Non-Transitive Games

Public summary: 

Some games really aren't fair. Even if your opponent knows your exact move you can still beat them. Explore some of these games and find out there might not be a sensible best choice.

What's the best option in games of chance?
Useful information
Kit List: 

A set of Grimes Dice (currently dots are drawn on, hopefully engraved/painted in future)
General Dice
(There's also a set of Effron Dice in the box too)

Packing Away: 

Live in the maths box, place dice sensibly in a bag.

Frequency of use: 

There should be a set of dice in the box with the following numerals
red - 9,4,4,4,4,4
blue - 7,7,7,2,2,2
olive - 5,5,5,5,5,0
(They're slightly larger than normal so we can move onto 5 dice later using the same ones)
Take turns with your opponent to pick a dice and see who can roll the highest number. Try a best of three or five to decide which dice to pick, if they pick first you can always beat them. What you'll find is Red > Blue > olive > Red. Get them to construct two of the chain and ask them to guess what happens with the third pairing. Most will think that it's going to be transitive and Red will beat olive, however it isn't! Easy way to remember the order is increasing word length.
Any easy way to relate this is to rock, paper, scissors. Here Rock > Paper > Scissors > Paper. When we pick at different times this games becomes very unfair. With the dice there's no certainty but you can explain by playing multiple times you make it more likely.
You can take some maths and try and work out the winning probability. Red beats Blue if we get a 9 straight away, that's probability 1/6. If we get a 4, 5/6 probability, then we win when Blue gets a 2, probability 1/2, these are 'independent events' so we can get a total probability of 1/6 + 5/12 =7/12 > 1/2. You find this is the same for all pairings.
If they've understood flip it around and let them beat you. Then move to two dice and keep going first. You'll want to increase the number of rolls to make it a best of 5 at least. You'll notice that in fact the order swaps around when you look at the totals. It should decrease to about 57%.
Lets add in 2 more dice. We want to be able to beat 2 players simultaneously
yellow - 8,8,3,3,3,3
magenta (pink) - 6,6,6,6,1,1
There's an alphabetical chain and a word length chain. Alphabetical chain has a higher win probability if there's only one player. If there's two you can find a unique dice to beat both.
This analogues Rock,Paper, Scissors, Lizard, Spock (from The Big Bang Theory TV show) with new rules
Scissors cuts Paper, Paper covers Rock, Rock crushes Lizard, Lizard poisons Spock, Spock smashes Scissors, Scissors decapitates Lizard, Lizard eats Paper, Paper disproves Spock, Spock vaporises Rock, (and as it always has) Rock crushes Scissors.
In this game the doubling of dice reverses the word length chain and the alphabetical chain remains more or less the same apart from the fact red and olive flips technically but remains very close to 50-50. Overall 59% win chance.
For beating two players we're at 44%. You may think that's bad as you still loose fairly often but you only loose to both 22.7% of the time. Consider a game where both opponents pay £1 and you'll pay out £1 to anyone who rolls higher. For 100 rolls you make £88 on the games you beat both on 44 rolls, loose £46 when you loose to both on 23 rolls and the rest of the time you make nothing (but don't lose anything either). So you're £42 in profit!

Inspired by the experiment here:

If you have a spare table this is a very easy experiment to float with by only going up to the Rock-Paper-Scissors part of the experiment. Demonstrate you can always win and sometimes "better than" isn't transitive.

Risk Assessment
Date risk assesment last checked: 
Wed, 05/02/2020
Risk assesment checked by: 
Andrew Sellek
Date risk assesment double checked: 
Wed, 05/02/2020
Risk assesment double-checked by: 
Beatrix Huissoon
Risk Assessment: 

Looking at probabilities and distributions for throwing dice.

Hazard Risk Affected Person(s) Likelihood Severity Overall Mitigation Likelihood Severity Overall
Dice Children swallowing dice. Public 2 5 10 Don't use with small children and keep the dice attended.
Call first aider if child swallows, if choking encourage child to cough.
1 5 5
Dice Dice could be a slip hazard if dropped on floor. All 2 2 4 Keep an eye on where any dice go, and try to confine them to a desk or fixed area. Do not let multiple unattended children use dice at the same time.
Call first aider in case of injury.
1 2 2