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### Course: Precalculus>Unit 7

Lesson 2: Using matrices to represent data

# Using matrices to represent data: Payoffs

Matrices are basically tables of numeric values. But you may be surprised at how many real-world situations you can model with this structure. Here, for example, we represent a game's payoffs: we represent the points each player gets in an elaborate version of rock, paper, scissors. Created by Sal Khan.

## Want to join the conversation?

• imagine watching this never playing rock, paper, scissors. "did I miss a video when did we learn paper beat rock?"
• Could you not just add the rows and see what option works best?
Row 1: 0 points
Row 2: 1 point
Row 3: -1 point
Since row two is positive and is the greatest paper would be best for Violet?
• I am confused by this matrix, but mainly the dot list. Could someone explain it please?

Also, I am the first to ask a question on this video
• You should go watch some videos or books about Game Theory, it will help you understand. Meanwhile, Game Theory is perhaps the subject that uses Matrix the most
• Would there be any strategy for Violet or Lennox if Lennox didn't have to pick at random?
• There wouldn't. There is, though, a bit of psychology involved. Violet is not incentivised to win with a certain choice, so her goal is to minimise Lennox's points. Lennox, on the other hand, has an incentive to win with rock. Lennox might play the lower numbers to catch Violet off guard, as she would be expecting him to play the choices that give him more points. This would give Lennox points off lots of wins. On the other hand, Lennox could play the higher numbers, because she might think that Violet is expecting him to catch her off guard. These are equally good reasons, so her strategy wouldn't really be a strategy, just an idea they chose to follow because all the options are equally good. Their "strategies" all depend on when they choose to give up debating.
• Why does Sal use averages and not just say that Violet always gets the same amount if she wins and Lennox gets the least points winning with scissors which beats paper?
• what a great real-life example!
and i feel Khan shines on this topic especially, regarding his former work experiences as a hedge fund analyst

here's a question by the way
is there anyone who knows the same or similar type of kiddo's game to rock paper scissor in your own country or culture? as far as i know and played, many east asian countries do have. and even with the same name with a tweak of paper to clothes in one case

anyhow many thanks to Khan as always
• Can someone explain how Sal's using expected values to figure out the outcomes?
(1 vote)
• Since Lennox chooses randomly between Rock, Paper and Scissors we could expect that over a large amount of games she would play each of the shapes roughly the same amount of times.

For example, if they played the game 6,000 times we could expect Lennox to have played Rock about 2,000 times, Paper about 2,000 times, and Scissors about 2,000 times.

Now, let's say that Violet always plays Paper.
Then, from the matrix we would see that after 6,000 games she could expect to have a total score of about
2,000⋅2 + 2,000⋅0 + 2,000⋅(−1) = 2,000

Thus, the expected value per game would be 2,000∕6,000 = 1∕3

– – –

We can generalize this idea by saying that they play 𝑛 games.

Then we could expect Lennox to play Rock 𝑛∕3 times, Paper 𝑛∕3 times, and Scissors 𝑛∕3 times.

Violet, always playing paper, could then expect a total score of
𝑛∕3⋅2 + 𝑛∕3⋅0 + 𝑛∕3⋅(−1)

And the expected value per game would be
(𝑛∕3⋅2 + 𝑛∕3⋅0 + 𝑛∕3⋅(−1))∕𝑛
= 1∕3⋅2 + 1∕3⋅0 + 1∕3⋅(−1)

This corresponds to
𝑃(Rock)⋅𝑆(Rock) + 𝑃(Paper)⋅𝑆(Paper) + 𝑃(Scissors)⋅𝑆(Scissors)
with 𝑃(𝑥) being the probability that Lennox plays 𝑥
and 𝑆(𝑥) being the score that Violet gets when Lennox plays 𝑥

If, for example, Lennox chose to play Rock 1∕2 of the time, Paper 1∕3 of the time, and Scissors 1∕6 of the time,
then Violet's expected score when playing Paper would be
1∕2⋅2 + 1∕3⋅0 + 1∕6⋅(−1)