CSCI 3210: Computational Game Theory Graphical Games: Structure - - PDF document
5/1/18 CSCI 3210: Computational Game Theory Graphical Games: Structure & Equilibria Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website: www.bowdoin.edu/~mirfan/CSCI-3210.html Questions u Relation
u Relation between connectivity & equilibria in
u Q1: What does a certain network structure tell
u Q2: What do the "rules of a game" tell us about
u Network formation games
u Provides a basis for understanding richer
u Powerful: Randomization abstracts a huge
u Context-free: independent of the story
u Static
u Given n nodes (n doesn’t grow over time)
u Variant I
u Inputs: number of nodes n and number of edges m u Create m edges uniformly at random out of nC2
total possible edges u Variant II: G(n, p)
u Inputs: number of nodes n and probability of
forming an edge = p
u Random payoff table for each player
u mk+1 payoff values for m actions and k neighbors u Payoff values are independent and identically
distributed according to some distribution (e.g., uniform) u 2 sources of randomness
u Graph G(n,p) is randomly created u Payoffs are randomly assigned
u Nonmonotonic relation w.r.t. average degree Empty graph Complete graph Probability of existence of pure-strategy NE prob = 1 prob = 1 – 1/e [Dresher (1970)] prob = 0 as nà+∞ [Daskalakis+ (2011)] Random graph with "medium connectivity"
Path Tree Star "Augmented" complete bipartite Complete bipartite
2-action random games Payoffs are assigned uniformly at random
u Is there a pure NE in this game? p q r Best-response tables (each is randomly chosen among all possibilities)
u Probability of pure-strategy NE <= (63/64)n-1
u Proof idea
u Probability of no pure NE in n-node path, given the
best response table of a boundary node = f(Probability of no pure NE in (n-1)-node path)
u Q2: What do the "rules of a game" tell us
u Examples
u Network infrastructure u Trading relationship u Political alliances u Professional collaborations u Friendships
u
Individual incentives to form or sever links
u
Overall societal welfare from a network
u
Equilibrium methods
u
Why do we see a particular type of network (e.g., why preferential attachment)?
u Tension between individual incentives and
u Price of stability (to be defined) u Price of anarchy (later)
u Nodes: {1, 2, ..., n} u Payoff of each node: function of network
u "connectedness" – cost of forming links
u Forming/severing links
u Mutual consent needed to form a link u Severing link needs the consent of only one player
u Does the traditional concept of Nash stability
u A network is pairwise stable if
1.
No player wants to unilaterally sever a link, and
2.
No two players both want to form a new link (i.e., in a pairwise stable network, if there does not exist an edge (i, j), it means that i or j or both will suffer from forming this edge)
u Allows deviations on only a single link at a
u Does not allow a group of more than 2
u NS: A node/player can unilaterally sever
u PS: A node cannot sever multiple links at once
u Depending on the level of benefits from
u Complete network u Star u Empty network
u Players are nodes u Action/strategy of a player u: Set of
u Other player(s) with whom u connects will not pay
u Cost (opposite of payoff) of player u: u Solution concept: traditional NE nu = Number of edges player u bought Cost of each edge dist(u, v) is +infinity if no path bet'n u & v
u alpha >= 1 è Star is NE u alpha <= 1 è Complete graph is NE Cost (opposite of payoff) of player u:
u The sum of players' costs Note: double counting dist(u, v) and dist(v, u)
u alpha >= 2 è Star is socially optimal u alpha <= 2 è Complete graph is optimal u Proof.
u Cost of a network with m edges
>= alpha * m + 2 * [n(n-1) - 2m] + 2m = (alpha – 2) m + 2 n (n - 1)
u Want to avoid disconnected graph (why?)
Any pair (i, j) not connected by an edge has distance >= 2
u Def.
u 1 < alpha < 2: PoS <= 4/3 (NE is star, social
u Proof. Simple ratio of costs for alpha à 1
u Otherwise: PoS = 1 (NE matches social opt)