1. Network Theory: Computational Phenomena and Processes Network Games Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale

2. Network Games: Basic Framework

3. Pure local effect Considering effects of neighbors only П i (s/g) ≡ ɸ ɳ i (g) (si, s Ni(g) ) Observation: Payoffs of two players with the same degree are identical.

4. Global effect П i (s/g) ≡ ɸ n-1 (s i, s -i ) Local + Global effects: П i (s/g) ≡ ɸ (S i, g ɳ i (S Ni(g) ), h ɳ i (S k Ni (s) U {i} ))

5. C UMULATIVE E FFECT

7. N ASH E QUILIBRIUM

8. N ETWORK G AME E XAMPLE 1: Dynamic Computer Network Configuration

9. F RACTIONAL NE (M IXED NE)

10. SINGLE SOURCE GAME A single source game is one where players share a common terminals and each player has exactly one other terminal t i. Theorem: in any single source game, there is a NE which purchase T*, am minimum cost Steiner tree on all player’s terminal nodes.

11. J OCAB S TEINER Jocab Steiner tree ≡ Given a set of vertices V, interconnect them by a network of shortest length. we are allowed to add Steiner part to the minimum spanning tree.

12. C ONTINUE

13. P URE NE / NP HARD

14. P RICE O F A NARCHY Most games have many NE and one must select for the best one. Some have no NE The Price of Anarchy = [The Worst NE (the most expensive) ] / [ the Centralized Optimum Equilibrium] Mechanism Design = Design a game such that players chose a desired outcome; that outcome is perceived a best outcome and strategies are selected to produce the design outcome.

15. A SSUMPTIONS

16. T HE C ONNECTION G AME Players connect their terminal to a network by purchasing links and costs are shared. Given G = (V,E)C(e) = Costs of an edge ≥ 0 P i (e) = Payment of player I for edge e. If ∑ i P i (e) ≥ C (e)  e is a purchased link edge. G p = graph of bought edges with payments P= NE is a payment function P such that no player has incentive to deviate function

17. A C ONNECTION GAME WITHOUT A NE

18. S TEINER T REE A LGORITHM

19. N ETWORK G AME E XAMPLE 2: MOBILE DEVICE TETHERING A mobile device (MD) can provide network interface for another; i.e., Wifi hotspot. MDs can be players in a game in Provider and Consumer roles. Strategies: Provider Cooperate/share connections Defect/reject connections Consumer Cooperate/accept Defect/reject

20. PAYOFF MATRIX Player 2 (Consumer) Player 1 (Provider) CooperateDefect Cooperate Defect Payoff matrix summarizes payoffs of a decision in a tabular form.

21. PRISONER’S DILEMMA version of matrix Player 2 Player 1 CooperateDefect Cooperate Defect Defect, Defect is a dominant equilibrium in a one shot game. In repeated interaction games, Coop, Coop is a social optimum.

22. HAWK-DOVE GAME version of matrix Player 2 Player 1 CooperateDefect Cooperate Defect

23. EVOLUTIONARY GAMES ON NETWORKS Beetle2 Beetle 1 SmallLarge Small large

24. EVOLUTIONARY STABLE STRATEGY Fitness Reproductive success in passing a strategy to offspring. Stability A strategy is evolutionary stable of the whole population uses it.

25. An Example Assume x fraction of population use the large option and 1-x fraction use the small option. A small beetle against another small beetle with possibility 1-x. A small beetle against a large beetle with possibility x. Which strategy is stable? Small or Large?

26. EXPECTED PAYOFFS

27. OPPOSITE ASSUMPTION

28. ANALYSIS Being large produces higher payoff and small beetles cannot affect them.