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Algorithmic game theory

Ruben Hoeksma October 15, 2019

Introduction

Course basics

Course basics

Ruben Hoeksma MZH 3320 Webpage: https://www.cslog.uni-bremen.de/teaching/winter19/agt/ Lectures: Monday 12:00 – 14:00 12:15 – 13:45 Tuesday 12:00 – 14:00 12:15 – 13:45 Exercises: ◮ 1 set per week ◮ 1 week to finish each ◮ 60%: +1/3 point ◮ All exercises treated during the lectures are part of the exam

Course basics

Ruben Hoeksma MZH 3320 Webpage: https://www.cslog.uni-bremen.de/teaching/winter19/agt/ Lectures: Monday 12:00 – 14:00 12:15 – 13:45 Tuesday 12:00 – 14:00 12:15 – 13:45 Examination: ◮ Oral exam (around 30 minutes) ◮ First question: Say something about your favorite topic/game from the course. ◮ Questions will include proofs and intu¨ ıtion ◮ Anything spoken about during lectures + any material on webpage

Introduction

Games, selfish behavior, and equilibria

What is a game?

What is a game?

Situations with multiple actors who make their own decisions. ◮ Situations of conflicting interests ◮ Situations of mutual interests ◮ Actors are called players ◮ Each player has some objective ◮ Each player has choices that influence both their own objective and that of others ◮ Each player is rational, i.e., they optimize for their objective ◮ If a player improves their objective by changing their strategy, we say that they have an incentive to do so

Simultaneous game

In a simultaneous game, all players, at the same time, choose a strategy from their own strategy space without knowledge about what the other players have done.

Definition (Simultaneous game)

A simultaneous game is defined by N: Set of n players Si: Set of strategies for each player i ∈ N S = S1 × S2 × . . . × Sn: set of strategy vectors ui: S → R Utility function for each player i ∈ N

Example: a routing game

• d

x 10 10 x ◮ Given this directed graph with origin o and sink d ◮ 10 players want to go from o to d ◮ Cost, c(x), for each arc depends on number of players that use it ◮ Cost of each player is the sum of cost of arcs they chose

Example: a routing game

• d

x 10 10 x Question: What do the players do? ◮ There are three routes {U, L, Z} ◮ If there are n players: ci(s) =

      

10 + #U(s) + #Z(s) if si = U, 10 + #L(s) + #Z(s) if si = L, #L(s) + #U(s) + 2#Z(s) if si = Z.

Equilibrium

Definition (Equilibrium)

An equilibrium is a state in which no player has an incentive to change their strategy.

Definition (Dominant strategy equilibrium (DSE))

A strategy vector s ∈ S is a DSE if for each player i ∈ N, all alternative strategies xi ∈ Si, and all strategies of the other players x−i ∈ S−i, we have ui(si, x−i) ≥ ui(xi, x−i) . The strategy si is called a dominant strategy for player i. s−i is the strategy vector s with player i’s strategy omitted. S−i = S1 × . . . × Si−1 × Si+1 × . . . × Sn.

Example: a routing game

• d

x 10 10 x Question: Does this game have a DSE? ci(s) =

      

10 + #U(s) + #Z(s) if si = U, 10 + #L(s) + #Z(s) if si = L, #L(s) + #U(s) + 2#Z(s) if si = Z.

Example: Battle of the sexes

◮ Two players N = {Man, Woman} ◮ Two strategies: F: go to the football match; T: go to the theater ◮ Woman prefers going to football and Man prefers going to theater ◮ Both prefer to go anywhere together over going anywhere alone Woman F T Man F (5, 6) (1, 1) T (2, 2) (6, 5) ◮ Normal form: explicit description of utility for all strategy combinations ◮ For two-player game: matrix ◮ Row/column player Question: Does this game have a dominant strategy equilibrium?

Pure Nash equilibrium

Answer: The battle of the sexes game does not have a DSE.

• Proof. Man and Woman have two strategies F and T. If Woman plays

F, Man prefers to play F. If Woman plays T, Man prefers to play T. So neither strategies is dominant for Man and no DSE exists.

Definition ((Pure) Nash equilibrium (NE))

A strategy vector s ∈ S is a NE if for each player i ∈ N and all alternative strategies of that player xi ∈ Si, we have ui(si, s−i) ≥ ui(xi, s−i) . si is a best response of player i to s−i.

Example: Battle of the sexes

◮ Two players N = {Man, Woman} ◮ Two strategies: F: go to the football match; T: go to the theater ◮ Woman prefers going to football and Man prefers going to theater ◮ Both prefer to go anywhere together over going anywhere alone Woman F T Man F (5, 6) (1, 1) T (2, 2) (6, 5) Question: What are the Pure Nash equilibria?

Example: Rock-paper-scissors

Definition (Zero-sum game)

A zero-sum game is a game where for any strategy vector s ∈ S the sum of the utilities of the players for that strategy vector is zero.

• i∈N

ui(s) = 0 Rock-paper-scissors R P S R (0, 0) (−1, 1) (1, −1) P (1, −1) (0, 0) (−1, 1) S (−1, 1) (1, −1) (0, 0) Q: Does RPS have a NE? A: No

• Proof. For any strategy of the row

player, there is a strategy for the column player that wins. Same the

• ther way around, so no pair of

strategies are each a best response to each other.

Mixed strategies

Definition (Mixed strategy)

A mixed strategy of player i ∈ N is a probability distribution over their strategy space Si. A mixed strategy vector is a vector of mixed strategies.

Definition (Mixed Nash equilibrium (MNE))

A mixed strategy vector s ∈ S is a MNE if for each player i ∈ N and all alternative strategies of that player xi ∈ Si, we have E[ui(si, s−i)] ≥ E[ui(xi, s−i)] .

Example: Rock-paper-scissors (cont.)

R P S R (0, 0) (−1, 1) (1, −1) P (1, −1) (0, 0) (−1, 1) S (−1, 1) (1, −1) (0, 0)

• Claim. Both players playing each strategy with probability 1

3 is a MNE.

• Proof. Let s be the mixed strategy

E[ur(sr, sc)] =

1 9(3 · −1 + 3 · 1 + 3 · 0) = 0

E[ur(xr, sc)] =

1 3(−1 + 1 + 0) = 0

∀ xr ∈ Sr

Two more. . .

Definition (Correlated equilibrium (CorEq))

Let p be a probability distribution over S. p is a CorEq if for each player i ∈ N and all strategies of that player si, xi ∈ Si, we have

• s−i∈S−i

p(si, s−i)ui(si, s−i) ≥

• s−i∈S−i

p(si, s−i)ui(xi, s−i) .

Definition (Coarse correlated equilibrium (CCE))

Let p be a probability distribution over S. p is a CCE if for each player i ∈ N and all alternative strategies of that player xi ∈ Si, we have

• s∈S

p(s)ui(s) ≥

• s∈S

p(s)ui(xi, s−i) .

CorEq and CCE

CorEq:

• s−i∈S−i

p(si, s−i)ui(si, s−i) ≥

• s−i∈S−i

p(si, s−i)ui(xi, s−i) . Let player i be the row player in the following representation s1

i

s2

i

. . . s1

−i

s2

−i

· · ·

CorEq and CCE

CCE:

• s∈S

p(s)ui(s) ≥

• s∈S

p(s)ui(xi, s−i) . Let player i be the row player in the following representation s1

i

s2

i

. . . s1

−i

s2

−i

· · ·

Example: Game of chicken (the traffic light)

D S D (−10, −10) (1, −1) S (−1, 1) (0, 0) DSE? No NE? (D, S) or (S, D) MNE? pi(D) = 1

10, pi(S) = 9 10 for i ∈ {1, 2}

CorEq? Traffic light {(S, S), (D, S), (S, D)}

Example: Rock-paper-scissors (again)

R P S R (0, 0) (−1, 1) (1, −1) P (1, −1) (0, 0) (−1, 1) S (−1, 1) (1, −1) (0, 0) CCE: (R, P), (P, R), (R, S), (S, R), (P, S), (S, P) all with probability 1

6.

Claim: The above probability distribution is not a CorEq.

• Proof. We consider the row player playing Rock. Given that the row

player plays Rock, the column player plays Paper and Scissors with probability 1

2 each and expected utility equal to 0. If the row player

plays Scissors instead their expected utility is 1

2.

Next time

◮ Existence of equilibria (Nash’s theorem) Exercise set 1 available today. Deadline 21.10.