Network Theory: Computational Phenomena and Processes Network Games

Network Theory:Computational Phenomena and ProcessesNetwork Games Dr. Henry HexmoorDepartment of Computer ScienceSouthern Illinois University Carbondale

Network Games:Basic Framework • A set of players A • Each player has a set of actions (i.e., strategies) • Network relationships among players G = (V,E) • Payoffs (i.e., utilities) Strategy Profile s = (s1, s2, …, sn) Пi : Sn g → R s-i ≡ (s1, s2, …, si-1, si+1, …, sn) ≡ Strategy profile of players minus player i Ni (g) ≡ neighbors of player i in g SNi(g) ≡ Straegy profile of players i’s neighbors

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

Global effect Пi (s/g) ≡ ɸ n-1 (si, s-i) Local + Global effects: Пi (s/g) ≡ ɸ (Si, g ɳ i(SNi(g)) , h ɳ i(Sk ϵ Ni (s) U {i}))

Cumulative Effect Пi (s/g) ≡ ɸ (Si, , k ) J k i (g) • Externalities(i.e. Indirect Effects) A game with pure local effects exhibits positive externality.

if k{0,1, 2,…, n-1}, siS , pair of neighbors with strategies S k , S’k S k implies that: ɸk (si, sk) ≥ ɸk(si,s’k). This game exhibits negative externality when: ɸk(si, sk) ≤ ɸk(si,s’k) A game exhibits strategic compliments / substitutes if the marginal retunes to own actions for player ian increasing o decreasing in the effects of her neighbors.

Nash Equilibrium A strategy profile S* = (S*1, S*2 , S*3 , … S*n ) is a Nash Equilibrium in networking if player i , gives the strategies of other players S*-i , S*i maximizes her payoffs. S* = (S*i , S*-i) is NE ing if Пi (S*i , S*-i /g) ≥ Пi (Si , S*-i /g) siS , i NE is strict if “≥” is “>”

Network Game Example 1:Dynamic Computer Network Configuration • S1 wishes content to terminal t1. • S2 wishes to content terminal t2. • Player 1 only contributes to edges a and b. • Player 2 only contributes to edges b and c. • Edges a and c must be bought full by player 1 adz, respectively. • At least one player must contribute for edge b. • NE since a player can buy d and be done.

Fractional NE (Mixed NE)

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

Jocab Steiner JocabSteiner 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.

Continue Ci (S)= α . + (s)  strategy profile α  # of edges purchased  # of sources  distance from i to j

Pure NE / NP hard • Pure NE is on S such that s & s’ may only differ in one component. Theorem : 1 NP hard to compose best response (Farbrikent. Et.al.2003)

Price Of Anarchy • 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.

Assumptions • The mechanism does not have power to enforce player choice • The mechanism does not have knowledge to detect if players disobey • Players have no private values. Values are common knowledge. Cost Sharing: a set of resources desired by players

The Connection Game • 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 • Pi(e) = Payment of player I for edge e. • If ∑iPi (e) ≥ C (e)  e is a purchased link edge. • Gp = graph of bought edges with payments • P= <P1 ,…, Pn> • NE is a payment function P such that no player has incentive to deviate function

A Connection game without a NE

Steiner Tree Algorithm

Network Game Example 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

PAYOFF MATRIX • Payoff matrix summarizes payoffs of a decision in a tabular form.

PRISONER’S DILEMMA version of matrix • Defect, Defect is a dominant equilibrium in a one shot game. • In repeated interaction games, Coop, Coop is a social optimum.

HAWK-DOVE GAME version of matrix • Mixed Strategy:

EVOLUTIONARY GAMES ON NETWORKS • Evolution of strategies over repeated games. • Example: • Population of beetles competing for food. • Not zero sum! • strategy choice. • NE is not applicable.

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

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?

EXPECTED PAYOFFS • Result: • Small is not a stable strategy.

OPPOSITE ASSUMPTION • Assume 1-x fraction of population use large and x fraction use small option. • Result: • Large is a stable strategy.

ANALYSIS • Being largeproduces higher payoff and small beetles cannot affect them.

Network Theory: Computational Phenomena and Processes Network Games