Language Games
Andreas Blume
October 2, 2015
Talk at NYU
Agenda
Understand the role of message spaces for strategic communication – in analyzing communication games we typically ignore that without
a common language communication is impossible, regardless of incentives.
Bigger question: relation between literal meaning and strategic meaning.
Modest intermediate goal: model the fact that meaning may be unknown, uncertain, not shared etc.
Language types: restrictions on which messages are available and which
messages are understood; these restrictions are private information.
Blume and Board [EMTA, 2013] use the language types apparatus to investigate misunderstandings in equilibrium – the focus is on commoninterest information-transmission games – results about optimal message
use 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 CommonInterest Communication Games” uses this framework to investigate the
fragility of information transmission in common-interest games
that results from higher-order uncertainty about message availability.
An early version of Blume and Board [2013] demonstrates possible benefits of uncertainty about availability and interpretation of messages in
information-transmission games.
The present project investigates which correlated equilibrium outcomes in static complete information games can arise as the result of misunderstandings.
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 distinguish (or understand)
Qi(m) – the partition element that contains message m
Player i must treat all members of Qi(m) identically (take the same action
in response; send with equal probability; and, make the same inference
when 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 understood
messages.
λ = (λ1, . . . , λI ) is a language state
Λi - player i’s set of language types
Λ = ×i=1Λi – the language state space
I
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; Posner [CWRLR, 1987] on statutory interpretation; Galison [Image and
Logic, 1997] on trading zones; Crémer, Garicano and Prat [QJE, 2007],
Jäger, 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 achieved
with communication – Aumann [JME, 1974] on correlated equilibria
• Cheap-talk extensions as a way to implement correlated
equilibria – Aumann, Maschler, Stearns [1968] on jointly controlled
lotteries; Bárány [MOR, 1992], Forges [EMTA, 1990], Ben-Porath [JET,
1998], Gerardi [JET, 2004] on conditions for getting the entire set of
correlated equilibria; Lehrer and Sorin [GEB, 1997] on one-shot public
mediated talk
An Example
L
R
U 4,4 2,5
D 5,2 0,0
This game of Chicken has three Nash equilibria, with payoffs (5, 2), (2, 5)
10
,
and 10
3 3 .
With face-to-face pre-play communication in any equilibrium of
any communication game on the equilibrium path players must be playing
a Nash equilibrium in the post-communication stage.
Hence, we cannot hope to induce more than the convex hull of the set
of Nash equilibria of the base game through face-to-face communication.
Jointly controlled lotteries
Suppose that each player’s message space is the interval [0, 1] and that
players send messages simultaneously.
Suppose also that given two numbers 0 ≤ α ≤ β ≤ 1 players use the
strategy 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 NE
otherwise.
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 set
of Nash equilibria.
The maximal symmetric payoff that can be achieved this way is
7 10
> .
2
3
We conclude that face-to-face communication can induce no less and
no more than the convex hull of the set of Nash equilibria.
Face-to-face communication cannot induce the correlated equilibrium distribution on the right:
L
R
L
R
U 4,4 2,5
U
1
3
1
3
D 5,2 0,0
D
1
3
0
This correlated equilibrium outcome is outside of the convex hull of the set
of Nash equilibrium outcomes.
This correlated equilibrium achieves a payoff of
11 7 10
> > .
3
2
3
I will show that this account is incomplete:
With independent private information about language constraints the game
with one round of simultaneous pre-play communication can have equilibrium outcomes outside the convex hull of the set of Nash equilibrium
outcomes.
For now define a language game as a complete-information strategicform game G augmented by a language structure L = (M, Λ, q) and preceded by one round of simultaneous exchange of messages.
I will construct a language game with independent language
types that has a Nash equilibrium that induces the distribution on the
right
L
R
L
R
U 4,4 2,5
U
1
3
1
3
D 5,2 0,0
D
1
3
0
The example with independent language types – continued
L
R
U 4,4 2,5
D 5,2 0,0
Players simultaneously send messages from a common message space M =
{∗, #, &, $} prior to playing Chicken.
“Row” has language type QRow
= (M ; {{∗}, {#}, {&, $}}) with proba1
bility 1/3 and language type QRow
= (M ; {{∗}, {#}, {&}, {$}}) other2
wise.
“Column” has language type QColumn
= (M ; {{∗, #}, {&}, {$}}) with
1
probability 1/3 and language type QColumn
= (M ; {{∗}, {#}, {&}, {$}})
2
otherwise.
Language types are drawn independently. This is commonly known.
Although not formally required, numbering messages facilitates the description of strategies.
Consider the numbering ∗ 7→ 1, # 7→ 2, & 7→ 3 and $ 7→ 4.
At the communication stage, Row randomizes uniformly over the
messages that she always understands, M̃1 = {∗, #}.
At the response stage, if she does understand the messages in M̃2, she
sent a message from M̃1 and Column sent a message from M̃2, she takes
action U if the sum of the messages is odd and takes action
D if the sum of the messages is even;
if she does not understand messages in M̃2 = {&, $}, she sent a
message from M̃1 and Column sent a message from M̃2, she takes action
U;
if either Row did not send a message from M̃1 or Column did not send a
message from M̃2, Row randomizes, taking action U with probability 32 .
Column’s strategy is the mirror image of Row’s strategy:
At the communication stage, Column randomizes uniformly over the messages that she always understands, M̃2 = {&, $}.
At the response stage, if she does understand the messages in M̃1, she sent
a message from M̃2 and Row sent a message from M̃1, she takes action
R if the sum of the messages is odd and takes action L if the sum of the
messages is even;
if she does not understand messages in M̃1 = {∗, #}, she sent a message
from M̃2 and Row sent a message from M̃1, Column takes action L;
if either Row did not send a message from M̃1 or Column did not send a
message from M̃2, Column randomizes taking action L with probability 32 .
L
U
D
Don’t, Don’t
1
1
1
3 × 3 = 9
Don’t, Even
1
2
1
1
×
×
=
3
3
2
9
Odd, Don’t
2
1
1
1
×
×
=
3
2
3
9
Even, Don’t
1
1
1
2
×
×
=
3
2
3
9
Even, Even
2
2
1
2
×
×
=
3
3
2
9
R
Don’t, Odd
2
1
1
3 × 3 × 2 =
Odd, Odd
2
2
1
×
×
3
3
2 =
1
9
2
9
L
R
U
4,4
2,5
D
5,2
0,0
L
U
D
Don’t, Don’t
1
1
1
3 × 3 = 9
Don’t, Even
1
2
1
1
×
×
=
3
3
2
9
Odd, Don’t
2
1
1
1
×
×
=
3
2
3
9
Even, Don’t
1
1
1
2
×
×
=
3
2
3
9
Even, Even
2
2
1
2
×
×
=
3
3
2
9
R
Don’t, Odd
2
1
1
3 × 3 × 2 =
Odd, Odd
2
2
1
×
×
3
3
2 =
1
9
2
9
L
R
U
4,4
2,5
D
5,2
0,0
There are three incentive constraints to consider for each player: Row
must “follow instructions” when she does not understand (U ), when she
does understand and the sum is odd (U ) and when she does understand
and the sum is even (D).
L
U
D
R
Don’t, Don’t
1
1
1
3 × 3 = 9
Don’t, Even
1
2
1
1
×
×
=
3
3
2
9
Odd, Don’t
2
1
1
1
×
×
=
3
2
3
9
Even, Don’t
1
1
1
2
×
×
=
3
2
3
9
Even, Even
2
2
1
2
×
×
=
3
3
2
9
Don’t, Odd
2
1
1
3 × 3 × 2 =
Odd, Odd
2
2
1
×
×
3
3
2 =
1
9
2
9
L
R
U
4,4
2,5
D
5,2
0,0
Row assigns probability 13 to Column playing L conditional on understanding and the sum being odd.
Row assigns probability
standing.
2
3
to Column playing L conditional on not under-
In both cases U is a best reply.
L
R
U
1
3
1
3
D
1
3
0
L
R
U
1
2
1
4
D
1
4
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 induce is
11 7 10
> > .
3
2
3
There is no simple modification of the construction we used to induce
the uniform distribution that would allow one to induce the nonuniform
distribution with a payoff of
15 11 7 10
>
> > .
4
3
2
3
In our construction there are two incentive constraints for playing U , one of
which is already tight – the constraint associated with not understanding.
Definition. A language game Γ(G, L) is a finite complete-information
strategic-form game G (the base game) augmented by a language structure L = (M, Λ, q) and preceded by one round of public pre-play communication.
Definition. A language equilibrium of the base game G is a Nash
equilibrium of a language game Γ(G, L) for some language structure
L.
Definition. An independent-language equilibrium of the base
game G is a Nash equilibrium of a language game Γ(G, L) for some
language structure L with independently drawn language types.
We focus on outcomes – distributions over action profiles – in the base
game.
Proposition. For every base game G the set of independent-language
equilibrium outcomes is convex.
Observation: Recall that the set of correlated equilibrium outcomes and
(obviously) the convex hull of Nash equilibrium outcomes are convex. The
set of independent-language equilibrium outcomes is a convex set wedged
between these convex sets, and as we saw in the example, sometimes strictly
larger that the convex hull of Nash outcomes.
Proposition. Every base game G with at least two strict Nash equilibria has an independent-language equilibrium outcome outside of the
convex hull of the set of Nash equilibrium outcomes of G.
Let V (G) denote the convex hull of the set of Nash equilibrium payoffs of
G.
For any two strategy profiles s and t and any player i let V (G; s, t, i) denote
the 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 and
V (G) ∩ V o(G; s, t, i) = ∅, then G has an independent-language equilibrium 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 with
ui > vi.
Define:
E(G) = {v ∈ V (G)|u > v ⇒ u 6∈ V (G)}: The efficient payoffs in the
convex hull of NE payoffs.
P (G) = {u ∈ RI |∃v ∈ E(G) with u > v}: The payoffs that dominate an
efficient payoff in the convex hull of NE payoffs.
Proposition. If s and t are two strict Nash equilibria of G and
V o(G; s, t, i) ⊂ P (G), then G has an independent-language equilibrium
outcome with payoffs that Pareto dominate a payoff in E(G).
A illustration of Pareto improvement over the convex hull of
Nash payoffs
Let U (s)U (t) ⊂ E(G). Let player 1 not understand with probability η.
U2 6
(s1, s2)
•@....U
@ ...
@ ..
@ ...
@
− )U (s1, s2) + (ηU (s1, t2) + (1 − η)U (t1, t2))
•...(1
@
.
....
@
@
....
@
....
@
@
.... •.. U (s1, t2)
@
.... ..
@
@
.....
@
@
•.. ηU (s1, t2) + (1 − η)U (t1, t2)
@
@
..
@
..
@
@
..
@
.
@
@ ..
@ .
@. U (t1 , t2 )
@
•
-
U1
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 the
set of Nash equilibrium payoffs is a singleton
6. correlated language types
Sequential Communication
L
R
U 4,4 2,5
D 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 type QColumn = (M ; {{∗}, {#}, {&}, {1}, {2}, {3}})
1
with probability 1 and therefore always can send and distinguish all messages.
In round 2, after observing Column’s message (filtered through her language
type), Row responds with a messages from M.
Row has language type QRow
= (M ; {{∗, #, &}, {1}, {2}, {3}}) with
1
probability p and language type QRow = (M ; {{∗}, {#}, {&}, {1}, {2}, {3}})
2
otherwise.
Sequential Communication Continued – Equilibrium
In round 1, Column randomizes uniformly over the messages in {∗, #, &}.
Row uses the rule
In round
2,
after
observing
Column’s
message,
type
Q
2
1
1
1
1
∗ 7→ 2 w/p 2 , 3 w/p 2 , # 7→ 1 w/p 2 , 3 w/p 2 ,
& 7→ 1 w/p 21 , 2 w/p 12 .
The type who understands avoids the histories (∗, 1), (#, 2) and (&, 3).
In round 2, after observing Column’s message, type QRow
randomizes
1
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 type QRow
plays D and type
2
QRow
plays U.
1
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 by
receiving her low pure-strategy equilibrium payoff.
The distribution that is induced by this strategy combination is:
L
R
L
R
2p
3
1p
3
U 4,4 2,5
U
D 5,2 0,0
D 1-p
0
When Row understands, she signals understanding by avoiding to match
and takes advantage of her understanding by playing D.
When Row fails to understand, there is a 32 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 actions
U and D.
It remains to check Column’s incentives.
L
R
L
R
2p
3
1p
3
U 4,4 2,5
U
D 5,2 0,0
D 1-p
0
When a match reveals that Row does not understand and therefore plays
U , it is uniquely optimal for Column to play R.
Following a mismatch player Column’s conditional probability of Row not
2p
3
understanding is (1−p)+
2p.
3
Hence, following a mismatch player Column is willing to play L as long as
4p 23 + 2(1 − p) ≥ 5p 23 , which is equivalent to p ≤ 34 .
With p = 34 , we get the distribution
L
R
L
R
U 4,4 2,5
U
1
2
1
4
D 5,2 0,0
D
1
4
0
In a language game with sequential communication there are
two rounds of communication, with one player sending messages in the
first round and another player sending messages in the second round.
Proposition. Every base game G with at least two strict Nash equilibria s and t and a player i with Ui(s) < Ui(t) and si 6= ti has an
independent-language equilibrium outcome of a language game with
sequential communication outside of the convex hull of the set of Nash
equilibrium outcomes of G.
Relaxation of the strict equilibrium requirement
Consider the game in which the Row first selects which 2 × 2 subgame to
play, A or B.
L
R
`
r
U 1,1 4,-2
u
4,4
2,5
D -2,4 6,6
d
5,2
0,0
A
B
The game in strategic form
L`
Lr
R`
Rr
AU
1,1
1,1
4,-2
4,-2
AD
-2,4
-2,4
6,6
6,6
Bu
4,4
2,5
4,4
2,5
Bd
5,2
0,0
5,2
0,0
There is an independent-language equilibrium that induces
the distribution in the lower left
L`
Lr
R`
Rr
AU
1,1
1,1
4,-2
4,-2
AD
-2,4
-2,4
6,6
6,6
2,5
4,4
2,5
0,0
5,2
0,0
1
3
Bu
4,4
1
3
1
3
Bd
5,2
There is also an independent-language equilibrium that induces this distribution in the lower left
L`
Lr
R`
Rr
AU
1,1
1,1
4,-2
4,-2
AD
-2,4
-2,4
6,6
6,6
2,5
4,4
2,5
0,0
5,2
0,0
1
2
Bu
4,4
1
4
1
4
Bd
5,2
Dummy players and public mediation.
1. Whenever the set of outcomes can be expanded, the addition of a dummy
player can be used to expand the payoff set.
2. Public mediation can be used to obtain the full set of correlated equilibria – Hungarian and Quechua example.
Pareto improvement when the Pareto frontier of the convex
hull of the set of Nash equilibrium payoffs is a singleton
X
Y
Z
A
4,4
0,0
2,5
B
0,0
3,3
3,0
C
5,2
0,3
0,0
This game has three Nash equilibria.
All equilibria are symmetric: (pX , pY, pZ ) =
6 7 3
30 30
2
1
10 10
,
,
with
payoff
,
;
(p
,
p
,
p
)
=
,
0,
with
payoff
X
Y
Z
16 16 16
16 16
3
3
3, 3 ;
and,(pX , pY , pZ ) = (0, 1, 0) with payoff (3, 3). There is a correlated equi1
on each of the cells (A, X), (A, Z) and (C, X)
librium with probability
3
11
and payoff 11
,
3 3 . This distribution is also supported by an independentlanguage equilibrium (both with simultaneous and sequential messages),
exactly as before.
X
Y
Z
A
4,4
0,0
2,5
B
0,0
3,3
3,0
C
5,2
0,3
0,0
This can be understood by our prior construction applied to the auxiliary
game in which the strategies B and Y are unavailable and then noticing
that when these strategies are admitted they are not better replies to any
of the beliefs players hold in the independent-language equilibrium of the
auxiliary game. This generalizes.
Correlated Language types
A typical language type of the row player:
f
•
∼
∗
D
U
U
U
#
U
U
D
U
&
U
D
U
U
%
U
U
U
D
L
R
U
1
2
1
4
D
1
4
0
Correlated Language types
Typical language types for both players:
f
•
∼
∗
D
U
U
R
#
R
U
D
U
&
U
D
R
U
%
U
R
U
D
L
R
U
1
2
1
4
D
1
4
0
Correlated Language types
In this manner (using the full range of partitions that can arise
with different language type structures) one can obtain the entire set of rational correlated equilibria (i.e. correlated equilibria
with 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 equilibria
outside 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).