Pragmatics & Game Theory Session 5: Introduction to Game ...

IntroductionIntroduction to Game Theory

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Pragmatics & Game Theory

Session 5: Introduction to Game Theory

Roland Mühlenbernd

WiSe 13/14

Roland Mühlenbernd Horn's principles and rules


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Table of Content

1 IntroductionHomeworks

2 Introduction to Game TheoryStrategic GamesSignaling Games

3 Outlook



OutlookHomeworks

Homeworks Question 1

What is the signi�cant bottleneck in the speed of humancommunication according to Levinson?

The bottleneck is constituted by the remarkably slowtransmission rate of human speech, with a limit in the range ofseven syllables or 18 segments per second.



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What is the logical problem Levinson mentioned? And what isaccording to Levinson necessary to diminish the logical problem?

In daily life we observe behavior and �gure out the underlyingintention by the same rules that we convert intentions into theactions that will e�ectuate them.

But this cannot work, since for all inference systems onecannot work backwards from conclusions to premises, becausethere is always an in�nite set of premises leading to the sameconclusion

for example you get the conclusion q from {p, p → q} or{p ∧ q} or {p ∨ q,¬p} or...Thus there have to be further constraints that limit the searchspace of sets of premises, called heuristics



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Why is it necessary that the �rst heuristic is restricted to salientcontrasts?

the �rst heuristic is called What isn't said, isn't the case

If the heuristic is unrestricted, then whatever one did notspecify would not be the case, and that would be such apowerful heuristic it would inhibit one from saying anything

�The �ag is blue� would infer: not completely blue but also:not dark blue, not light blue, not medium blue, not a big �ag,not a small �ag...



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What are Levinson's three heuristics and in which way do theymatch to Gricean maxims?

1 What isn't said, isn't the case.

Covered by the �rst quantity maxim: Make your contributionas informative as is required

2 What is simply described is stereotypically exempli�ed.

Covered by the second quantity maxim: Do not make yourcontribution more informative than is required.

3 What is said in an abnormal way, isn't normal.

Covered by the manner maxims M1 and M4: Avoid obscurityof expression, Be orderly (and probably M3: be brief)

Note that while Grice describes how to produce expressions(speaker's point of view), Levinson gives heuristics how to get theright interpretation (hearer's point of view).



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What is Levinson's solution for the projection problem with respectto his three heuristics?

The impicatures triggered by the three heuristics aresystematically resolved by an ordered set of priorities:

Q > M > I (Q-clausal > Q-scalar)



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In what ways are Q- and M-inferences similar, in what ways do theydi�er? And what does it mean that both implicatures aremetalinguistic?

Both are metalinguistic in the sense that they can only berecovered by reference to what else might have been said butwas not

Q and M are similar in the way that both essentially induce ametalinguistic mode of inference, and it is in each case anegative inferences

they di�er in the kind of metalinguistic contrast they rely on:

Q relies on sets of alternates of essentially similar form withcontrastive semantic contentwhereas M relies on sets of alternates that contrast in form butnot in inherent semantic content



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Strategic GamesSignaling Games

Gricean Pragmatics & Rationality

As one of my avowed aims is to see talking as a special

case or variety of purposive, indeed rational, behaviour, it

may be worth noting that the speci�c expectations or

presumptions connected with at least some of the

foregoing maxims have their analogues in the sphere of

transactions that are not talk exchanges. (Grice 1989)

I am enough of a rationalist to want to �nd a basis that

underlies these facts (i.e. the way people in fact

communicate), undeniable though they may be; I would

like to be able to think of the standard type of

conversation practice not merely as something that all or

most do in fact follow but as something that is reasonable

for us to follow, that we should not abandon. (Grice 1989)



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Game Theoretic Pragmatics

Why Game Theory?

Game Theory o�ers mathematical models of interactivedecision making of (mostly: idealized and rational) agents

Pragmatics competence can be modeled as behavior ofidealized agents in a game situation

Game Theory has been applied to the study of implicatures inmany forms, especially from a rationalistic point of view(e.g. Parikh 1991, Benz and van Rooij 2007)

By making explicit the role of belief formation and reasoning inan abstract interactive situation we can reasonably implementempirically attested and introspectively plausible assumptionabout the psychology of reasoners in general and languageusers in particular (Franke 2011, p. 13)



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What is a game?

A game

in its technical sense is a mathematical structure thatabstractly represents a decision situation of several agents

where the outcome of the decisions of each agent depends onthe choices of the other agentsis not a model of interactive reasoning or decision making, butonly a model of the situations in which agents engage in this

kind of deliberation and choice (a solution concept describesactual reasoning and/or decision making)

S R

S 2;2 0;1R 1;0 1;1

Table: The stag huntgame

C D

C 2;2 0;3D 3;0 1;1

Table: The prisoner'sdilemma

B S

B 2;1 0;0S 0;0 1;2

Table: The battle ofthe sexes

How will the players behave?Roland Mühlenbernd Horn's principles and rules


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Solution concepts

S R

S 2;2 0;1R 1;0 1;1

Table: Stag hunt

C D

C 2;2 0;3D 3;0 1;1

Table: P dilemma

B S

B 2;1 0;0S 0;0 1;2

Table: BoS

How will the players behave?

choosing randomly

choosing the dominating strategy

choosing the strategy with highest expected utility

choosing the risk-dominant strategy

choosing a Nash equilibrium (a Pareto-optimal one)

choosing by learning (update dynamics, repeated games)

choosing after pre-play communication

choosing the best response to a rational belief



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Kind of Games

Games are traditionally classi�es in two dimensions:

1 players choose simultaneously or in sequence

static (strategic) games: simultaneous choicedynamic (sequential) games: choices are made in sequence

2 players have complete or incomplete information

complete information: knowing all decision relevant details ofthe game, except the other player's choiceperfect information: knowing the action choices of the otherplayer

Note: the stag hung, prisoner's dilemma and battle of the sexesgame is each an instance of a strategic game with completeinformation.



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Strategic game with incomplete information

Formal de�nition: 〈N,Ai∈N ,<i∈N〉

N is a set of players

Ai are actions available for player i

<i is player i's preference relation over possible outcomes ofthe game

Example:

N = {R,C}AR = {c, d}, AC = {c, d}<R : 〈d , c〉 < 〈c, c〉 < 〈d , d〉 < 〈c, d〉<C : 〈c, d〉 < 〈c, c〉 < 〈d , d〉 < 〈d , c〉

c d

c 23;23 0;35d 35;0 1;1

Table: P dilemma



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The Nash Equilibrium

The Nash equilibrium is a possible solution concept for a game:

Nash Equilibrium

A Nash equilibrium of a strategic game is an action pro�le a∗ suchthat for all i ∈ N there is no ai ∈ Ai for which:

(a∗−i , ai ) <i a∗



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Strategic game with incomplete information

Example: The prisoner's dilemma 〈N,Ai∈N ,<i∈N〉

N = {R,C}AR = {c, d}, AC = {c, d}<R : 〈d , c〉 < 〈c, c〉 < 〈d , d〉 < 〈c, d〉<C : 〈c, d〉 < 〈c, c〉 < 〈d , d〉 < 〈d , c〉

c d

c 2;2 0;3d 3;0 1;1

Table: P dilemma

What is the only Nash equilibrium? Show in two ways: with thetable and the formal de�nition.



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Rationazibility

The Nash equilibrium as a solution concept for strategic gamesdoes not crucially appeal to a notion of rationality in a playersreasoning

another concept that is more explicitly linked to a reasoningprocess of players in a one-shot game is called ratiolalizability

the set of rationalizable actions can be found by the iteratedstrict dominance1 algorithm:

1 Given a strategic game G2 Do the following step until there aren't dominated strategies

left:3 for all players: remove all dominated strategies in game G

1A strategy ai strictly dominates a strategy aj , if for all opponent's strategiesb: U(ai , b) > U(aj , b)



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Iterated Strict Dominance Algorithm

Example

L C R

U 1;1 2;0 2;2

M 3;3 1;5 2;6

D 2;4 3;6 3;0

Table: P dilemma

D strictly dominates U → PR would never choose U

now C strictly dominates L → PC would never choose L

now D strictly dominates M → PR would never choose M

now C strictly dominates R → PC would never choose R

the only rationalizable strategy is 〈D,C 〉



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Game Models and Solutions

Any application of game theory generally requires to decide on

a proper game model that captures all contingencies that arerelevant for the phenomenon

an adequate solution concept that �t the overall description orexplanatory purpose of the phenomenon

Which game and which solution concept should we consult toanalyze and formalize Gricean pragmatics?



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Signaling Games

static games with complete information are not the best choicefor a model of language use and interpretation

dynamics games are able to capture the sequential nature ofutterance and subsequent reception/reaction

incomplete information captures the fact that the hearerprobably lacks an information that the speaker wants tocommunicate with him

a signaling game (Lewis 1969) is a dynamic game withincomplete information that models language use andinterpretation



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Example: Red Wine or White Wine?

Wine-Choice Scenario:

Alice is preparing dinner for her visitor Bob

Depending of Alice prepares beef or �sh, Bob would like tobring red wine or white wine

Both share the same interest, namely that the wine matchingthe dinner

Alice knows what she is preparing, Bob does not

To make the right choice Alice should simply tell Bob �I ampreparing �sh/beef.�



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The Signaling Game

Formal de�nition: 〈{S ,R},T ,Pr ,M, ‖ · ‖,A,US ,UR〉

S is the sender, R the receiver

T is the set of information states

Pr ∈ ∆(T ) is probability distribution over T , which usuallyrepresents the receiver's uncertainty which state is actual

M is a set of messages the sender can send

‖ · ‖ : M → P(T )\∅ is a denotation function that gives theprede�ned semantic meaning of a message

A is the set of response actions available for the receiver

US,R : T ×M × A→ R are utility functions expressing theplayers' (both, sender and receiver) desirability of each possibleplay of the game



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Modeling the Wine-Choice Scenario

How should the wine-choice scenario be modeled as a signal game,in the case that Bob believes that Alice is slightly more likelypreparing beef than �sh (3:2)?

(Formal de�nition: 〈{S ,R},T ,Pr ,M, ‖ · ‖,A,US ,UR〉)

Pr(t) ared awhite mbeef m�sh

tbeef 3/5 1,1 0,0√

-t�sh 2/5 0,0 1,1 -

√

Table: Wine-Choice Scenario



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Extensive Form of the Wine-Choice Scenario

N

S

R

1 0

R

1 0

S

R

0 1

R

0 1

.6 .4tbeef t�sh

m′beef ′ m′�sh′ m′beef ′ m′�sh′

abeef a�sh abeef a�sh abeef a�sh abeef a�sh

Figure: 'extensive form' Wine-Choice-ScenarioRoland Mühlenbernd Horn's principles and rules


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Modeling a Scalar Implicature

The wine-choice scenario is not really interesting forpragmatics.

How should a situation be modeled, that captures the scalarimplicature for the scale 〈 all, some 〉 (let's assume the case'all' is expected to happen with probability p)?

Pr(t) a∃¬∀ a∀ msome mall

t∃¬∀ 1− p 1,1 0,0√

-t∀ p 0,0 1,1

√ √

Table: The scalar implicature



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Extensive Form of the Scalar Implicature

N

S

R

1 0

R

1 0

S

R

0 1

R

0 1

1− p pt∃¬∀ t∀

m′some′ m′all ′ m′some′ m′all ′

a∃¬∀ a∀ a∃¬∀ a∀ a∃¬∀ a∀ a∃¬∀ a∀

Figure: 'extensive form' scalar implicatureRoland Mühlenbernd Horn's principles and rules


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Resume and Outlook

Resume

games are mathematical structures that abstractly representdecision situations of several agents

games are not models of interactive reasoning or decision making;here a solution concept is needed

games are traditionally classi�es in two dimensions

static (strategic) VS. dynamic (sequential) gamescomplete VS. incomplete information

signaling games are sequential games with incomplete information

and a promising tool for modeling and analyzing pragmaticphenomena

Homework

Read Signal to Act (Franke, 2009), Chap. 1.2.1 and 1.2.2 (p. 13-24)

Answer 6 questions that guide you through the chapter


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