Network Theory:Computational Phenomena and Processes

Network Games

Dr. Henry HexmoorDepartment of Computer Science

Southern 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 effectПi (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}, si S ,

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 i an 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) si S , 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

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.

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

1. The mechanism does not have power to enforce player choice

2. The mechanism does not have knowledge to detect if players disobey

3. 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 ∑i Pi (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

Player 2 (Consumer)

Player 1 (Provider)

Cooperate Defect

Cooperate , ,

Defect , ,

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

𝑃 (𝑥 , 𝑦 )=𝐵𝑒𝑛𝑒𝑓𝑖𝑡 (𝑥 ) −𝐶𝑜𝑠𝑡 (𝑥)

PRISONER’S DILEMMA version of matrix

Player 2

Player 1 Cooperate Defect

Cooperate , ,

Defect , ,

• 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

Player 2

Player 1 Cooperate Defect

Cooperate , ,

Defect , ,

• 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.

Beetle2

Beetle 1Small Large

Small , ,

large , ,

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?

Result: Small is not a stable strategy.

EXPECTED PAYOFFS

Assume 1-x fraction of population use large and x fraction use small option.

Result: Large is a stable strategy.

OPPOSITE ASSUMPTION

ANALYSIS

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