Noncooperative Game And Competitive Influence DingZhu Du University
- Slides: 39
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. 6/8/2021 27
Example 2 6 1 5 3 4 Step 2: 1 --3, 6 --2. 3 becomes red. 6/8/2021 28
Example 2 6 1 5 3 4 Step 3: 1 --2, 3 --4, 6 --4. 4 becomes red or green. 6/8/2021 29
Example 2 6 1 5 3 4 Step 4: 1 --3, 3 --2, 6 --4, 4 --5. 5 is protected. 6/8/2021 30
Example 2 6 1 5 3 4 end: no more node can be activated. 6/8/2021 31
Lemma 32
Lemma This lemma seems useless? It can be used later! 6/8/2021 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! 6/8/2021 39
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