadle point, and there is no dominance or favor for either
player. Chapter 2 explains that Row will play rows R, P, S with probabilities (x,
y, 1-x-y). For further explanation of this strategy please refer to Display 3
and/or Chapter 2 in the text. It is checked that Row’s and Column’s strategies
do have a solution that equals 0. This is proven in Display 4. A 3×3 matrix is the
key to discovering an expectation – equalizing strategy. It is important to realize
that this method of equalizing expectations will fail if the solution to be 3×3 game
Well there you have it. A ten page analysis of the day old game of Rocks,
Paper, Scissors. My game varies slightly from the traditional but the score
system was essential towards discovering strategy. A matrix was reasonably the
easiest way to break the game down. The complexity of this game makes it go
around infinitely. There are no sure ways to win. No one strategy can
guarantee a victory. No saddle points and no dominance makes this game
playable forever. Both players are equal because the game matrix has no favor
towards any certain player. My great grandparents have played this day old
game, and will continue to solve everyday dilemmas like “who gets the extra
slice” for many years to come. There is one rule, however, that mush never be
forgotten. It is a game so have fun with it.

Row Paperx(2) + y(0) + (1-x-y)(-3)
Row Scissorsx(-5) + y(3) + (1-x-y)(0)
Column will be unable to take advantage of Row’s mixed strategy if these
Row’s EV = (3/10, 5/10, 2/10)
Thus Row’s expectation – equalizing mixed strategy is (3/10, 5/10, 2/10).
By the symmetry of the matrix, this is also Column’s expectation – equalizing
Legend:P= PaperS= ScissorsR= Rock
Bibliography:
THis is a paper that I had to write for my Game Theory math class my sophemore year in college. It is a break down and analysis of the ever popular ‘Rock Paper Scissors’.