Midterm 2

Stat155 Lecture 21: Midterm 2 Review “This is an open book exam: you can use any printed or written material, but you cannot use a laptop, tablet, or phone (or any device that can communicate). There are three questions, each consisting of three parts. Peter Bartlett Each part of each question carries equal weight. Answer each question in the space provided.”

November 8, 2016

1 / 59 2 / 59 Topics Series and parallel games

Series and parallel games. A series game: Both players first play G1, then both play G2. Two-player general-sum games Value is V1 + V2. Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games A parallel game: Both players simultaneously decide which game to functions, Nash equilibria. Nash’s Theorem play, and an action in that game. Congestion games and potential games If they choose the same game, they get the payoff from that game. Congestion games: Every has a pure If they choose different games, the payoff is 0. Potential games: Every congestion game is a Value is1 /(1/V1 + 1/V2). Criticisms of Nash equilibria Evolutionarily stable strategies Electrical networks ESS and Nash equilibria Multiplayer ESS of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

3 / 59 4 / 59 Topics Cooperative versus noncooperative games

Series and parallel games. Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem Noncooperative games Congestion games and potential games Players play their strategies simultaneously. Congestion games: Every congestion game has a pure Nash equilibrium Potential games: Every congestion game is a potential game They might communicate (or see a common signal; e.g., a traffic Criticisms of Nash equilibria signal), but there’s no enforced agreement. Evolutionarily stable strategies Natural solution concepts: ESS and Nash equilibria Nash equilibrium, correlated equilibrium. Multiplayer ESS No improvement from unilaterally deviating. Correlated equilibrium Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

5 / 59 6 / 59 General-sum games General-sum games

Dominated pure strategies Notation A pure e for Player I is dominated by e in payoff matrix A if, for A two-person general-sum game is specified by two payoff matrices, i i 0 m n all j 1,..., n , A, B R × . ∈ { } ∈ aij ai j . Simultaneously, Player I chooses i 1,..., m and the Player II ≤ 0 ∈ { } chooses j 1,..., n . Similarly, a pure strategy ej for Player II is dominated by ej0 in payoff ∈ { } matrix B if, for all i 1,..., m , Player I receives payoff aij . ∈ { } Player II receives payoff bij . b b . ij ≤ ij0

7 / 59 8 / 59 General-sum games General-sum games

Nash equilibria

Safety strategies A pair( x , y ) ∆m ∆n is a Nash equilibrium for payoff matrices ∗m n∗ ∈ × A safety strategy for Player I is an x ∆m that satisfies A, B R × if ∗ ∈ ∈

min x>Ay = max min x>Ay. max x>Ay = x>Ay , y ∆n ∗ x ∆m y ∆n x ∆m ∗ ∗ ∗ ∈ ∈ ∈ ∈ max x>By = x>By . x maximizes the worst case expected gain for Player I. y ∆n ∗ ∗ ∗ ∗ ∈ Similarly, a safety strategy for Player II is a y ∆n that satisfies ∗ ∈ If Player I plays x and Player II plays y , neither player has an min x>By = max min x>By. ∗ ∗ x ∆m ∗ y ∆n x ∆m incentive to unilaterally deviate. ∈ ∈ ∈ x is a to y , y is a best response to x . y maximizes the worst case expected gain for Player II. ∗ ∗ ∗ ∗ ∗ In general-sum games, there might be many Nash equilibria, with different payoff vectors.

9 / 59 10 / 59 Comparing two-player general-sum and zero-sum games Comparing two-player general-sum and zero-sum games

Zero-sum games Zero-sum games 1 A pair of safety strategies is a Nash equilibrium ( theorem) 4 If each player has an equalizing mixed strategy 2 Hence, there is always a Nash equilibrium. (that is, x>A = v1> and Ay = v1), 3 If there are multiple Nash equilibria, they form a convex set, and the then this pair of strategies is a Nash equilibrium. expected payoff is identical within that set. (from the principle of indifference) Thus, any two Nash equilibria give the same payoff. General-sum games General-sum games 4 If each player has an equalizing mixed strategy 1 A pair of safety strategies might be unstable. for their opponent’s payoff matrix (Opponent aims to maximize their payoff, not minimize mine.) (that is, x>B = v21> and Ay = v11), 2 There is always a Nash equilibrium (Nash’s Theorem). then this pair of strategies is a Nash equilibrium. 3 There can be multiple Nash equilibria, with different payoff vectors.

11 / 59 12 / 59 Topics Multiplayer general-sum games

Series and parallel games. Two-player general-sum games Notation Payoff matrices, dominant strategies, safety strategies, Nash eq. A k-person general-sum game is specified by k utility functions, Multiplayer general-sum games uj : S1 S2 Sk R. Utility functions, Nash equilibria. Nash’s Theorem × × · · · × → Player j can choose strategies s S . Congestion games and potential games j ∈ j Congestion games: Every congestion game has a pure Nash equilibrium Simultaneously, each player chooses a strategy. Potential games: Every congestion game is a potential game Player j receives payoff uj (s1,..., sk ). Criticisms of Nash equilibria Evolutionarily stable strategies ESS and Nash equilibria k = 2: u1(i, j) = aij , u2(i, j) = bij . Multiplayer ESS For s = (s1,..., sk ), let s i denote the strategies without the ith one: Correlated equilibrium − Price of anarchy s i = (s1,..., si 1, si+1,..., sk ). Braess’s paradox − − The impact of adding edges And write( si , s i ) as the full vector. Classes of latencies − Pigou networks

13 / 59 14 / 59 Multiplayer general-sum games Multiplayer general-sum games

Definition A sequence( x ,..., x ) ∆ ∆ (called a strategy profile) is a 1∗ k∗ ∈ S1 × · · · × Sk Definition Nash equilibrium for utility functions u1,..., uk if, for each player j 1,..., k , A vector( s ,..., s ) S S is a pure Nash equilibrium for utility ∈ { } 1∗ k∗ ∈ 1 × · · · × k functions u1,..., uk if, for each player j 1,..., k , ∈ { } max uj (xj , x∗ j ) = uj (xj∗, x∗ j ). xj ∆S − − ∈ j max uj (sj , s∗ j ) = uj (sj∗, s∗ j ). sj Sj − − ∈ Here, we define

uj (x∗) = Es1 x1,...,sk xk uj (s1,..., sk ) If the players play these sj∗, nobody has an incentive to unilaterally ∼ ∼ deviate: each player’s strategy is a best response to the other players’ = x (s ) x (s )u (s ,..., s ). 1 1 ··· k k j 1 k strategies. s S ,...,s S 1∈ 1X k ∈ k

If the players play these mixed strategies xj∗, nobody has an incentive to unilaterally deviate: each player’s mixed strategy is a best response to the other players’ mixed strategies.

15 / 59 16 / 59 Multiplayer general-sum games Multiplayer general-sum games

Nash’s Theorem (1951) Every finite general-sum game has a Nash equilibrium. Theorem Consider a strategy profile x ∆ ∆ . Let Proof Idea (two players) ∈ S1 × · · · × Sk T = s S : x (s) > 0 . The following statements are equivalent. We find an “improvement” map M(x, y) = (ˆx, yˆ), so that i { ∈ i i } 1 x is a Nash equilibrium. 1 xˆ>Ay > x>Ay (orˆx>Ay = x>Ay, if x was a best response to y),

2 For each i, there is a ci such that 2 x>Byˆ > x>By (or x>Byˆ = x>By, if y was a best response to x),

1 For all si Si , ui (si , x i ) ci . No better response outside Ti 3 ∈ − ≤ ←− M is continuous. 2 For si Ti , ui (si , x i ) = ci . Indifferent within Ti ∈ − ←− It’s easy to find a map like this. For instance, take a step in the direction of increasing payoff: Compare with the principle of indifference in the zero-sum case. xˆ = (x + ηAy),ˆ y = (y + ηB x). P∆m P∆n > A Nash equilibrium is a fixed point of M. The existence of a Nash equilibrium follows from Brouwer’s fixed-point theorem.

17 / 59 18 / 59 Nash’s Theorem Topics

Series and parallel games. Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem Congestion games and potential games Brouwer’s Fixed-Point Theorem Congestion games: Every congestion game has a pure Nash equilibrium A continuous map f : K K from a convex, closed, bounded K Rd Potential games: Every congestion game is a potential game → ⊆ has a fixed point, that is, an x K satisfying f (x) = x. Criticisms of Nash equilibria ∈ Evolutionarily stable strategies ESS and Nash equilibria Multiplayer ESS Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

19 / 59 20 / 59 Congestion games: games with a pure Nash equilibrium Congestion games

Definition A congestion game has Example k players A m facilities 1,..., m (e.g., edges) { } (1,2,4) (2,3,5) For player i, there is a set Si of strategies that are sets of facilities, s 1,..., m (e.g., paths) ⊆ { } k I, II, III S (1,2,6) T For facility j, there is a cost vector cj R , where cj (n) is the cost of ∈ facility j when it is used by n players.

(2,3,5) (2,4,8) For a sequence s = (s1,..., sn), the of the players are defined by

B cost (s) = u (s) = c (n (s)), i − i j j j s X∈ i where n (s) = i : j s is the number of players using facility j. j |{ ∈ i }|

21 / 59 22 / 59 Congestion games Congestion games

Proof We define a potential function Φ: S1 Sk R as For a sequence s = (s1,..., sn), the utilities of the players are defined by × · · · × →

m nj (s) costi (s) = ui (s) = cj (nj (s)), − Φ(s) = cj (l). j si X∈ Xj=1 Xl=1 Fix strategies for the k players s = (s1,..., sk ). where nj (s) = i : j si is the number of players using facility j. |{ ∈ }| What happens when Player i changes from si to si0? Egalitarian: The utilities depend on how many players use each ∆costi = costi (si0, s i ) costi (si , s i ) facility, and not on which players use it. − − − = c (n (s) + 1) c (n (s)) j j − j j j (s s ) j (s s ) Theorem ∈Xi0− i ∈Xi − i0 Every congestion game has a pure Nash equilibrium. increased cost decreased cost = Φ(si0, s i ) Φ(si , s i ) | − {z− −} | {z } = ∆Φ.

23 / 59 The potentialΦ reflects how a player’s costs change. 24 / 59 Congestion games Potential games

Proof If we start at an arbitrary s, and update one player’s choice to Definition decrease that player’s cost, the potential must decrease. Consider a multiplayer game G: Continuing updating other player’s strategies in this way, we must k players eventually reach a local minimum (there are only finitely many For player i, there is a set Si of strategies. strategies). For player i, there is costi : S1 Sk R. × · · · × → Since no player can reduce their cost from there, we have reached a We sayΦ: S1 Sk R is a potential for game G if ∆Φ = ∆costi , pure Nash equilibrium. × · · · × → that is, for all i, s and si0, Φ(si0, s i ) Φ(si , s i ) = costi (si0, s i ) costi (si , s i ). − − − − − − This gives an algorithm for finding a pure Nash equilibrium: Update We say that G is a potential game if it has a potential. the choice of one player at a time to improve their cost.

25 / 59 26 / 59 Potential games Topics

Series and parallel games. Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem In considering congestion games, we proved two things: Congestion games and potential games Congestion games: Every congestion game has a pure Nash equilibrium Theorem Potential games: Every congestion game is a potential game 1 Every congestion game is a potential game. Criticisms of Nash equilibria 2 Every potential game has a pure Nash equilibrium. Evolutionarily stable strategies ESS and Nash equilibria Multiplayer ESS Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

27 / 59 28 / 59 What’s wrong with Nash equilibria? Evolutionarily stable strategies

Will all players know everyone’s utilities? Maximizing expected utility does not (explicitly) model . Will players maximize utility and completely ignore the impact on There is a population of individuals. other players’ utilities? There is a game played between pairs of individuals. How can the players find a Nash equilibrium? Each individual has a pure strategy encoded in its genes. How can the players agree on a Nash equilibrium to play? The two players are randomly chosen individuals. Will players actually randomize? A higher payoff gives higher reproductive success. Alternative equilibrium concepts This can push the population towards stable mixed strategies. Correlated equilibrium Evolutionary stability Equilibria in perturbed games.

29 / 59 30 / 59 Evolutionarily stable strategies Evolutionarily stable strategies

Suppose that x is invaded by a small population of mutants z: x is replaced by (1 )x + z. − Will the mix x survive?

Consider a two-player game with payoff matrices A, B. x’s utility: x>A (z + (1 )x) = x>Az + (1 )x>Ax Suppose that it is symmetric (A = B>). − − Consider a mixed strategy x. z’s utility: z>A (z + (1 )x) = z>Az + (1 )z>Ax − − Think of x as the proportion of each pure strategy in the population.

Definition Mixed strategy x ∆ is an evolutionarily stable strategy (ESS) if, for any ∈ n pure strategy z, 1 z Ax x Ax (x, x) is a Nash equilibrium. > ≤ > ←− 2 If z>Ax = x>Ax then z>Az < x>Az.

31 / 59 32 / 59 ESS and Nash equilibria ESS and Nash equilibria

Theorem Every ESS is a Nash equilibrium. Definition A strategy profile x = (x ,..., x ) ∆ ∆ is a strict Nash This follows from the definition: 1∗ k∗ ∈ S1 × · · · × Sk equilibrium for utility functions u1,..., uk if, for each player Definition j 1,..., k , for all xj ∆Sj that is different from xj∗, A strategy x ∆ is an evolutionarily stable strategy (ESS) if, for any ∈ { } ∈ ∈ n pure strategy z = x, uj (xj , x∗ j ) < uj (xj∗, x∗ j ). 6 − − 1 z Ax x Ax > ≤ > 2 If z>Ax = x>Ax then z>Az < x>Az. A Nash equilibrium has uj (xj , x∗ j ) uj (xj∗, x∗ j ). − ≤ − Proof: If every pure strategy z satisfies z>Ax x>Ax, then every By the principle of indifference, only a pure Nash equilibrium can be a ≤ mixed strategy z (mixture of pure strategies) must obey the same strict Nash equilibrium. inequality. Hence,( x, x) is a Nash equilibrium. The converse is true for strict Nash equilibria.

33 / 59 34 / 59 ESS and Nash equilibria ESMS: evolutionarily stable against mixed strategies

Pure versus mixed Theorem An ESS is a Nash equilibrium( x , x ) satisfying, for all ei = x , if Every strict Nash equilibrium is an ESS. ∗ ∗ 6 ∗ ei>Ax = x>Ax , then ei>Aei < x>Aei . ∗ ∗ ∗ ∗ What about invasion by a mixed strategy? Proof: A strict Nash equilibrium has z>Ax < x>Ax for z = x, so both 6 Say that a symmetric strategy( x , x ) is evolutionarily stable against conditions defining an ESS are satisfied. ∗ ∗ mixed strategies (ESMS) if it is a Nash equilibrium and, for all mixed strategies z = x , if z>Ax = x>Ax , then z>Az < x>Az. Definition 6 ∗ ∗ ∗ ∗ ∗ (ESS is sometimes defined this way, e.g., Leyton-Brown and Shoham) A strategy x ∆ is an evolutionarily stable strategy (ESS) if, for any ∈ n pure strategy z = x, Clearly, every ESMS strategy is an ESS. 6 1 z>Ax x>Ax ≤ Theorem 2 If z Ax = x Ax then z Az < x Az. > > > > For a two-player2 2 , every ESS is ESMS. ×

35 / 59 36 / 59 Multiplayer evolutionarily stable strategies Multiplayer evolutionarily stable strategies

Consider a symmetric multiplayer game (that is, unchanged by relabeling the players). Definition Suppose that a symmetric mixed strategy x is invaded by a small (Suppose, for simplicity, that the utility for player i depends on si and on population of mutants z: x is replaced by (1 )x + z. the set of strategies played by the other players, but is invariant to a − permutation of the other players’ strategies.) Will the mix x survive? A strategy x ∆ is an evolutionarily stable strategy (ESS) if, for any ∈ n pure strategy z = x, 6 1 u1(z, x 1) u1(x, x 1) x is a Nash equilibrium. x’s utility: u1(x, z + (1 )x, z + (1 )x) − ≤ − ←− − − 2 If u1(z, x 1) = u1(x, x 1) then for all j = 1, 2 − − 6 = u1(x, z, x) + u1(x, x, z) + (1 2)u1(x, x, x) + O( ) u1(z, z, x 1, j ) < u1(x, z, x 1, j ). − − − − − z’s utility: u (z, z, x) + u (z, x, z) + (1 2)u (z, x, x) + O(2). 1 1 − 1

37 / 59 38 / 59 Topics Correlated equilibria: A driving example

Series and parallel games. Payoff Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Go Stop Multiplayer general-sum games Go (-100,-100) (1,-1) Utility functions, Nash equilibria. Nash’s Theorem Stop (-1,1) (-1,-1) Congestion games and potential games Congestion games: Every congestion game has a pure Nash equilibrium Potential games: Every congestion game is a potential game Nash equilibria Criticisms of Nash equilibria 2 99 2 99 (Go, Stop), (Stop, Go), 101 , 101 , 101 , 101 . Evolutionarily stable strategies (because we want indifference: 100p + 1 p = p (1 p)) ESS and Nash equilibria −  −  − − − Multiplayer ESS Better solution? Correlated equilibrium Price of anarchy A traffic signal: Pr((Red, Green)) = Pr((Green, Red)) = 1/2, and Braess’s paradox both players agree: Red means Stop, Green means Go. The impact of adding edges After they both see the traffic signal, the players have no incentive to Classes of latencies deviate from the agreed actions. Pigou networks

39 / 59 40 / 59 Correlated equilibrium Correlated equilibrium

Definition Definition For a two player game with strategy sets S1 = 1,..., m and A correlated strategy pair for a two-player game with payoff matrices A { } S2 = 1,..., n , a correlated strategy pair is a pair of random variables and B is a correlated equilibrium if { } (R, C) with some joint probability over pairs of actions 1 for all i, i 0 S1, Pr(R = i) > 0 E [ai,C R = i] E ai 0,C R = i . (i, j) S1 S2. ∈ ⇒ | ≥ | ∈ × 2 for all j, j0 S2, Pr(C = j) > 0 E [bR,j C = j] E bR,j C = j . ∈ ⇒ | ≥  0 |  Example   c.f. a Nash equilibrium In the traffic signal example, Pr(Stop, Go) = Pr(Go, Stop) = 1/2. A strategy profile( x, y) ∆ ∆ is a Nash equilibrium iff ∈ Sm × Sn 1 for all i, i 0 S1, Pr(R = i) > 0 E [ai,C ] E ai ,C . c.f. a pair of mixed strategies ∈ ⇒ ≥ 0 2 for all j, j S , Pr(C = j) > 0 [b ] b . If we have x ∆ and y ∆ , then choosing the two actions( R, C) 0 2 E R,j E R,j0 ∈ Sm ∈ Sn ∈ ⇒ ≥ independently gives Pr(R = i, C = j) = xi yj . When R and C are independent, these expectations and the conditional In the traffic signal example, we cannot have Pr(Stop, Go) > 0 and expectations are identical. Pr(Go, Stop) > 0 without Pr(Go, Go) > 0. Thus, a Nash equilibrium is a correlated equilibrium.

41 / 59 42 / 59 Correlated equilibria Correlated equilibria

CE vs NE Nash’s Theorem implies there is always a correlated equilibrium. They are easy to find, via linear programming. It is not unusual for correlated equilibria to achieve better solutions Combinations of correlated equilibria for both players than Nash equilibria. Given any two correlated equilibria, you can combine them to obtain Implementation another: Imagine a public random variable that determines which of the correlated equilibria will be played. Knowing which correlated We can think of a correlated equilibrium being implemented in two equilibrium is being played, the players have no incentive to deviate. equivalent ways: The payoffs are convex combinations of the payoffs of the two CEs. 1 There is a random draw of a correlated strategy pair with a known distribution, and the players see their strategy only. 2 There is a draw of a random variable (‘external event’) with a known probability distribution, and a private signal is communicated to the players about the value of the random variable. Each player chooses a mixed strategy that depends on this private signal (and the dependence is ).

43 / 59 44 / 59 Topics Braess’s paradox

Series and parallel games. Example: before Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem Congestion games and potential games Congestion games: Every congestion game has a pure Nash equilibrium Potential games: Every congestion game is a potential game Criticisms of Nash equilibria Evolutionarily stable strategies Example: after ESS and Nash equilibria Multiplayer ESS Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

45 / 59 (Karlin and Peres, 2016)46 / 59 Price of anarchy Price of anarchy

Definition For a routing problem, define Example: Nash equals socially optimal average travel time in worst Nash equilibrium price of anarchy = . minimal average travel time

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.

47 / 59 48 / 59 Price of anarchy Price of anarchy

Example: price of anarchy=4 /3

Theorem 1 For linear latency functions, the price of anarchy is1. 2 For affine latency functions, the price of anarchy is no more than4 /3.

(Karlin and Peres, 2016) 49 / 59 50 / 59 Price of anarchy Nash equilibrium flows exist (and they’re easy to find)

Traffic flow Theorem For a DAG with latency functions ` that are continuous, non-decreasing, A flow from source s to destination t in a directed graph is a mixture e and non-negative, if there is a path from source to destination, there is a of paths from s to t, with mixture weight f for path P. P Nash equilibrium unit flow. We write the flow on an edge e as Proof idea F = f . e P This is the non-atomic version of a congestion game. P e X3 For the atomic version (finite number of players), we showed that there is a pure Nash equilibrium that can be found by descending a Latency on an edge e is a non-decreasing function of F , written e potential function. `e (Fe ). The same approach works here. The potential function is Latency on a path P is the total latency, LP (f ) = e P `e (Fe ). ∈ Fe Average latency is L(f ) = P fP LP (f ) = e Fe `e (PFe ). Φ(f ) = `e (x) dx. e 0 A flow f is a Nash equilibrium if, for all P and P0, if fP > 0, X Z P P If f is not a Nash equilibrium flow, thenΦ( f ) is not minimal. LP (f ) LP0 (f ). ≤ Φ is convex, on a convex, compact set, so it has a minimum. 51 / 59 52 / 59 Price of anarchy The impact of adding edges

Theorem

For linear latencies, that is, `e (x) = ae x, with ae 0, if f is a Nash Theorem ≥ equilibrium flow and f ∗ is a socially optimal flow (that is, L(f ∗) is Consider a network G with a Nash equilibrium flow fG and average latency minimal), then LG (fG ), and a network H with additional roads added. Suppose that the L(f ) = L(f ∗). price of anarchy in H is no more than α. Then any Nash equilibrium flow f has average latency L (f ) αL (f ). H H H ≤ G G Theorem Proof For affine latencies, that is, ` (x) = a x + b , with a , b 0, if f is a e e e e e ≥ Nash equilibrium flow and f ∗ is a socially optimal flow (that is, L(f ∗) is LH (fH ) αLH (fH∗) αLH (fG∗) = αLG (fG∗) αLG (fG ). minimal), then ≤ ≤ ≤ 4 L(f ) L(f ∗). ≤ 3

53 / 59 54 / 59 Classes of latency functions Pigou networks and price of anarchy

Price of anarchy A Pigou network Suppose we allow latency functions from some class . L For example, we have considered

= x ax : a 0 , L { 7→ ≥ } = x ax + b : a, b 0 , L { 7→ ≥ } What about d = x ad x : ad 0 ? L ( 7→ ≥ ) Xd 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 is at most the price of anarchy in a L certain small network with these latency functions: a Pigou network. (Karlin and Peres, 2016)

55 / 59 56 / 59 Pigou networks and price of anarchy Nonlinear latency functions

d Example: nonlinear latency `e (x) = x Theorem

d Define the Pigou price of anarchy as the price of anarchy for this network with latency function ` and total flow r:

r`(r) (why x 0?) αr (`) = . ≥ minx 0(x`(x) + (r x)`(r)) ≥ − Price of anarchy For any network with latency functions from and total flow1, the price L of anarchy is no more than Nash equilibrium flow: all through top edge. L(f ) = 1. Socially optimal flow: max max αr (`). 0 r 1 ` d+1 (d+1)/d ≤ ≤ ∈L L(f ∗) = min(1 x + x ) = 1 d(d + 1)− . x − − Price of anarchy: 1 d . 1 d(d + 1) (d+1)/d  ln d − − 57 / 59 58 / 59 Topics

Series and parallel games. Two-player general-sum games Payoff matrices, dominant strategies, safety strategies, Nash eq. Multiplayer general-sum games Utility functions, Nash equilibria. Nash’s Theorem Congestion games and potential games Congestion games: Every congestion game has a pure Nash equilibrium Potential games: Every congestion game is a potential game Criticisms of Nash equilibria Evolutionarily stable strategies ESS and Nash equilibria Multiplayer ESS Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

59 / 59