W-S model prediction, Game theory. CS 249B: Science of Networks Week 06: Monday, 03/03/08 Daniel Bilar Wellesley College Spring 2008

Size: px

Start display at page:

Download "W-S model prediction, Game theory. CS 249B: Science of Networks Week 06: Monday, 03/03/08 Daniel Bilar Wellesley College Spring 2008"

Anissa Hodges
5 years ago
Views:

1 W-S model prediction, Game theory CS 249B: Science of Networks Week 06: Monday, 03/03/08 Daniel Bilar Wellesley College Spring

2 Goals this lecture Watts-Strogatz (1998) s Small World model Regular (Ordered) Lattice Random Lattice Properties W-S predictions: Some claims for dynamical systems Infectious disease spread Iterated n-player Prisoner s Dilemma New concept (course weak link ): Game theory 2

3 Review: Network Models (Good) models and reality should have features in common! Features of real life networks that we have encountered so far Power law distributions on degree k i Fat tail, Megavalues do occur Short average path length l Small world, Six degrees of separation High average clustering C My friends are friends with each other and a few other properties we ll see later on 3

4 Review: Erdos-Renyi (1959) For <k> >1 C rand ~ 1/N (if the average degree <k> is held constant) <k> pn p = 0.0 ; k = 0 p = 0.09 ; k = 1 Short average path length l High average clustering C Power law distributions on degree k i p = 1.0 ; k N-1 4

5 Watts-Strogatz Small World (1998) WS constructed a model in which networks can have both short path lengths l like ER and high clustering C Result of W-S model in English: a few random links in an otherwise clustered graph give an average shortest path close to that of a random graph D. J. Watts and S. H. Strogatz, Collective dynamics of smallworld networks, Nature, 393 (1998), pp

6 W-S model investigation Effects of rewiring probability p Does small shortest path mean small clustering? And large shortest path mean large clustering? Watts figured it out through numerical simulation As he increased p from 0 to 1. Fast decrease of mean distance l(p) Slow decrease in clustering C(p) Exist range of p which generated graphs with short path lengths l(p) and high clustering C(p) 6

7 Original W-S model Each node has K>=4 nearest neighbors (local) Probability p of rewiring to randomly chosen nodes p small gives Regular lattice (also called Ordered ) p large gives Random graph Tunable with parameter p What is a Regular Lattice? 7

8 Constructing W-S Select a fraction p of edges Reposition on of their endpoints Add a fraction p of additional edges leaving underlying lattice intact No loops or multiple edges allowed 8

9 C(p) for Ordered lattice (p =0) Recall Clustering Coefficient C k neighbors, who can have k*(k-1)/2 pairwise connections between them Some of the connections between them are present in the lattice Here, for K=4: C i = 3/ ((4*3)/2) = ½ <C i > roughly ½ as well 9

10 C(p) for Ordered, K-connected lattice i Which nodes do i and i s neighbours have in common? K/2 hops away from i can connect to (K/2 1) of i s neighbors K/2-1 hops away from i can connect to (1 + K/2 1) neighbors The number of connections between neighbors is given by K/2 2 hops away from i (2 + K/2 1) neighbors. 1 hop away from i 2*(K/2 1) K 1 2 j= 0 K ( 2 + i 1) = 3 8 K( K 2) Sum this up Note: We have to multiply by factor of 2 because i has neighbors on both sides but also have to divide by a factor of 2 because edges are undirected -> no net effect on summation 10

11 C(p) for ordered, k-connected lattice (cont) The number of connections between neighbors is given by K 1 2 j= 0 K ( 2 + i 1) = 3 8 K( K 2) The maximum number of connections is k*(k-1)/2 Clustering coefficient is C = 3( K 4( K 2) 1) 11

12 l(p) for Ordered lattice Recall: Geodesic distance l i,j between vertices i and j is the shortest path connecting i and j We are also interested in <l>, the average geodesic distance between vertex pairs in G Also called sometimes average path Average node is N/k hops away (for k=4, a quarter of the way around the ring), and you can hop over k/2 nodes at a time l N 2 K >> 1 12

13 C(p), l(p) for Random Lattice (p >> 0) There are an average of K links per node. The probability that any two nodes are connected is p = K/N The probability that two nodes which share in a neighbor in common are connected themselves is the same as any two random nodes: K/N (actually (K-1)/N because they have already expended one edge on their common neighbor. l C ln ln K N N K small small As p=1, W-S is (almost) like Erdos-Renyi random graphs 13

match the properties expected from real networks! Want BOTH!

14 Summary: Regular vs Random Graph Ordered Graphs (p approx 0) Have a high clustering coefficient but high path lengths Random Graphs (p >> 0) have low path length but a low clustering coefficient Each match the properties expected from real networks! Want BOTH! Ordered Graph (k=4) Long paths L ~ n/(2k) Highly clustered C~3/4 Random Graph (k=4) Short path length L~log k N Almost no clustering C~k/n 14

15 Tune p Change in clustering coefficient and average path length as a function of the proportion of rewired edges Exact analytical solution C(p)/C(0) l(p)/l(0) No exact analytical solution 1% of links rewired 10% of links rewired 15

16 Small-World is Sweet Spot Does this model real life networks? Is there still a fly (or two) in the ointment? 16

17 Fly #1: Degree distribution Watts-Strogatz <k> approx. K P(k) ~ Poisson(K) Real-Life P(k) ~ k -α 17

18 Fly #2: Mechanism W-S assume 1. Fixed N But networks grow and shrink 2. Equal (rewiring/addition of link) probability p Each link is equally probable of being required? Seems too crude.. what about the rich getting richer? We ll see more sophisticated Barabasi-Albert (BA) model Two mechanism: Growth and Preferential Attachment 18

19 W-S claims for dynamical systems Disease spread Spreads easily in SW (Cellular automata) (Coupled oscillators) Game theory (iterated PD n-player) Cooperation less likely with addition of shortcuts and increasing p in population with generalized tit for tat strategy 19

20 Dynamical systems:disease spread in WS Simulation results for a simple model of disease spreading a: Critical infectiousness r half, at which the disease infects half the population, decreases with p. b: The time T(p) required for a maximally infectious disease (r = 1) to spread throughout the entire population has essentially the same functional form as the characteristic path length L(p) What does this mean/seem to imply? 20

21 Game theory A mathematical theory designed to model: How rational individuals should behave when individual outcomes are determined by collective behavior Rational usually means only out for themselves --- but not always Rational players develop strategies to play the game Traditionally viewed as a subject of economics, but subsequently applied by many fields evolutionary biology, social psychology, actual games (chess, checkers, tic-tac-toe, Poker (people win in tournaments in Vegas today!)), auctions, foreign policy (Cold war) There exists conflict of interest almost everywhere In all man s written record there has been a preoccupation with conflict of interest; possibly the topics of God, love, and inner struggle have received comparable attention. Game theory, which separately originated from Borel in 1921 and von Neumann in 1928 and formalized by von Neumann and Morgenstern in 1944, is one of the best mathematical theories to analyze situations of conflict of interest 21

22 Some history Emile Borel (1921) French mathematician, published several papers on the theory of games Used poker as an example and addressed the problem of bluffing and second-guessing the opponent in a game of imperfect information. Borel envisioned game theory as being used in economic and military applications Von Neumann and Morgenstern (1944) Mathematician, Hungarian genius: "Theory of Parlor Games (1928) Teamed up with Oskar Morgenstern, an Austrian economist at Princeton, to develop his theory Book Theory of Games and Economic Behavior (1944) was milestone: Finding optimal solutions for two-person cooperative zero-sum games John Nash (1950) US Mathematician Expanded VN-M results to nonzero-sum games Harsanyi (1964) Hungarian economist, Nobel Prize economics winner (1994) with Nash, Selten Expanded to incomplete information, Bayesian games Bayesian game is one in which information about characteristics of the other players (i.e. payoffs) is incomplete. Can convert any incomplete-information game into an equivalent complete-information game containing random moves, thereby significantly expanding the applicability of game theory to political and economic conflicts 22

23 Types of games Zero-sum games Player benefits only at expense of another Net result of all outcomes, for every combination of strategies, always adds to zero Examples: dividing a cake, chess, matching pennies Nonzero-sum games Win-win situation exist for both players Some outcomes have net results greater or less than zero Example: PD, most economic situations like trade are non-zero-sum, since valuable goods and services can be created, destroyed, or badly allocated, and any of these will create a net gain or loss. 23

24 Prisoner s dilemma (non-zero sum) cooperate defect Claim that (defect, defect) is an equilibrium: If I am definitely going to defect, you choose between -10 and -8 so you will also defect Same logic applies to me Equilibrium No player has anything to gain by changing only his or her own strategy Pareto-optimal No-one can be made better off with making at least one other person worse off No other pair of strategies is an equilibrium in PD But is it NOT the (Pareto-)optimal result they both should (cooperate, cooperate) Paradox: Individually rational decision lead collectively to worse results In a non-zero sum game the Pareto optimum and the Nash Equilibrium can be opposite In other words, a non-zero sum game may not have a solution that is both optimal and stable; the Pareto optimum can be unstable and the Nash equilibrium can be sub-optimal cooperate -1, , -10 defect -10, , -8 cooperate = deny the crime defect = confess guilt of both 24

25 Multi-player PD Also known as Tragedy of the commons (Hardin, UC Santa Barbara, 1968) Hardin argued (using the example of a pasture shared by shepherds) that the problem of human population growth and Earth's natural resources is a multiperson Prisoner s Dilemma (PD) Family size: In many third world countries, it is in a person s interest to have many kids but, collectively, it spells resource overload Overfishing ( ) Water supply (LA, Las Vegas, see ) Can you find other examples of this in every day life? 25

26 Penny matching (zero sum) One player will be "even" and the other will be "odd" Each one then shows a penny. heads heads 1, -1 tails -1, 1 tails -1, 1 1, -1 What are the equilibrium strategies now? There are none! if I play heads then you will of course play tails but that makes me want to play tails too which in turn makes you want to play heads Etc.. But what if we can each (privately) flip coins? the strategy pair (1/2, 1/2) is an equilibrium Such randomized strategies are called mixed strategies 26

27 Game-theoretical strategies If you have more than one action, mixed strategy is a distribution on them e.g. 1/3 rock, 1/3 paper, 1/3 scissors A general mixed strategy is a vector P = (P[1], P[2], P[n]): P[i] is a distribution over the actions for player i assume everyone knows all the distributions P[j] but the coin flips used to select from P[i] known only to i P is an equilibrium if: for every i, P[i] is a best response to all the other P[j] 27

28 Nash equilibrium A general mixed strategy P = (P[1], P[2], P[n]) is an equilibrium if: for every i, P[i] is a best response to all the other P[j] Nash great result (1950) every game, zero-sum and general-sum, has at least one mixed strategy equilibrium no matter how many rows and columns there are in fact, no matter how many players there are Thus known as a Nash equilibrium A major reason for Nash s Nobel Prize in economics, and the 2005 Nobel prize for Shelling and Aumann 28

29 Example: Multiple Nash Equilibria hawk dove hawk dove (V-C)/2, (V-C)/2 0, V V, 0 V/2, V/2 Two parties confront over a resource of value V May fight or not Cost of losing a fight: C > V Assume parties are equally likely to win or lose There are three Nash equilibria: (hawk, dove), (dove, hawk),(v/c hawk, V/C hawk) Alternative interpretation for C >> V: Chicken game (hawk = speed through intersection, dove = yield) 29

30 Consequence of Existence of Equilibrium Let s take chess, a two player, zero-sum game Imagine an absurdly large game matrix for chess: each row/column represents a complete strategy for playing strategy = a mapping from every possible board configuration to the next move for the player number of rows or columns is huge --- but finite! Thus, an equilibrium for chess exists! There is a theoretical solution to chess.. You would not have to play it anymore, since the game s outcome would be completely determined after white s first move It s just completely infeasible to compute it Note: This equilibrium result for zero sum, 2-player games is due to von Neumann (1928).. Nash extended this for non-zero sum games In July 2007, Checkers was solved After 18 years and 500 billion billion checkers positions, Schaeffer (U Alberta) built a checkers-playing computer program Chinook that cannot be beaten. May be played to a draw but will never be defeated See 30

31 Repeated Games Nash equilibrium analyzes one-shot games We meet for the first time, play once, and separate forever Natural extension -> Repeated games Play the same game (e.g. Prisoner s Dilemma) many times in a row Like a board game, where the state is the history of play so far Strategy = a mapping from the history so far to your next move So repeated games also have a Nash equilibrium may be different from the one-shot equilibrium (and may move to optimal solution) 31

32 Repeated PD If we play for R rounds, and both know R: (always defect, always defect) can be shown by backwards induction to still be the only Nash equilibrium If we do not know R, e.g. how many times we are playing? Cooperation and Tit-for-tat strategies can become equilibria World-wide yearly competition started by Axelrod(U mich, 1980) Tit for tat winning generalized strategy since 1981 (proved in 1991) Tit-for-two-tat introduced early on but loses against more aggressive competitors Master-Slave in 2004 wins (two like players recognize each other by a sequence of 5-10 moves, then slave sacrifice themselves for master!) see 32

33 Game theory and Networks What does game theory have to do with networks? Both can leverage one another 1. Networks as a representation of games A game between two players can be modeled by two vertices with a single link between them A link is then a much richer interaction than just info transmission, messages 2. Games capturing /exploiting/examining structured interaction that arise in networks Local interaction Shared information: economies, financial markets Voting systems, evolutionary games Now.. Back to W-S claim: Cooperation less likely with addition of shortcuts and increasing p in population with generalized tit for tat strategy What do you think this means? 33

34 For Thursday min: Alberich (2002) Marvel Universe looks almost like a real social network Write down terms, concepts that are new/unclear Draw concept map to hand in to me 34

Introduction to (Networked) Game Theory. Networked Life NETS 112 Fall 2016 Prof. Michael Kearns

Introduction to (Networked) Game Theory Networked Life NETS 112 Fall 2016 Prof. Michael Kearns Game Theory for Fun and Profit The Beauty Contest Game Write your name and an integer between 0 and 100 Let