Noncooperative Game And Competitive Influence DingZhu Du University

Non-cooperative Game And Competitive Influence Ding-Zhu Du University of Texas at Dallas

Outline • Non-cooperative Game • Competitive Influence • Approximate Nash Equilibrium 2

“Decisions are made by a set of noncooperative agents whose action spaces are subsets of an underlying groundset. The actions of the agents induce some social utility, measured by a set function. The goal of the agents, though, is not to maximize the overall social utility; rather, they seek to maximize their own private utility functions. ” The only assumptions we make are • The social utility and private utility functions are measured in the same standard unit 3

Mathematical Formulation 4

groundset actions acts 5

Notations 6

Notations 7

Utility Functions 8

Nash Equilibrium Theorem (Nash, 1951) 9

A Beautiful Mind- John Nash 10

Utility System 11

Valid Utility System 12

Basic Utility System Theorem 13

Proof of basic => valid Submodularity basic 14

Remark 15

Lemma 16

Union and subtraction 17

Proof Submodular 18

Theorem 19

Proof utility system Valid utility system 20

Theorem 21

Proof 22

Outline • Non-cooperative Game • Competitive Influence • Approximate Nash Equilibrium 23

Independent Cascade (IC) Model • When node v becomes active, it has a single chance of activating each currently inactive neighbor w. • The activation attempt succeeds with probability pvw. • The deterministic model is a special case of IC model. In this case, pvw =1 for all (v, w).

Example Y 0. 6 Inactive Node 0. 3 0. 2 X 0. 4 0. 5 w 0. 2 U 0. 1 0. 3 0. 2 0. 5 v Stop! Active Node Newly active node Successful attempt Unsuccessful attempt

IC Model (Competitive Version) • each of b players selects a color and a set Si of at most ki nodes. • A node activated by players with one color will take the color. • A node activated by players with multi-color will take the color of one of the players uniformly at random. • Process ends until no new activations occur. 26

Example 2 6 1 5 3 4 Two players red and green. Step 1: 1 --2, 6 --2. 2 becomes red or green. 10/27/2020 27

Example 2 6 1 5 3 4 Step 2: 1 --3, 6 --2. 3 becomes red. 10/27/2020 28

Example 2 6 1 5 3 4 Step 3: 1 --2, 3 --4, 6 --4. 4 becomes red or green. 10/27/2020 29

Example 2 6 1 5 3 4 Step 4: 1 --3, 3 --2, 6 --4, 4 --5. 5 is protected. 10/27/2020 30

Example 2 6 1 5 3 4 end: no more node can be activated. 10/27/2020 31

Lemma 32

Lemma This lemma seems useless? It can be used later! 10/27/2020 33

Game • • Each node is an act. Each action for player i is a subset of ki nodes. Private utility function Social utility function 34

Valid Utility system Theorem Proof Nondecreasing 35

Outline • Non-cooperative Game • Competitive Influence • Approximate Nash Equilibrium 36

Theorem 37

Corollary 38

Thank you! 10/27/2020 39