Gaming Routines in the Slot Machine Industry

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Gaming Routines in the Slot Machine Industry

• Noelia Oses, Ph.D.
• http//www.noeliaoses.charitydays.co.uk

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Agenda

• Introduction to the Slot Machine Industry
• Brief summary of evolution
• Analytical overview of gaming routines
• Conclusions

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The Slot Machine Industry

• Slot machines gambling machines that
• Have 3 or more reels.
• Reels spin at the start of the game (after the
  player has paid).
• If visible symbols match a winning pattern the
  machine pays out a prize.
• Controlled or Random.

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2. Brief Summary of Evolution

• Mechanical machines
• Electro-Mechanical machines
• Sound and light effects.
• Computerised machines
• Random number generator.
• Video machines
• Flexibility

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3. Analytical overview of gaming routines

• Mathematical model necessary to determine
• the percentage of the collected money it will
• return to the players in the long term.
• Main features
• Line wins
• Scatters
• Gambles
• Bonus games
• Free spins

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3.i. Line wins
SkyBetVegas Wheel of Fortune
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3.i. Line wins

• The outcome of reel spin is completely random
• No recommended strategy for players
• Economic assessment is straight forward

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3.ii. Scatters
Three or more instances of a predetermined symbol
are visible anywhere in view (not necessarily on
a payline).

• Clayvision.coms Rock n Roll

• Outcome of spin random gt no strategy
• Economic assessment straight forward

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3.iii. Gambles

• Gamble a win to double or nothing.
• does not alter the percentage return
• Hi-Lo gamble guess whether the next number in a
  sequence of positive random integers is higher or
  lower than the last
• The economic and playing analysis of the Hi-Lo
  gamble has been reported in several papers by Dr
  Jim Freeman

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3.iii. Gambles (Hi-Lo)
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3.iv. Bonus games

• Triggered by an event in the reels
• Played on the top box or in another screen
• Bingo, Poker
• General Probability
• Trails
• Markov Chains
• Sequences of offers
• Stochastic Dynamic Programming
• Markov chains (Optimal stopping of)
• Chases
• Competing random walks
• Strategy games

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3.iv. Bonuses Poker

• Poker hand contribution (2 joker deck)
• The number of hands that are a Two Pairs win
  with no jokers is
• The probability is

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3.iv. BonusesExtra Balls Bingo
m Hits
Probability P( m hits )

• N Bingo numbers.
• Player selects K.  
• D numbers drawn at random.
• Prizes paid according to the win plan.
• If prize is not the best or worst prize then an
  additional 2 balls are drawn.
• Player pays 1 credit per game.
• Distribution Version of hypergeometric

Extra matches P( xm m )
Total Probability of m hits
Contribution ()
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3.iv. Bonuses Trails

• Modelled as Markov Chains
• Prize can only increase play till the end
• Throw die
• Advance along
• Until falling on collect
• Or maximum number of moves

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3.iv. Bonuses Sequences of offers

• Several prizes are offered in sequential order
• Player, at each stage, decides whether to take
  the prize on offer or reject it.
• The player does not have the option of choosing
  to take previously rejected prizes.
• The maximum number of offers that the player can
  have is limited and predetermined.
• The last offer, if reached, is auto-collected.
• The prize on offer can decrease therefore player
  needs strategy

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3.iv. Sequences of offers (I)

• E.g. Top Dollar
• 4 offers X1 , , X4 I.I.D
• At each stage
• the player either accepts the offer and the game
  ends
• or goes on (in the 4th stage he doesnt have a
  choice)
• Maximise expected value
• Stochastic dynamic programming.

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3.iv. Sequences of offers (II)

• E.g. Bitz Pizzas
• One selected at random
• 4 spins of the reel
• Plus One
• Plus Two
• Change One
• Change All
• Markov Chain (optimal stopping Bather)

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3.iv. Bonuses Chases

• Player vs. something
• Competing Random Walks (non-Markovian)
• Player throws die first
• Monster next
• Finishes when
• one catches the other or
• max number of moves completed
• Bonus prizes catch, start and max moves

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3.vi. Bonuses Strategy games

• Player must make decisions at each stage
• E.g.
• Battleships
• Stochastic Dynamic Programming

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3.v. Free Spins

• When free spins can be won from within free spins
• Where
• p1 is the total probability of initiating free
  games in a normal spin
• n1 is the expected number of free games given
  that the player has won free games
• p2 is the probability of re-triggering free spins
  inside the free spins
• n2 is the expected number of free games given
  that the re-trigger has occurred

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3.v. Free Spins (II)

• If the total number of free spins that can be won
  in one go is limited then
• Where (n1n2N) Total number of
• free spins that can be won in one go (constant of
  the game).

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3.v. Free Spins (III)

• The total percentage return ( ) is
• is the percentage return of the base game,
  calculated without considering the value of the
  free games.
• is the percentage return of the bonus
  (free spin) games.

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Note

• Monte-Carlo simulation is used to double-check
  the results of the calculations
• Graphical display
• Fast simulation

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4. Conclusions

• The slot machine industry provides practical
  examples of Operational Research and Stochastic
  Processes applications.
• As the industry evolves, the games will become
  more sophisticated and, almost certainly, more
  interesting to study.
• Video Slots can have virtually any game as a
  feature
• Must be possible to calculate the expected value