EVOLUTIONARY GAMES WITH GROUP SELECTION
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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
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