Chris Hazard - Modeling, Math and Science for Building Strategic Serious Games

Modeling, Math and Sciencefor Building Games that Improve

Organization Operation,and Workforce Effectiveness

Christopher J. Hazard, PhD

Christopher J. Hazard, PhD August, 2016 2

Hazardous Software Serious Games


Image from user boysdean on Flickr.com

Image from user BLANCOBILL on TripAdvisor.com

Who is a Gamer?


Play = immersion + learning +

minimized actual risk + time travel


Simulation-Based Serious Games

Close Combat – Modern Tactics, Matrix Games

CyberCIEGE, NPS & Rivermind

EteRNA, CMU


More Serious GamesWith OR Aspects

Code of Everand, UK Department for TransportMMORPG, 2009-2011

Cargo Dynasty, Serious Games Interactive,TSU, TUR

Wildfire game, Lincoln Labs


Not aboutVirtual Worlds &

Chocolate Covered Broccoli

Second Life


How Different From M&S?

• Have human-centric interfaces• Focus on usability• Focus on exercise deployability• Focus on creating & managing reusable

scenarios• Focus on realistic communication &

controls• Have AAR, AI, help, and tutorials

integrated/embedded


Implicit Grinding(and optimal downtime)

Just Cause 2

Niel de la Rouviere, Stellenbosch University


Adaptive vs Choice vs Fixed Content

Choice of content Adaptive content

D. Sharek PhD dissertation at NCSU, 2012. Investigating Real-time Predictors of Engagement:Implications For Adaptive Video Games and Online Training.


Humans Are Rational*

*given limited computational bounds, strong heuristics, poor probabilistic reasoning, unfounded beliefs of others, inaccurate capability assessments, inexplicable valuations, and some level of [im]patience


Utility & Currency

• Common currency: average-player time– Skilled players & devoted players have most

• Find exchange rates for everything– If items purchasable in $, find exchange

between player time and $

• Find amortization / discount rate


Skill, Strategy, & Information Gain• Skill

– Driven by capabilities, signaling, reputation– Measured using statistics, hindsight

• Strategy– Driven by preferences (valuations),

sanctioning, trust– Solved using game theory, foresight

• Information Gain– Driven by immersion, curiosity, relevance– Provided via narrative, setting, instruction, cues


Keynesian Beauty Pageant:Guess 2/3 the average

• Everyone choose number [1,100]• Closest to 2/3 the average wins

Image from thedigeratilife.com


A Simple Game...

• Strategist• Negotiator• Artist• Logician (e.g., programmer/lawyer)• Impulsivist or risk seeker• Risk avoider


Bidding Game Rules

Card is cost:

A: 1

2: 2

3: 3

…

J: 11

Q: 12

K: 13

• Bid each round• Winning bidder gets

price – cost• Highest profit wins


Bidding Game Results

• 3-4 rounds to "convergence"• Generally considered "unfair"• Bayesian Nash Equilibrium!

– Big reveal of same card: surprise– Lack of reveal: anchoring and bias hook


NASCAR: Drafter's Dilemma

• Red ahead, Blue behind, leave line together

• Payoff = number of cars passed

• Cooperate = allow other to jump back in line

• Defect = jump back in line without the other

Ronfeldt, First Monday J., '00

Cooperate Defect

Cooperate 3 3 -5 3

Defect 2 -5 1 1


Mixed Strategy & Risk

• Intransitivity• “Every unit overpowered”• Forced risk

P S

R 0, 0 -1, 1 1, -1P 1, -1 0, 0 -1, 1S -1, 1 1, -1 0, 0

Street Fighter 4


Payoff, Risk, Commitment

Stag Hare

Stag 10 10 0 8

Hare 8 0 7 7

Swerve Straight

Swerve 0 0 -1 +1

Straight +1 -1 -1000 -1000


Creeping Sniper's Dilemma

Original image from ShadowShield.com


Creeping Sniper's Dilemma

Single sniper position, σ, as a function of time:

• Multiple sniper: match quickest visible discount strategy unless too risky

Pos

ition

of

Sni

per

Near Target

Far Target


Operations Research: Lanchester's LawsGang of N units vs 1, all with sufficient action range

X DPS, Y health

N each retain Y (1 – 1/N^2)

Original image from XCOM: Enemy Unknown


Balancing With Game Theory: Strength and Utility

Hammer Spear Curse

Hammer 1 3 0.5

Spear 0.33 1 0.5

Curse 2 2 1 Hammer Spear Curse

Hammer 0.000 -0.043 0.095

Spear 0.043 0.000 -0.070

Curse -0.095 0.070 0.000Cost

Hammer 0.23

Spear 0.56

Curse 0.21

S (strength: # of player 1 to defeat player 2)

C (cost)

U (utility)

One player loses all utility, another fractionSpear vs Hammer:

gain - loss0.23 - (1/3 * 0.56)

Symmetric!


Balancing With Game Theory: Probabilities

Hammer Spear Curse

Hammer 0.000 -0.043 0.095

Spear 0.043 0.000 -0.070

Curse -0.095 0.070 0.000

U (utility)Probability

Hammer 0.336

Spear 0.456

Curse 0.208

P (probability)

Probability

Hammer 0.333

Spear 0.334

Curse 0.333

P (probability)Cost

Hammer 0.255

Spear 0.545

Curse 0.200

C (cost)Hammer Spear Curse

Hammer 0.000 -0.073 0.073

Spear 0.073 0.000 -0.073

Curse -0.073 0.073 0.000

U (utility)


Ambiguity as an Interestingness Measure

• Find Nash equilibrium– 20% sniper rifle, 30% machine gun, 50% shotgun– 33% sniper rifle, 33% machine gun, 34% shotgun

• Control tightness– Ambiguity vs predictability of next game states

(discounted)

• Difficulty of puzzles & optimal strategy ascertainment– Some ambiguity good, too much boring


Learning & Information Gain

• Measure information gain between player strategy and optimal

• Mixed strategy Nash equilibria– 1/3 rock, 1/3 paper, 1/3 scissors

• How much information left to teach player?– 1/4 rock, 1/4 paper, 1/2 scissors– Info gain to achieve desired Nash equilibrium


Complexity of Behavior


Information Conveyance


Corpse PartyChapter 1 Infirmary


Infirmary Flow

take match from furnace

try door

try door

try match

try match

get rubbing alcohol

try door

exit

• Actual branching factor: 12• Perceived branching factor: 11• Exaggerated expectation

[Hilbert, PSYCHOL BULL '12]

– P(progress | revisit item) higher than anticipated


Infirmary Surprisal• Player unsure of what to do, so assume

uniform distribution over new possibilities:Q(X) ≈ 1/11, Q(Repeat) ≈ 0 => ~3.5 bits

• Correct distribution over possibilities, minimizing assumptions: P(X) = 1/12

•

Q(repeat) ≈ 0 means1/12 * ln( (1/12) / 0) = 1/12 * ln(∞) = ∞

Massive surprisal if assume no repeat actions advance game


Measuring Difficulty By Decision Information Rate

X X

X3 out of 6 paths lose

1

11

0

0 No loss, no information

Average 1 bit of information

Average 0.5 bits of information

1.5 bits of total information to win

1.5 bits / 2 steps = 0.75 bits per step to win


Mutual Exclusion Between Mechanical and Social Reasoning(Jack et al., Neuroimage, 2012)

Working Memory Capabilities & Affective Control(Schweizer et al., J Neuroscience, 2013)


Time Manipulation in Gaming• Time zones • Reverse time

ChronoTrigger Braid

• Fixed jump back• Time loop

Ratchet & Clank

Majora'sMask


Time Manipulation Transforms Gameplay

Obstacle/Combat Course (FF12) Maze

Gran Turismo Sudoku (Optimization)

+Undo

+Timeline


Time Manipulation & Causality• Dynamically correct plans: blur hypothetical &

committed

• Long-term thinking about decisions

• Just in time vs redundancy

• Minmax & Nash equilibria

• Qualitative sensitivity analysis: plan fragility

• “Newton's Method” of strategy


Time Manipulation Game Mechanics• “Chronoenergy”: causality as a resource

– Locality & change magnitude– What is a unit of causality?

• Player's intention vs low-level control

• AI to assist “when you're not then”

• Collaborative planning


Desirability Index• Desirability Index (geometric mean of

conflicting metrics) in multicriteria optimization:

• Used for optimization in chemistry, chemical engineering, mechanical engineering

• Related to Shannon Entropy Maximization• Easy to relate to output as a score, hard to

“game”


Understanding Probability Distributions


Little’s Law, MDPs & More OR

• L = λW to measure expected length of queue by wait time

• MDPs for modeling, visualization into process

• MILP, Pareto Frontier


Questions?

[email protected]

Chris Hazard - Modeling, Math and Science for Building Strategic Serious Games

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