The use of random numbers

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‘It's not just about the math!’

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V1.1

• Keith Martin-Smith

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• HexWar Games

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• Code Wizards

What is the point of our games?

• Model real events and revue past options

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• Predict futures outcomes

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• Test intellectual ability

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• Test a players decision making

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• Develop team skills

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• All this and much more …..

What is the objective of a playing a particular game?

• To teach a particular lesson?

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• Possibly, anything else is superfluous; luck, real life and even the facts, provided the target audience learns the lesson.

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• The use of randomness in games can get in the the way of the learning objective.

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Common themes in games

Excessive

Random

Impact?

Randomness

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• Random numbers are a key part of a game’s design and outcome

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• A challenge is the player’s ability to understand random numbers

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• A bigger issue is the impact of randomness on player behaviour

Do we want this in our games

Cognitive Bias

Facts

Understanding randomness

5 Questions

(1) Countries that play in the football world cup

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(A) Is this more than 175 or less than 175?

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(B) How many is it?

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(2) Powerball

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• What are the odds to win the Powerball?

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• What are the odds to win any prize in the Powerball?

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(3) Horse Racing – Combinations

Let’s play an accumulator. You have to win every race to win the bet.

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• You can play 3 horses in 3 races

- One at 4:1, one at: 6:1 and one at 8:1

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OR

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(B) You can play an 8 horse accumulator

- Two at 2:1. two at 3:2, two at 5:4 and two at 4:3

OR

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(C) A 5 horse accumulator

- All at 3:1

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(4) D6

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You are shooting at a target with 3 strength points

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(A) You can have 3 dice rolling 5 or higher to hit

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OR

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(B) You can have 2 dice rolling 4 or higher to hit

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(5) D6, D8 & D10

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You need 4 hits on a target, where you need a 3 or less to hit.

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(A) You can have 10 x 8 sided dice

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OR

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(B) You can have 5 x 6 sided dice and 5 x 10 sided dice

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5 Answers

(1) Countries that play in the football world cup

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(A) Is this more than 175 or less than 175

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(B) How many is it? 206

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By setting a base point most players will settle for an answer around that base point. This is a form of ‘Anchoring’

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Bonus fact:

There are only 195 sovereign states in the world!

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(2) Powerball – Irrational Odds

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(A) What are the odds to win the Powerball? 292M : 1

(B) What are the odds to win any prize in the Powerball 25 : 1

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• But when Powerball is played, you really believe it could be you!
• When you buy the ticket, when you think about the game, if you watch or listen to the draw, you generate dopamine.

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• Do not underestimate the impact of ‘It could be you!’.
• There is an irrational understanding of the odds.

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(3) Horse Racing - Combinations

(A) 3 horses in a 3 game accumulator 192 : 1

• One at 4:1, one at: 6:1 and one at 8:1

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(B) An 8 horse accumulator 25:1

- Two at 2:1. two at 3:2, two at 5:4 and two at 4:3

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(C) A 5 horse accumulator 243:1

- All at 3:1

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• It’s hard to work out the real odds!
• Our ‘belief systems’ get in the way
• Three horses may win, five horse winning is an outside chance, but getting 8 horse race correctly, that’s impossible

(40 D6

You are shooting at a target with 3 strength points

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(A) You can have 3 dice rolling 5 or higher to hit

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(B) You can have 2 dice rolling 4 or higher to hit

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• The odds of any hit are the same for both, but the 3 dice option is the only one that could wipe out the target at 216:1. The impact of concurrency.
• A simple odds’ calculation but very hard to do on quickly

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(5) D6, D8 & D10

You need 4 hits on a target, where you need a 3 or less to hit.

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(A) You can have 10 x 8 sided dice 3.75 hits

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(B) You can have 5 x 6 sided dice and 5 x 10 sided dice 4 hits

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These are simple examples;

• Impractical to work out the outs so risk comparisons are just a guess
• Small tweaks e.g. re-rolls, make this almost impossible

Back to Horse Racing

Something where we can all understand the odds … or do we?

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• 2:1, 3:1, 4:1 … clear and easy to understand
• So why play at 100:1 … It could be me, and we know of past winners!
• So why does no one play at 200:1 … that never happens

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Here is a balance of;

• Clearly understanding the odds
• Taking an irrational chance that you know could happen
• Taking an irrational decision to dismiss the odds

Cards vs. Dice – Incomprehensible odds!

Cards

• 52 cards
• 4 suites
• 13 cards in each suite
• 4 Royal cards in each suite
• Odds or evens
• Black or red

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Dice

• 4, 6, 8, 10, 12, 16, 20 etc. sides
• Rolled in multiples
• A mix of types in a single role
• Some numbers have special effect’s e.g. re-roll 6’s

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Unfamiliar territory?

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• Custom cards and custom dice = confusion and stress.

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Ideal?

• Keep the number system within the players comprehension and belief system.

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• Use a single D6 or D10

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• Best of all 2 x D10 as a percentage, everyone understands that

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OR

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• Deliberately operate outside of this number range, but be very aware it will cause player stress, and most players will become irrational in their decision making and in the risk levels they will take.

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Heuristics

Heuristics: How Our Minds Betray Us

• Heuristics are conscious or unconscious mental shortcuts we use to make snap decisions, freeing up cognitive resources for more complex tasks

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• We make countless small decisions every day while under pressure and/or working with limited information

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• They can mislead us into acting on unfair biases

Monte Carlo Fallacy a.k.a. Representativeness or The Gamblers Fallacy

• We are all guilty of chasing a sequence, but it is very irrational

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• In Monte Carlo in 1913, after black appeared 13 times bets piled in on red … Black repeated 26 times

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• Every time a 6 is rolled players truly believe it is very unlikely to happen again!

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• Players are basing their future risk/odds on previous events, irrationally discounting the real odds

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• The impact is an increase their acceptable risk levels

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Sunk-Cost Bias

• Sunk-cost bias occurs when a person makes increasing commitments to a failing course of action in order to justify the time and effort they’ve invested.

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• Typically, one more attempt at a failed attack!

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• This encourages irrational and increasing levels of risk taking.

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Compulsive Fear

• If a player follows a pattern, no matter how unsuccessful that pattern is, they fear changing the pattern

e.g. Trying to roll a 6 three times and failing, then changing to a different action and they finally roll the 6 that is now of no use to them.

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• Fear of this outcome drives repeating previously unsuccessful actions.

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• We have the same outcome of repeated risk from three very motivations;
• Monte – Carlo - An irrational understanding of the odds
• Sunk Cost – Inability to write off the time and effort expended
• Compulsive Fear – Inability to walk away from a failed event in case it does happen after walking away

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Anchoring

• A individual's decisions are influenced by a particular reference point or 'anchor’

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• The sample need not be based on facts. We looked at the the Football World Cup

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• Lottery draws that quote the rareness of a number see that number being played more often

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• Lottery draws that quote the high frequency of a number see that number being played more often

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• Increasing players awareness of ‘successful’ random encourages players to repeat the same risk taking
• Without any statistical basis

Diversification

• This heuristic describes how people tend to demonstrate greater diversity when confronted with simultaneous rather than sequential choices

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• Allow a player a list of choices they have to make together e.g. as a written list, and they will tend to make a list of different choices

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• Do the same task from the same list as a sequential e.g. telling the other players. They will tend to make a narrow selection and repeated choices

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• The method of asking for a choice impacts the choice to be made

Availability Bias

• A tendency to attach greater significance to events that leave the strongest impression or are easier to recall.

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• We know a horse will win a race; someone will win the lottery. It has been done so we know it can be done.

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• We hear of other players winning an outrageous attack succeeding.

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• These ‘success’ messages encourage irrational and increasing levels of risk taking.

The Theory of a Near Miss

• A near miss is where the player almost got the random event they wanted but just failed

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• Not quite the same reaction as succeeding, but near miss events generate just about the same amount of dopamine as a success

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• Creating a sense of ‘almost there’, increases the player’s propensity to try again

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• A random number system generating lots of near miss events will drive increasing levels of irrational risk taking by players

Illusion of Control

• Illusion of control occurs when a person overestimates the influence of their choices on outcomes, even when those outcomes are left to chance

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• Allowing a choice of the random number system to use such as D6, D8 or D10. This may have a valid point in your game play but can also create an ‘Illusion of control’ and irrational risk taking

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• Slot machines create an illusion of control by allowing you to hold numbers, for no statistical benefit.

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• Allowing the players a choice around random events creates a confidence in that choice with no statistical basis

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Cognitive Dissonance

• Cognitive Dissonance is the feeling of mental discomfort or stress a person may experience when they hold two or more contradictory beliefs.

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• When a player realizes they are losing or doing badly and aren’t actually as good as they think, it creates a conflict between actual reality and what the player thinks.

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• Instead of accepting this truth and just making better decisions next time, players will instead start to backwards rationalize or make impulsive and emotional responses instead of rational responses.

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• In our case typically blaming the random events in the game

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Common themes when dealing with randomness

Paired/Duplicate Games

• In sports it is one game at home and one game away against the same opponent, and the two scores are combined to get to the overall result

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• Paired/Duplicate games significantly should reduce irrational behaviour

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• In a single game:
• Players increase risk driving for an elusive win
• Players increase risk when facing defeat

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• In a paired game:
• Players will tend to avoid unnecessary risks
• Players are encouraged to make the best of a bad job when losing, rather than escalate risk levels

Conclusion

• One strategy is to use randomness to sow confusion, increase player stress, and drive irrational player behaviour and decision making

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Alternatively

• Use paired games where possible.
• Keep randomness within an easily understood range e.g. percentages
• Reduce the impact of random events upon player decision making
• Keep player decision making within the realms of strategy and tactics and not a decision to ‘play the odds!’
• Try to measure the impact of randomness in player decision making

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Confusing bad luck with bad planning

Joe Abercombie

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‘Luck is indeed where preparation meets opportunity’

Randy Pausch