1. 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

2. 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

3. 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 CooperatesDefects Bob 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 Payoffs Strategies Player

4. 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

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

6. …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

7. EXAMPLES “A series of interesting choices” = 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?

8. LET’S BE CLEAR… Alice Strategy 1Strategy 2Strategy 3 Bob 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 Strategy C Bob: 3 points Alice: 3 points Bob: 1 point Alice: 1 point Bob: 2 points Alice: 2 points 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) What will happen?

9. LET’S BE CLEAR… What will happen? 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.

10. INTERESTING? Non-interesting choices are often fun Run straight  0 points Jump  1 point

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

12. COMPETETIVENESS Cooperative game Alice RightLeft Bob 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 Fire Truck (Atari, 1978) 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

13. COMPETETIVENESS Semi-cooperative game Wizard CooperateDefect Valkyrie 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) Players have an incentive to cooperate but a temptation to defect Cooperation will be unstable

14. COMPETETIVENESS Competitive game Alice PeacefulAggressive Bob 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 Spacewar (Russel et al, 1962) Players cannot rely on each other to be peaceful There will be no cooperation

15. COMPETETIVENESS Fully competitive Semi-collaborative Fully collaborative Collective action problems e.g. 2-player non-constant sum (Prisoner’s Dilemma) Players directly opposed e.g. 2-player constant sum Players directly aligned e.g. coordination games (traffic)

16. 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

17. 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/ www.jonassmith.dk/weblog/