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Evolutionary Game Algorithm for continuous parameter optimization Alireza Mirian

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Evolutionary Computation presentation, 2012 A system in which a number of rational players make decision in a way that maximize their utility. 2 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms What is a Game?

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Alireza Mirian Evolutionary Computation presentation, 2012 Each player (agents) has a set of possible actions (strategies) to choose from Each player have their Utility Function that determines the profit/outcome of any decision Agents are rational self-interested decision makers, i.e. they make decision upon their view of utility. Players doesn’t have full control over outcome. That is, a person’s success is based upon the choices of others 3 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms What is a Game?

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Alireza Mirian Evolutionary Computation presentation, 2012 Games have wide range, from parlor games (chess, poker, bridge) to various economic, political, military or biological situations. 4 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms What is a Game?

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Alireza Mirian Evolutionary Computation presentation, 2012 Game theory: the study of mathematical models of games John von Neumann & John Nash Has lots of applications in economics, political science, and psychology, and other, more prescribed sciences, like logic or biology. tries to find a “solution” for game 5 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms What is Game Theory?

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Alireza Mirian Evolutionary Computation presentation, 2012 Decision Theory: A special case of Game with one player 6 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms What is Game Theory?

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Alireza Mirian Evolutionary Computation presentation, 2012 In non-cooperative games the goal of each player is to achieve the largest possible individual gain (profit or payoff) In cooperative games the action of players are directed to maximize the gain of “collectives” (coalitions) without subsequent subdivision of the gain among the players within the coalition 7 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Non-cooperative and cooperative games

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Alireza Mirian Evolutionary Computation presentation, 2012 Non-cooperative: Two player Hokm Cooperative: Four player Hokm 8 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Non-cooperative and cooperative games

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Alireza Mirian Evolutionary Computation presentation, 2012 Let I denote the set of players Let S i denote the set of all possible actions for player i (strategies of player i) |S i | > 1 (why?) At each “round” of the game, each player chooses a certain strategy s i S i So, after each round: (s 1,s 2,…,s n ) = s is put together. This system is called a situation In each situation, each player gets a profit S = S 1 ×…×S n = ∏ iI S i (strategy profile). 9 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Non-cooperative game

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Alireza Mirian Evolutionary Computation presentation, 2012 Definition of Non-cooperative Game: G=[ I, { S i } iI, {U i } iI ] I = {1,2, …, n} : set of players S i : strategy set for player i (set of possible actions) Ui : Utility function defined on set S=∏ iI S i 10 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Non-cooperative game

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Alireza Mirian Evolutionary Computation presentation, 2012 Example: 4-barg! I = {1,2} S 1 = {,,, } S 2 = {,,, } U 1 ( s ) = U 1 ({, }) = 2 U 2 ( s ) = U 2 ({, }) = 0 11 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Non-cooperative game 2 1 s ={, }

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Alireza Mirian Evolutionary Computation presentation, 2012 s = {s 1, …,s i-1, s i, s i+1, …, s n } s || s ΄ i = {s 1, …,s i-1, s ΄ i, s i+1, …, s n } That is, s || s ΄ i is a situation that differs from s, only in s i Admissible situation: a situation s is called admissible for player i if any other strategy s ΄ i for this player we have: U i (s || s ΄ i ) ≤ U i (s) 12 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Admissible situation

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Alireza Mirian Evolutionary Computation presentation, 2012 A situation s, which is admissible for all the players is called an equilibrium situation That is, in a equilibrium situation, no player is interested to change their strategy. (why?) Solution of a non-cooperative game: determination of an equilibrium situation 13 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Equilibrium point

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Alireza Mirian Evolutionary Computation presentation, 2012 An optimization problem: arg max f(x) x ∈ D where x = (x 1,x 2,...,x n ) ∈ R n, xi ∈ [x i l, x i u ], i = 1,2,...,n, is n-dimensional real vector, f(x) is the objective function, D = [x i l, x i u ] ⊆ R n defines the search space, and x ∗ that satisfies f(x ∗ )= max { f (x) | x ∈ D } is the optimal solution of problem 14 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Optimization problem

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Alireza Mirian Evolutionary Computation presentation, 2012 In EGA the optimization problem maps into a non-cooperative Optimum will find by exploring the equilibrium situations in corresponding game Global convergence property of the algorithm is proofed 15 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Optimization problem and game

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Alireza Mirian Evolutionary Computation presentation, 2012 x = (x 1,x 2,...,x n )G = ( I, { S i } iI, {U i } iI ) Variable x is mapped to strategy profile of game agents Objective function f is mapped to game agents΄ utility function N x :the number of agents that their strategy profile will represent a variable x i |I| = n * n x | S i | = m Size of strategy profile of n x agent: m n x -1 Precision of this mapping: (x i u – x i l )/(m n x -1) 16 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Mapping between strategy profile and x i

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Alireza Mirian Evolutionary Computation presentation, 2012 Decoding function φ: x i = φ(s i ) = x i l + decimal(s i ) × (x i u – x i l )/(m n x -1) Example: f(x) = x 1 + x 2 where x i [-2.048, 2/048], i = 1,2 x n = 10, m = 2 overall strategy profile of n I = n × n x =20 agent is a binary string with length of 20: S: x 1 = decimal( ) 2 ×4.096/( ) x 2 = decimal( ) 2 ×4.096/( ) x 1 = , x 2 = What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Mapping between strategy profile and x i x1x1 x2x2

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Alireza Mirian Evolutionary Computation presentation, 2012 All the agents have the same utility function which is just objective function u = { u i (s) ≡ f(φ(s)), i є I} where I = {1, 2, 3, …, n I } s is the strategy profile of n I = n × n x In the previous example: s = ( ) u(i) = f(φ(s)) = f(x 1, x 2 ) = x 1 + x 2 = i = 1, 2, 3, …, What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Utility function

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Alireza Mirian Evolutionary Computation presentation, 2012 At the start of EGA each agent randomly selects a strategy from its strategy set {0, 1,..., m − 1} with a probability 1/m After that, In each loop: Random perturb: current strategy of each agent is replaced with a random strategy with a probability 1/m for each strategy agents will do a deterministic process to reach an equilibrium point s e (t) 19 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Procedure of EGA

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Alireza Mirian Evolutionary Computation presentation, 2012 Procedure EGA t = 0; randomly initialize s (0) and set it as current solution; while termination condition is not satisfied do perform a random perturb on current solution s (t) ; do a deterministic process to reach an equilibrium point s e (t) ; if utility of s e (t) ≥ utility of current solution current solution = s e (t) end t = t + 1; end 20 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Procedure of EGA

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Alireza Mirian Evolutionary Computation presentation, 2012 How to reach the equilibrium point? Coalition: n x agents that represent the same component x i of variable x are defined as one coalition In out example: agent 1, 2,..., 10 that represent x 1 is a coalition, and agent 11, 12,..., 20 that represent x 2 is another coalition. BRC: the strategy profile of a coalition that maximizes its utility while strategy profile of other coalitions are fixed is called the Best-Response Correspondence (BRC) of that coalition. Process of reaching equilibrium: While equilibrium point is not reached, all coalitions replace their strategy profile with their BRC in sequence 21 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Reaching equilibrium point

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Alireza Mirian Evolutionary Computation presentation, 2012 Pseudo code of reaching equilibrium point: while equilibrium state is not achieved for agent coalition i = 1, 2,...,n agent coalition i replaces its strategy profile with its BRC; end Now two other thing: How to decide whether an equilibrium point is achieved? How does an agent coalition find out its BRC 22 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Reaching equilibrium point

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Alireza Mirian Evolutionary Computation presentation, 2012 How to decide whether an equilibrium point is achieved? when r (the number index of BRC rounds) reaches a predefined number R the utility has not improved in d r consecutive rounds How does an agent coalition find out its BRC? Exact BRC ~> have to compute the utilities of all possible strategy profiles within its strategy profile space Cardinality of the strategy profile set of a coalition ( = m n x ) usually is a very large number inner level optimization is used to find an approximate BRC. 23 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms Two remaining problem

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Alireza Mirian Evolutionary Computation presentation, 2012 Inner level optimization for approximating BRC has two phases: first phase: with a perturb probability p d, the current strategy of each agent in a coalition is replaced with a new strategy with a probability 1/m for each strategy. Second phase: each agent in the coalition replaces its current strategy with an optimal strategy selected from its strategy set { 0,1,...,m − 1 } which maximizes its utility in sequence. inner level optimization process has the same structure as the main loop of EGA itself if we regard one agent as a coalition (except that the inner process only has one loop i.e. one BRC round) 24 What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms inner level optimization

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Alireza Mirian Evolutionary Computation presentation, What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms inner level optimization

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Alireza Mirian Evolutionary Computation presentation, What is Game Theory? Non-cooperative and cooperative games Equilibrium point Evolutionary Game Algorithm Mapping between strategy profile and x i Procedure of EGA Results and comparison with other algorithms inner level optimization

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Alireza Mirian Evolutionary Computation presentation, 2012 Y. Jun a, L. Xiande, H. Lu, “Evolutionary game algorithm for continuous parameter optimization”, Information Processing Letters, 2004 N. N. Vorob’ev, “Game Theory Lectures for Economists and Systems Scientists”, Springer-verlag,1977 R. D. Luce, H. Raiffa, “Games and Decision”, J. Wiley & sons, 1957 R. Cressman, “The Stability Concept of Evolutionary Game Theory”, Springer-verlag, 1992 E. V. Damme, “non-cooperative Games” TILEC and CentER, Tilburg University, 2004 Y. Jun, L. Xiande, H. Lu, “Evolutionary game algorithm for multiple knapsack problem”, Proc. of 2003 IEEE/WIC International Conference on Intelligent Agent Technology, Ross, Don, "Game Theory", The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), Edward N. Zalta (ed.), 2011 D. K. Levine, “What is Game Theory?”, Department of Economics, UCLA 27 References

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Alireza Mirian Evolutionary Computation presentation, 2012 Thanks for your attention :D 28

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