Outline for today Stat155 Game Theory Lecture 19:.. Peter Bartlett Recall: Linear and affine latencies Classes of latencies Pigou networks Transferable versus nontransferable utility November 1, 2016 1 / 23 2 / 23 Definition For a routing problem, define price of anarchy = average travel time in worst Nash equilibrium. minimal average travel time Example: Linear latency; Nash equals socially optimal The minimum is over all flows. The flow minimizing average travel time is the socially optimal flow. The price of anarchy reflects how much average travel time can decrease in going from a Nash equilibrium flow (where all individuals choose a path to minimize their travel time) to a prescribed flow. 3 / 23 4 / 23
Example: Affine latency; price of anarchy = 4/3 Theorem 1 For linear latency functions, the price of anarchy is 1. 2 For affine latency functions, the price of anarchy is no more than 4/3. Classes of latency functions (Karlin and Peres, 2016) 5 / 23 6 / 23 Suppose we allow latency functions from some class L. For example, we have considered A Pigou network What about L = {x ax : a 0}, L = {x ax + b : a, b 0}, { } L = x a d x d : a d 0 d? We ll insist that latency functions are non-negative and non-decreasing. It turns out that the price of anarchy in an arbitrary network with latency functions chosen from L is at most the price of anarchy in a certain small network with these latency functions: a Pigou network. (Karlin and Peres, 2016) 7 / 23 8 / 23
d Theorem Define the Pigou price of anarchy as the price of anarchy for this network with latency function l and total flow r: α r (l) = rl(r). (why x 0?) min x 0 (xl(x) + (r x)l(r)) For any network with latency functions from L and total flow 1, the price of anarchy is no more than max max α r (l). 0 r 1 l L Proof Write f for a NE flow and f for the socially optimal flow. By considering a Pigou network with total flow F e and latency function l = l e, we have L(f ) = e F e l e (F e ) = α Fe (l e ) min (xl e(x) + (F e x)l e (F e )) x 0 e ( max α r (l) Fe l e (Fe ) + ) (F e Fe )l e (F e ) r,l e e max α r (l)l(f ). r,l 9 / 23 Nonlinear latency functions 10 / 23 Example: nonlinear latency l e (x) = x d Theorem For any network with latency functions from L, the price of anarchy is no more than the worst price of anarchy in the Pigou network (with one constant latency edge l(r) and one variable latency edge l(x)). Nash equilibrium flow: all through top edge. L(f ) = 1. Socially optimal flow: : L(f ) = min x (1 x + x d+1 ) = 1 d(d + 1) (d+1)/d. 1 1 d(d + 1) (d+1)/d d ln d. 12 / 23 11 / 23
Outline Cooperative versus noncooperative games Recall: Linear and affine latencies Classes of latencies Pigou networks Transferable versus nontransferable utility Noncooperative games Players play their strategies simultaneously. They might communicate (or see a common signal; e.g., a traffic signal), but there s no enforced agreement. Natural solution concepts: Nash equilibrium, correlated equilibrium. No improvement from unilaterally deviating. 13 / 23 14 / 23 Cooperative versus noncooperative games Players can make binding agreements. e.g.: prisoner s dilemma. Both players gain from an enforceable agreement not to confess. Two types: Transferable utility The players agree what strategies to play and what additional side payments are to be made. Nontransferable utility The players choose a joint strategy, but there are no side payments. Example Nontransferable utility 1 2 3 1 2 Both players need to agree on what they will play. It might involve randomness (playing a correlated strategy pair). Try it! Did anyone agree to play I:1 and II:1 (with payoff )? Did anyone agree to play I:2 and II:1 (with payoff )? 15 / 23 16 / 23
Example Pure strategy payoffs 1 2 3 1 2 Payoff vectors: NTU The set of payoff vectors that the two players can achieve is called the feasible set. With nontransferable utility, it is the convex hull of the entries in the payoff bimatrix. A feasible payoff vector (v 1, v 2 ) is Pareto optimal if the only feasible payoff vector (v 1, v 2 ) with v 1 v 1 and v 2 v 2 is (v 1, v 2 ) = (v 1, v 2 ). 17 / 23 18 / 23 Pure strategy payoffs Feasible payoffs (NTU) Example 1 2 3 1 2 Transferable utility Assume the payoff is in $. The two players need to agree on what they will play and on who pays what to whom. What is the best total payoff that can be shared? How should it be shared? Try it! Where is the Pareto optimal boundary? Is there a pure Nash equilibrium? So who can threaten what? 19 / 23 20 / 23
Feasible payoff vectors: transferable utility Feasible payoffs (TU) The players can choose to shift a payoff vector. Payoffs For example, suppose a pure strategy pair gives payoff (a ij, b ij ). Suppose the players agree to play it, and Player I will give Player II a payment of p. The payment shifts the payoff vector from (a ij, b ij ) to (a ij p, b ij + p). The players can choose any (positive or negative) payment, and any correlated strategy pair. Thus, with transferable utility, the players can choose any payoff vector in the convex hull of the set of lines {(a ij p, b ij + p) : i {1,..., m}, j {1,..., n}, p R}. 21 / 23 Where is the Pareto optimal boundary? 22 / 23 Outline Recall: Linear and affine latencies Classes of latencies Pigou networks Transferable versus nontransferable utility 23 / 23