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Science of Fun. Ten Billion $/year from Six Thousand Slot Machines. Foxwoods Casino and Resort. What makes slots fun?. Pared-down Poker: Cutting to the Core of Command and Control. Proc. of IEEE Symposium on Computational Intelligence and Games .

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Ten Billion $/year from Six Thousand Slot Machines

Foxwoods Casino and Resort

What makes slots fun?

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Pared-down Poker: Cutting to the Core of Command and Control.

Proc. of IEEE Symposium on Computational Intelligence and Games.

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Analyzing humor (and fun) is like dissecting a frog. Control.

Few people are interested and the frog dies of it....

E.B. White

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Expected Utility Control.≡ Probability * Utility

“fair slots”

1*A = P*J

A = Anted

J = Jackpot

P = Prob.

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The $20 Question Control.

You have a choice between:

(s) a sure-thing of getting $20;

(g) a gamble with 20% chance of getting $100

and 80% chance of getting nothing.

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The $80 Question Control.

You have a choice between:

(s) a sure-thing of getting $80;

(g) a gamble with 80% chance of getting $100

and 20% chance of getting nothing.

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De Martino et al. (2006). “Frames, Biases and Rational Decision

Making in the Human Brain”. Science, Vol. 313, pp 684-687.


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Kahneman and Tversky (1979). “Prospect Theory: An Analysis of

Decision Under Risk”. Econometrica, Vol. 47, No. 2, pp 263-291.

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Apparent Contribution from Aesthetic Utility = W - P of

On a crusade in search of the Holy Grail… It’s Fun!

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The Atoms of EVE’: A Bayesian Basis for Aesthetic Analysis…

Artificial Intelligence for Engineering Design, Analysis and Manufacturing (AIEDAM)

Expectation – Violation – Explanation

Set-up – Punchline – Get-it?

Garden – Eaten – Tragedy?

Analyzing humor

is like dissecting a frog.

Few people are interested…

and the frog dies of it.


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Per EVE’, Analysis…Fun (S) is the sum of:

pleasure (p) from success in forming

Expectations (E)


pleasure (p’) from success in forming

Explanations (E’)

S = G*E + G’*E’

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Expectation (E) Analysis…

Win (P=Prob)

Loss (Q=1-P)

E = P * log P + Q * log Q

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Entropy of an event with Prob. P is defined as: Analysis…

Unpredictability = Probability * Unexpectability

P * -log P

Entropy = (-P * log P) + (-Q * log Q)

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Measure of Expectation (E) Analysis…

This E is a negative entropy

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Violation (V) Analysis…

V = -E

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Explanation (E’) Analysis…

E’ = - [H+ * P * log P * R+ ] + [H- * Q * log Q * R-]

H+/H- = sense of humor

R = Bayesian resolution

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Bayesian Belief ( Analysis…ρ is a Prob.)

posterior = prior * likelihood

R = ρ (Y|e) = ρ (Y) * ρ (e|Y)

e is evidence: win or loss

Y is hypothesis: “good luck” or “bad luck”

For slots, posterior = 1; R=+1 (win) or -1 (loss)

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Measure of Explanation (E’) Analysis…

H+/H- = 2

H+/H- = 1

Σ signed/weighted entropies

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Fun = Analysis…S = “S-thetic” utility

S = (G * E) + (G’ * E’)

G’/G = call to adventure

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Fun = Analysis…S = “S-thetic” utility

Goldilocks Function: S = G*E + G’*E’

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Assumed by Prospect Theory Analysis…

Derived by EVE’s Theory

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Twenty-one Bell Three Wheel Nickel Slot Machine (Rake = 6%). Analysis…

Has 8 different payoff combinations/probabilities, w/ Pagg = 13%.

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Game of Skill (Punch Out) Analysis…

WPI Data

EVE’ Model

Relating Cognitive Models of Computer Games to User Evaluations of Entertainment.

P. Piselli, Masters Thesis, WPI Department of Computer Science, 2006.

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EVE’s Fun Functions Analysis…

E = log P

V = -log P

E’ = -log P * H * R

EVE’s Fun Factors

H+/H- = sense of humor

G’/G = call to adventure

F = price for pleasure

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Shlomo Dubnov Analysis…

“Thoughts About Memex”.

Memex Music

Memex, the machine (Bush 1945), was a futuristic device,

For creating and recalling associations –

In the form of memory trails.

Memex, the music (Dubnov 2006), is an algorithmic composition,

Designed to create new music from old music –

By associations along probabilistic trails.

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Let’s say that the current note in Memex is G - taken from Bach. To get the next note:

The machine will step forward with probability Q - or jump backward with probability P

Where jump backward is to the same note (different song) with “most similar” history.

Bach: … C D F E C G C…



Beethoven: …A F E C GA C…


Mozart: … D C G B B A …


If it steps, the next note is C.

If it jumps, the next note is A.

Beethoven’s 4 > Mozart’s 2.

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Fun (flow) function computed by EVE’ with same G’/G as for slots.

All E’ was assumed to be positive and Resolution (R) was set to Q2.

P was tweaked by the human creator until the machine composition

sounded “best”, which turned out to be much like slots – a P of 13%.

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EVE’s Entropy: A Formal Gauge of Fun in Games for slots.. In:

Advanced Intelligent Paradigms in Computer Games.


Imagine a community with thousands of people sitting at machines playing games for hours. What makes it fun? Is it virtual reality? Is it engaging narrative? Is it multiplayer interaction? Actually, it’s none of the above. The community is Foxwoods and the machines are slots. I’ll bet that slots are the most popular and profitable machine game of all time – more than any modern computer game. I also think that research and development in digital media has not done much to advance a scientific understanding of fun in any game. So I cut to the chase, dissecting the aesthetic experience using mathematical analyses and psychological experiments. I look at gambling, music and artwork. I show how formal notions of Bayesian probability and Shannon entropy can explain and predict feelings of pleasure. I have some demos to make the math fun. I guarantee you have never seen fun like this before.

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Historical for slots.

Ch 1: “Slots of Fun, Slots of Trouble:

An Archaeology of Arcade Gaming”


“Rise of Aesthetics”


Ch 24: “Games as the Play of Pleasure”

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Psychological for slots.

“Optimal Experience”




Pg 46: “Fun is just another word for learning”.

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Informational for slots.

Shannon Theory


Bayesian Theory


Prospect Theory

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Mathematical for slots.



Computing Comedy


Causality and Probability

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On TRACS: Dealing with a Deck of Double-sided Cards for slots.

Proc. IEEE Symposium on Comp. Intelligence and Games.