Game Theory Conflict Robin Burke GAM 224 Fall

Game Theory Conflict Robin Burke GAM 224 Fall 2005

Outline ¢ ¢ Admin Quiz Game Theory l Utility theory l Zero-sum and non-zero sum games l Decision Trees l Degenerate strategies Conflict l Types of goals l Systems of conflict

Admin ¢ Due Monday Rule paper l Hard copy in class l turnitin. com l ¢ Assassin starts midnight Friday pick up player kits from Prof. Wilcox or l wait until Monday l

Ernest Adams ¢ ¢ ¢ Talk l 10: 30 – 12 noon l go to talk and write reaction paper l substitute for any other reaction paper Game design workshop l 1 – 4 pm l CTI students only, preference to GAM majors l work with other students / faculty l apply his techniques to develop game ideas l personal feedback from the expert For workshop l send email to jtrinkle@cs. depaul. edu l must attend talk to go to workshop

Quiz ¢ 30 minutes

Game Theory ¢ ¢ ¢ A branch of economics Studies rational choice in a adversarial environment Assumptions l l rational actors complete knowledge • in its classic formulation l l known probabilities of outcomes known utility functions

Utility Theory ¢ Utility theory a single scale l value with each outcome l ¢ Different actors may have different utility valuations l but all have the same scale l

Expected Utility ¢ Expected utility what is the likely outcome l of a set of outcomes l each with different utility values l ¢ Example l Bet • $5 if a player rolls 7 or 11, lose $2 otherwise l Should you take this bet?

How to evaluate ¢ Expected Utility l for each outcome • reward * probability l ¢ Meaning l ¢ (1/6) * 5 + (1/18) * 5 + (7/9) (-2) = -2/9 If you made this bet 1000 times, you would probably end up $222 poorer. Doesn't say anything about how a given trial will end up l Probability says nothing about the single case

Game Theory Examine strategies based on expected utility ¢ The idea ¢ l a rational player will choose the strategy with the best expected utility

Example ¢ ¢ ¢ Non-probabilistic Cake slicing Two players l l cutter chooser Cutter's Utility Choose bigger piece smaller piece Cut cake evenly ½ - a bit ½ + a bit Cut unevenly Small piece Big piece

Rationality ¢ Choose bigger piece smaller piece Cut cake evenly (-1, +1) (+1, -1) Cut unevenly (-10, +10) (+10, -10) Rationality l l ¢ Both Utilities each player will take highest utility option taking into account the other player's likely behavior In example l if cutter cuts unevenly • he might like to end up in the lower right • but the other player would never do that • -10 l if the current cuts evenly, • he will end up in the upper left • -1 • this is a stable outcome • neither player has an incentive to deviate

Zero-sum ¢ Note l for every outcome • the total utility for all players is zero ¢ Zero-sum game l l something gained by one player is lost by another zero-sum games are guaranteed to have a winning strategy • a correct way to play the game ¢ Makes the game not very interesting to play l to study, maybe

Non-zero sum ¢ A game in which there are nonsymmetric outcomes l ¢ better or worse for both players Classic example l Prisoner's Dilemma Hold Out Confess Hold Out [-1, -1] [-3, 0] Confess [-5, -5] [0, -3]

Degenerate Strategy ¢ A winning strategy is also called l ¢ Because l l ¢ a degenerate strategy it means the player doesn't have to think there is a "right" way to play Problem l l game stops presenting a challenge players will find degenerate strategies if they exist

Nash Equilibrium ¢ Sometimes there is a "best" solution l ¢ A Nash equilibrium is a strategy l l ¢ Even when there is no dominant one in which no player has an incentive to deviate because to do so gives the other an advantage Creator l l l John Nash Jr "A Beautiful Mind" Nobel Prize 1994

Classic Examples ¢ Car Dealers l l Why are they always next to each other? Why aren't they spaced equally around town? • Optimal in the sense of not drawing customers to the competition ¢ Equilibrium l l because to move away from the competitor is to cede some customers to it

Prisoner's Dilemma ¢ Nash Equilibrium l ¢ Because l ¢ Confess in each situation, the prisoner can improve his outcome by confessing Solution l l l iteration communication commitment

Rock-Paper-Scissors Player 2 Player 1 Rock Paper Scissors Rock [0, 0] [-1, +1] [+1, -1] Paper [+1, -1] [0, 0] [-1, +1] Scissors [-1, +1] [+1, -1] [0, 0]

No dominant strategy ¢ Meaning l there is no single preferred option • for either player ¢ Best strategy (single iteration) l choose randomly l "mixed strategy" l

Mixed Strategy ¢ ¢ Important goal in game design Player should feel l all of the options are worth using l none are dominated by any others Rock-Paper-Scissors dynamic l is often used to achieve this Example l Warcraft II • • Archers > Knights > Footmen > Archers must have a mixed army

Mixed Strategy 2 ¢ ¢ Other ways to achieve mixed strategy Ignorance l If the player can't determine the dominance of a strategy • a mixed approach will be used • (but players will eventually figure it out!) ¢ Cost l Dominance is reduced • if the cost to exercise the option is increased • or cost to acquire it ¢ Rarity l Mixture is required • if the dominant strategy can only be used periodically or occasionally ¢ Payoff/Probability Environment l Mixture is required • if the probabilities or payoffs change throughout the game

Mixed Strategy 3 ¢ In a competitive setting l l ¢ mixed strategy may be called for even when there is a dominant strategy Example l l l Football third down / short yardage highest utility option • • • ¢ But l if your opponent assumes this • l ¢ running play best chance of success lowest cost of failure defense adjusts to prevent a run increasing the payoff of a long pass But l if you are behind in a close game and time is short • • l payoff changes because an incomplete pass stops the clock opponent may not defend the run

Degeneracies Are not always obvious ¢ May be contingent on game state ¢

Example ¢ Liar's Dice l l roll the dice in a cup state the "poker hand" you have rolled stated hand must be higher than the opponent's previous roll opponent can either • accept the roll, and take his turn, or • say "Liar", and look at the dice l if the description is correct • opponent pays $1 l if the description is a lie • player pays $1

Lie or Not Lie ¢ Make outcome chart for next player l assume the roll is not good enough l ¢ Roller l ¢ lie or not lie Next player l accept or doubt

Expectation ¢ Knowledge the opponent knows more than just this l the opponent knows the previous roll that the player must beat l • probability of lying

Note ¢ The opponent will never lie about a better roll l ¢ Outcome cannot be improved by doing so The opponent cannot tell the truth about a worse roll l Illegal under the rules

Expected Utility ¢ ¢ What is the expected utility of the doubting strategy? l P(worse) - P(better) When P(worse) is greater than 0. 5 l doubt Probabilities l pair or better: 95% l 2 pair or better: 71% l 3 of a kind or better: 25% So start to doubt somewhere in the middle of the twopair range l maybe 4 s-over-1 s

BUT ¢ There is something we are ignoring

Repeated Interactions Roll 1 doubt Truth Lie doubt accept Win Lose Roll 2 doubt Lie doubt Truth accept doubt Roll 1 Truth Lie accept Roll 2

Decision Tree ¢ ¢ ¢ Examines game interactions over time Each node l Is a unique game state Player choices l create branches Leaves l end of game (win/lose) Important concept for design l usually at abstract level l question • can the player get stuck? ¢ Example l tic-tac-toe

Future Cost ¢ There is a cost to "accept" l l ¢ To compare doubting and accepting l ¢ I may be incurring some future cost because I may have to lie and get caught we have to look at the possible futures of the game In any case l the game becomes degenerate • the player must doubt

Conflict ¢ Somewhat obvious l "artificial conflict" is part of the definition

Struggle to achieve a goal ¢ Single player vs. single player l ¢ Group vs. group l ¢ Basketball, Soccer, Battlefield 1942 One against many l ¢ Chess, Boxing, Warcraft II Tag, Most action-adventure and FPS games Every man for himself l Marathon, Risk, Mario Kart

Struggle to achieve a goal ¢ Single player vs. computer l ¢ Group of single players vs. game l ¢ Tetris Blackjack Group against the game Lord of the Rings board game l Cooperative mode in Star Wars Battlefront, etc. l

Game goal May not be fixed ¢ May not be obvious from the game's premise ¢ Single game may support multiple goals ¢ l from the player's point of view

Example: Centipede ¢ Single player mode l l l ¢ Two player mode l l ¢ Do well against the game Beat your personal best Get on the high score list Beat your opponent Get on high score list Compete against friend to get on the high score list

Example: Joust ¢ Single player mode l ¢ Players compare scores Two player mode One player against the other l Battle mode l Cooperation mode players vs. computer l

Example: Gauntlet Single player mode ¢ Two player mode ¢ Cannot attack each other l Real time display of scores l Players compete for “food” l Allow most needy to eat or be greedy l Compete for spending money l

Competitive vs. cooperative All games are competitive ¢ Can they be cooperative at the same time? ¢ Adherence to the Magic Circle l Players on a Basketball team l Conflict within a cooperative framework l

Systems of conflict ¢ Games exhibit conflict at multiple levels l ¢ Local goals contribute to larger goals Conflicts at different levels work together to create a system l the interrelatedness of the conflicts determines if any given conflict seems meaningful

Example: Legend of Zelda: The Wind Waker ¢ Largest conflict l ¢ Intermediate conflict l ¢ evil sorcerer, Gannon defeat level Local conflict solve puzzle l defeat particular enemy l master particular skill l

Wind Waker, cont'd ¢ Conflict system l each level is crafted to introduce new skills • which are essential to succeeding in this and later levels l each level features enemies with certain common characteristics • essential to defeating the boss ¢ Game logic links conflicts l l l hero must collect pearls to defeat Gannon pearls are obtained by defeating a boss enemy boss enemies are found at the innermost room of each dungeon exploration of the dungeon requires defeating enemies and solving puzzles exploration also requires weapons, tools and power-ups items found by exploring the dungeon

Conflict system may fail ¢ Game lacks meaningful play l ¢ if players do not perceive the links between conflicts Example l Kingdom Hearts • goal is to protect various "worlds" • travel between worlds is done through a 3 -D space shooting game • no coherence between this local conflict and the global one • it feels tacked on and rather pointless

Cooperative games? ¢ "New Games" Movement l Dragon • conflict between head and tail • mediated by players with uncertain stakes l Stand-up • conflict with gravity / stability ¢ UN Food Force game l l mission = help refugees conflict with terrain, elements, limited budgets, logistics

Monday ¢ Analysis Case Study l Legend of Zelda: Wind Waker