Language Games

Andreas Blume

October 2, 2015

Talk at NYU

Agenda

Understand the role of message spaces for strategic communica-tion – in analyzing communication games we typically ignore that withouta common language communication is impossible, regardless of incentives.

Bigger question: relation between literal meaning and strategic mean-ing.

Modest intermediate goal: model the fact that meaning may be un-known, uncertain, not shared etc.

Language types: restrictions on which messages are available and whichmessages are understood; these restrictions are private information.

Blume and Board [EMTA, 2013] use the language types apparatus to in-vestigate misunderstandings in equilibrium – the focus is on common-interest information-transmission games – results about optimal messageuse suggest that meanings are better thought of as distributions than sets(as, for example, in truth-conditional semantics).

Blume [ask me] “Failure of Common Knowledge of Language in Common-Interest Communication Games” uses this framework to investigate thefragility of information transmission in common-interest gamesthat results from higher-order uncertainty about message availability.

An early version of Blume and Board [2013] demonstrates possible bene-fits of uncertainty about availability and interpretation of messages ininformation-transmission games.

The present project investigates which correlated equilibrium out-comes in static complete information games can arise as the result of mis-understandings.

Language Types

M – the universe of messages

Mi ⊆M – messages available to be sent by player i

Qi, a partition of M – a description of which messages player i can distin-guish (or understand)

Qi(m) – the partition element that contains message m

Player i must treat all members of Qi(m) identically (take the same actionin response; send with equal probability; and, make the same inferencewhen receiving)

λi = (Mi;Qi) – player i’s language type

An example are language types of the form

λi = (Mi; {{m}m∈Mi,M i}),

where the set of available messages coincides with the set of understoodmessages.

λ = (λ1, . . . , λI) is a language state

Λi - player i’s set of language types

Λ =×Ii=1Λi – the language state space

q : common knowledge distribution over the language state space

L = (M,Λ, q) – language structure

Literature

•Coarseness of and disagreement about meaning – Arrow[The Limits of Organization, 1974] on organizational codes; Pos-ner [CWRLR, 1987] on statutory interpretation; Galison [Image andLogic, 1997] on trading zones; Cremer, Garicano and Prat [QJE, 2007],Jager, Metzger and Riedel [GEB, 2011] and Sobel [Gerzensee conference,2015] on optimal uses of finite message spaces.

•Correlated equilibria as a bound on what can be achievedwith communication – Aumann [JME, 1974] on correlated equilib-ria

•Cheap-talk extensions as a way to implement correlatedequilibria – Aumann, Maschler, Stearns [1968] on jointly controlledlotteries; Barany [MOR, 1992], Forges [EMTA, 1990], Ben-Porath [JET,1998], Gerardi [JET, 2004] on conditions for getting the entire set ofcorrelated equilibria; Lehrer and Sorin [GEB, 1997] on one-shot publicmediated talk

An Example

U

D

L R

4,4 2,5

5,2 0,0

This game of Chicken has three Nash equilibria, with payoffs (5, 2), (2, 5)and

(103 ,

103

).

With face-to-face pre-play communication in any equilibrium ofany communication game on the equilibrium path players must be playinga Nash equilibrium in the post-communication stage.

Hence, we cannot hope to induce more than the convex hull of the setof Nash equilibria of the base game through face-to-face communica-tion.

Jointly controlled lotteries

Suppose that each player’s message space is the interval [0, 1] and thatplayers send messages simultaneously.

Suppose also that given two numbers 0 ≤ α ≤ β ≤ 1 players use thestrategy of randomizing uniformly over their messages spaces [0, 1]and playing one NE if the sum of messages (modulo 1) is in [0, α),another NE if the if the sum of messages is in [α, β) and the third NEotherwise.

For each pair (α, β) this is an equilibrium in the communication game.

Note that players cannot gain by deviating at the communication stage.Regardless of which message a player sends, the distribution of the sum(modulo 1) is always uniform on [0, 1].

By varying α and β one can induce the entire convex hull of the setof Nash equilibria.

The maximal symmetric payoff that can be achieved this way is

7

2>

10

3.

We conclude that face-to-face communication can induce no less andno more than the convex hull of the set of Nash equilibria.

Face-to-face communication cannot induce the correlated equilibrium dis-tribution on the right:

U

D

L R

4,4 2,5

5,2 0,0

U

D

L R13

13

13 0

This correlated equilibrium outcome is outside of the convex hull of the setof Nash equilibrium outcomes.

This correlated equilibrium achieves a payoff of

11

3>

7

2>

10

3.

I will show that this account is incomplete:

With independent private information about language constraints the gamewith one round of simultaneous pre-play communication can have equi-librium outcomes outside the convex hull of the set of Nash equilibriumoutcomes.

For now define a language game as a complete-information strategic-form game G augmented by a language structure L = (M,Λ, q) and pre-ceded by one round of simultaneous exchange of messages.

I will construct a language game with independent languagetypes that has a Nash equilibrium that induces the distribution on theright

U

D

L R

4,4 2,5

5,2 0,0

U

D

L R13

13

13 0

The example with independent language types – continued

U

D

L R

4,4 2,5

5,2 0,0

Players simultaneously send messages from a common message space M ={∗,#,&, $} prior to playing Chicken.

“Row” has language type QRow1 = (M ; {{∗}, {#}, {&, $}}) with proba-

bility 1/3 and language type QRow2 = (M ; {{∗}, {#}, {&}, {$}}) other-

wise.

“Column” has language type QColumn1 = (M ; {{∗,#}, {&}, {$}}) with

probability 1/3 and language typeQColumn2 = (M ; {{∗}, {#}, {&}, {$}})

otherwise.

Language types are drawn independently. This is commonly known.

Although not formally required, numbering messages facilitates the de-scription of strategies.

Consider the numbering ∗ 7→ 1, # 7→ 2, & 7→ 3 and $ 7→ 4.

At the communication stage, Row randomizes uniformly over themessages that she always understands, M1 = {∗,#}.

At the response stage, if she does understand the messages in M2, shesent a message from M1 and Column sent a message from M2, she takesaction U if the sum of the messages is odd and takes actionD if the sum of the messages is even;

if she does not understand messages in M2 = {&, $}, she sent amessage from M1 and Column sent a message from M2, she takes actionU ;

if either Row did not send a message from M1 or Column did not send amessage from M2, Row randomizes, taking action U with probability 2

3.

Column’s strategy is the mirror image of Row’s strategy:

At the communication stage, Column randomizes uniformly over the mes-sages that she always understands, M2 = {&, $}.

At the response stage, if she does understand the messages in M1, she senta message from M2 and Row sent a message from M1, she takes actionR if the sum of the messages is odd and takes action L if the sum of themessages is even;

if she does not understand messages in M1 = {∗,#}, she sent a messagefrom M2 and Row sent a message from M1, Column takes action L;

if either Row did not send a message from M1 or Column did not send amessage from M2, Column randomizes taking action L with probability 2

3.

U

D

L R

Don’t, Don’t13 ×

13 = 1

9

Don’t, Even13 ×

23 ×

12 = 1

9

Odd, Don’t23 ×

12 ×

13 = 1

9

Don’t, Odd13 ×

23 ×

12 = 1

9

Odd, Odd23 ×

23 ×

12 = 2

9

Even, Don’t23 ×

12 ×

13 = 1

9

Even, Even23 ×

23 ×

12 = 2

9U

D

L R

4,4 2,5

5,2 0,0

U

D

L R

Don’t, Don’t13 ×

13 = 1

9

Don’t, Even13 ×

23 ×

12 = 1

9

Odd, Don’t23 ×

12 ×

13 = 1

9

Don’t, Odd13 ×

23 ×

12 = 1

9

Odd, Odd23 ×

23 ×

12 = 2

9

Even, Don’t23 ×

12 ×

13 = 1

9

Even, Even23 ×

23 ×

12 = 2

9U

D

L R

4,4 2,5

5,2 0,0

There are three incentive constraints to consider for each player: Rowmust “follow instructions” when she does not understand (U), when shedoes understand and the sum is odd (U) and when she does understandand the sum is even (D).

U

D

L R

Don’t, Don’t13 ×

13 = 1

9

Don’t, Even13 ×

23 ×

12 = 1

9

Odd, Don’t23 ×

12 ×

13 = 1

9

Don’t, Odd13 ×

23 ×

12 = 1

9

Odd, Odd23 ×

23 ×

12 = 2

9

Even, Don’t23 ×

12 ×

13 = 1

9

Even, Even23 ×

23 ×

12 = 2

9U

D

L R

4,4 2,5

5,2 0,0

Row assigns probability 13 to Column playing L conditional on understand-

ing and the sum being odd.

Row assigns probability 23 to Column playing L conditional on not under-

standing.

In both cases U is a best reply.

U

D

L R13

13

13 0

U

D

L R12

14

14 0

The distribution we did induce and the one we might want to induce.

Recall that the payoff from the correlated equilibrium we managed to in-duce is

11

3>

7

2>

10

3.

There is no simple modification of the construction we used to inducethe uniform distribution that would allow one to induce the nonuniformdistribution with a payoff of

15

4>

11

3>

7

2>

10

3.

In our construction there are two incentive constraints for playing U , one ofwhich is already tight – the constraint associated with not understanding.

Definition. A language game Γ(G,L) is a finite complete-informationstrategic-form game G (the base game) augmented by a language struc-ture L = (M,Λ, q) and preceded by one round of public pre-play com-munication.

Definition. A language equilibrium of the base game G is a Nashequilibrium of a language game Γ(G,L) for some language structureL.

Definition. An independent-language equilibrium of the basegame G is a Nash equilibrium of a language game Γ(G,L) for somelanguage structure L with independently drawn language types.

We focus on outcomes – distributions over action profiles – in the basegame.

Proposition. For every base game G the set of independent-languageequilibrium outcomes is convex.

Observation: Recall that the set of correlated equilibrium outcomes and(obviously) the convex hull of Nash equilibrium outcomes are convex. Theset of independent-language equilibrium outcomes is a convex set wedgedbetween these convex sets, and as we saw in the example, sometimes strictlylarger that the convex hull of Nash outcomes.

Proposition. Every base game G with at least two strict Nash equi-libria has an independent-language equilibrium outcome outside of theconvex hull of the set of Nash equilibrium outcomes of G.

Let V (G) denote the convex hull of the set of Nash equilibrium payoffs ofG.

For any two strategy profiles s and t and any player i let V (G; s, t, i) denotethe convex hull of the payoffs U(s), U(t) and U(si, t−i).

Let V o(G; s, t, i) denote the relative interior of V (G; s, t, i).

Proposition. If s and t are two strict Nash equilibria of G andV (G)∩ V o(G; s, t, i) = ∅, then G has an independent-language equilib-rium outcome with payoffs outside of V (G).

Let u > v if and only if u` ≥ v` for all ` ∈ I and there is an i ∈ I withui > vi.

Define:

E(G) = {v ∈ V (G)|u > v ⇒ u 6∈ V (G)}: The efficient payoffs in theconvex hull of NE payoffs.

P (G) = {u ∈ RI|∃v ∈ E(G) with u > v}: The payoffs that dominate anefficient payoff in the convex hull of NE payoffs.

Proposition. If s and t are two strict Nash equilibria of G andV o(G; s, t, i) ⊂ P (G), then G has an independent-language equilibriumoutcome with payoffs that Pareto dominate a payoff in E(G).

A illustration of Pareto improvement over the convex hull ofNash payoffs

Let U(s)U(t) ⊂ E(G). Let player 1 not understand with probability η.

-

6

@@@@

@@@

@@@

@@

@@@

@@@

@@@

@@@

@@@

.......................................

...................

•

•

•

•

• (1− ε)U(s1, s2) + ε(ηU(s1, t2) + (1− η)U(t1, t2))

ηU(s1, t2) + (1− η)U(t1, t2)

U(s1, s2)

U(t1, t2)

U(s1, t2)

U1

U2

Extensions

1. sequential communication

2. relaxation of the strictness requirement

(a) extensive-form games

(b) mixed equilibria

3. dummy players (who have no choices)

4. public mediation

5. Pareto improvement when the Pareto frontier of the convex hull of theset of Nash equilibrium payoffs is a singleton

6. correlated language types

Sequential Communication

U

D

L R

4,4 2,5

5,2 0,0

Two communication rounds precede the play of Chicken.

In round 1 Column sends a message from M = {∗,#,&, 1, 2, 3}.Column has language typeQColumn

1 = (M ; {{∗}, {#}, {&}, {1}, {2}, {3}})with probability 1 and therefore always can send and distinguish all mes-sages.

In round 2, after observing Column’s message (filtered through her languagetype), Row responds with a messages from M.

Row has language type QRow1 = (M ; {{∗,#,&}, {1}, {2}, {3}}) with

probability p and language typeQRow2 = (M ; {{∗}, {#}, {&}, {1}, {2}, {3}})

otherwise.

Sequential Communication Continued – Equilibrium

In round 1, Column randomizes uniformly over the messages in {∗,#,&}.

In round 2, after observing Column’s message, type QRow2 uses the rule

∗ 7→{

2 w/p 12, 3 w/p 1

2

}, # 7→

{1 w/p 1

2, 3 w/p 12

},

& 7→{

1 w/p 12, 2 w/p 1

2

}.

The type who understands avoids the histories (∗, 1), (#, 2) and (&, 3).

In round 2, after observing Column’s message, type QRow1 randomizes

uniformly over the messages in {1, 2, 3}

At the action stage Column uses the rule (∗, 1) 7→ R, (#, 2) 7→ R, (&, 3) 7→R; otherwise she plays L. At the action stage typeQRow

2 plays D and type

QRow1 plays U.

A player who deviates by sending a message outside her designated set (e.g.Column sending one of the messages 1,2 or 3) of messages is punished byreceiving her low pure-strategy equilibrium payoff.

The distribution that is induced by this strategy combination is:

U

D

L R

4,4 2,5

5,2 0,0

U

D

L R23p

13p

1-p 0

When Row understands, she signals understanding by avoiding to matchand takes advantage of her understanding by playing D.

When Row fails to understand, there is a 23 chance of her succeeding in

avoiding a match, and thus inducing Column to play left.

When Row fails to understand there is a 13 chance of her matching and thus

revealing that she fails to understand and plays up.

Conditional on failing to understand Row is indifferent between the actionsU and D.

It remains to check Column’s incentives.

U

D

L R

4,4 2,5

5,2 0,0

U

D

L R23p

13p

1-p 0

When a match reveals that Row does not understand and therefore playsU , it is uniquely optimal for Column to play R.

Following a mismatch player Column’s conditional probability of Row not

understanding is23p

(1−p)+23p.

Hence, following a mismatch player Column is willing to play L as long as4p23 + 2(1− p) ≥ 5p23, which is equivalent to p ≤ 3

4.

With p = 34, we get the distribution

U

D

L R

4,4 2,5

5,2 0,0

U

D

L R12

14

14 0

In a language game with sequential communication there aretwo rounds of communication, with one player sending messages in thefirst round and another player sending messages in the second round.

Proposition. Every base game G with at least two strict Nash equi-libria s and t and a player i with Ui(s) < Ui(t) and si 6= ti has anindependent-language equilibrium outcome of a language game withsequential communication outside of the convex hull of the set of Nashequilibrium outcomes of G.

Relaxation of the strict equilibrium requirement

Consider the game in which the Row first selects which 2× 2 subgame toplay, A or B.

U

D

L R

1,1 4,-2

-2,4 6,6

A

u

d

` r

4,4 2,5

5,2 0,0

B

The game in strategic form

AU

AD

Bu

Bd

L` Lr R` Rr

1,1 1,1 4,-2 4,-2

-2,4 -2,4 6,6 6,6

4,4 2,5 4,4 2,5

5,2 0,0 5,2 0,0

There is an independent-language equilibrium that inducesthe distribution in the lower left

AU

AD

Bu

Bd

L` Lr R` Rr

1,1 1,1 4,-2 4,-2

-2,4 -2,4 6,6 6,6

4,4 2,5 4,4 2,5

5,2 0,0 5,2 0,0

13

13

13

There is also an independent-language equilibrium that in-duces this distribution in the lower left

AU

AD

Bu

Bd

L` Lr R` Rr

1,1 1,1 4,-2 4,-2

-2,4 -2,4 6,6 6,6

4,4 2,5 4,4 2,5

5,2 0,0 5,2 0,0

14

12

14

Dummy players and public mediation.

1. Whenever the set of outcomes can be expanded, the addition of a dummyplayer can be used to expand the payoff set.

2. Public mediation can be used to obtain the full set of correlated equi-libria – Hungarian and Quechua example.

Pareto improvement when the Pareto frontier of the convexhull of the set of Nash equilibrium payoffs is a singleton

A

B

C

X Y Z

4,4 0,0 2,5

0,0 3,3 3,0

5,2 0,3 0,0

This game has three Nash equilibria. All equilibria are symmetric: (pX, pY , pZ) =(616,

716,

316

)with payoff

(3016,

3016

); (pX, pY , pZ) =

(23, 0,

13

)with payoff

(103 ,

103

);

and,(pX, pY , pZ) = (0, 1, 0) with payoff (3, 3). There is a correlated equi-librium with probability 1

3 on each of the cells (A,X), (A,Z) and (C,X)and payoff

(113 ,

113

). This distribution is also supported by an independent-

language equilibrium (both with simultaneous and sequential messages),exactly as before.

A

B

C

X Y Z

4,4 0,0 2,5

0,0 3,3 3,0

5,2 0,3 0,0

This can be understood by our prior construction applied to the auxiliarygame in which the strategies B and Y are unavailable and then noticingthat when these strategies are admitted they are not better replies to anyof the beliefs players hold in the independent-language equilibrium of theauxiliary game. This generalizes.

Correlated Language typesA typical language type of the row player:

∗

#

&

%

f • ∼ �

D U U U

U U D U

U D U U

U U U D

U

D

L R12

14

14 0

Correlated Language typesTypical language types for both players:

∗

#

&

%

f • ∼ �

D U U R

R U D U

U D R U

U R U D

U

D

L R12

14

14 0

Correlated Language types

In this manner (using the full range of partitions that can arisewith different language type structures) one can obtain the en-tire set of rational correlated equilibria (i.e. correlated equilibriawith rational coefficients.)

This set is dense in the entire set of correlated equilibria.

One can also get the entire interior of the set of correlated equilibria.

Summary

Demonstration that the “language type” machinery is productive.

Sufficient conditions for the existence of independent-language equilibriaoutside the convex hull of Nash equilibria.

Conditions for both simultaneous and sequential communication.

Conditions for extensive form games.

(Nearly) the entire set of correlated equilibria with public mediation (cute,trivial) or correlated language types (not quite so trivial).