Evolutionary Game Theory
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Evolutionary Game Theory

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Introduction to Game Theory

Introduction to Game Theory
Evolutionary Game Theory

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  • There is a set of participants, whom we call the playersIn our example, you and yourpartner are the two players.(ii) Each player has a set of options for how to behave; we will refer to these as the player'spossible strategiesIn the example, you and your partner each have two possiblestrategies: to prepare for the presentation, or to study for the exam.(iii) For each choice of strategies, each player receives a payo that can depend on thestrategies selected by everyoneThepayos will generally be numbers, with eachplayer preferring larger payos to smaller payosIn our current example, the payoto each player is the average grade he or she gets on the exam and the presentation

Evolutionary Game Theory Evolutionary Game Theory Presentation Transcript

  • DATA MINING AND MACHINE LEARNING IN A NUTSHELLEVOLUTIONARY GAME THEORY Mohammad-Ali Abbasi http://www.public.asu.edu/~mabbasi2/ SCHOOL OF COMPUTING, INFORMATICS, AND DECISION SYSTEMS ENGINEERING ARIZONA STATE UNIVERSITY Arizona State University http://dmml.asu.edu/Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 1
  • What is a Game • Game theory is concerned with situations in which decision-makers interact with one another, • and in which the happiness of each participant with the outcome depends not just on his or her own decisions but on the decisions made by everyone. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 2 2
  • What is game theory? • Formal way to analyse interactions between agents who behave strategically • Mathematics of decision making in conflict situations • Usual to assume players are “rational” • Widely applied to the study of economics, warfare, politics, animal behaviour, sociology, business, ecology and evolutionary biology Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 3
  • Basic elements of a Game • Players – Everyone who has an effect on your earnings • Strategies – Actions available to each player – Define a plan of action for every contingency • Payoffs – Numbers associated with each outcome – Reflect the interests of the players Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 4
  • Nash Equilibrium • A Nash equilibrium is a situation in which none of them have dominant Strategy and each player makes his or her best response – (S, T) is Nash equilibrium if S is the best strategy to T and T is the best strategy to S • John Nash shared the 1994 Nobel prize in Economic for developing this idea! Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 5
  • Prisoner’s Dilemma Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 6
  • Evolutionary Game Theory Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 7
  • Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 8
  • evolutionary stable strategy • Taller trees get more light, so taller trees reproduce more. • Taller trees have to consume more resources to be tall. • System converges to a state where only the tallest trees are present. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 9
  • Evolutionary stable strategy Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 10
  • Evolutionary Game Theory 1 x1 x1 1 1/ r r…relative fitness of new mutant n n…population size 1 1/ r For a neutral mutant, r=1, the fixation probability is 1/n. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 11
  • Evolutionary Game Theory • Is the attempt to invent and study mathematical equations describing how population change over time due to mutation and selection (Learning). Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 12
  • GT vs. EGT • In GT, one assumes that agents are perfectly rational. • In EGT, trial and error process gives strategies that can be selected for by some force (evolution - biological, cultural, etc…). • This lack of rationality is the point of departure between EGT and GT. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 13
  • Evolutionary game theory • population of players • follow different strategies • frequencies of strategies change over time, dependent on success relative to other strategies • genetic inheritance (mutation) or learning (innovation) – Irrational Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 14
  • Evolutionary Biology Evolutionary biology is based on the idea that an organisms genes largely determine its observable characteristics, and hence its fitness in a given environment. Organisms that are more fit will tend to produce more offspring, causing genes that provide greater fitness to increase their representation in the population. In this way, fitter genes tend to win over time, because they provide higher rates of reproduction. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 15
  • Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 16
  • Competition for food • When beetles of the same size compete, they get equal shares of the food • When a large beetle competes with a small beetle, the large beetle gets the majority of the food. • In all cases, large beetles experience less of a fitness benefit from a given quantity of food, since some of it is diverted into maintaining their expensive metabolism Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 17
  • Body Size Game Beetle 2 Small Large Small 5, 5 1, 8 Beetle 1 Large 8, 1 3, 3 Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 18
  • • Small portion of Large Beetles: x• Small Beetles: 1-x• Expected Pay off in a population that small is majority – Small beetle • 5(1-x) + 1.X = 5- 4x – Large beetle • 8 (1- x) + 3.x = 8-5x• Small is not evolutionary stable! Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 19
  • • Expected Pay off in a population that Large is majority• Large Beetle: 3 * (1-x) + 8 * x = 3 + 5x• Small Beetle: 1 * (1-x) + 5 * x = 1 + 4x• Large is evolutionary stable Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 20
  • Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 21
  • Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 22
  • Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 23
  • General Description Organism 2 S T S a, a b, c Organism 1 T c, b d, d Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 24
  • • X -> T• (1-x) -> S• Play S – Expected Payoff: a (1-x) + bx• Play T – Expected Payoff: C ( 1- x) + dx A ( 1- x) + bx > c ( 1-x ) + dx Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 25
  • • In a two-player, two-strategy, symmetric game, S is evolutionarily stable precisely when either a>c or a = c and b > d. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 26
  • Relationship between evolutionary and NashEquilibria • (S, S) is a Nash equilibrium when S is a best response to the choice of S by the other player a >= c • The condition for S to be evolutionarily stable a>c Or a = c and b > d • If strategy S is evolutionarily stable, then (S, S) is a Nash equilibrium Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 27
  • • Other direction – (S, S) is a Nash equilibrium -> S is not ESS a = c and b < d Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 28
  • Strict Nash Equilibrium Hunt Hunter 2 Stag or Hare S H S 4, 4 0, 3 Hunter 1 H 3, 0 3, 3 Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 29
  • Strict Nash Equilibrium Hunt Hunter 2 Stag or Hare S H S 4, 4 0, 4 Hunter 1 H 4, 0 3, 3 Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 30
  • Strict Nash Equilibrium choice of strategies is a strict Nash equilibrium if each player is using the unique best response to what the other player is doing for symmetric two-player, two-strategy games, the condition for (S, S) to be a strict Nash equilibrium is that a > c the set of evolutionarily stable strategies S is a subset of the set of strategies S for which (S, S) is a Nash equilibrium if (S, S) is a strict Nash equilibrium, then S is evolutionarily stable Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 31
  • Nash Equilibrium and Evolutionary Stability • In a Nash equilibrium – we consider players choosing mutual best responses to each others strategy – This equilibrium concept places great demands on the ability of the players to chose optimally and to coordinate on strategies that are best responses to each other. • Evolutionary stability – no intelligence or coordination on the part of the players – strategies are viewed as being hard-wired into the players, perhaps because their behavior is encoded in their genes Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 32
  • Evolutionarily Stable Mixed Strategies Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 33
  • General Description Player 2 S T S a, a b, c Player 1 T c, b d, d Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 34
  • Evolutionarily Stable Mixed Strategies Organism 1: Play S with probability p and T with (1-p) Organism 2: Play S with probability q and T with (1-q) V (p, q) = pqa + p(1-q)b + (1-p)qc + (1-p)(1-q)d For p to be ESMS (1-x)V(p, p) + xV(p, q) > (1-x) V(q, p) + xV(q, q) Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 35
  • Evolutionarily Stable Mixed Strategies • In the General Symmetric Game, p is an evolutionarily stable mixed strategy if there is a (small) positive number y such that when any other mixed strategy q invades p at any level x < y, the fitness of an organism playing p is strictly greater than the fitness of an organism playing q. Arizona State University Data Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 36
  • Mohammad-Ali Abbasi (Ali), Ali, is a Ph.D student at Data Mining and Machine Learning Lab, Arizona State University. His research interests include Data Mining, Machine Learning, Social Computing, and Social Media Behavior Analysis. http://www.public.asu.edu/~mabbasi2/ Arizona State UniversityData Mining and Machine Learning Lab Data Mining and Machine Learning- in a nutshell Evolutionary Game Theory 37