EVOLUTIONARY GAMES WITH GROUP SELECTION

Size: px
Start display at page:

Download "EVOLUTIONARY GAMES WITH GROUP SELECTION"

Transcription

1 EVOLUTIONARY GAMES WITH GROUP SELECTION Martin Kaae Jensen Alexandros Rigos Department of Economics University of Leicester Controversies in Game Theory: Homo Oeconomicus vs. Homo Socialis ETH Zurich 12/09/2014 M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

2 OUTLINE 1 INTRODUCTION MOTIVATION 2 THE MODEL Setup Solution Concepts Relationship with Trait-group models 3 RESULTS 4 APPLICATIONS 5 CONCLUSION M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

3 WHAT IS OUR AIM? To provide an equilibrium concept that captures arbitrary nonrandom matching in evolutionary models. Look into welfare under non-random matching. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

4 EVOLUTION Darwin (1859) The Origin of Species. Survival of the fittest. Fitness = Expected number of children. More children higher fraction of the population in the long run. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

5 THE PUZZLE WITH COOPERATION If natural selection favors the fittest, then cooperative behaviour/traits should have been wiped out of existence long ago. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

6 THE PUZZLE WITH COOPERATION If natural selection favors the fittest, then cooperative behaviour/traits should have been wiped out of existence long ago. We observe cooperative behaviour. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

7 THE PUZZLE WITH COOPERATION If natural selection favors the fittest, then cooperative behaviour/traits should have been wiped out of existence long ago. We observe cooperative behaviour. How did cooperation survive? M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

8 THE PUZZLE WITH COOPERATION If natural selection favors the fittest, then cooperative behaviour/traits should have been wiped out of existence long ago. We observe cooperative behaviour. How did cooperation survive? ESS is a NE refinement so it cannot explain any departures from NE behaviour (e.g. cooperation in PD, the existence of altruism or spite). M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

9 THE PUZZLE WITH COOPERATION If natural selection favors the fittest, then cooperative behaviour/traits should have been wiped out of existence long ago. We observe cooperative behaviour. How did cooperation survive? ESS is a NE refinement so it cannot explain any departures from NE behaviour (e.g. cooperation in PD, the existence of altruism or spite). One answer: Nonrandom/assortative matching. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

10 DIFFERENT TYPES OF ASSORTATIVE MATCHING Kin selection: Interact mostly with your relatives. (Hamilton, 1964) Local interactions: Interact mostly with your neighbours. (Boyd and Richerson, 2002; Grund, Waloszek, and Helbing, 2013) Green beard effect: Recognise others types. (Dawkins, 1976) Alger and Weibull (2012): Assortative matching by homophily (social preference assortativity). Meritocratic matching: Mechanism that assorts actions according to cooperation (Nax, Murphy, and Helbing, 2014). Bergström (2003): Formal model of assortativity. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

11 OUR CONTRIBUTION Introduce broad equilibrium concepts that capture non-random matching. (Nash Equilibrium with Group Selection, Evolutionarily Stable Strategy with Group Selection) M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

12 OUR CONTRIBUTION Introduce broad equilibrium concepts that capture non-random matching. (Nash Equilibrium with Group Selection, Evolutionarily Stable Strategy with Group Selection) Provide a welfare theorem for evolutionary models with nonrandom matching. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

13 OUTLINE 1 INTRODUCTION MOTIVATION 2 THE MODEL Setup Solution Concepts Relationship with Trait-group models 3 RESULTS 4 APPLICATIONS 5 CONCLUSION M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

14 MODEL INGREDIENTS A normal-form game: Dictates how individuals get payoffs. A matching rule: Dictates how individuals form groups. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

15 THE NORMAL FORM GAME A large population (I = [0, 1]) of agents play a game. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

16 THE NORMAL FORM GAME A large population (I = [0, 1]) of agents play a game. n-player, m-strategy symmetric normal form game. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

17 THE NORMAL FORM GAME A large population (I = [0, 1]) of agents play a game. n-player, m-strategy symmetric normal form game. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

18 THE NORMAL FORM GAME A large population (I = [0, 1]) of agents play a game. n-player, m-strategy symmetric normal form game. Symmetric game payoff of an individual following strategy j M will depend only on the number of opponents in his/her group who play each of the m strategies (not on who these opponents are). M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

19 THE NORMAL FORM GAME (CONT D) There are γ(n, m) = (n+m 1)! n!(m 1)! different types of n-sized groups that can be formed with m different strategies. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

20 THE NORMAL FORM GAME (CONT D) There are γ(n, m) = (n+m 1)! n!(m 1)! different types of n-sized groups that can be formed with m different strategies. n = 2, m = 2 i n 1 n n = 2, m = 3 i n 1 n 2 n M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

21 THE NORMAL FORM GAME (CONT D) There are γ(n, m) = (n+m 1)! n!(m 1)! different types of n-sized groups that can be formed with m different strategies. The payoff of agents following strategy j that find themselves in an i-type group will be denoted as A i j. n = 2, m = 2 i n 1 n n = 2, m = 3 i n 1 n 2 n M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

22 THE NORMAL FORM GAME (CONT D) There are γ(n, m) = (n+m 1)! n!(m 1)! different types of n-sized groups that can be formed with m different strategies. The payoff of agents following strategy j that find themselves in an i-type group will be denoted as A i j. The set of the various A i j is the payoff matrix of the game. n = 2, m = 2 i n 1 n n = 2, m = 3 i n 1 n 2 n M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

23 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

24 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

25 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Agents, do not choose their strategies. They just behave the way their genes instruct them to behave. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

26 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Agents, do not choose their strategies. They just behave the way their genes instruct them to behave. In each of the groups each agent follows the strategy of his/her type. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

27 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Agents, do not choose their strategies. They just behave the way their genes instruct them to behave. In each of the groups each agent follows the strategy of his/her type. Depending on the payoff matrix of the game, the agents get fitness. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

28 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Agents, do not choose their strategies. They just behave the way their genes instruct them to behave. In each of the groups each agent follows the strategy of his/her type. Depending on the payoff matrix of the game, the agents get fitness. Fitness is interpreted as expected number of children of the agent. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

29 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Agents, do not choose their strategies. They just behave the way their genes instruct them to behave. In each of the groups each agent follows the strategy of his/her type. Depending on the payoff matrix of the game, the agents get fitness. Fitness is interpreted as expected number of children of the agent. All children of an agent adopt their parent s type/behaviour. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

30 DYNAMICS State of the world is x = (x 1, x 2,..., x m ) S m. At each date t all agents in the population are drawn into groups of n agents each. Agents, do not choose their strategies. They just behave the way their genes instruct them to behave. In each of the groups each agent follows the strategy of his/her type. Depending on the payoff matrix of the game, the agents get fitness. Fitness is interpreted as expected number of children of the agent. All children of an agent adopt their parent s type/behaviour. The children of date t s population form the population of date t + 1. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

31 THE MATCHING RULE How are the individuals being drawn into the groups? M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

32 THE MATCHING RULE How are the individuals being drawn into the groups? Matching Rule: f i gives the proportion of total groups that are of type i. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

33 THE MATCHING RULE How are the individuals being drawn into the groups? Matching Rule: f i gives the proportion of total groups that are of type i. DEFINITION (MATCHING RULE) A matching rule is a function that maps population distributions to group distributions. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

34 THE MATCHING RULE How are the individuals being drawn into the groups? Matching Rule: f i gives the proportion of total groups that are of type i. DEFINITION (MATCHING RULE) A matching rule is a function that maps population distributions to group distributions. DEFINITION (CONSISTENT MATCHING RULE) A matching rule is consistent if the proportion of individuals of each type found in the group distribution is equal to the proportion of the individuals of that type in the population. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

35 THE MATCHING RULE How are the individuals being drawn into the groups? Matching Rule: f i gives the proportion of total groups that are of type i. DEFINITION (MATCHING RULE) A matching rule is a function that maps population distributions to group distributions. DEFINITION (CONSISTENT MATCHING RULE) A matching rule is consistent if the proportion of individuals of each type found in the group distribution is equal to the proportion of the individuals of that type in the population. DEFINITION A Group Selection Model consists of a symmetric normal form game and a matching rule. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

36 MATCHING RULES & FITNESS STRUCTURE EXAMPLE i group Proportion f 1 (x) f 2 (x) f 3 (x) of i-kind groups Complete x 1 0 x 2 Segregation Random Matching x1 2 2x 1 x 2 x2 2 Fitness A 1 1 A 2 1 Fitness A 2 2 A 3 2 M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

37 EXPECTED PAYOFFS OF j-types Expected payoff of a j-type: π j (x) = i n i j f i(x) nx j A i j w i j (x) = ni j f i (x) : Proportion of j-types in i-kind groups. nx j nj i : Number of j-types in an i-kind group. A i j : Fitness of each j-type in an i-kind group. Full M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

38 EQUILIBRIUM CONCEPTS Decision problem facing an individual: max y Sm y π(x) M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

39 EQUILIBRIUM CONCEPTS Decision problem facing an individual: max y Sm y π(x) DEFINITION (NEGS) A point x S m is a (symmetric) Nash Equilibrium for the group selection game < I, G, f > if for any individual in the population it is a best response to use the mixed strategy x when they are taking for given that every other indivdual in the population is also using x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

40 EQUILIBRIUM CONCEPTS Decision problem facing an individual: max y Sm y π(x) DEFINITION (NEGS) A point x S m is a (symmetric) Nash Equilibrium for the group selection game < I, G, f > if for any individual in the population it is a best response to use the mixed strategy x when they are taking for given that every other indivdual in the population is also using x DEFINITION (ESSGS) A (mixed) strategy x S m is an Evolutionarily Stable Strategy for the group selection game < I, G, f > if for any possible invasion of the population using x from a population using a mixed strategy y, individuals using x get strictly higher payoffs than those using y. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

41 GROUP SELECTION MODELS Main idea: Groups that have higher total payoffs will outperform those with lower. Haystack model: Introduced by Maynard Smith (1964). Trait-group model: Introduced by Wilson (1975). Also analysed by Cooper and Wallace (2004). M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

42 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

43 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

44 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

45 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

46 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. They get payoffs in terms of number of children. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

47 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. They get payoffs in terms of number of children. The children remain in the trait-group and get pooled without mixing with other trait-groups. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

48 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. They get payoffs in terms of number of children. The children remain in the trait-group and get pooled without mixing with other trait-groups. They are randomly matched in pairs and get payoffs in terms of fitness and according to the same PD payoff matrix. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

49 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. They get payoffs in terms of number of children. The children remain in the trait-group and get pooled without mixing with other trait-groups. They are randomly matched in pairs and get payoffs in terms of fitness and according to the same PD payoff matrix. The same process is repeated for T generations. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

50 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. They get payoffs in terms of number of children. The children remain in the trait-group and get pooled without mixing with other trait-groups. They are randomly matched in pairs and get payoffs in terms of fitness and according to the same PD payoff matrix. The same process is repeated for T generations. After T generations the whole population is pooled together. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

51 TRAIT-GROUP MODEL LAYOUT A continuum population I = [0, 1]. Individuals can be either Cooperators or Defectors (x: proportion of cooperators). They are randomly matched to form trait-groups of size 2. They interact in the trait-groups and get payoffs according to a PD payoff matrix. They get payoffs in terms of number of children. The children remain in the trait-group and get pooled without mixing with other trait-groups. They are randomly matched in pairs and get payoffs in terms of fitness and according to the same PD payoff matrix. The same process is repeated for T generations. After T generations the whole population is pooled together. The process repeats itself. Equations M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

52 TRAIT-GROUP MODEL (CONT D) PROPOSITION Consider a T -period trait-group model with a symmetric payoff matrix (A i j ). Consider also the normal form game G with payoff matrix (Ai j ). Then there is a matching rule f such that the dynamics and steady states of the group selection model < I, G, f > coincide with the dynamics and steady states of the trait-group model. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

53 OUTLINE 1 INTRODUCTION MOTIVATION 2 THE MODEL Setup Solution Concepts Relationship with Trait-group models 3 RESULTS 4 APPLICATIONS 5 CONCLUSION M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

54 THEOREMS (CONSISTENCY WITH NE AND ESS) THEOREM (EXISTENCE OF EQUILIBRIA) Let G =< I, G, f > be a group selection game and assume that f is continuous and that the (upper) partial derivatives + j f i (x) exist whenever x j = 0 (for all j M and i supp(j)). Then G has a NEGS. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

55 THEOREMS (CONSISTENCY WITH NE AND ESS) THEOREM (EXISTENCE OF EQUILIBRIA) Let G =< I, G, f > be a group selection game and assume that f is continuous and that the (upper) partial derivatives + j f i (x) exist whenever x j = 0 (for all j M and i supp(j)). Then G has a NEGS. THEOREM (EQUILIBRIA AND STEADY STATES) Let G =< I, G, f > be a group selection game under a matching rule. Then, all NEGS of G are steady states of the associated dynamical system. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

56 THEOREMS (CONSISTENCY WITH NE AND ESS) THEOREM (EXISTENCE OF EQUILIBRIA) Let G =< I, G, f > be a group selection game and assume that f is continuous and that the (upper) partial derivatives + j f i (x) exist whenever x j = 0 (for all j M and i supp(j)). Then G has a NEGS. THEOREM (EQUILIBRIA AND STEADY STATES) Let G =< I, G, f > be a group selection game under a matching rule. Then, all NEGS of G are steady states of the associated dynamical system. THEOREM (RANDOM MATCHING AND NASH) Under the random matching rule, all equilibria of a population game G coincide with the (symmetric) Nash equilibria of the underlying normal form game. Random Matching Rule M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

57 MORE RESULTS THEOREM (BOOK-KEEPING) Let G =< I, G, f > be a group selection game and assume that f satisfies the assumptions of the existence theorem and consider the evolutionary steady states of the associated dynamical systems. Then, 1 Any NEGS is a steady state of the discrete time replicator dynamics as well as the continuous time replicator dynamics. 2 If x is the ω-limit of an orbit x(t) of the replicator dynamics that lies everywhere in the interior of S m, then x is a NEGS. 3 If x is Lyapunov stable for the replicator dynamics, then x is a NEGS. 4 Assume that f is of class C 1. Then if x is an ESSGS, it is asymptotically stable under the replicator dynamics. Dynamics M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

58 WELFARE DEFINITION (EVOLUTIONARY OPTIMUM) Let G be a normal form game. A population state x S m together with a matching rule f F n,m is said to be an evolutionary optimum if π f (x ) π f (x) for all (x, f) E = {(x, f) S m F n,m : x is a steady state of I, G, f }. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

59 WELFARE DEFINITION (EVOLUTIONARY OPTIMUM) Let G be a normal form game. A population state x S m together with a matching rule f F n,m is said to be an evolutionary optimum if π f (x ) π f (x) for all (x, f) E = {(x, f) S m F n,m : x is a steady state of I, G, f }. THEOREM (WELFARE THEOREM) Let (x, f ) be an evolutionary optimum. Then there exists a matching rule h F n,m which satisfies the continuity assumptions of the existence theorem, such that x is a NEGS under h, and such that (x, h) is an evolutionary optimum (in particular, π h (x ) = π f (x )). M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

60 OUTLINE 1 INTRODUCTION MOTIVATION 2 THE MODEL Setup Solution Concepts Relationship with Trait-group models 3 RESULTS 4 APPLICATIONS 5 CONCLUSION M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

61 CONSTANT INDEX OF ASSORT/TY MATCHING RULE: EQUILIBRIA & DYNAMICS A A2 2 < A1 1 + A C D C 16,16 4,18 D 18,4 12, ẋ α = 0.0 α = 0.2 α = 0.4 α = 0.6 α = 0.8 α = 1.0 x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

62 TRAIT-GROUP MODEL MATCHING RULE: EQUILIBRIA & DYNAMICS A A2 2 < A1 1 + A C D C 16,16 4,18 D 18,4 12, ẋ T = 1 T = 2 T = 3 x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

63 CONSTANT INDEX OF ASSORT/TY MATCHING RULE: COMPARATIVE STATICS 1 A A2 2 < A1 1 + A3 2 ESSGS NEGS (not ESSGS) x 0 0 A 2 2 A1 1 A 2 2 A3 2 α A 3 2 A2 1 A 1 1 A2 1 1 M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

64 EQUILIBRIUM WELFARE A A2 2 < A1 1 + A3 2 A 1 1 Equilibrium Welfare Normal Form Game Welfare W A x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

65 CONSTANT INDEX OF ASSORT/TY MATCHING RULE: EQUILIBRIA & DYNAMICS A A2 2 > A1 1 + A C D C 11,11 2,18 D 18,2 5, ẋ α = 0.0 α = 0.2 α = 0.4 α = 0.6 α = 0.8 α = 1.0 x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

66 TRAIT-GROUP MODEL MATCHING RULE: EQUILIBRIA & DYNAMICS I A A2 2 > A1 1 + A C D C 55,55 38,64 D 64,38 40, ẋ T = 1 T = 2 x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

67 TRAIT-GROUP MODEL MATCHING RULE: EQUILIBRIA & DYNAMICS II A A2 2 > A1 1 + A C D C 11,11 2,18 D 18,2 5, ẋ T = 1 T = 2 T = 3 x M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

68 CONSTANT INDEX OF ASSORT/TY MATCHING RULE: COMPARATIVE STATICS A A2 2 > A1 1 + A3 2 ESSGS 1 x 0 0 A 3 2 A2 1 A 1 1 A2 1 α A 2 2 A1 1 A 2 2 A3 2 1 M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

69 EQUILIBRIUM WELFARE A A2 2 > A1 1 + A3 2 A 1 1 W A Equilibrium Welfare Normal Form Game Welfare x A 2 1 +A2 2 2A3 2 2(A 2 1 +A2 2 A1 1 A3 2 ) 1 M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

70 OUTLINE 1 INTRODUCTION MOTIVATION 2 THE MODEL Setup Solution Concepts Relationship with Trait-group models 3 RESULTS 4 APPLICATIONS 5 CONCLUSION M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

71 WHAT DID WE DO? We generalised the concept of ESS (Maynard Smith and Price, 1973) to allow for more complex population interactions (n players, m strategies). Our formulation captures standard group selection models (such as the haystack/trait-group model). We proved a welfare theorem for evolutionary models with nonrandom matching: An evolutionary optimum can be attained with an appropriately chosen matching rule. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

72 SO WHAT? Our model is overarching: It gives a unified way to study nonrandom/assortative matching whether this is a result of natural/biological evolution or of institutions in place. If we are able to change the mathcing rule in effect ( change the rules ), we can obtain welfare-maximising outcomes. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

73 REFERENCES I Alger, I. and J. W. Weibull (2012). A generalization of Hamilton s rule Love others how much? Journal of Theoretical Biology 299, pp Bergström, T. C. (2003). The algebra of assortative encounters and the evolution of cooperation. International Game Theory Review 5.3, pp Boyd, R. and P. J. Richerson (2002). Group Beneficial Norms Can Spread Rapidly in a Structured Population. Journal of Theoretical Biology 215.3, pp Cooper, B. and C. Wallace (2004). Group selection and the evolution of altruism. Oxf. Econ. Pap. 56.2, p Darwin, C. R. (1859). The Origin of Species. London: John Murray. Dawkins, R. (1976). The Selfish Gene. 1st. New York: Oxford University Press. M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

74 REFERENCES II Grund, T., C. Waloszek, and D. Helbing (2013). How natural selection can create both self-and other-regarding preferences, and networked minds. Scientific reports 3. Hamilton, W. D. (1964). The genetical evolution of social behaviour. II. Journal of Theoretical Biology 7.1, pp Maynard Smith, J. (1964). Group selection and kin selection. Nature , pp Maynard Smith, J. and G. R. Price (1973). The logic of animal conflict. Nature , pp Nax, H. H., R. O. Murphy, and D. Helbing (2014). Meritocratic Matching Stabilizes Public Goods Provision. Available at SSRN Wilson, D. S. (1975). A theory of group selection. Proceedings of the National Academy of Science of the U.S.A. 72.1, p M.K. JENSEN, A. RIGOS (LEICESTER) EV. GAMES WITH GROUP SELECTION ETH 12/09/ / 38

Computational Evolutionary Game Theory and why I m never using PowerPoint for another presentation involving maths ever again

Computational Evolutionary Game Theory and why I m never using PowerPoint for another presentation involving maths ever again Computational Evolutionary Game Theory and why I m never using PowerPoint for another presentation involving maths ever again Enoch Lau 5 September 2007 Outline What is evolutionary game theory? Why evolutionary

More information

Reciprocity and Trust under an Evolutionary Context

Reciprocity and Trust under an Evolutionary Context Reciprocity and Trust under an Evolutionary Context Francisco Poggi UdeSA September 24 Francisco Poggi (UdeSA) Reciprocity and Trust... September 24 1 / 33 Motivation The economist is usually not interested

More information

BELIEFS & EVOLUTIONARY GAME THEORY

BELIEFS & EVOLUTIONARY GAME THEORY 1 / 32 BELIEFS & EVOLUTIONARY GAME THEORY Heinrich H. Nax hnax@ethz.ch & Bary S. R. Pradelski bpradelski@ethz.ch May 15, 217: Lecture 1 2 / 32 Plan Normal form games Equilibrium invariance Equilibrium

More information

Evolution & Learning in Games

Evolution & Learning in Games 1 / 27 Evolution & Learning in Games Econ 243B Jean-Paul Carvalho Lecture 2. Foundations of Evolution & Learning in Games II 2 / 27 Outline In this lecture, we shall: Take a first look at local stability.

More information

Evolutionary Game Theory

Evolutionary Game Theory Evolutionary Game Theory ISI 330 Lecture 18 1 ISI 330 Lecture 18 Outline A bit about historical origins of Evolutionary Game Theory Main (competing) theories about how cooperation evolves P and other social

More information

Other-Regarding Preferences: Theory and Evidence

Other-Regarding Preferences: Theory and Evidence Other-Regarding Preferences: Theory and Evidence June 9, 2009 GENERAL OUTLINE Economic Rationality is Individual Optimization and Group Equilibrium Narrow version: Restrictive Assumptions about Objective

More information

arxiv: v2 [q-bio.pe] 13 Apr 2016

arxiv: v2 [q-bio.pe] 13 Apr 2016 Assortment and the evolution of cooperation in a Moran process with exponential fitness * aniel ooney 1 Benjamin Allen 2,4 arl Veller 3,4,5 arxiv:1509.05757v2 [q-bio.pe] 13 Apr 2016 Abstract: We study

More information

MATCHING STRUCTURE AND THE EVOLUTION OF COOPERATION IN THE PRISONER S DILEMMA

MATCHING STRUCTURE AND THE EVOLUTION OF COOPERATION IN THE PRISONER S DILEMMA MATCHING STRUCTURE AN THE EVOLUTION OF COOPERATION IN THE PRISONER S ILEMMA Noureddine Bouhmala 1 and Jon Reiersen 2 1 epartment of Technology, Vestfold University College, Norway noureddine.bouhmala@hive.no

More information

Conjectural Variations in Aggregative Games: An Evolutionary Perspective

Conjectural Variations in Aggregative Games: An Evolutionary Perspective Conjectural Variations in Aggregative Games: An Evolutionary Perspective Alex Possajennikov University of Nottingham January 2012 Abstract Suppose that in aggregative games, in which a player s payoff

More information

Population Games and Evolutionary Dynamics

Population Games and Evolutionary Dynamics Population Games and Evolutionary Dynamics (MIT Press, 200x; draft posted on my website) 1. Population games 2. Revision protocols and evolutionary dynamics 3. Potential games and their applications 4.

More information

Computation of Efficient Nash Equilibria for experimental economic games

Computation of Efficient Nash Equilibria for experimental economic games International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 197-212. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Computation of Efficient Nash Equilibria for experimental economic games

More information

Irrational behavior in the Brown von Neumann Nash dynamics

Irrational behavior in the Brown von Neumann Nash dynamics Irrational behavior in the Brown von Neumann Nash dynamics Ulrich Berger a and Josef Hofbauer b a Vienna University of Economics and Business Administration, Department VW 5, Augasse 2-6, A-1090 Wien,

More information

Essays in Game Theory

Essays in Game Theory Essays in Game Theory Alexandros Rigos Department of Economics University of Leicester A thesis submitted for the degree of Doctor of Philosophy at the University of Leicester. September 2015 To my mother

More information

Institut für Höhere Studien (IHS), Wien Institute for Advanced Studies, Vienna

Institut für Höhere Studien (IHS), Wien Institute for Advanced Studies, Vienna Institut für Höhere Studien (IHS), Wien Institute for Advanced Studies, Vienna Reihe Ökonomie / Economics Series No. 67 Siblings, Strangers, and the Surge of Altruism Oded Stark Siblings, Strangers, and

More information

Game Theory -- Lecture 4. Patrick Loiseau EURECOM Fall 2016

Game Theory -- Lecture 4. Patrick Loiseau EURECOM Fall 2016 Game Theory -- Lecture 4 Patrick Loiseau EURECOM Fall 2016 1 Lecture 2-3 recap Proved existence of pure strategy Nash equilibrium in games with compact convex action sets and continuous concave utilities

More information

Game interactions and dynamics on networked populations

Game interactions and dynamics on networked populations Game interactions and dynamics on networked populations Chiara Mocenni & Dario Madeo Department of Information Engineering and Mathematics University of Siena (Italy) ({mocenni, madeo}@dii.unisi.it) Siena,

More information

Game Theory, Evolutionary Dynamics, and Multi-Agent Learning. Prof. Nicola Gatti

Game Theory, Evolutionary Dynamics, and Multi-Agent Learning. Prof. Nicola Gatti Game Theory, Evolutionary Dynamics, and Multi-Agent Learning Prof. Nicola Gatti (nicola.gatti@polimi.it) Game theory Game theory: basics Normal form Players Actions Outcomes Utilities Strategies Solutions

More information

Introduction to game theory LECTURE 1

Introduction to game theory LECTURE 1 Introduction to game theory LECTURE 1 Jörgen Weibull January 27, 2010 1 What is game theory? A mathematically formalized theory of strategic interaction between countries at war and peace, in federations

More information

Graph topology and the evolution of cooperation

Graph topology and the evolution of cooperation Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Graph topology and the evolution of cooperation Author(s) Li, Menglin

More information

Static and dynamic stability conditions for structurally stable signaling games

Static and dynamic stability conditions for structurally stable signaling games 1/32 Static and dynamic stability conditions for structurally stable signaling games Gerhard Jäger Gerhard.Jaeger@uni-bielefeld.de September 8, 2007 Workshop on Communication, Game Theory, and Language,

More information

Complexity in social dynamics : from the. micro to the macro. Lecture 4. Franco Bagnoli. Lecture 4. Namur 7-18/4/2008

Complexity in social dynamics : from the. micro to the macro. Lecture 4. Franco Bagnoli. Lecture 4. Namur 7-18/4/2008 Complexity in Namur 7-18/4/2008 Outline 1 Evolutionary models. 2 Fitness landscapes. 3 Game theory. 4 Iterated games. Prisoner dilemma. 5 Finite populations. Evolutionary dynamics The word evolution is

More information

Evolutionary Games and Periodic Fitness

Evolutionary Games and Periodic Fitness Evolutionary Games and Periodic Fitness Philippe Uyttendaele Frank Thuijsman Pieter Collins Ralf Peeters Gijs Schoenmakers Ronald Westra March 16, 2012 Abstract One thing that nearly all stability concepts

More information

Evolutionary Dynamics and Extensive Form Games by Ross Cressman. Reviewed by William H. Sandholm *

Evolutionary Dynamics and Extensive Form Games by Ross Cressman. Reviewed by William H. Sandholm * Evolutionary Dynamics and Extensive Form Games by Ross Cressman Reviewed by William H. Sandholm * Noncooperative game theory is one of a handful of fundamental frameworks used for economic modeling. It

More information

Costly Signals and Cooperation

Costly Signals and Cooperation Costly Signals and Cooperation Károly Takács and András Németh MTA TK Lendület Research Center for Educational and Network Studies (RECENS) and Corvinus University of Budapest New Developments in Signaling

More information

EVOLUTIONARILY STABLE STRATEGIES AND GROUP VERSUS INDIVIDUAL SELECTION

EVOLUTIONARILY STABLE STRATEGIES AND GROUP VERSUS INDIVIDUAL SELECTION 39 EVOLUTIONARILY STABLE STRATEGIES AND GROUP VERSUS INDIVIDUAL SELECTION Objectives Understand the concept of game theory. Set up a spreadsheet model of simple game theory interactions. Explore the effects

More information

Deterministic Evolutionary Dynamics

Deterministic Evolutionary Dynamics 1. Introduction Deterministic Evolutionary Dynamics Prepared for the New Palgrave Dictionary of Economics, 2 nd edition William H. Sandholm 1 University of Wisconsin 1 November 2005 Deterministic evolutionary

More information

Cooperation Achieved by Migration and Evolution in a Multilevel Selection Context

Cooperation Achieved by Migration and Evolution in a Multilevel Selection Context Proceedings of the 27 IEEE Symposium on Artificial Life (CI-ALife 27) Cooperation Achieved by Migration and Evolution in a Multilevel Selection Context Genki Ichinose Graduate School of Information Science

More information

ABSTRACT: Dissolving the evolutionary puzzle of human cooperation.

ABSTRACT: Dissolving the evolutionary puzzle of human cooperation. ABSTRACT: Dissolving the evolutionary puzzle of human cooperation. Researchers of human behaviour often approach cooperation as an evolutionary puzzle, viewing it as analogous or even equivalent to the

More information

April 29, 2010 CHAPTER 13: EVOLUTIONARY EQUILIBRIUM

April 29, 2010 CHAPTER 13: EVOLUTIONARY EQUILIBRIUM April 29, 200 CHAPTER : EVOLUTIONARY EQUILIBRIUM Some concepts from biology have been applied to game theory to define a type of equilibrium of a population that is robust against invasion by another type

More information

The coevolution of recognition and social behavior

The coevolution of recognition and social behavior The coevolution of recognition and social behavior Rory Smead 1 and Patrick Forber 2 May 4, 2016 Affiliations 1 (Corresponding author) Department of Philosophy and Religion, Northeastern University, Holmes

More information

Game Theory, Population Dynamics, Social Aggregation. Daniele Vilone (CSDC - Firenze) Namur

Game Theory, Population Dynamics, Social Aggregation. Daniele Vilone (CSDC - Firenze) Namur Game Theory, Population Dynamics, Social Aggregation Daniele Vilone (CSDC - Firenze) Namur - 18.12.2008 Summary Introduction ( GT ) General concepts of Game Theory Game Theory and Social Dynamics Application:

More information

Gains in evolutionary dynamics. Dai ZUSAI

Gains in evolutionary dynamics. Dai ZUSAI Gains in evolutionary dynamics unifying rational framework for dynamic stability Dai ZUSAI Hitotsubashi University (Econ Dept.) Economic Theory Workshop October 27, 2016 1 Introduction Motivation and literature

More information

Assortative matching with inequality in voluntary contribution games

Assortative matching with inequality in voluntary contribution games Assortative matching with inequality in voluntary contribution games Stefano Duca Dirk Helbing Heinrich H. Nax February 11, 2016 Abstract Voluntary contribution games are a classic social dilemma in which

More information

Other Equilibrium Notions

Other Equilibrium Notions Other Equilibrium Notions Ichiro Obara UCLA January 21, 2012 Obara (UCLA) Other Equilibrium Notions January 21, 2012 1 / 28 Trembling Hand Perfect Equilibrium Trembling Hand Perfect Equilibrium We may

More information

Natural selection, Game Theory and Genetic Diversity

Natural selection, Game Theory and Genetic Diversity Natural selection, Game Theory and Genetic Diversity Georgios Piliouras California Institute of Technology joint work with Ruta Mehta and Ioannis Panageas Georgia Institute of Technology Evolution: A Game

More information

Game Theory and Evolution

Game Theory and Evolution Game Theory and Evolution Toban Wiebe University of Manitoba July 2012 Toban Wiebe (University of Manitoba) Game Theory and Evolution July 2012 1 / 24 Why does evolution need game theory? In what sense

More information

Evolution of motivations and behavioral responses! Integrating the proximate and ultimate causes of behavior!

Evolution of motivations and behavioral responses! Integrating the proximate and ultimate causes of behavior! Evolution of motivations and behavioral responses! Integrating the proximate and ultimate causes of behavior! Erol Akçay! National Institute for Mathematical! and Biological Synthesis (NIMBioS)! University

More information

Evolutionary stability of games with costly signaling

Evolutionary stability of games with costly signaling 1/25 Evolutionary stability of games with costly signaling Gerhard Jäger Gerhard.Jaeger@uni-bielefeld.de December 17, 2007 Amsterdam Colloquium 2/25 Overview signaling games with costly signaling Evolutionary

More information

Observations on Cooperation

Observations on Cooperation Introduction Observations on Cooperation Yuval Heller (Bar Ilan) and Erik Mohlin (Lund) PhD Workshop, BIU, January, 2018 Heller & Mohlin Observations on Cooperation 1 / 20 Introduction Motivating Example

More information

Belief-based Learning

Belief-based Learning Belief-based Learning Algorithmic Game Theory Marcello Restelli Lecture Outline Introdutcion to multi-agent learning Belief-based learning Cournot adjustment Fictitious play Bayesian learning Equilibrium

More information

Darwinian Evolution of Cooperation via Punishment in the Public Goods Game

Darwinian Evolution of Cooperation via Punishment in the Public Goods Game Darwinian Evolution of Cooperation via Punishment in the Public Goods Game Arend Hintze, Christoph Adami Keck Graduate Institute, 535 Watson Dr., Claremont CA 97 adami@kgi.edu Abstract The evolution of

More information

arxiv: v1 [q-bio.pe] 22 Sep 2016

arxiv: v1 [q-bio.pe] 22 Sep 2016 Cooperation in the two-population snowdrift game with punishment enforced through different mechanisms André Barreira da Silva Rocha a, a Department of Industrial Engineering, Pontifical Catholic University

More information

A finite population ESS and a long run equilibrium in an n players coordination game

A finite population ESS and a long run equilibrium in an n players coordination game Mathematical Social Sciences 39 (000) 95 06 www.elsevier.nl/ locate/ econbase A finite population ESS and a long run equilibrium in an n players coordination game Yasuhito Tanaka* Faculty of Law, Chuo

More information

Meaning, Evolution and the Structure of Society

Meaning, Evolution and the Structure of Society Meaning, Evolution and the Structure of Society Roland Mühlenbernd November 7, 2014 OVERVIEW Game Theory and Linguistics Pragm. Reasoning Language Evolution GT in Lang. Use Signaling Games Replicator Dyn.

More information

Spatial Economics and Potential Games

Spatial Economics and Potential Games Outline Spatial Economics and Potential Games Daisuke Oyama Graduate School of Economics, Hitotsubashi University Hitotsubashi Game Theory Workshop 2007 Session Potential Games March 4, 2007 Potential

More information

Distributional stability and equilibrium selection Evolutionary dynamics under payoff heterogeneity (II) Dai Zusai

Distributional stability and equilibrium selection Evolutionary dynamics under payoff heterogeneity (II) Dai Zusai Distributional stability Dai Zusai 1 Distributional stability and equilibrium selection Evolutionary dynamics under payoff heterogeneity (II) Dai Zusai Tohoku University (Economics) August 2017 Outline

More information

Evolutionary Game Theory Notes

Evolutionary Game Theory Notes Evolutionary Game Theory Notes James Massey These notes are intended to be a largely self contained guide to everything you need to know for the evolutionary game theory part of the EC341 module. During

More information

arxiv:math/ v1 [math.oc] 29 Jun 2004

arxiv:math/ v1 [math.oc] 29 Jun 2004 Putting the Prisoner s Dilemma in Context L. A. Khodarinova and J. N. Webb Magnetic Resonance Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, England NG7 RD, e-mail: LarisaKhodarinova@hotmail.com

More information

Pairwise Comparison Dynamics for Games with Continuous Strategy Space

Pairwise Comparison Dynamics for Games with Continuous Strategy Space Pairwise Comparison Dynamics for Games with Continuous Strategy Space Man-Wah Cheung https://sites.google.com/site/jennymwcheung University of Wisconsin Madison Department of Economics Nov 5, 2013 Evolutionary

More information

The Stability of Strategic Plasticity

The Stability of Strategic Plasticity The Stability of Strategic Plasticity Rory Smead University of California, Irvine Kevin J.S. Zollman Carnegie Mellon University April 30, 2009 Abstract Recent research into the evolution of higher cognition

More information

Understanding and Solving Societal Problems with Modeling and Simulation

Understanding and Solving Societal Problems with Modeling and Simulation Understanding and Solving Societal Problems with Modeling and Simulation Lecture 8: The Breakdown of Cooperation ETH Zurich April 15, 2013 Dr. Thomas Chadefaux Why Cooperation is Hard The Tragedy of the

More information

Quantum Games. Quantum Strategies in Classical Games. Presented by Yaniv Carmeli

Quantum Games. Quantum Strategies in Classical Games. Presented by Yaniv Carmeli Quantum Games Quantum Strategies in Classical Games Presented by Yaniv Carmeli 1 Talk Outline Introduction Game Theory Why quantum games? PQ Games PQ penny flip 2x2 Games Quantum strategies 2 Game Theory

More information

Distributed Learning based on Entropy-Driven Game Dynamics

Distributed Learning based on Entropy-Driven Game Dynamics Distributed Learning based on Entropy-Driven Game Dynamics Bruno Gaujal joint work with Pierre Coucheney and Panayotis Mertikopoulos Inria Aug., 2014 Model Shared resource systems (network, processors)

More information

Evolutionary Game Theory: Overview and Recent Results

Evolutionary Game Theory: Overview and Recent Results Overviews: Evolutionary Game Theory: Overview and Recent Results William H. Sandholm University of Wisconsin nontechnical survey: Evolutionary Game Theory (in Encyclopedia of Complexity and System Science,

More information

EC319 Economic Theory and Its Applications, Part II: Lecture 2

EC319 Economic Theory and Its Applications, Part II: Lecture 2 EC319 Economic Theory and Its Applications, Part II: Lecture 2 Leonardo Felli NAB.2.14 23 January 2014 Static Bayesian Game Consider the following game of incomplete information: Γ = {N, Ω, A i, T i, µ

More information

For general queries, contact

For general queries, contact PART I INTRODUCTION LECTURE Noncooperative Games This lecture uses several examples to introduce the key principles of noncooperative game theory Elements of a Game Cooperative vs Noncooperative Games:

More information

The Limits of ESS Methodology

The Limits of ESS Methodology The Limits of ESS Methodology Simon M. Huttegger Kevin J. S. Zollman December 31, 2010 Abstract In this paper we show that there are certain limits as to what applications of Maynard Smith s concept of

More information

N-Player Prisoner s Dilemma

N-Player Prisoner s Dilemma ALTRUISM, THE PRISONER S DILEMMA, AND THE COMPONENTS OF SELECTION Abstract The n-player prisoner s dilemma (PD) is a useful model of multilevel selection for altruistic traits. It highlights the non zero-sum

More information

How Altruism Can Prevail Under Natural Selection

How Altruism Can Prevail Under Natural Selection How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version: March 22, 2002 How Altruism Can Prevail Under Natural Selection

More information

Axiomatic bargaining. theory

Axiomatic bargaining. theory Axiomatic bargaining theory Objective: To formulate and analyse reasonable criteria for dividing the gains or losses from a cooperative endeavour among several agents. We begin with a non-empty set of

More information

Mohammad Hossein Manshaei 1394

Mohammad Hossein Manshaei 1394 Mohammad Hossein Manshaei manshaei@gmail.com 1394 2 Concept related to a specific branch of Biology Relates to the evolution of the species in nature Powerful modeling tool that has received a lot of attention

More information

Near-Potential Games: Geometry and Dynamics

Near-Potential Games: Geometry and Dynamics Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo September 6, 2011 Abstract Potential games are a special class of games for which many adaptive user dynamics

More information

DARWIN: WHICH MATHEMATICS?

DARWIN: WHICH MATHEMATICS? 200 ANNI DI DARWIN Facoltà di Scienze Matemtiche Fisiche e Naturali Università del Salento 12 Febbraio 2009 DARWIN: WHICH MATHEMATICS? Deborah Lacitignola Department of Mathematics University of Salento,,

More information

Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks

Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks Pierre Coucheney, Corinne Touati, Bruno Gaujal INRIA Alcatel-Lucent, LIG Infocom 2009 Pierre Coucheney (INRIA)

More information

arxiv: v1 [q-bio.pe] 12 Dec 2013

arxiv: v1 [q-bio.pe] 12 Dec 2013 Evolutionary game theory and the tower of Babel of cooperation: Altruism, free-riding, parasitism and the structure of the interactions in a world with finite resources. Rubén J. Requejo-Martínez a, a

More information

First Prev Next Last Go Back Full Screen Close Quit. Game Theory. Giorgio Fagiolo

First Prev Next Last Go Back Full Screen Close Quit. Game Theory. Giorgio Fagiolo Game Theory Giorgio Fagiolo giorgio.fagiolo@univr.it https://mail.sssup.it/ fagiolo/welcome.html Academic Year 2005-2006 University of Verona Summary 1. Why Game Theory? 2. Cooperative vs. Noncooperative

More information

Computing Solution Concepts of Normal-Form Games. Song Chong EE, KAIST

Computing Solution Concepts of Normal-Form Games. Song Chong EE, KAIST Computing Solution Concepts of Normal-Form Games Song Chong EE, KAIST songchong@kaist.edu Computing Nash Equilibria of Two-Player, Zero-Sum Games Can be expressed as a linear program (LP), which means

More information

A generalization of Hamilton s rule Love thy sibling how much?

A generalization of Hamilton s rule Love thy sibling how much? A generalization of Hamilton s rule Love thy sibling how much? Ingela Alger and Jörgen W. Weibull May 9, 2011 Abstract According to Hamilton s (1964) rule, a costly action will be undertaken if its fitness

More information

Survival of Dominated Strategies under Evolutionary Dynamics. Josef Hofbauer University of Vienna. William H. Sandholm University of Wisconsin

Survival of Dominated Strategies under Evolutionary Dynamics. Josef Hofbauer University of Vienna. William H. Sandholm University of Wisconsin Survival of Dominated Strategies under Evolutionary Dynamics Josef Hofbauer University of Vienna William H. Sandholm University of Wisconsin a hypnodisk 1 Survival of Dominated Strategies under Evolutionary

More information

Policing and group cohesion when resources vary

Policing and group cohesion when resources vary Anim. Behav., 996, 52, 63 69 Policing and group cohesion when resources vary STEVEN A. FRANK Department of Ecology and Evolutionary Biology, University of California at Irvine (Received 5 January 996;

More information

Emergence of Cooperation and Evolutionary Stability in Finite Populations

Emergence of Cooperation and Evolutionary Stability in Finite Populations Emergence of Cooperation and Evolutionary Stability in Finite Populations The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation

More information

DETERMINISTIC AND STOCHASTIC SELECTION DYNAMICS

DETERMINISTIC AND STOCHASTIC SELECTION DYNAMICS DETERMINISTIC AND STOCHASTIC SELECTION DYNAMICS Jörgen Weibull March 23, 2010 1 The multi-population replicator dynamic Domain of analysis: finite games in normal form, G =(N, S, π), with mixed-strategy

More information

Correlated Equilibria of Classical Strategic Games with Quantum Signals

Correlated Equilibria of Classical Strategic Games with Quantum Signals Correlated Equilibria of Classical Strategic Games with Quantum Signals Pierfrancesco La Mura Leipzig Graduate School of Management plamura@hhl.de comments welcome September 4, 2003 Abstract Correlated

More information

Population Games and Evolutionary Dynamics

Population Games and Evolutionary Dynamics Population Games and Evolutionary Dynamics William H. Sandholm The MIT Press Cambridge, Massachusetts London, England in Brief Series Foreword Preface xvii xix 1 Introduction 1 1 Population Games 2 Population

More information

Brief history of The Prisoner s Dilemma (From Harman s The Price of Altruism)

Brief history of The Prisoner s Dilemma (From Harman s The Price of Altruism) Brief history of The Prisoner s Dilemma (From Harman s The Price of Altruism) 1948 The RAND Corporation. a civilian nonprofit think tank of an elite cadre of physicists, mathematicians, economists, and

More information

Irrational behavior in the Brown von Neumann Nash dynamics

Irrational behavior in the Brown von Neumann Nash dynamics Games and Economic Behavior 56 (2006) 1 6 www.elsevier.com/locate/geb Irrational behavior in the Brown von Neumann Nash dynamics Ulrich Berger a,, Josef Hofbauer b a Vienna University of Economics and

More information

Assortative Matching with Inequality in Voluntary Contribution Games

Assortative Matching with Inequality in Voluntary Contribution Games Comput Econ (2018) 52:1029 1043 https://doi.org/10.1007/s10614-017-9774-5 Assortative Matching with Inequality in Voluntary Contribution Games Stefano Duca 1 Dirk Helbing 1 Heinrich H. Nax 2 Accepted:

More information

Divided We Stand: The Evolution of Altruism. Darwin 101. The Paradox of Altruism. Altruism: Costly behavior that helps others

Divided We Stand: The Evolution of Altruism. Darwin 101. The Paradox of Altruism. Altruism: Costly behavior that helps others Predator inspection Divided We Stand: The Evolution of Altruism Karthik Panchanathan buddha@ucla.edu Territorial defense Hunting Grooming Altruism: Costly behavior that helps others Food sharing Warfare

More information

Mechanism Design: Basic Concepts

Mechanism Design: Basic Concepts Advanced Microeconomic Theory: Economics 521b Spring 2011 Juuso Välimäki Mechanism Design: Basic Concepts The setup is similar to that of a Bayesian game. The ingredients are: 1. Set of players, i {1,

More information

Efficient social contracts and group selection

Efficient social contracts and group selection Biol Philos DOI 10.1007/s10539-011-9265-3 Simon M. Huttegger Rory Smead Received: 16 November 2010 / Accepted: 17 March 2011 Ó Springer Science+Business Media B.V. 2011 Abstract We consider the Stag Hunt

More information

Université Libre de Bruxelles

Université Libre de Bruxelles Université Libre de Bruxelles Institut de Recherches Interdisciplinaires et de Développements en Intelligence Artificielle Evolutionary Game Dynamics of Intrademic Multilevel Selection Tom Lenaerts, Anne

More information

Near-Potential Games: Geometry and Dynamics

Near-Potential Games: Geometry and Dynamics Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo January 29, 2012 Abstract Potential games are a special class of games for which many adaptive user dynamics

More information

Play Locally, Learn Globally: The Structural Basis of Cooperation

Play Locally, Learn Globally: The Structural Basis of Cooperation Play Locally, Learn Globally: The Structural Basis of Cooperation Jung-Kyoo Choi 1 University of Massachusetts, Amherst, MA 01003 November 2002 1 I would like to thank the Santa Fe Institute for the Computational

More information

ALTRUISM OR JUST SHOWING OFF?

ALTRUISM OR JUST SHOWING OFF? ALTRUISM OR JUST SHOWING OFF? Soha Sabeti ISCI 330 April 12/07 Altruism or Just Showing Off? Among the many debates regarding the evolution of altruism are suggested theories such as group selection, kin

More information

University of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): /jeb.

University of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): /jeb. Okasha, S., & Martens, J. (2016). Hamilton's rule, inclusive fitness maximization, and the goal of individual behaviour in symmetric two-player games. Journal of Evolutionary Biology, 29(3), 473-482. DOI:

More information

Evolutionary Games and Computer Simulations

Evolutionary Games and Computer Simulations Evolutionary Games and Computer Simulations Bernardo A. Huberman and Natalie S. Glance Dynamics of Computation Group Xerox Palo Alto Research Center Palo Alto, CA 94304 Abstract The prisoner s dilemma

More information

Evolution of Cooperation in the Snowdrift Game with Incomplete Information and Heterogeneous Population

Evolution of Cooperation in the Snowdrift Game with Incomplete Information and Heterogeneous Population DEPARTMENT OF ECONOMICS Evolution of Cooperation in the Snowdrift Game with Incomplete Information and Heterogeneous Population André Barreira da Silva Rocha, University of Leicester, UK Annick Laruelle,

More information

Static (or Simultaneous- Move) Games of Complete Information

Static (or Simultaneous- Move) Games of Complete Information Static (or Simultaneous- Move) Games of Complete Information Introduction to Games Normal (or Strategic) Form Representation Teoria dos Jogos - Filomena Garcia 1 Outline of Static Games of Complete Information

More information

Index. Causality concept of, 128 selection and, 139, 298. Adaptation, 6, 7. See also Biotic adaptation. defining, 55, 133, 301

Index. Causality concept of, 128 selection and, 139, 298. Adaptation, 6, 7. See also Biotic adaptation. defining, 55, 133, 301 Index Adaptation, 6, 7. See also Biotic adaptation altruistic, 60-61 defining, 55, 133, 301 group, 136-137 group -related, 52, 53 at level of multicellular vehicle, 172 meaning of, 238 at organismallevel,

More information

M any societal problems, such as pollution, global warming, overfishing, or tax evasion, result from social

M any societal problems, such as pollution, global warming, overfishing, or tax evasion, result from social SUBJECT AREAS: SOCIAL EVOLUTION SOCIAL ANTHROPOLOGY COEVOLUTION CULTURAL EVOLUTION Received 8 February 2013 Accepted 4 March 2013 Published 19 March 2013 Correspondence and requests for materials should

More information

An Introduction to Evolutionary Game Theory

An Introduction to Evolutionary Game Theory An Introduction to Evolutionary Game Theory Lectures delivered at the Graduate School on Nonlinear and Stochastic Systems in Biology held in the Department of Applied Mathematics, School of Mathematics

More information

Outline for today. Stat155 Game Theory Lecture 16: Evolutionary game theory. Evolutionarily stable strategies. Nash equilibrium.

Outline for today. Stat155 Game Theory Lecture 16: Evolutionary game theory. Evolutionarily stable strategies. Nash equilibrium. Outline for today Stat155 Game Theory Lecture 16: Evolutionary game theory Peter Bartlett October 20, 2016 Nash equilibrium 1 / 21 2 / 21 A strategy profile x = (x1,..., x k ) S 1 Sk is a Nash equilibrium

More information

Complex networks and evolutionary games

Complex networks and evolutionary games Volume 2 Complex networks and evolutionary games Michael Kirley Department of Computer Science and Software Engineering The University of Melbourne, Victoria, Australia Email: mkirley@cs.mu.oz.au Abstract

More information

Applications of Game Theory to Social Norm Establishment. Michael Andrews. A Thesis Presented to The University of Guelph

Applications of Game Theory to Social Norm Establishment. Michael Andrews. A Thesis Presented to The University of Guelph Applications of Game Theory to Social Norm Establishment by Michael Andrews A Thesis Presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Mathematics

More information

Mathematical Economics - PhD in Economics

Mathematical Economics - PhD in Economics - PhD in Part 1: Supermodularity and complementarity in the one-dimensional and Paulo Brito ISEG - Technical University of Lisbon November 24, 2010 1 2 - Supermodular optimization 3 one-dimensional 4 Supermodular

More information

C31: Game Theory, Lecture 1

C31: Game Theory, Lecture 1 C31: Game Theory, Lecture 1 V. Bhaskar University College London 5 October 2006 C31 Lecture 1: Games in strategic form & Pure strategy equilibrium Osborne: ch 2,3, 12.2, 12.3 A game is a situation where:

More information

Gains in evolutionary dynamics: unifying rational framework for dynamic stability

Gains in evolutionary dynamics: unifying rational framework for dynamic stability Gains in evolutionary dynamics: unifying rational framework for dynamic stability Dai ZUSAI October 13, 2016 Preliminary 1 Abstract In this paper, we investigate gains from strategy revisions in deterministic

More information

EVOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOVE GAMES

EVOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOVE GAMES ROCKY MOUNTAIN JOURNAL OF MATHEMATICS olume 25, Number 1, Winter 1995 EOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOE GAMES R. CRESSMAN ABSTRACT. Although two individuals in a biological species often interact

More information

Measures of Assortativity

Measures of Assortativity University of California, Santa Barbara From the SelectedWorks of Ted C Bergstrom 013 Measures of Assortativity Ted C Bergstrom, University of California, Santa Barbara Available at: https://works.bepress.com/ted_bergstrom/10/

More information