The Game Theory of Game Theory Ruben R. Puentedura, Ph.D.
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1 The Game Theory of Game Theory Ruben R. Puentedura, Ph.D.
2 Why Study Game Theory For Game Creation? Three key applications: For general game design; For social sciences-specific game design; For understanding human interaction in game spaces.
3 Basic Definitions: The Two-Player, Zero-Sum Game Player B Player A Strategies i ii iii I II III Strategy Profile: (III, ii) Payoff: A gains 3, B loses 3
4 A Simple Example: A Throw Fingers -Type Game Player B # of fingers thrown Player A
5 Definition: Dominating Strategy Strategy I dominates strategy II if the player s payoff for strategy I is always greater than or equal to the payoff for strategy II If the payoff is always strictly greater, strategy I is said to strongly dominate strategy II Otherwise, strategy I is said to weakly dominate strategy II
6 Definition: Nash Equilibrium A strategy profile is a Nash equilibrium if no player can unilaterally improve their payoff by changing their strategy, while the rest of the profile remains the same More than one Nash equilibrium may exist for a given game
7 A Political example: Should Ethanol Be Subsidized? Democrats Debate Strategies Republicans Favor Oppose Dodge Favor 45% 50% 40% Oppose 60% 55% 50% Dodge 45% 55% 40%
8 Another Political example: Should Head Start Be Subsidized? Democrats Debate Strategies Republicans Favor Oppose Dodge Favor 35% 10% 60% Oppose 45% 55% 50% Dodge 40% 10% 65%
9 An Example of Multiple Equilibria: Lord Marshes and Duke Meadows Duke Meadows Each Chooses: Grass Mud Track Hounds Horses Lord Marshes
10 Another Throw Fingers -Type Game Player B # of fingers thrown Player A
11 We Need To Construct a Mixed Strategy
12 A Mixed Strategy Example: The Convict and the Sheriff Sheriff Route Chosen Highway Forest Highway 0 1 Forest Convict
13 What Should the Convict and the Sheriff Do?
14 The Minimax Theorem Every finite two-person zero-sum game has at least one optimal mixed strategy, and therefore has a value V, which is the average amount that one player can expect to win, and the other expect to lose, if both players act sensibly. (John von Neumann, 1928)
15 Non-Zero Sum Games Player B Player A Strategies i ii iii I (1,-1) (-3,4) (1,-2) II (2,2) (5,4) (-1,-2) III (0,0) (3,5) (-2,-1) Strategy Profile: (III, ii) Payoff: A gains 3, B gains 5
16 Example: Big Ape and Little Ape Bobo King Kong Climbs tree Waits Climbs tree (5,3) (4,4) Waits (9,1) (0,0)
17 Each Ape Sees Their Payoff As: King Kong Bobo Bobo Climbs tree King Kong Bobo Climbs tree Waits Climbs tree 3 4 Waits 1 0 Waits Climbs tree 5 4 Waits 9 0 King Kong A Third (Mixed Strategy) Equilibrium: King Kong climbs 1/2 the time, waits 1/2 the time - Payoff 4.5 Bobo climbs 1/2 the time, waits 1/2 the time - Payoff 2
18 The Nash Existence Theorem Every finite n-player game has at least one Nash equilibrium in mixed strategies. (John Nash, 1950)
19 Speeding and the Community (Hamburger, 1979) Community Enforce Violate (-190,-25) Ignore (10,-5) Driver Comply (0,-20) (0,0)
20 What If the Community Uses a Mixed Strategy? Community Enforce Violate (-190,-25) Ignore (10,-5) Driver Comply Violate (0,-20) Community Community Community 6% Enforce 94% Ignore 5% Enforce 95% Ignore 4% Enforce 96% Ignore (-2,-6.2) Driver Violate (0,-6) Driver Comply (0,0) (0,-1.2) Violate (2,-5.8) Comply (0,-0.8) Driver Comply (0,-1)
21 The Prisoner s Dilemma: A Political Example City B City A Support A s Bond Issue Do Not Support A s Bond Issue Support B s Bond Issue (8,8) (-1,9) Do Not Support B s Bond Issue (9,-1) (0,0)
22 Tit-For-Tat Is Not Enough (Puentedura 2002) 100 AllD ATfT Random 75 TfT AllC time } } } Noncoop Strategy Random Strategy Coop. Strategy
23 What Does Work In Generating Cooperation? There exist several possibilities: Group Selection : operates if groups with more cooperators do better than groups with more noncooperators (Price 1970) Tagging : allows cooperators to recognize noncooperators
24 How Do People Actually Play? People respond more to what they perceive in their opponent, than to equilibrium strategies (e.g., Brayer 1964) In isolated scenarios, noncooperation frequently wins out over cooperation (e.g., Scodel et al. 1959) In social scenarios, excessive greed is punished (e.g. ultimatum game - Henrich et al. 2001)
25 Results From Henrich et al. (2001) For industrial societies: mean offer: 0.44 modal offer: % rejections:
26 MMORPG Player Types (Bartle 1996) Acting Killers (less common) Achievers (more common) Players World Socializers (more common) Interacting Explorers (less common)
27 How Do They Interact? Increase Decrease
28 Four Stable Game Configurations Result: Type 1: Killers and Achievers in equilibrium, with hardly any Socializers and Explorers Type 2: Socializers dominate, with everyone else only in bit parts Type 3: A balance between all four types, with enough Explorers to control the Killers Type 4: An empty world Type 3 is the most stable; types 1 and 2 eventually go to type 4
29 Two Good Reference Books Game Theory: A Nontechnical Introduction, by Morton D. Davis The examples on slides 7, 8, 12, 19, and 21 are adapted from Davis book. Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction, by Herbert Gintis The example on slide 16 is adapted from Gintis book.
30 Hippasus Talk URL: Site URL:
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