Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Game Theory for Linguists

Fritz Hamm, Roland Mühlenbernd

9. Mai 2016

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Overview Overview

1. Review: Session 1-3 2. Rationalizability, Beliefs & Best Response 3. Signaling Games

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Conclusions Conclusion of Sessions 1-3 I a game describes a situation of multiple players I that can make decisions (choose actions) I that have preferences over ‘interdependent’ outcomes (action profiles) I preferences are defined by a utility function I games can be defined by abstracting from concrete utility values, but describing it with a general utility function (c.f. synergistic relationship, contribution to public good) I a solution concept describes a process that leads agents to a particular action I a Nash equilibrium defines an action profile where no agent has a need to change the current action I Nash equilibria can be determined by Best Response functions, or by ‘iterated elimination’ of Dominated Actions

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Rationalizability Rationalizability

I The Nash equilibrium as a solution concept for strategic games does not crucially appeal to a notion of rationality in a players reasoning.

I Another concept that is more explicitly linked to a reasoning process of players in a one-shot game is called ratiolalizability. I The set of rationalizable actions can be found by the iterated strict dominance algorithm:

I Given a strategic game G I Do the following step until there aren’t dominated strategies left:

I for all players: remove all dominated strategies in game G

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Rationalizability Nash Equilibria and Rationalizable Actions 1. Apply the Iterated Strict Dominance Algorithm 2. Apply the Best Response Function

LCR U 1,1 3,3∗ 4,2 M 0,2 5,16 ∗,3∗ D 2∗,26 ∗,2 0,3∗ Tabelle: The ‘RNE’ game

I L is dominated by R I B1(L) = {D}, B1(C) = {D}, I U is dominated by M B1(R) = {M}

I C is dominated by R I B2(U) = {C}, B2(M) = {R}, B (R) = {R} I D is dominated by M 2

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Rationalizability Nash Equilibria and Rationalizable Actions 1. Apply the Iterated Strict Dominance Algorithm 2. Apply the Best Response Function

DBS D -1,-1 -2,0∗ -2,0∗ B 0∗,-22 ∗,1∗ 0,0 S 0∗,-2 0,0 1∗,2∗ Tabelle: The ‘Bach-Stravinsky-Dentist’ game

I D is dominated by B and S

I B1(D) = {B, S}, B1(B) = {B}, B1(S) = {S}

I B2(D) = {B, S}, B2(B) = {B}, B2(S) = {S}

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Rationalizability Nash Equilibria and Rationalizable Actions

Exercises: 1. Apply the Iterated Strict Dominance Algorithm 2. Apply the Best Response Function

EFGH A 0,7∗ 2,5 7∗,0 0,1 B 5,2 3∗,3∗ 5,2 0,1 C 7∗,0 2,5 0,7∗ 0,1 D 0,0∗ 0,-2 0,0∗ 10∗,-1 Tabelle: The ‘non-RNE’ game

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Rationalizability Nash Equilibria and Rationalizable Actions

I ratiolalizability can lead to a unique solution (‘RNE’ game),

I but generally does only restrict the number of rational actions (‘Bach-Stravinsky-Dentist’ game)

I or does not help at all, even when there is a unique Nash equilibrium (‘non-RNE’ game) Note: if the Iterated Strict Dominance algorithm (rationalizability) leads to one unique strategy profile, then it is the only Nash equilibrium1 of the game, but not the other way around

1and it is a strict Nash equilibrium

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Beliefs Beliefs

I ratiolalizability solely makes use of the standard assumption, namely that each player knows the utilities of the game, and assumes the other player(s) i) to know the utilities of the game, and ii) to be rational

I but players can be assumed to have more knowledge about the other player(s), c.f. such as a belief about how the other player would behave

I a belief can be represented as a mixed strategy

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Beliefs Mixed Strategies

I a mixed strategy δi of player i is a probability distribution over the set of the player’s actions Ai : δi ∈ ∆(Ai ) I ∆ : A → R has the following properties: I ∀a ∈ A : 0 ≤ ∆(a) ≤ 1 P I a∈A ∆(a) = 1 I Exercise: given an action set A = {r, p, s}. Which of the following probability distribution is a mixed strategy for A? √ I δ = (0.2, 0.3, 0.5) (∆(r) = 0.2, ∆(p) = 0.3, ∆(s) = 0.5) I δ = (0.3, 0.4, 0.2) F I δ = (0.1, 1.2, −0.3) F I δ = (0.4, 0.4, 0.3) F I δ = (1.0, 0.0) F √ I δ = (1.0, 0.0, 0.0)

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Beliefs Beliefs as Mixed Strategies

I a mixed strategy δi of player i is a probability distribution over the set of the player’s actions Ai : δi ∈ ∆(Ai )

I example: the mixed strategy δi = (0.3, 0.7) for the prisoner’s dilemma action set {C, D} represents that player i chooses C with probability 0.3 and D with probability 0.7

I a mixed strategy δi can represent: I player i’s uncertainty how to act I player j’s belief about player i how to act

I problem: actual behavior can only be represented by pure strategies, not by mixed strategies

I but it is reasonable to represent beliefs by mixed strategies

I where do such beliefs come from?

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Beliefs Expected Utility Definition

Let G = ({1, 2}, A1,2, u1,2) be a two-player game and δi ∈ ∆Ai a mixed strategy of player i. Then the expected utility EUi of player i for playing ai with a belief δj is given as: X EUi (ai |δj ) = δj (aj ) × ui (ai , aj ) (1)

aj ∈Aj

CD I Compute EU1(C|(0.3, 0.7)): 0.6 C 2;2 0;3 I Compute EU1(D|(0.1, 0.9)): 1.3 D 3;0 1;1

I Compute EU1(C|(0.1, 0.9)): 0.2 pris. dilemma

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Beliefs Best Response

Definition

Let G = ({1, 2}, A1,2, u1,2) be a two-player game and δi ∈ ∆Ai a mixed strategy of player i. Then the best response BRi of player i for a belief δj is given as:

BRi (δj ) = arg max EUi (ai |δj ) (2) ai ∈Ai

BS Compute BR ((0.5, 0.5)): {B} I 1 B 3;2 1;1 I Compute BR1((0.3, 0.7)): {S} S 1;1 2;3 1 2 I Compute BR1(( 3 , 3 )): {B, S} lifted BoS

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

Coordination & Signaling Coordination & Signaling

RL aL aS R 1 0 tL 1 0 L 0 1 tS 0 1

Messages: One or two lanterns?

tL m1 tL m1 tL m1 tL m1 s1: s2: s3: s4: m tS m2 tS m2 tS m2 tS 2

m1 aL m1 aL m1 aL m1 aL r1: r2: r3: r4: a m2 aS m2 aS m2 aS m2 S

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

The Lewis Game The Lewis Game: Definition

I a signaling game is a tuple SG = h{S, R}, T , Pr, M, A, Ui

I a Lewis game is defined by:  1 if i = j T = {t , t } I U(t , a ) = I L S i j 0 else I M = {m1, m2} a a I A = {aL, aS} L S tL 1 0 I Pr(tL) = Pr(tS) = .5 tS 0 1

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

The Lewis Game The Lewis Game: Extensive Form

N

tL .5 .5 tS

S S

m1 m2 m1 m2

R R R R

aL aS aL aS aL aS aL aS

1 0 1 0 0 1 0 1

Abbildung: Extensive form of the Lewis game Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

The Lewis Game Pure strategies

Pure strategies are contingency plans, players act according to.

I sender strategy: s : T → M

I receiver strategy: r : M → A

tL m1 tL m1 tL m1 tL m1 s1: s2: s3: s4: m tS m2 tS m2 tS m2 tS 2

m1 aL m1 aL m1 aL m1 aL r1: r2: r3: r4: a m2 aS m2 aS m2 aS m2 S

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

The Lewis Game Signaling Systems

I signaling systems are combinations of pure strategies. The Lewis game has two: L1 = hs1, r1i and L2 = hs2, r2i tL m1 aL tL m1 aL L1: L2: tS m2 aS tS m2 aS I signaling systems are strict Nash equilibria of the EU-table:

r1 r2 r3 r4 s1 1 0 .5 .5 s2 0 1 .5 .5 s3 .5 .5 .5 .5 s4 .5 .5 .5 .5 I in signaling systems messages associate states and actions uniquely I signaling systems constitute evolutionary stable states

Game Theory for Linguists Overview Review: Session 1-3 Rationalizability, Beliefs & Best Response Signaling Games

The Lewis Game Summary & Outlook

I Rationalizability

I returns rationalizable actions I solely makes use of the standard assumptions I does not return a unique solution in most cases I Best Response BR(s) to a belief s

I does return a unique solution in almost all cases I needs an additional assumption: a belief (mixed strategy) about the other player(s)

I A Signaling Game is a dynamic/sequential game of imperfect information and models an act of communication

I Outlook: signaling games and best response to a belief are fundamental concepts in game-theoretic Pragmatics

Game Theory for Linguists