Designing a Game

Introduction

By the end of the lesson, you will hopefully be able to design a game and calculate the relevant probabilities to decide on whether it is a fair game or not. Let's begin with a few games first:

Game 1

• Roll two 4-sided dice, multiply the numbers

• You get $0 if you roll a 2 or 4

• You get $2 if you roll a multiple of 3

• You get $5 if you roll a 8 or 16

• You get $10 if you roll a 1

1. How would you work out the probability of the above events?

2. What would be the expected return of the game?

3. Was it a fair game? What would you charge to make this a fair game?

4. What is the relative frequency/experimental probability for playing this game?

5. Use this worksheet to help you understand all the concepts above.

Game 2

• Roll 2 dice, add the numbers up

• You get $1 if you get a prime

• You give me $1 if you don’t

• We play it 36 times

• Who wants to gamble with me?

1. What is the relative frequency/experimental probability?

2. Draw a sample space, what is the theoretical probability?

3. What is the expected value for playing 36 games? What is the expected value for playing one game?

4. Was it a fair game? How can you make it fair?

Game 3

• Roll 2 dice, multiply the numbers up

• You get $2 if you get an odd

• You give me $1 if you don’t

• We play it 36 times

• Who wants to gamble with me?

1. What is the relative frequency/experimental probability?

2. Draw a sample space, what is the theoretical probability?

3. What is the expected value for playing 36 games? What is the expected value for playing one game?

4. Was it a fair game? How can you make it fair?

Game 4

• 2 players

• Rock, Paper, Scissors

• You pay $5 to play

• You get $6 if we play different things e.g. R and S

• You get $3 if we play the same thing e.g. R and R

• We play it 36 times

• Who wants to gamble with me?

1. What is the relative frequency/experimental probability?

2. Draw a sample space, what is the theoretical probability?

3. What is the expected value for playing 36 games? What is the expected value for playing one game?

4. Was it a fair game? How can you make it fair?

Game 5

• 3 players

• Rock, Paper, Scissors

• You pay $5 each to play

• You get $18 each if we all play the same things e.g. R, R and R

• You get $9 each if we all play different things e.g. R, P and S

• You get $3 each if two of us play the same thing e.g. R, R and S

• We play it 36 times

• Who wants to gamble with me?

1. What is the relative frequency/experimental probability?

2. Draw a sample space, what is the theoretical probability?

3. What is the expected value for playing 36 games? What is the expected value for playing one game?

4. Was it a fair game? How can you make it fair?

Now you design your own game

• You can:

  • Roll at least 2 dice

  • Spin at least 2 coins

  • Roll 1 die and toss 1 coin

  • Pick up to 2 cards from from 4Js 4Qs 4Ks

• You can charge up to $10

1. Write down the rules, including how much each pays and how much they get

2. Calculate the relative frequency/experimental probability

3. Draw a sample space, and calculate the theoretical probability

4. Calculate the expectation

5. Show whether it is a fair game or not and make it fair if it isn’t

Further Questions and Challenges

• What makes a good game in terms of the win/loss ratio for a potential player?

• What’s good for the bank?

• What makes a popular game in terms of the odds?

Extension

You may wish to take a look at this activity, an experiment conducted by two investment fund managers Victor Haghani and Rich Dewey. I the experiement, they invited 61 people, a combination of college-age students in finance and economics and some young professionals at finance firms (including 14 who worked for fund managers), to take a test. They were each given a stake of $25 and then asked to bet on a coin that would land heads 60% of the time. The prizes were real, although capped at $250. What would your strategy be?

Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment. Neither approach is in the least bit optimal. The best strategy was devised by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956. If you are interested, google it!