I THINK THAT YOU THINK THAT I THINK - GAME THEORY AND VIDEO GAMES Jonas Heide Smith (smith@itu.dk) Game Analysis ITU, 18.04.2006

“If I had my way, a solid grounding in economics would be required of anyone seeking to learn about game design” - Greg Costikyan

Shall we play… a game? Two players • Each player chooses either ”Nice” or ”Nasty” • Points: – Nice/Nasty: Nice gets 0 points, Nasty gets 5 points – Nice/Nice: Both get 3 points – Nasty/Nasty: Both get 1 point

• Variation: 10 round version

THE PRISONER’S DILEMMA The most famous of all game theory “games” The object of thousands of studies Models a “social dilemma”; a situation where one is tempted to do something which would lead to disaster if everyone made the same choice Alice

Player

Cooperates

Defects

Cooperates

Bob: 3 points Alice: 3 points

Bob: 0 points Alice: 5 points

Defects

Bob: 5 points Alice: 0 points

Bob: 1 point Alice: 1 point

Bob

Payoffs Strategies

WHAT IS GAME THEORY? A series of techniques for modeling strategic interaction (not a theory and not about games) Models social situations as if they were games Long history but formalized within economics and later spread to many fields Where (neo)classical economics deals with situations in which individuals simply act on their preferences, game theory deals with situations where the choices of others affect outcomes

MODELING GAMES Two model types: Strategic or extensive form Two game types: Parametric (games against nature) or strategic Two sub-areas: Cooperative and noncooperative game theory Two sum types: Zero-sum and non-zero sum Two basic theory types: Social and evolutionary Two approaches: Analytical and behavioural

…AND VIDEO GAMES Game theory is • a systematic version of influential game design thinking • über-ludological (focuses on conflict and choice at the expense of other perspectives) • able to provide a general understanding of strategic interaction as well as more directly applicable tools for tweaking game dynamics

A B

EXAMPLES â€œA series of interesting choicesâ€? = A game should have neither dominant nor dominated strategies (Rollings and Morris) = The game designer should not include options that will not be used (given standard economics assumptions) = Games should have no pure strategy Nash equilibria (only mixed strategy equilibria) What do you think?

LETâ€™S BE CLEARâ€Ś Alice

Bob

Strategy 1

Strategy 2

Strategy 3

Strategy A

Bob: 1 point Alice: 5 points

Bob: 0 points Alice: 1 point

Bob: 1 point Alice: 3 points

Strategy B

Bob: 5 points Alice: 0 points

Bob: 0 points Alice: 0 points

Bob: 3 points Alice: 1 point

Bob: 3 points Alice: 3 points

Bob: 1 point Alice: 1 point

Bob: 2 points Alice: 2 points

Strategy C

What will happen?

1. No matter what Alice plays, Bob would be better of playing C than A. 2. It is clear to Bob that Alice will never play her Strategy 2 (since 3 is always better). 3. Bob can confidently choose Strategy B as it always yields a better result than C 4. Alice chooses 3 and Bob chooses B (A and 2 are dominated by C and 3)

LETâ€™S BE CLEARâ€Ś When there are only two agents on a patch of grass, each of the agents tries to eat a certain amount of the grass. There are fair agents (red), modest agents (brown), and greedy agents (blue). Fair agents try to eat half the grass, modest agents try to eat a third of the grass, and greedy agents try to eat two-thirds of the grass. If the total amount requested by both agents is greater than 100%, then both agents die. Otherwise, each agent gets his requested share of the patch's resources.

What will happen?

INTERESTING?

Non-interesting choices are often fun Run straight 0 points Jump 1 point

DOMINANCE Only applies to games where players are interdependent These fall into three types which shape/determine behavior

COMPETETIVENESS Cooperative game Alice

Right

Left

Right

Bob: 1 point Alice: 1 point

Bob: 0 points Alice: 0 points

Left

Bob: 0 points Alice: 0 points

Bob: 1 points Alice: 1 points

Bob

•Players’s utility functions are identical (they even have a collective score count). •From the perspective of either, hurting the other player means hurting oneself •Players will cooperate fully

Fire Truck (Atari, 1978)

COMPETETIVENESS Semi-cooperative game Wizard

Valkyrie

Cooperate

Defect

Cooper ate

V: 2 points W: 2 point

V: 0 points W: 3 points

Defect

V: 3 points W: 0 points

V: 1 points W: 1 points

Gauntlet (Atari, 1982) â€˘Players have an incentive to cooperate but a temptation to defect â€˘Cooperation will be unstable

COMPETETIVENESS Competitive game Alice

Peaceful

Aggressive

Peaceful

Bob: 0,5 point Alice: 0,5 point

Bob: 0 points Alice: 1 points

Aggressive

Bob: 1 points Alice: 0 points

Bob: 0,5 points Alice: 0,5 points

Bob

•Players cannot rely on each other to be peaceful •There will be no cooperation

Spacewar (Russel et al, 1962)

COMPETETIVENESS

Players directly opposed e.g. 2-player constant sum

Fully competitive

Collective action problems e.g. 2-player non-constant sum (Prisonerâ€™s Dilemma)

Players directly aligned e.g. coordination games (traffic)

Semi-collaborative

Fully collaborative

THE BIGGER PICTURE Game theory systematically addresses a crucial aspect of games (and has been all but ignored by game studies) Economics can inform game design and vice versa Game theory formalizes game design principles and can be used to test them

FURTHER READING Binmore, K. (1991). Fun and Games: A Text on Game Theory. Lexington-Toronto: D.C. Heath. Axelrod, R. (1984). The Evolution of Co-operation. London: Penguin Books. Smith, J. H. (Awaiting publication). The games economists play - implications of economic game theory for the study of computer games. Draft at: www.jonassmith.dk/weblog/