From D. Rothschild's online study on Grice's implicature
"Much of philosophy of language and linguistics is concerned with showing
what is special about language."
"One of Grice’s (1967/1989) contributions, against this tendency, was to treat speech as a form of rational activity, subject to the same sorts of norms and expectations that apply to all such activity."
--- and he was doing that in 1966 in Oxford already, "Logic and Conversation: The Oxford lectures" 1966, deposited in the Grice Papers, UC/Berkeley, Bancroft Library. Principle of conversational self-love, principle of conversational benevolence, etc.
"This general perspective has proved very fruitful in pragmatics."
"However, it is rarely explicitly asked whether a particular pragmatic phenomenon should be understood entirely in terms of rational agency or whether positing special linguistic principles is necessary."
Rothschild's essay "is concerned with evaluating the degree to
which a species of simple pragmatic inferences, ... implicatures, should be
viewed as a form of rational inference."
"A rigorous answer to this last question
requires us to use the theoretical resources of game theory."
Rothschild argues that
"weak-dominance reasoning, a standard form of game-theoretic reasoning, allows
us cash out scalar implicatures as rational inferences in a large class of communicative
situations."
"This account of the derivation of scalar implicatures is
more principled and robust than other explanations in the game theory and
pragmatics literature."
2 Deriving implicatures
"Consider this paradigmatic example of an implicature:
( Rothschild's essay is "a distant descendent of his "first, tentative foray into these issues, “Grice, Rationality, and Utterance”." Rotschild would like to thank Dirk Bergemann, Emmanuel Chemla, Vince Crawford,
Michael Rothschild, and Joel Sobel for very useful discussion of these topics. I am also grateful to the hardy participants at a seminar in Oxford.
1
Some/All Case There was a small cake in the refrigerator the night before.
Mary says to Kay, the next morning, “I ate some of the cake last night.”
"There is an obvious suggestion that Mary conveys the information that
she ate some but not all of the cake last night."
"For the example to work, we need to make certain background assumptions,
but when these are in place (and it is not hard to imagine them in place) then
Mary’s words will carry the meaning suggested. Despite the seeming triviality
of this phenomenon, scalar implicatures play a lively role in current contemporary
linguistics, both formal and experimental (e.g. Horn, 1972; Chierchia,
2004; Noveck, 2001), .
It might seem that the kind of reasoning in the Some/All Case and others
like it is simply rational inference. Let us try to tease out this thought: We
can think of communicative situations of this kind as ones in which there is
asymmetric information: Mary knows something Kay does not, namely how
much cake she (Mary) ate. Mary wants Kay to know what she knows. Now
suppose Mary, wanting to speak only once, has to choose between uttering the
following two sentences:
(1) a. I ate some of the cake.
b. I ate all of the cake.
Let us assume, as is standard, that (1-b) strictly entails (1-a): if someone ate
all the cake, then ipso facto they ate some of the cake. Consider the choice of
utterance from Mary’s perspective: If Mary ate all of the cake then she should
assert (1-b) rather than (1-a) as that would convey more information to Kay,
given the entailment relations. On the other hand, if she ate only some of the
cake then (1-b) would give Kay false information which is undesirable. So, if
Mary is reasonable she will utter (1-a) if she ate just some of the cake, and (1-b)
if she ate all of the cake. Now think of this from Kay’s perspective: she can
also reason, as we did, about Mary’s behavior, so she can conclude that Mary
will only utter (1-a) if and only if she only ate some of the cake. So, when Mary
utters (1-a) Kay will be able to work out that that Mary only ate some of the
cake, and hence that (1-b) is false. This last inference, is the scalar implicature.
The informal reasoning, plausible as it may sound, is flawed. To see this
note that in the reasoning above, when Mary considers what information she
is conveying to Kay she is not taking account of the fact that Kay will read
more into her utterance than just the literal message. If she does take this into
account, the entailment relation between (1-a) and (1-b) does not carry over to
an entailment relation between the information conveyed by each of the sentences.
In other words,Mary cannot assume that just because one sentence has
a stronger literal meaning than another that it will convey more information.
But if we cannot assume that, the informal reasoning above does not get off the
ground.
To put the point another way: All that we can assume about Mary’s preferences
is that she wants to convey as much (true) information as possible. We
cannot determine which of (1-a) or (1-b) will convey more information based
on their literal meaning alone since we need to assume that sentences may
convey more than their literal meaning. So, from the fact that Mary prefers
to convey as much information as possible and the fact that (1-b) is literally
weaker than (1-a), it does not follow that Mary will say (1-b) rather than (1-a).
Something more is needed.
What we have in a communicative situation like the Some/All Case is a decision
problem involving two rational agents. Such problems are known to be
complex, and there is a special branch of economics and decision theory, game
theory, devoted to analyzing them. Most of this paper is concerned with making
the case that game theory allows us view scalar implicatures, like the one
above, as a form of rational inference. There is already a substantial literature
on this question, but my approach and conclusions differ from the standard
ones in that literature, and I will try to mark these differences as I go along.
3 Plan
Here is the plan of this Rothschild's essay:
§4 discusses Grice’s maxims and the treatment of scalar implicatures in contemporary linguistics.
§5–§8 presents the game-theoretic background necessary for the rest of the essay.
§5 introduces the signaling game which is the standard model for communicative
situations in game theory.
§6 discusses the Nash equilibrium notion and the standard refinements of it used for thinking about games like the signaling game.
§7 discusses what constraints, outside of equilibrium notions, there are on how rational player should play games.
§8 discusses in general terms what we should hope for in a game-theoretic model of scalar implicatures.
§9 introduces a barebones game formodeling scalar implicatures and argues that standard equilibriumconcepts cannot be used to argue that scalar implicatures
are a rational inference.
§10 argues that iterative weak-dominance reasoning can capture the derivation of scalar implicatures.
§11 briefly considers (and rejects) the proposal that Gricean reasoning should be captured by means of the Pareto-Nash equilibrium.
§12 discusses and criticizes what Rothschild calls reasoning-based accounts, popular models in the current literature for explaining the derivation of implicatures.
§13–§16 discusses how to apply weak-dominance arguments to different types of scalar and relevance implicatures.
§17 discusses disclosure cases from economics literature and assesses their connection to scalar implicatures.
There are twomain aims to Rothschild's essay, one theoretical and one methodological.
The theoretical aim is to show that iterative weak dominance provides a
good way of explaining the derivation of scalar implicatures within a gametheoretic
framework.
To a significant extent this vindicates the commonly held
view that scalar implicatures—in idealized cases—are a form of rational inference.
Rothschild's methodological aim is less specific.
There has been a significant amount of work in game-theoretic pragmatics in the last decade or so.
This work provides a variety of different techniques for deriving implicatures of many types.
This is an important project because simple
explanations of even the most basic implicatures in terms of rationality face
serious conceptual hurdles.
We must stay aware, though, that game theory, as a collection of different techniques for modeling situations, is a very powerful
tool.
Merely providing some model in game theory that makes a set of behavioral
predictions is of little interest, since, for any precisely characterized and
vaguely reasonable behavior, we should antecedently expect to be able to provide
some game-theoretic model.
We need to evaluate game-theoretic models of
pragmatic reasoning along a variety of dimensions, asking questions like:
How robust is the model against slight changes in one’s description of the situation?
How standard is the sort of reasoning being attributed to the players?
Can these models extend to parallel non-linguistic cases?
Rothschild argues that many of the prominent theories in the literature do not fare well in the face of this scrutiny.
I should note, however, that this paper is by no means intended as a review of the current literature, and many important contributions go without mention.1
4 Maxims and scalar implicatures
Grice (1967) first systematically identified and analyzed conversational phenomena along the line of the Some/All Case.
Grice attempted to account for these by positing a set of maxims governing conversation.
One of these maxims, Quantity, enjoins the speaker to
“Make your contribution as informative as is required (for the current purposes of exchange).”
Another maxim, Quality, enjoins the speaker to tell the truth.
Implicatures consist, essentially, of inferences people can make about what the speaker must have meant to convey given that he was following Grice’s maxims.
So, in the Some/All case when we assume that Mary obeys Quantity and Quality and that Kay knows this, we seem to be able to get the right inference:
Quantity enjoins Mary to say she ate all of the cake if she did while Quality prevents her fromdoing so if she doesn’t.
So if Mary says she ate some of the cake, the speaker can reasonably
infer that she didn’t eat all of it.
This inference is the implicature.
Grice wanted to ground these maxims in human rationality rather than to
merely put them forward as behavioral generalizations.
He discusses the possibility that the maxims are valid because
“it is just a well-recognized empirical
fact that people do behave in these ways”
but suggests that he would prefer to
have them more solidly grounded in human rationality. Grice writes,
"So I would like to be able to show that observance of the Cooperative
Principle and maxims is reasonable (rational) along the following
lines: that anyone who cares about the goals that are central
to conversation/communication (such as giving and receiving
information, influencing and being influenced by others) must be
expected to have an interest, given suitable circumstance, in participation
in talk exchanges that will be profitable only on the assumption
that they are conducted in general accordance with the
Cooperative Principle and the maxims."
Grice’s views here are more nuanced than this one quote might suggest, but I
think it is clear that Grice was interested in trying to ground either the maxims themselves
(1Most notably, as my interest is in reconstructing inferences speakers canmake, I do not discuss attempts to explain implicatures in terms of evolutionary game theory.)
---
(or the behavior they require) in human rationality and, more specifically, in the commonality of interests in speaker and audience in certain communicative situations.
Grice himself did not consider cases of exactly the form of the Some/All
Case—though he considered very similar ones in his discussion of disjunction.
Horn (1972) is widely regarded as the first person to give a systematic consideration
of scalar implicatures.2
Horn suggests that certain classes of linguistic expressions can be arrayed on a scale of strength.
Examples include some/all,
one/two/three. . . , and few/none.
The critical feature of such paradigmatic scales
is that the scale is ordered by logical strength: the higher members of the scale
entail the lower members.
So, I ate all the cake entails I ate some of the cake, I ate three hotdogs entails I ate two hotdogs, I saw none of the students entails I saw
few students.
This excludes, for instance, short and tall from being on a scale
together since I am tall does not entail I am short or vice versa (similarly with
few and all—though a few is on a scale with all).
Horn observed that the use of scalar terms systematically leads to the derivation
of scalar implicatures parallel to that in the Some/All Case.
This can be explained by supposing that choice between scalar terms is governed by the maxims of quantity and quality.
Since Grice and Horn, the exact nature and logic of these implicatures has been developed extensively (e.g., Gazdar, 1979; Soames, 1982; Sauerland, 2004; Spector, 2006).
Some linguists have recently argued that some aspects of the phenomenon are best captured by grammatical devices for generating implicatures rather than simply making reference to the Gricean maxims (Chierchia, 2004; Chierchia et al., forthcoming; Fox, 2006).
Others have argued that the Gricean maxims, properly formalized, can capturemost
observed scalar implicatures (Sauerland, 2004; Schulz and van Rooij,
2006; Spector, 2006).
What I want to highlight is that in this debate over the nature of scalar implicatures, it is generally assumed by both parties that an explanation in terms
of the Gricean maxims would, all else equal, be more principled than one that
makes reference to other special linguistic rules.
Those who argue for special grammatical rules think the Gricean story is not capable of capturing all the relevant linguistic data.
I take it the default preference for the Gricean view
(2There are also substantial non-Gricean—though Gricean-inspired—schools for explaining implicature such as relevance theory (Sperber and Wilson, 1986).
---
of implicatures gets some of its plausibility from the common belief that the
maxims are not special linguistics rules, but rather follow from basic rationality
assumptions. As I suggested earlier, game theory allows us to assess this
assumption directly.
5 Signaling games
David Lewis’s Convention (1969) gave the basic model of the signaling game.
These games were introduced to the economics literature by Spence (1973) and
have been extensively studied since (see Sobel, 2009, for a recent review) as an
important tool for modeling situations with asymmetric information.
The basic situation of a (two-person) signaling game can be described as
follows.
There are two players, a speaker and a hearer.
The speaker has private information.
We can describe this private information by saying there are different
types the speaker might have and that the speaker’s type is assignedwith
some probability distribution by nature.
Each type of speaker has available to him the option to send a message in a set M.
Each hearer, having seen the message, chooses an an actions from a set A.
For each triplet of type, message and action we get a payoff U.3
Signaling games are extensive form games, which is to say they are represented
as a series of sequential moves by different players.
Figure 1 gives a standard representation of a simple signaling game.
This chart can be explained as follows:
N represents “nature” which is taken to decide the type of the sender S with probability p for type 1 and 1 − p for type 2.
The speaker of either type has two message available to him a or b.
The dashed line represents the information state of the hearer: indicating that he cannot tell what
(3So we can represent a (finite) signaling game as a {T, p,M,A,U} where T is the set of types p
is a probability distribution over them, M is the set of message, A is a function from messages to
actions, and U is a function from triplets of types, messages and actions to pairs of payoffs from
the speaker and hearer.)
p N 1 − p
b
a
S1
b
a
S2
H
R
k, l
L
i, j
H
R
c, d
L
a, b
R
g, h
L
e, f
R
o, p
L
m, n
Fig. 1: A signaling game
the speaker type is but only what signal the speaker used.
L and R represent the two moves available to the hearer in any given state.
The fact that the two speaker types are not connected by dashed lines means that the speaker knows which type he is.
The pair of numbers at the end of each final branch represent the speaker and hearer payoff respectively.
Note that we will discuss variations on this game where there are limitations on S or H’s action based either on the type (in the case of S) or the message received (in the case of H).
The normal way of thinking about strategies in extensive-form games is as fully specific dispositions to play the game in a particular way (no matter how
the other person plays).
So, for example, in the game above we can identify the following pure (i.e. non-probabilistic) speaker strategies.
S1 plays a and S2 plays a (which we’ll write 1a2a), S1 plays a and S2 plays b (1a2b), 1b2a, 1b2b.
The pure hearer strategies, on the other hand, are responses to the different
signals
aLbL, aLbR, aRbL, aRbR.
Note that hearer strategies cannot depend on the type of speaker because the hearer does not have that information available to him.
Mixed strategies are ones in which responses are chosen with a probability distribution rather than deterministically.
We can also talk about the expected payoff for a pair of strategies in the normal way.
A natural question is how extensive-form games, such as signaling games, relate to the standard representation of games by tables, such as the prisoner’s
dilemma game represented in figure 2.
We call representations these tabular representations, normal or strategic forms.
Corresponding to each extensive form game is a normal-form game, in which we think of each player as (at once)
C D
C 3, 3 1, 4
D 4, 1 2, 2
Fig. 2: Prisoner’s Dilemma
choosing complete pure strategies.
I will give an example of a normal-form representation of an extensive-form game in the next section.
For a signaling game the normal form can be given as a mapping from all the possible combinations of pure strategies to the expected payoff (given the probability distribution over states of nature).4
Note that the correspondence between extensive form games and normal-form games is not one-to-one.
For any normal-form game there are many extensive-form games compatible with it.
6 Nash equilibria and refinements
Game theory provides various solution concepts for assessing howplayers should
play games.
Themost standard solution concept is the Nash equilibrium.
A little notation.
Let Ui(s1, S2) be the expected payoff to player i of playing strategy
s1 against an opponent who plays strategy s2.
A Nash equilibrium is a pair of strategies for player 1 and player 2, (s1, s2),
such that for all strategies s′1 possible for player 1, U1(s1, s2) U1(s′
1, s2) and for all strategies s′2 possible for player 2, U2(s1, s2) U2(s1, s′
2).
In other words, a Nash equilibrium is pair of strategies that neither player has an incentive to deviate alone from.
In the prisoner’s dilemma the only Nash equilibrium is the one where both players defect: there is an incentive for both players to deviate together but not for just one to.
In extensive-formgameswith sequentialmoves, such as the signaling game
there are Nash equilibria that are not very plausible.
To see, this consider the simple multi-stage game in figure 3.
In this game the first player chooses between L and R, so he has two pure strategies (L and R), whereas the second player has a different choice of move depending on what the first player has done.
Thus, there are four full, non-probabilistic strategies for him: choose a
in response to L, and c in response to R (which we’ll write LaRc), LaRd, LbRc
4Alternatively, but less standardly, we can treat nature as another “player” also and have a three-person game with its normal form.
L R
1
b
1, 2
a
2, 1
2
d
3, 1
c
0, 0
2
Fig. 3: Extensive form game
and LbRd. We can use this listing of strategies to give the normal-form representation of the game in figure 4.
The strategy pair in which player 1 plays
LaRc LaRd LbRc LbRd
L 2, 1 2, 1 1, 2 1, 2
R 0, 0 3, 1 0, 0 3, 1
Figure 4: Normal form of game in figure 3
strategy L and player 2 plays LbRc is a Nash equilibrium. Note, however, that
if this equilibrium is played, player 2 will never have a chance to play c in response to R.
However, the strategy pair being a Nash Equilibrium depends on
this disposition on his part, as otherwise player 1 would prefer to play R. It
seems that, in fact, were player 1 to play R, player 2, if he were rational (in the
sense of utility maximizing), would have to play d in response to maximize his
payoff. However, if player 1 knows that, and is himself rational, then he will
know that he can maximize his payoff by playing R. So while (L,LbRc) is a
Nash equilibrium, it is not one which we should expect rational agents to play.
There are a set of refinements of the Nash equilibrium concept designed to
deal with this inadequacy. Indeed, there are different refinements for games
with perfect information such as the one in figure 3 from those for games with
imperfect information such as signaling games. Since we will focus on games
of imperfect information, I will only present refinements that apply to them.5
What we can do is ensure, relative to a pair of strategies, rationality of the
strategy at every point where a player might make a decision. The most standard
equilibrium refinement for games of imperfect information that captures
this sequential rationality is the perfect Bayesian equilibrium (PBE). As one
might guess from the name, a PBE is modeled on the idea that both players
(5Thus I will not discuss subgame perfection, the most common refinement to deal with games like that of figure 3).
are Bayesian-updaters trying to maximize expected utility. When a strategy is
a PBE it means, essentially, that it could be played by two players exhibiting
such Bayesian rationality who each believe the other is playing the equilibrium
strategy. Informally, a pair of strategies is a PBE if at every decision point, each
move at that point maximizes expected utility relative to the credence function
of the player, which itself is constrained by the belief that the other player is
playing the PBE andwhatever other information is available at that point given
the game structure. If the belief that the other player is playing the PBE is not
possible at a decision point, then the only condition put on the players move is
it maximize utility relative to some belief about how the other player is acting.
In the game in figure 3, the move Rc fails to maximize utility on any beliefs and
so is ruled out.
We can give the definition of a PBE for signaling games more explicitly.
There are two conditions for a pair of strategies, (sS, sH), to be a PBE: one
condition on the speaker strategy and one condition on the hearer strategy.
The speaker strategy must simply be such that for each type it maximizes expected
utility on the assumption that the hearer plays sH. The hearer strategy
must meet a more complex condition. We assume the hearer starts (before the
game play) with a graded belief about the speaker’s type based on the probability
distribution associated with N’s move. We then assume that in response
to each message the hearer updates his belief about the speaker’s type in accordance
with Bayes’ rule based on the assumption that the speaker is playing
sS if this is possible. If this is not possible, we simply assume he forms
some consistent belief about how the hearer will play. The condition on the
hearer’s response to a message m is that there is a credence function satisfying
the conditions above on which the hearer’s response maximizes his expected
utility. Essentially this is two separate conditions: for so-called “on the path”
messages—messages that given sS and the game structure, the speaker has a
positive probability of receiving—Bayes’ rule determines hearers credences after
receiving the message and the equilibrium condition is that the response
is rational on for those credences. For so-called “off-the-path” messages—
messages the hearer would assigns probability 0 to given the game structure
and that the speaker is playing sS—the hearer’s response must be optimal for
some belief about the speaker’s type that is consistent with the hearer receiving
an off-the-path message. Refinements of the PBE typically take the form
of a restriction on beliefs in response to off-the-path messages (Cho and Kreps
(1987) give a notable example of such a refinement).
7 Rationality and dominance
The equilibrium notions discussed above are just that: pairs of strategies that
are self-reinforcing. Once you know you are in an equilibrium you don’t have
any incentive to get out of it. However, as many have noted, when we think of
games as one-off events without prior coordination these notions tell us little
about how players should or will play: for in these cases the players cannot
be assumed to have gotten into an equilibrium already. There are two related
problems: 1) Even if there is only one Nash (or perfect Bayesian) equilibrium
available in a game we have not explained why rational agents should play
it. 2) Even if we could argue that rational agents should only play strategies
that form part of some Nash equilibrium, most games provide more than one
equilibria so we still are left with the question of how players should choose
an equilibrium.6
In many cases, there simply is nothing to say about what rational agents
should do. As an example consider the simple coordination game in figure 5.
In this game, the players have a mutual interest in meeting at one of the two
pure Nash equilibria: the upper left (AC) and the lower right squares (BD). It
C D
A 1, 1 0, 0
B 0, 0 1, 1
Fig. 5: Coordination game
should be obvious that there is no privileged rational way to play the game.
To summarize: Nash equilibria (and their refinements) are strategy pairs
that one player does not want to deviate from if he knows the other player
will play his part. The existence of a Nash equilibria does not mean a player
should play it, a point amply proved by the fact that there can bemultiple Nash
equilibria. In certain cases, equilibrium refinements may help eliminate some
equilibria that are in some way defective, but in symmetric situations like figure 5
(6There is a large literature on equilibrium selection (most notably, Harsanyi and Selten, 1988), but the considerations there do not obviously connect up with simple considerations of rationality)
it follows a priori that equilibrium concepts will not dictate a unique way
rational players should play.
There are, also, clear cases where we can say how rational agents should
play the game. To delineate these cases there is a tradition of using nonequilibriumnotions, such as dominance reasoning to capture theways inwhich
simple rationality might dictate play. Dominance relations, by definition, obtain
between strategies in a normal-form representation. A given strategy, s,
for player, i, strictly dominates another strategy, s′, if no matter what s guarantees i a strictly higher payoff than s′. By no matter what, I mean no matter what strategy the other playermakes and no matterwhat the state of nature is (if nature is considered as another player). We can reasonably assume that rational
agentswill not play strictly dominated strategies, based on the simple idea that
rational agents want to maximize expected utility. The notion of strict dominance
alone, in some cases, determine how rational agents will play a game.
For instance, in the prisoner’s dilemma, since defection is a strictly dominant
move for both players, we expect rational agents to defect.7
Another form of dominance reasoning that will be particularly important
for us is weak dominance. A given strategy s for player i weakly dominates another
move s′ if no matter what s guarantees i at least as high a payoff as s′ and
for some state of affairs (i.e. some possible opponent strategy and/or state of
nature) s gives i a strictly higher payoff then s′. Consider, for instance, the
game in figure 6. Here, itmight seemthat a rational player 2 will choose C over
C D
A 1, 1 1, 0
B 1, 1 1, 1
Fig. 6: Simple game
D as that guarantees him at least as high a payoff as D no matter what player 1
does and a higher pay off in one case. Note that weakly dominated strategies,
unlike strongly dominates strategies can form part of a Nash equilibrium (as
well as PBEs). Take for instance the game in figure 6: here there are three pure
Nash equilibria: (A,C), (B,C), (B,D) and the last of these includes theweakly
(7As a note, it is easy to show that if there is a strategy pair determined by strict domination it is the unique Nash equilibrium. Also it’s worth noting that we can view survival of rounds of dominance reasoning as an equilibrium refinement. However, I call these non-equilibrium notions since they don’t depend on considering an equilibrium).
dominated move D.
In both normal-form and extensive-form games we can also consider iterative
applications of dominance reasoning. The informal idea is this: player 1
and player 2 are both rational and both know they know this, and know they
know they know this, and so on. In other words, they have common knowledge
of rationality. In this case, it would seem, we can iteratively eliminate
dominated strategies: i.e. in player 1’s own strategic reasoning he can assume
that player 2 will not play any strictly dominated strategies, and evaluate dominance
on that assumption. A simple example is in figure 7. Note that no move
C D
A 1, 1 2, 2
B 2, 2 1, 3
Fig. 7:
Solvable by iterative dominance
is dominant for player 1 without any assumptions about player 2’s behavior.
However, for player 2, D strictly dominates C. If player 1 knows that player 2
will not play strictly dominated strategies, then, on this assumption, move A
strictly dominates B for player 1.
Iterative dominance reasoning, in general, works as follows: we eliminate
a strategy or move based on the fact that it is dominated by another move or
strategy. Then we update our understanding of the game to reflect this elimination
by considering a new game in which the dominated strategy or move is
not allowed. We continue until there are no more dominated strategies in the
game we have left.
With respect to weak dominance such iterative reasoning can be problematic.
Iterative weak-dominance reasoning is possible, but in many cases the
result of such reasoning depends on the order in which it is done. Take, for,
example the case in figure 8. Here different orders of elimination of weakly
L R
A 2, 3 3, 3
B 1, 0 0, 1
C 0, 1 1, 0
Fig. 8:
Order of weak dominance elimination matters
dominated strategies result in either (A,L) or (A,R) being the strategy pair left
after elimination. What this shows is that merely demonstrating that some
strategy pair is be reached by a chain of iterated dominance reasoning cannot
be an argument in itself that rational players should play the strategies in the
pair. Despite this, there are some well-known results that in a restricted class
of games the order of iterated weak dominance does not affect the outcome
(at least in terms of payout).8 In particular, in games with identical payoffs
the order of iterative elimination by weak dominance does not affect strategic
reasoning. This will be useful for us, as all the games we will consider have
identical payoffs for both players. Another way of ensuring that iteratedweakdominance reasoning results in a unique result is to ensure that at each stage of reasoning one eliminates all strategies/moves that are weakly dominated.
This special form of iterative weak dominance is called iterative admissibility.
Given that the order of eliminating strategiesmatters, it cannot be a requirement
on rationality that players do not play iteratively dominated strategies
(for this could eliminate all strategies, as in 8). However many authors, most
influentially Kohlberg and Mertons (1986), posit that rational players should
not play a strategy unless it survives some complete series of iterative elimination
of weakly-dominant strategies.9
I should note that iterative-weak dominance reasoning is a much more useful
a tool than strict dominance reasoning for extensive-form games. Consider,
for instance, the extensive form game in figure 3. Here there is not sufficient
strict domination of any strategy over another to determine what we think of
as rational play. For instance, the strategy for second player of playing b if
the first player plays L and c if the first player plays R is not strictly dominated.
For this strategy will do just as well as playing b and d as long as the
first player plays L. Nonetheless LbRd weakly dominates LbRc since there
it always does as well, but sometimes does better. Thus, in this game, as in
many extensive-formgames, weak-dominance reasoning is necessary to single
out rational lines of play. In extensive-form games, we can speak of a move
rather than a strategy being weakly dominated, by which we will mean that
every strategy that includes that move is weakly dominated. For instance, in
the game in figure 3 player 2’s moves a and c are weakly dominated, since any
strategy that includes those moves will be weakly dominated.
(8The earliest such result (which is good enough for our purposes) is Rochet (1980) (see also Marx and Swinkles, 1997)).
(9By ‘complete series’ I mean a series of eliminations such that at end no further elimination by weak dominance is possible).
What I have done here is present two related ways of reasoning about
games in absence of an equilibrium: iterative strict and weak dominance. Both
of theseways of reasoning about games apply to extensive-formgames through
their normal form. On the surface, both seem plausible as ways rational agents
should think about games: after all, why should one ever play a dominated
strategy?
However, spelling out a formal concept of rationality and common knowledge
of rationality that justifies these forms of reasoning (in both extensive
and normal-form games) is a non-trivial task. Since the eighties many gametheorists
have tried to spell out constraints on rational strategic reasoning that
capture inter alia these forms of dominance reasoning.10 I will not review this
extensive and complex literature on the epistemic foundations of game theory.11
However, it is worth noting that conceptions of common knowledge (or belief)
of rationality that require players to play only those strategies reached by iterative elimination of strictly dominated strategies have long been known, while
conceptions of common knowledge of rationality the force iterative elimination
of weakly dominated strategies have, more recently, been explored (Brandenburger,
2007; Brandenburger et al., 2008). For our purposes it will be enough to
say that iterative dominance reasoning (in both forms) is prima-facie a natural
form of rational inference in games where common knowledge of rationality
obtains.
8 Game-theoretic account of implicatures
The goal of Rothschild's study is to provide an analysis of the communicative situation in which scalar implicatures occur that explains the derivation of the implicatures as a sort of rational inference.
The hope would be that, at least for idealized cases, we could view the speaker’s following of the Gricean maxims as well as the hearer’s derivation of the scalar implicatures as simply rationally compelling behavior, given the set-up of the game. If we succeed, we show why in idealized but still useful models rational agents should behave as the Gricean maxims dictate.
If we can show this then we need not view the
(10The starting point is the notion of rationalizable strategies independently proposed by Pearce (1984) and Bernheim (1984)).
(11See Battigalli and Bonanno (1999) and Brandenburger (2007) for introduction to program)
maxims as mere useful empirical generalizations but rather as generalizations
about what how rational agents should act—at least for idealized cases.
Any attempt at carrying out this project will include two elements (not always
entirely separate):
(a) a model of the communicative situation (i.e. the game
itself) and
(b) an analysis of why rational agents should choose the Gricean reasoning.
The plausibility of such a model will depend on both these components.
The model itself is meant to describe in a formally tractable way, the communicative
problem involved in using scalar terms.
To do this, we need to abstract away from much of the complexity of real-world communication.
Thus, themodels we usewill be idealized and simplistic.
This itself is not a problem if the idealizations and simplifications made do not change the basic structure of the communicative problem we are focusing on.
Modeling is a delicate art because the space of possibilities is so wide.
It’s not interesting to find some model that vindicates Gricean reasoning, one needs to show that that the model is, at least, a simple, plausible representation of the real-world situation.
In this section I will discuss some basic components of the scalar implicature
situation that a game-theoretic account should include and give some indications about how signaling games can capture these.
8.1 Asymmetric information
The classic Gricean situations include asymmetric information:
the speaker knows something that the hearer does not.
Signaling games are a natural way of modeling this asymmetry, of course, which is why they are often used to model language in a game-theoretic setting.
8.2 Cooperativeness
Gricean reasoning works by assuming that the speaker and hearerwant to help
each other, and they both know this.
The natural way of modeling this in a game-theoretic setting is to say that the payoffs for both players are aligned.
Games with this property are called games of coordination (Schelling, 1960;
Lewis, 1969).
8.3 Relevance
It is well known that scalar implicatures are only possible when the differences
between the different states of affairs is relevant to the hearer.
The most natural way to model such relevance is to have the payoffs dependent on the hearer action in such a way that the hearer benefits from knowing the private information of the speaker.
If we are interested in pure communication then the hearer “action” is only a sort of nominal aspect of the game intended to capture the fact that it is useful for the hearer (and hence the speaker) for him to know what the speaker knows.
8.4 Background beliefs
It is standard in game theory to assume that speaker and hearer have common
knowledge of the structure of the game.
This will be a useful assumption for us to incorporate as well:
we need to assume that the payoffs are identical for both players and that they know this (and that they know they know this. . . ).
8.5 Message costs
We assume that there are not high costs for uttering one sentence rather than
another: after all, different sentences require only slightly different muscle
movements.
In game theory, a signaling game with no message costs is called
a cheap talk game.12
8.6 Meaning
The most vexed issue about modeling implicatures using signaling games is
how to build in the Gricean account of the literal meaning of sentences.
On the Gricean view, the sentence I ate some of the cake is literally compatible with
eating all of the cake.
It is only once the implicature is drawn that it comes to have the stronger meaning which is incompatible with eating all of the cake.
In order to explain this inference, we need to provide a model in which we can
say that the sentence has this weaker literal meaning.
This is a considerable theoretical challenge.
(12The classic paper on cheap talk games is Crawford and Sobel (1982), see also Farrell and Rabin (1996)).
Standardly in the economics literature on signaling games it is assumed
that all messages (in a cheap talk game) are “inherently” meaningless signals
which only get meaning in the context of an equilibrium.
This view is largely inherited from Lewis’s seminal treatment of signaling games in Convention.
What I will call a Lewis signaling game is very similar to the ones we want
to discuss.
Lewis also assume costlessmessages and identity of payoff between
speaker and hearer.
However, Lewis was focused on the question of how messages get meanings to start with, not the question of how, given the message
meaning, extra inferences can be inferred.
Lewis’s suggestion is that meaning derives fromthe use ofmessages in certain repeated signaling gameswhere the same equilibrium was repeatedly played.
Take for instance a simple Lewis signaling game like that in figure 9.
There are the two separating equilibria where
p N 1 − p
b
a
S1
b
a
S2
H
R
0, 0
L
1, 1
H
R
0, 0
L
1, 1
R
1, 1
L
0, 0
R
1, 1
L
0, 0
Fig. 9:
A Lewis game
the speaker plays either 1a2b or 1b2a and the hearer responds either aLbR or
aRbL, respectively. These equilibria result in real communication, in Lewis’s
picture.
Other equilibria such as the babbling equilibria where speakers choose
signals randomly and pooling equilibria where speakers always use the same
message do not result in any “communication” in the usual sense.
Within each
separating equilibria we can speak of the “meaning” of a or b, but the notion
of meaning of the signals is only defined relative to the equilibria that is being
played.13 The meaning of a signal in an equilibrium is just defined by what
types of speakers uses that signal (or what actions it induces, if the meaning is
viewed “imperatively”).
This is the notion of meaning most often implicitly or
(13Lewis was essentially trying to give a game-theoretic reconstruction of Carnap’s notion of truth-in-model explicitly assumed in the game theory literature).
That this conception of meaning will not be adequate in a model of scalar
implicatures should be obvious.
The entire point of the Gricean reasoning is
that with implicatures there is a divergence between how a sentence is used
and what its literal meaning is. So defining the meaning of a term by how it
is actually used in the game model will preclude the possibility of capturing
scalar implicatures within the model.
The only way we can use Lewis’s equilibrium conception of meaning is by
considering an equilibrium as a sort of starting point and thinking of implicatures
as rationally motivated deviations from the equilibrium.
In a sense, this
is exactly the approach taken in much of the game theory and pragmatics literature
such as Benz (2006); Benz and van Rooij (2007); Franke (2009), theories
which I will discuss later.
I think there is some promise to this approach, but
it is conceptually difficult.
If people do not play some according to some equilibrium
in the end, and if it can be derive by some sort of reasoning that they
will not, then it is irrational to take as one’s starting point in rational deliberation the proposition that they will play this way.
Franke (2009) embraces the irrationality and claims his model is a model of bounded rationality.
This is an interesting tact, but it is a bit strange to think that we only can model literal meaning in a model of bounded rationality as It would be desirable to model
meaning in a way that does not constitutively depend on bounded rationality
assumptions.
One obvious way of modeling meaning is by the constraint that speakers
can only use messages when they are literally true.
To effect this we can use non-standard signaling games where speakers are restricted to use only those messages that are literally true given their type.
For examples supposing in the
Lewis game in figure 9 that the signal a literally means the speaker is in state 1,
whereas the signal b literally means the speaker is in state 1 or in state 2.
If it is common knowledge that speakers only usemessages that are literally true then
we canmodel this situation with the game in figure 10.
The critical point is that the meaning assumptions are not strong enough to fully determine speaker actions.
Thus, how the speaker acts within the confines of these assumptions
may allow the hearer to make strategic inferences.
Restricting speakers to send only truemessages is certainly themost natural
and most common way of treating Gricean literal meaning in a game-theoretic
p N 1 − p
b
a
S1
b
S2
R
0, 0
L
1, 1
H
R
0, 0
L
1, 1
R
1, 1
L
0, 0
Fig. 10:
Lewis game with built-in meanings
setting.
However, there are some problems with using this notion.
One is that it seems to exclude from the start the idea that speech may be either nonliteral (i.e. metaphorical in some way) or intentionally deceitful.
This does not seem like a serious criticism.
The game is a model of how a speaker and hearer conceptualize their situation.
This does not mean that every assumption in the model needs to be considered an unrevisable assumption of the speaker or hearer.
The speaker or hearer do not need to always conceptualize speech-act
situations as constraining speech to literal meaning, but for the purposes of
scalar implicatures (where speakers are assumed to speak truly, in the standard
Gricean model) this seems like a natural assumption.
Criticizing this model for not allowing for non-literal or strategically deceitful utterances is simply criticizing the model for not doing something it is not intended to do.
Franke (2009, pp. 35–37) motivates his choice not to stipulate truthful utterances
in an account of scalar implicatures by two further arguments. First,
he argues, for reasons we will see below, that there is no unique equilibrium
requiring Gricean play in these models. So these models simply are not structured
correctly to account for scalar implicature.
Most of this paper is concerned
with trying to demonstrate that we can make a good argument for
Gricean behavior in this model.
Besides the methodological problem this argument
has of assuming that we should be able to derive Gricean play in our
model, I also think this point is substantively wrong. The main argument of
this paper is that such models do allow derivation of Gricean play by using
weak-dominance reasoning, rather than standard equilibrium refinements.
Second, Franke suggests that there is a conceptual problem with this model
of literal meaning. He writes as follows:
I can very well say whatever I like, whenever I like to whomever I
like. I may have to face social or even legal consequences fromtime
to time, but it is not as if the semantics of my language restricts the
muscles of my jaw and vocal track, regulating what I possibly can
and what I cannot utter.
I think this argument is not compelling: Game models surely do not need to
providemoves corresponding to all physically possible actions. We usemodels
to capture players assumptions about how certain situations are structured—
what reasonable possibilities players consider. Even if the assumptions turn
out to be false in some instances it does not mean that speakers and hearers do
not make them.
This is not to say I think that forcing speakers to say only true messages is
the only way to model natural language meanings in a game-theoretic setting.
The literature on credibility in game theory provides some interesting other possibilities
(Crawford and Sobel, 1982; Farrell, 1993; Rabin, 1990; Stalnaker, 2005).
I hope to address these other approaches and their relations to scalar implicatures
in another paper.
8.7 Rationality
Besides the model itself, a game-theoretic grounding for scalar implicatures
will include an argument that rational players in the situation modeled should
play in the Gricean way. The plausibility of such an argument depends on
the use of appropriate tools. The game theory literature contains a plethora of
equilibrium refinements andmore ad hoc refinements can be invented. Merely
showing that there is some refinementwhich justifies Gricean behavior does not
show much about the rationality of Gricean reasoning. We do not necessarily
need to use standard game-theoretic tools to explain the rationality of Gricean
implicatures, but whatever tools we use, we need to convincingly argue for
their appropriateness.
8.8 Robustness
Whatever model of the game and of player rationality we use its plausibility
and explanatory value depends on it lacking arbitrary restrictions. A robust
model should not depend, in order to get results, on relatively arbitrary assumptions.
For instance, a signaling game model that depended on a very
specific payoff structure to yield the Gricean result would not provide a robust
account of scalar implicatures. Similarly, very strong constraints on the
reasoning patterns of players will not give a plausible grounding of Gricean
reasoning.
Many of the game-theoretic derivations of implicatures that one can find
in the current literature are not robust against small changes. For instance,
as I will argue, some versions of the Iterated Best Response model and the
Optimal Answer model need strong assumptions about players beliefs about
the probability of various states in the game. We do not want our solution
concept to depend on such assumptions.
8.9 Relation to non-linguistic reasoning
Game theory was developed to deal with strategic interaction generally rather
than language use. Indeed, systematic treatment of linguistic communication
in mainstream game theory is a relative recent phenomenon. One hope in using
game theory to model pragmatic inferences is to relate the underling reasoning
driving these inferences to reasoning in non-linguistic cases. (This idea
of relating pragmatics to non-linguistic behavior was also one of Grice’s major
goals.)
The explanatory value of a model of scalar implicatures, thus, depends to
some degree on whether it is sufficiently general to also capture non-linguistic
behavior with similar structures to that of pragmatic inferences. If we can do
this, we can provide an argument that pragmatic inferences are grounded in
general practical rationality rather than some specific linguistic mechanism.
I will argue, in section 17 that the particular reasoning I use here extends to
parallel cases from the economics literature.
9 Simple Gricean game
In this section I present and discuss a very simple model of the Some/All Case
in a game-theoretic framework that is meant to capture its essential features
in accord with the principles about modeling given above. I then show that
the standard equilibrium concept for signaling games, perfect Bayesian equilibrium
(PBE), fails to single out the Gricean strategy in this game.
In this basic model, nature determines whether we are in a some situation
or an all situation with a certain probability distribution. Intuitively we think
of the some situation as the one where the speaker ate just some of the cake and
knows it and the all situation as the one where he ate all of the cake and knows
it.
If the speaker is in the some situation (he is of type Ss) then he can and must
send themessage ms. If he is in the all situation (Sa) then he can send either the
message ms or ma. It follows that if S sends message ma then H will knows
the speaker-type directly by knowledge of the game structure, but if S sends
message ms then H does not know the speaker type directly, but rather must
infer it.
If the hearermakes this inference then this corresponds to deriving the
scalar implicature. Figure 11 represents this partial specification of the game.
p N 1 − p
ms
Ss
ma
ms
Sa
H
H
Fig. 11:
A simple Griceian game
We have not yet specifiedwhat H does once S has sent a message, nor what
the payoffs are. How we do this will affect our assessment of the rationality of
different strategies.
One natural way of modeling the fact that the S’s type
matters to H is to suppose that the H makes some sort of choice after hearing
S’s message, and thatH wants tomake the choice oneway if S is of type Ss and
a different way if S is of type Sa. We assume a cooperative situation in which
the speaker and hearer’s interests are aligned so that the payoffs are identical.
I represent this situation with the complete extensive game in figure 12.
p N 1 − p
ms
Ss
ma
ms
Sa
H
H
R
0, 0
L
1, 1
R
1, 1
L
0, 0
R
1, 1
L
0, 0
Fig. 12:
A simple Griceian game with recipient's intended response
I will call strategies Gricean when they intuitively accord with Grice’s maxims
and Grice’s posited implicatures.
The Gricean strategy profile for this game is clear.
For S it is to send mS when in state Ss and ma when in state Sa, for
H it is the best response to this, i.e., to play L in response to ms and to play
R in response to ma.
This combination of strategies will ensure payoffs of 1 in each play, the best that can be hoped for, as indicated by the normal-form description of the game in figure 13.
Since both players are guaranteed their
msLmaR msLmaL msRmaL msRmaR
SsmsSama 1 p 0 1 − p
SsmsSams p p 0 1 − p
Fig. 13:
Normal form of a Griceian game
highest possible payoffs if they play this pair of strategies, the pair is a PBE.
Unfortunately, the Gricean strategy pair is not the only PBE.
Suppose p .5.
Consider the pooling strategy for both players: S sends ms no matter what and
H responds R no matter. It is easy to see that this pair of strategies is also
a PBE: neither type of S has any incentive to change his behavior (type Sa
gets his maximum payoff, whereas Ss has no other options) and H, getting
no usable information from the S, is strictly maximizing his expected payoff
by playing R. For similar reasons, the pooling equilibrium also satisfies many
standard equilibrium refinements such as the Intuitive Criterion of Cho and
Kreps (1987).14
(14Depending on the probabilities their can also be babbling PBEs in this game: one’s where I think it can be safely said that at an intuitive level the Gricean strategy seems compelling. It seems like even without prior discussion either player can safely assume the other playerwill play according to it. The question is howwe
cash out the intuitive rationality of the strategy by means of plausible gametheoretic tools. All we have seen so far is that the standard equilibrium concepts cannot do this.15
10 Dominance arguments
We saw above how the Simple Gricean Model captures the basic conversational
situation in which the speaker chooses between ‘some’ and ‘all’.
However, the standard equilibrium notions do not single out the Gricean strategy.
Here I’ll show that iterative weak-dominance reasoning does single it out.
If we look at the normal-form representation in figure 13 we can immediately
read off the weak dominance relations.
Note first that any hearer strategy which includes the response of L to ma is weakly dominated.
If we eliminate
these two hearer strategies we get the normal form representation in
figure 14. Here each player has two choices of strategies: the Gricean strategy
and the pooling strategy. In this reduced game the speaker’s pooling
msLmaR msRmaR
SsmsSama 1 1 − p
SsmsSams p 1 − p
Figure 14:
Normal form after first removal of dominated strategies
strategy (SsmsSams) is weakly dominated by the speaker’s Gricean strategy
(SsmsSama). So we can eliminate that pooling strategy to get the game in figure
15. A final application of strict dominance eliminates the hearer’s pooling
msLmaR msRmaR
SsmsSama 1 1 − p
Figure 15:
Normal form after second removal of dominated strategies
speaker S chooses messages randomly (when he has a choice) and H ignores the message.
(15This conclusion might seemin tensionwith van Rooij’s claims thatGricean behavior in various games arises because it is the only Nash equilibrium (van Rooij, 2009; de Jager and van Rooij, 2007)). Van Rooij’s work uses substantive assumption about the structure of the games beyond the ones here. For this reason I do not find his results robust enough to support the conclusion that scalar implicatures are generally derived because Gricean behavior is the unique Nash equilibrium.
strategy, and we are left with the Gricean speaker strategy.16)
It is not very intuitive to think of Gricean games by way of their normal
form.
We normally think of players choosing moves as they go rather than opting
for total strategies.
For this reason it will be helpful to redo the reasoning
using the extensive-form representation.
We can think of each of our applications
of weak dominance as an elimination of one possible move from the
extensive form.
We start with the representation in figure 11. Since the move L
in response to message ma will always result in a lower payoff it can be safely
assumed that H will not make that move. This gives us the reduced game tree
in figure 16. In this tree, the speaker in state Sa will guarantee himself the highp
N 1 − p
ms
Ss
ma
ms
Sa
H
H
R
0, 0
L
1, 1
R
1, 1
L
0, 0
R
1, 1
Fig. 16:
A simple Griceian game
after first removal of dominated strategies
est payoff by sending ma, so this move is weakly dominant. If we adjust the
tree to reflect this we now get the game in figure 17. In this tree a rationalH can
only play L in response to ms, so we can conclude that the full Gricean strategic
behavior is what we should expect. Intuitively, these successive removal
of weakly dominated moves are justified on the assumption that it is common
knowledge that both players will not play weakly dominated moves.17
To summarize:
for the Simple Gricean game, iterative elimination ofweakly
dominated straggles leaves only the Gricean strategy pair. So, in this model it
seems that we do not need to stipulate Gricean maxim over and above player
(16Note that the dominance argument does not depend on choosing the normal-form representation of signaling games in which nature is factored in by expectations rather than treating it as a separate player. The same reasoning works if we treat the Simple Gricean game as a three person game with nature being the third player, however in this version the last step is another instance of weak dominance not strict dominance.)
As I noted earlier, actually modeling this kind of common knowledge of rationality in a explicit way is difficult.
Brandenburger et al. (2008) discuss how to overcome these conceptual problems.
p N 1 − p
ms
Ss
ma
Sa
H
R
0, 0
L
1, 1
R
1, 1
Fig. 17:
A simple Griceian game
after second removal of dominated strategies
rationality. In the next two sections, I discuss and criticize alternative ways
of capturing scalar implicatures from the recent game theory and pragmatics
literature.
11 Pareto-Nash equilibrium and payoff dominance
In his well-known work on game theory and pragmatics Prashant Parikh (1991,
2001) argues that we should capture Gricean reasoning using the Pareto-Nash-
Equilibriumsolution concept.
That iswe should assume that rational agents (in
these sorts of games)will only play strategies that are on Nash equilibria which
provide at least as high payoffs as any other Nash equilibria.18
In the Simple Gricean game there are only two perfect Bayesian equilibria the Gricean one and the Pooling one.
The Gricean one gives higher expected payoffs then the
pooling one so it is the only Pareto-Nash Equilibrium.
The critical question here is whether choosing Pareto-Nash Equilibria are
really a legitimate constraint on equilibrium choice. Parikh notes that without
such constraints we cannot explain obviously compelling behavior in simple
games with coordinative pay-off structure. For example consider the coordination
game in figure 18. It is arguable that two rational agents playing this
game would choose the A and C equilibrium.19 Standard equilibrium refine-
(18In other words, in the terminology of Harsanyi and Selten (1988) rational players never choose payoff-dominated equilibria).
(19Of course, I am assuming that the ratios between payoffs rather than just the strict ordering matter, not a standard assumption. Regardless I am just making the intuitive point that where there are great differences in payoffs the Pareto-Nash equilibrium seems quite compelling).
C D
A 1000, 1000 0, 0
B 0, 0 1, 1
Fig. 18:
Lopsided coordination game
ments or conceptions of rationality cannot capture this.
It is not clear, however, that small payoff-dominance has such a stark effect.
For instance, if the difference is just a small one, it is not clear thatwe should argue that rationality compels players to choose that one: the goal of each player
is just to play what the other plays. In most cases, payoff dominance makes
one solution salient in the sense of Schelling (1960) and Lewis (1969), and since
it also pays more it is natural way to play (and to assume that the other play
will too).
The Gricean strategy is also salient in being the unique Pareto-Nash
equilibriumin the Simple Gricean game (in addition to whatever other salience
it may have) its salience may well supports its choice as an equilibrium. But
the conclusion that the Gricean strategy is played because it is salient, would
not seem to be a very satisfying account of its basis in rationality, and in game
theory it is rarely assumed that rational players only play strategies that are
part of payoff dominant equilibria.
12 Reasoning-based accounts
For comparative purposes, in this section, I will discuss the treatment of the
simple Gricean game by the reasoning-based strategies. These strategies,which
include the Iterated Best Response models of J ¨ager (2007) and Franke (2009)
and the Optimal Answermodel of Benz (Benz, 2006; Benz and van Rooij, 2007),
specify a certain form of reasoning and argue that this form of reasoning leads
to Gricean behavior.
They differ from the treatment above in that less needs
to be built into the game structure itself and more into the assumptions about
how the players will play the game.
12.1 Iterated Best Response models
The idea behind Iterated Best Response models is that strategic reasoning involves
a hierarchy of increasingly sophisticated thinking terminating at the
point where further strategic sophistication is otiose, i.e. a fixed point, if such
a point can be reached. In the game model itself we make no constraints based
on the literalmeaning, as in figure 19. This game is, thus, a simple coordination
p N 1 − p
ma
ms
Ss
ma
ms
Sa
H
R
0, 0
L
1, 1
H
R
0, 0
L
1, 1
R
1, 1
L
0, 0
R
1, 1
L
0, 0
Fig. 19:
Model with no built-in meaning
game. Literal meanings instead of being part of the game structure, go rather
into the reasoning of the players.20
I will analyze a variation of Franke’s model.21 Here is a simplified version
of the model: The most basic first assumption is that there is a sort of default or
focal behavior for S that consist in simply sending random signals that accord
with the literal meaning. Call an S who behaves this way an S0. Let H1 be an H
who plays the best strategy he can on the assumption that he is playing against an
S0. Let an S2 be an S who plays the best strategy he can assuming he is playing
against an H1. And so on.22 This gives us a hierarchy of types of plays for both
S and H.
These definition are not quite as precise as Franke’s but they are a good
starting point. Let us see what they do for us in the game in figure 19. The
behaviors for the hierarchy of types, when p = .5 is in figure 20. For any n > 1,
if n is odd then Hn = H1 and n is even then SN = S2. So, the series reaches
a fixed point immediately. According to the IBR theory (in this simple form)
(20This feature is for the cases we consider here inessential: We could start with the more articulated game structure forcing truthfulness and start the chain of reasoning within that structure.)
(21The reader should look at Franke (2009, ch. 2) for more details. Franke’s model and J ¨ager’s model are quite similar, though J ¨ager intended hismodel as an evolutionary one, whereas Franke’s is meant to be a model of bounded rationality. An evolutionary model does not give a rational grounding for Gricean implicatures in the usual sense, and thus is outside the purview of my discussion here).
(22I am keeping things quicker by not discussing the parallel sequence starting with H0 as an H who has a literal interpretation of the message from S).
Type Strategy
S0 some ! s all : .5 ! a.5 ! s
H1 s ! R a ! L
S2 some ! s all ! a
H3 s ! R a ! L
. . .
Fig. 20:
Hierarchy of utterer and addressee types, p = .5
Type Strategy
S0 some ! s all : .5 ! a.5 ! s
H1 s ! R a ! R
S2 any any
. . .
Fig. 21:
Hierarchy of utterer and addressee types, p = .1
playerswill play those fixed points.
This is perfectly Gricean play, so themodel predicts the basic scalar implicatures.
As Franke (2009) makes clear this is a theory of bounded rationality.
We are considering a hierarchy of increasingly sophisticated players playing against
each other.
However, none of the reasoning used in determining the hierarchy itself is consistent with genuine rationality in the usual sense.
It is not rational—in the usual sense—to suppose that the person you are playing against is less sophisticated than you.23
The predictions of thismodel differ depending on the underlying probabilities
in the game structure.
Consider, for instance, the game in figure 19 inwhich
p = .1.
The Iterated Best Response sequence is given in figure 21. The problem
is that when H1 is responding to S0, the ms message does not sufficiently alter
his beliefs to make H1 want to act any differently from how he would in the
absence of any information. This means that there is no unique best response
for S2 to H1, since H1 ignores the message given. Thus, the basic characterization
of the model I gave above does not extend to cover this case since there
(23This is not to say that it is not rational to play the strategies that are the result of such reasoning: it is just that the reasoning itself is not easily characterized as what we should expect from two players with common knowledge of rationality)
is no unique best response. We can, though, easily extend the model to handle
cases in which there is not a unique best response. For instance we can
assume that the player chooses randomly among eligible responses (as Franke
does) or we can allow sets of responses into the model (as Jager does). Either
way, in the case considered here, we will not get a sequence that converges to
Gricean strategy. Thus, the simple IBR model fails to deal with a large class of
Some/All Cases.
I should note that Franke (2009, §2.2.4, 3.1) openly admits to some of the
shortcomings of the IBR approach for some initial probabilities. Franke argues
that the probabilities in hismodels should not be thought of as real-world probabilities but rather as “condensed and simplified representations of generally
accessible meaning associations.” In other words, we should think of some aspects
of the representation of the scalar implicature situation not as relating to a
real-world situation but rather as something that comes as part of the meaning
of the words ‘some’ and ‘all’. Moves like this, however, make the model seem
like less of a rational reconstruction and more of a substantive psychological
account of meaning.24
12.2 Optimal Answer model
A related model is the Optimal Answer model of Benz (2006) and Benz and van
Rooij (2007). This model is, in essence, a version of the IBR model, though it is
framed rather differently. Essentially S’s action is predicated on the assumption
that H simply update his beliefs by means of the literal meaning of the
message. Then, H’s actual response is based on the belief that S will act in
the way just stated (and a faith in S’s expertise). Without going into detail it
is worth noting that a model with this structure makes very strong bounded
rationality assumptions. For the model to work S needs to solve a decision
problem based on the idea that H will act in a na¨ıve way in which he does not
actually act (i.e. S thinks that H will just update his beliefs according to the
literal message).
There are predictive problems which are similar in character to those facing
the IBR model. In essence, predictive success requires H to have the right
(24Franke also suggests an entirely different way of dealing with the problem: by assuming that speakers and hearers always assign low probabilities to any available strategies. This will work, but note that it is analogous to the standard underpinnings of iterative weak-dominance in the epistemic game theory literature (Brandenburger et al., 2008)).
background beliefs about the probability of the different states. As in the IBR
model, if the H thinks that the some state is very unlikely and that S is merely
speaking literally then H’s action will not be affected by S’s message. So, in
this case, S with his assumption of a na¨ıve H will not have any incentive to use
the Gricean strategy.
I should note that these worries (for both IBR and Optimal Answermodels)
can be assuaged by structuring the payoffs in a different way. Indeed, many of
the derivations in the literaturewithin Optimal Answermodel depend on payoffs
being determined by degrees of H’s belief in the true state of the S, rather
than a discrete choice on H’s part. This treatment of payoffs may eliminate the
models failings in the face of skewed probabilities, but we must then take it on
as yet another substantive assumption. Iterative weak dominance reasoning
will work with either way of modeling payoffs.
12.3 Critique of reasoning-based models
Reasoning-based accounts, both the IBR and the Optimal Answer models, do
not provide a credible reconstruction of scalar implicatures as a species of rational
inference. Letme summarize the two (related) points that leadme to this
conclusion:
Non-Standard Framework Both these accounts use non-standard frameworks:
they propose particular, intricate chains of reasoning to account for scalar
implicatures. It is important to note that these models are not given as a
heuristic to get a result also achievable by assumptions of real rationality.
In that sense the title of bounded rationality ismisleading. In fact, on these
models literalmeanings are defined in terms of theses chains of reasoning,
so it is not even in principle possible to reconstruct a non-bounded, truly
rational model.
Since it is not a priori surprising that some chain of reasoning can be
given to derive Gricean reasoning, it is hard to know how to evaluate
the (limited) positive results of these models. Franke attempts to defend
his model by pointing to empirical evidence in favor of level-k strategic
reasoning, which he employs. Even limited evidence for this style
of bounded-rationality model, does not give evidence for the particular
implementation.
Parameter Setting
I noted that, unlike weak dominance reasoning, both the
IBR model and the Optimal Answer model only worked over a limited
range of assumptions about the prior probabilities. Both also rely on a
particular specification of what naive behavior is to get the reasoning going
(i.e. the starting point of the level-k reasoning embodied in the hierarchy
of speaker and hearer types). Not only do we need to assume a
of specific pattern of reasoning, but we also need to assume a particular
initial set-up not common to all (real-world) cases of scalar implicature
derivation. This is not to say this assumption is unpalatable: after all,
perhaps we do as a matter of fact assume a kind of even distribution of
probabilities of underlying states when calculating scalar implicatures.
Even if the assumption has some plausibility it is a substantive assumption
about how these situations are structured. The more such assumptions
one makes, the less robust the model is.
For these reasons it seems to me that we must view the reasoning-based
models not as credible explanations of why implicatures are rational inferences
but rather as strong empirical hypotheses (statedwithin a game-theoretic
framework) about how implicatures are calculated. To make this a credible hypothesis
these models need to provide more empirical coverage than standard
Gricean theories can.25 This is an interesting line of research but I think it is important
to distinguish what we get from them from the project here of trying
to capture scalar-implicature derivation as rational inference.
13 Non-expertise
Effectively the Simple Gricean game assumes speaker expertise.
The two types
of speakers both have all of the relevant information about the state of the
world: the speaker either knows that some of the cake was eaten or that all of
the cake was eaten. However, we can imagine that speakers might only have
partial information.
(25Franke (2009), in particular, tries to cover a range of cases beyond simple scalar implicatures including manner implicatures and free-choice implicatures with his model. Weak-dominance reasoning will not derive free-choice implicatures, but this might not be a bad result as there is considerable empirical evidence that they do not pattern with normal scalar implicatures (Chemla, 2009).
Type Knowledge
Sa {wa}
Ss {ws¬a}
Sas {wa,ws¬a}
Fig. 22:
Speaker types
Let us model the two complete situations as worlds: wa is the world where
all was eaten, ws¬a is the world where just some was eaten. Now consider the
simple Some/All game where the speaker can have any level of knowledge
compatible with truthfully saying ‘some’ or ‘all’. Figure 22 lists the three types
of speakers in this situation, classified according to their states of knowledge.
Ignoring payoffs, but assuming that the speaker can only utter sentences he
believes to be true, we can draw this as an extensive-form game in figure 23.
Assuming that the payoffs are aligned and H benefits from having confidence
r
p
q
N
ms
ma
Sa
ms
Ss
ms
Ssa
Fig. 23:
A simple Griceian game with non-expert
in the true state of theworld, any strategy for Sa that does not have himplaying
ma with probability one will be weakly dominated by the variant in which Sa
sendsma with probability 1. IfH knows that S will not playweakly dominated
strategies, then H can infer that if he receives signal ms then S is not of type ta.
Thus, whatweak-dominance reasoning tells us that the hearer can infer that
if the message ‘some’ is received then the speaker does not know the world is
wa (though he might not have ruled it out either). This is the standard ‘epistemic’ inference assumed in the literature on pragmatics.26 So, merely adding
uncertainty to the simple model does not cause any problem for the kind of
reasoning we were using before.
14
Three-point scales
So far, for simplicity’s sake,we have concentrated on a two-point scale, ‘some’/‘all’.
However, classic Gricean reasoning based on the maxim of Quantity extends
to scales of arbitrary size.
Consider, for instance, a
three-point scale,
such as
{‘some’, ‘most’, ‘all’},
where ‘all’ entails ‘most’ which entails, in turn, ‘some’—
e.g. ‘I ate all the cake’ ! ‘I ate most of the cake’ ! ‘I ate some of the cake’. If
a speaker chooses between these three expression guided by the the Maxim of
Quantity then he will only say ‘some’ when he does not know ‘most’ or ‘all’,
and hewill only say ‘most’ when does not know ‘all’. Thus, the hearer can infer
from an utterance of ‘most’ that the speaker does not know ‘all’, and from an
utterance of ‘some’ that the speaker does not know ‘most’ or ‘all’. (Assuming,
as always, that these differences are known to be relevant.)
Let us consider the case in which we are dealing with an expert speaker,
and thus limit ourselves to three speaker types. The extensive-form representation
(without H’s responses) is given in figure 24. The full set of possible
pure strategies for S is in figure 25. In this game the hearer has three possible
responses A, B, C and the payoffs (which are dependent just on S’s type and
H’s response) are coordinative: both players are best off if the hearer can determine
the speaker’s type and act appropriately. Figure 26 gives all the possible
pure hearer strategies in this game (with the first round of weakly-dominated
strategies removed).
In this game there will always be more than one PBE. While the Gricean
strategies will always be a PBE, either the pair (PP2,NG3) or (PP2, NG4) will
also be a PBE (which one depends on the initial probabilities of the different
speaker types).
Despite there being at least two possible PBEs, iterative weak-dominance
reasoning will again pick out the Gricean strategy pair. First, consider the
speaker of type Sa. For him the use of any message but ma is weakly dom-
(26The point that epistemic inferences are the only one’s licensed without knowledge assumption was first emphasized in a formal framework in Soames’ (1982) critique of Gazdar’s (1979) treatment of implicatures).
r
p
q
N
ms
ma
Sa
C
0
A
1
B
H 0
C
0
A
1
B
0
mm
mm
C
0
A
0
B
1
C
0
A
1
B
0
ms
Sm
C
0
A
0
B
1
ms
Ss
C
1
A
0
B
0
H
H
Fig. 24:
Three-point scale
Type Sa Sm Ss
Grice ma mm ms
PP1 mm mm ms
PP2 mm ms ms
PP3 ms mm ms
Pooling ms ms ms
Fig. 25:
Speaker pure strategies
ma mm ms
Grice A B C
NG1 A B B
NG2 A B A
NG3 A A C
NG4 A A B
NG5 A A A
Fig. 26:
Hearer pure strategies
inated: ma will always get him the highest payoff no matter what strategy
the hearer chooses, whereas using and mm and ms will not always do so. If
the hearer knows that the speaker will always use ma when he is of type Sa,
then he can infer that if he receives another message the speaker cannot be of
type Sa. If the speaker knows this, in turn, then for the speaker of type Sm
sending message mm weakly dominates sending message ms, its only alternative.
Thus, the hearer will send that message, which means he will play the
Gricean strategy in all instances. If the hearer, in turn, knows this, then the
only strategy left is the pure Gricean interpretation strategy. So iterative weak
dominance reasoning forces Gricean behavior. It should be clear, too, that the
reasoning here generalizes to n-point scales for any finite n.
15 Expanding alternative utterances
A persistent criticism of the Gricean account of scalar implicatures is it depends
on a very narrow limitation on the number of messengers a speaker is
able to send. A version of this problem, recently dubbed “the symmetry problem”,
can be explained as follows: Suppose we have an expert speaker in in
the Some/All problem. Recall the informal reasoning that gets us from the fact
that the speaker says ‘some’ to the implicature that he knows that the state of
the world is the some state and not the all state: If the speaker had known that
the ‘all’ sentence was true he would have said it. He did not, so he must not
know it to be true. Therefore, he knows that the speaker knows ‘some’ is true
(since he is an expert).
Unfortunately, we can give a symmetric line of reasoning if the speaker
has utterance ‘some but not all’ available to him. If the speaker had known
that ‘some but not all’ was true he should have said it (by Quantity). Since he
didn’t say it he doesn’t know it. Therefore he must know that ‘all’ is true (since
he is an expert).
In truth, then, an expert speaker who has available to him the utterances
‘some’, ‘some but not all’, and ‘all’ who is governed just by themaximof Quantity
(and truthfulness constraints) should always say either ‘some but not all’
or ‘all’: he should not ever say ‘some’ alone. This is a bad prediction: in real
life, expert speakers often use ‘some’ alone.
A parallel problem faces a game-theoretic treatment once we allow speakers
to use a ‘some but not all’ message in the Simple Gricean game. This is
represented in figure 27. As the symmetry of the game there makes clear, there
is no way to argue on rational grounds in this model that ms should be interpreted
in any particularway. Thuswe again seemto require thatwell-informed
speakers should never say ‘some’ as opposed to ‘some but not all’.
p N 1 − p
msbna
ms
Ss
ma
ms
Sa
Fig. 27:
Gricean game with some-but-not-all
I will discuss two ways of dealing with this problem. One way is to restrict
the alternatives available in a given speech-act situation. The other way is to
add message costs.
Restrictions on alternatives
It is common to simply posit that there are a restricted set of messages that
we consider when we evaluate scalar implicatures, and that these messages do
not include ‘some but not all’. If we do this we can simply keep our previous
models in which ‘some but not all’ was not a recognized speaker option. This
may not be as ad hoc as it seems. We might think that there are constraints
in the grammar on which lexical items compete against each other, and that
these constraints facilitate Gricean inferences. Since Horn (1972) there has been
some effort in linguistics to give principled conditions for two lexical items to
compete against each other (Matsumoto, 1995; Fox and Katzir, 2009).
Message costs
A trick common in the game theory and pragmatics literature for dealing with
this problem is to suppose that the ‘some but not all’ has a small cost.27 This
cost should not be of the same magnitude as the cost of failing to convey information:
cooperative speakers do not sacrifice relevant communicative content
by saying something shorter (rather monosyllabic teenagers do this to display
uncooperativeness). There are different ways of modeling such small costs in
game theory, but for here simply using small numbers will be sufficient.
Take the case of an expert speaker who has available to him three messages
ma, ms, and msbna. We will assume that msbna incurs a small cost. Here, there
are only two types of speakers but three messages: the basic game structure
is in figure 28. Assuming the S will not use weakly dominate strategies we
can prune the game to get the new game in figure 29. Now, of course, msbna
becomes a dominated move for the S. Thus, assuming speaker expertise and
marginal cost to saying the longer form, we get the desired result that ms is
used to indicate the speaker knows some.
Note that this reasoning has limited generality. As the reader can verify, we
still are forced to make the prediction that a speaker who is not assumed to be
an expert will never say ‘some’ to mean ‘some but not all’. To overcome this
limitation we need to makemore substantive (and less plausible) assumptions.
(27For examples in the IBR tradition see Franke (2009) and J ¨ager and Ebert (2008))
p N 1 − p
msbna
ms
Ss
ma
ms
Sa
R
0, 0
L
.9, .9
R
0, 0
L
1, 1
R
1, 1
L
0, 0
R
1, 1
L
0, 0
Fig. 28:
Some but not all with costs
p N 1 − p
msbna
ms
Ss
ma
Sa
L
.9, .9
L
1, 1
R
1, 1
Fig. 29:
Some but not all with costs
Relevance implicatures
So far, I have only considered cases of scalar implicatures. However, in this
section I will suggest that another kind of implicature, a relevance implicature,
might be captured using weak-dominance reasoning.
A famous example from Grice goes as follows.
The hearer stops in the car and asks the speaker if there’s anywhere to get petrol.
The speaker responds that there’s petrol station down the road.
This response takes this to implicate that the station is open and operating.
We can give a model of this situation in which the expert speaker has two
possible states of knowledge,
1) that he knows where the petrol station is and knows it is open (call a speaker who knows this a So) ,
2) that he knows where one is but knows it is closed (call a speaker who knows this a Sc).
This is a rather crude simplification but a game-theoretic model of this situation will need to make some sort of assumption like this.
We can also think of two
signaling options available to him:
1) giving information about the location of the petrol station (g),
2) not giving information (¬g).
If the hearer is told
where the petrol station is he has the choice of either going to the station (action
G), or continuing his journey without going there (action C). If he does not
receive that information he must continue his journey. The payoffs are ordered
as follows: the payoffs are highest if the gas station is open and the hearer
goes there. The medium payoff is that the hearer just continues his search
(regardless of whether the gas station is there or not). And the lowest payoff is
if the gas station is closed and the hearer goes there. Figure 30 is the extensiveform representation of this game.
p N 1 − p
¬g
1, 1
g
So
¬g
1, 1
g
Sc
H
C
1, 1
G
2, 2
C
1, 1
G
0, 0
Fig. 30:
Petrol station game
It is immediately clear from this model that for a speaker (Sc) who knows
the petrol station is closed not providing any information is aweakly dominant
move. For doing so ensures that the speaker continues his journey which is
the highest payoff possible. If this is common knowledge, then then it is also
common knowledge that if the the speaker gives the location of the gas station
then he is of type So. In this case, when hearer receives the location, he will
know that it is open (and thus go to it to maximize his utility).
So in this simple model of Grice’s example, weak dominance reasoning
again yields the correct implicature. This is a very open-ended kind of example
however, so it is hard to say if this is the right way of modeling it. My main
point here is just to demonstrate that the iterative weak dominance reasoning
plausibly extends to cover other examples of implicatures besides scalar implicatures.
However the open-endedness inherent in how we model such examples
makes it hard to give a convincing argument that this is how we should
think of such implicatures.
A non-linguistic case
In this section, I will consider a situation, analogous to a commonly discussed
one in the economics literature, which has the same structure as some of our
scalar implicature cases.28
I show that the iterative weak-dominance reasoning
provides a plausible explanation of how people act in this case, just as it does
in the scalar implicature cases. This provides further support for the idea that
Gricean reasoning need not be thought of as a special sort of reasoning: rather
the exact same patterns of reasoning can be observed in non-linguistic contexts.
First, I will sketch the standardmodel in the economics literature on disclosure
of verifiable information (Grossman, 1981; Milgrom, 1981). In this situation,
a seller has some asset, which he knows the value of, and the seller has the
option to release evidence that proves the asset is at least a certain worth. For
simplicity assume there are two (total) pieces of evidence the seller can release
e1 and e2. Assume the buyer knows there are three possible states of value the
asset can have v0 < v1 < v2. In state v0 there is no evidence the seller can
release, in state v1, the seller can release e1, and in state v2 the seller can release either e1 or e2. The situation works as follows: first the seller the releases some
evidence, and then the buyer performs an action (i.e. makes purchase or a bid
for a purchase). We assume the payoffs are such that the buyer can make the
right decision only if he knows the value of the asset. The seller, on the other
hand, prefers buyer actions corresponding to higher beliefs about the value,
regardless of the actual value.
Regardless of the buyer’s initial beliefs about the likelihood of the asset
having each of the values (as long as he thinks each value is possible), there is
a direct argument that the seller will release all the information he can: i.e. the
seller with an asset of value v2 will release e2 and the seller with an asset of
value v1 will release e1. The argument for this view goes as follows: the seller
of an asset v2 guarantees himself his highest possible return by releasing e2,
(28Dirk Bergemann first suggested this connection to me; Sobel (2010) discusses this connection).
so he will do this. Since this is known to the buyer, then the seller of an asset
with value v1 will guarantee himself his highest return by releasing e1. This
reasoning can easily be extended to any case in which there is a finite range
of values and evidence. This result is sometimes known as the full-disclosure
theorem.
To capture this reasoning in game-theoretic terms we can represent the situation
as a signaling game such as that in figure 31. As the payoffs indicate,
the seller wants the buyer to think the item has as high a value as possible,
whereas only can guarantee his best payoff if he knows the value of the item.
It is easy to see that in this situation, there is only one PBE, the one described
above.29
The PBE concept works here better than it does for the similar scalar implicature
case, the three-point scale discussed in §14, because of the way the
seller’s incentives are structured. The seller wants to fetch the highest bid possible, not just a bid that reflects the true price. This eliminates the pooling
equilibrium in which the seller of v2 releases just e1 and the other two kinds
of sellers release no information: this cannot be part of a PBE since the seller
of v1 will have an incentive to release e1 and be mistaken for a seller of value
v2. So despite the similarity between the the three-point scale game in figure
24 and the disclosure game here, the PBE concept only works to pick out the
“Gricean” style behavior in the disclosure game.
Thus, there is a significant disanalogy between the economic case of disclosure
and the Gricean signaling games I discussed, even the structurally similar
one of the three-point scale.
However, note that in both games iterative elimination
of weakly dominated strategies will serve to pick out the the fully communicative,
“Gricean” behavior. Thus, unlike the PBE notion, iterative weak
dominance captures the commonality between these cases.
We can consider economic situations slightly different from the standard
disclosuremodelwherewewill need iterative elimination of dominated strategies
to pick out the intuitive solution. Consider this variation: A company is
being sold, and, again, there is a mutually known range of possible values,
v0 < v1 < v2, aswell as the same range of evidence that can be released e1 < e2.
The difference here is we suppose that the manager of the company is disclos-
29This observation is related closely to theMilgrom’s argument (1981), though he uses the notion of sequential equilibrium. As it happens, the two equilibrium concepts are equivalent in cases of this
form (Fudenberg and Tirole, 1991, discuss the relationship between the two concepts).
r
p
q
N
;
e2
Sv2
L
0,0
H
2,2
M
1,1
L
0,0
H
2,2
M
1,1
e1
e1
L
0,1
H
2,1
M
1,2
L
0,0
H
2,2
M
1,1
;
Sv1
L
0,1
H
2,1
M
1,2
;
Sv0
L
0,2
H
2,0
M
1,1
R
R
----
Three-state disclosure
----
ing the information, and he does not want the buyer to pay any more than the
actual value, but he does want him to pay as much as possible up to that value.
More specifically Iwill assume that themanager’s least desired outcome comes
frominducing a belief on the part of the buyer that the company is worth more
than it is.
In this case there will always be a PBE with less than full disclosure. After
all, there is a PBE in which the seller of v2 releases e1 and the other two types of
sellers will release no information. This is a PBE because no type of seller has
an incentive to deviate: the seller of v2 gets his best possible payoff, whereas
the seller of v1 does not want to release e1 because he would induce an undesired
action corresponding to the buyer’s belief that the company is of value
v2. However, it seems to me that we would not expect this type of behavior.
Rather, I think, we should expect, as before, full-disclosure. To derive this we
can use iterated weak dominance reasoning here, just as we did in the case of
the three-point scale. So we can see that the kind of reasoning needed in that
case, iterative elimination of weakly dominated strategies, is also needed in
non-linguistic cases.
I take it as a distinct virtue of the iterative dominance approach that it supports
such structural analogies between linguistic problems and non-linguistic
problems. This analogy is one the reasoning-based approaches I discussed
above cannot capture: For those views, as I argued, are sensitive to the initial
probability distribution. But in the disclosure case, the behavior is not contingent
on this distribution. Ratherwe should think (by iterativeweak dominance
reasoning) that regardless of the initial probability distribution we expect the
Gricean behavior on the part of seller and buyer.
---
Rothschild's main purpose in his study is to show that for a variety of idealized
cases game-theoretic accounts of rationality do compel speakers to be behave in
a Gricean manner (i.e. in obeying the Quantity maxim).
We assumed discrete
models taken out of context with mutually known and understood payoffs—
something we rarely see in real speech-act situations.
We also assumed completely
cooperativeness which we modeling by absolute identity of payoffs.
We also hard-wired into the model the semantic conventions by refusing to let
speakers say sentences they do not believe to be true.
Even with these assumptions the most standard solution concepts, such as Nash equilibrium or perfect Bayesian equilibrium, do not serve to pick out the Gricean behavior.
However, we saw that in many cases iterative elimination of weakly dominated strategies serves to single out the Gricean strategy.
References
Battigalli, P. and G. Bonanno,
"Recent results on belief,
knowledge and the epsitemic
foundations of game theory."
---- Research in Economics, 53.
Benz, A.
"Utility and relevance of answers."
In Benz Anton, Gerhard
Jaeger, and Robert van Rooij,
Game Theory and Pragmatics. Breheney.
--- & van Rooij, Robert
"Optimal assertions and
what they implicate."
Topoi, 26.
Bernheim, B. D.
"Rationalizable strategic behaviour."
Econometrica, 52.
Brandenburger, A.
"The power of paradox: some
recent developments
in interactive epistemology."
International Journal of Game Theory, 35
---- Friedenberg, Amanda, and Keisler, H. Jerome
"Admissibility in games."
Econometrica, 76
Chemla, E.
"Universal implicatures and
free choice effects:
Experimental
data.
Semantics and Pragmatics, 2
Chierchia, G.
"Scalar implicatures, polarity
phenomenon, and the
syntax/pragmatic interface.
In A. Belleti (ed.), Structures and Beyond.
Oxford University Press.
---- & Fox, Danny, and Spector, Benjamin
"The
grammatical view of implicature
and the relationship between semantics
and pragmatics."
In Paul Portner, Claudia Maienborn, and Klaus
von Heusinger (eds.), Handbook of Semantics. Mouton.
Cho, In-Koo and Kreps, David M.
"Signaling games and stable equilbiria."
Quarterly Journal of Economics, 102
Crawford, Vincent and Sobel, Joel
"Strategic information transmission."
Econometrica, 50
Farrell, J.
"Meaning and credibility in cheap-talk games."
Games and Economic Behavior, 5
---- & Rabin, Matthew
"Cheap talk."
Journal of Economic Perspectives, 10
Fox, Danny.
"Free choice and the theory of implicature."
Unpublished manuscript.
---- & Katzir, Roni (2009). On the characterization of alternatives.
Unpublished manuscript, MIT.
Franke,Michael.
Signal to Act.
Ph.D. thesis, Institute for Logic, Language
and Computation, University of Amsterdam.
Fudenberg, Drew and Tirole, Jean.
Perfect bayesian equilibrium and
sequential equilbrium.
Journal of Economic Theory, 53
Gazdar, Gerald.
Pragmatics: Implicature, Presupposition and Logical Form.
Academic Press.
Grice, H. P. 1941. Personal identity, Mind.
--. 1948. Meaning, repr. in WoW
--. 1961. The causal theory of perception, repr. in WoW
--. 1966. Logic and conversation: The Oxford lectures.
--. 1967. Logic and conversation. Repr. in a revised form in Studies in the Ways of Words. Harvard University Press, 1989.
--. 1987. Retrospective epilogue, to WoW.
--. 2001. Aspects of reason.
Grossman, S.
"The informational role of warranties and private
disclosure about product quality."
Journal of Law and Economics, 24
Harsanyi, John and Selten, Reinhard
A General Theory of Equilibrium
Selection in Games. MIT Press.
Horn, L. R.
The Semantics of the Logical Operators in English.
Ph.D. thesis, UCLA.
--- A brief history of negation.
Jaeger, G.
Game dynamics connects semantics with pragmatics.
In Ahti-Veikko Pietarinen (ed.), Game Theory and Linguistic Meaning. Elsevier.
---- & and Ebert, Christian.
"Pragmatic rationalizability",
In
Arndt Riester and Torgim Solstad (eds.), Proceedings of SuB13.
de Jager, Tikitu and van Rooij, Robert.
"Explaining implicature."
In Proceedings of the 11th conference on
"Theoretical Aspects of Rationality".
Kohlberg, Elon and Mertons, Jean-Francois.
On the strategic stability of
equilibria.
Econometrica, 54
Lewis, David.
Convention. Harvard University Press.
Marx, Leslie and Swinkles, Jeroen
Order independence for iterated
weak dominance.
Games and Economic Behavior, 18
Matsumoto, Yo
The conversational condition on Horn scales. Linguistics
and Philosophy, 18
Milgrom, Paul (1981).
Good news and bad news: Representation theorems and
applications. Bell Journal of Economics, 12:380–391.
Noveck, Ira (2001).
When children are more logical than adults:
experimental investigations of implicature.
Cognition, 78:165–188.
Parikh, Prashant
Communication and strategic inference.
Linguistics
and Philosophy, 14(473–531).
— (2001). The Use of Language. CSLI.
Pearce, David (1984).
Rational strategic behavior and the problem of perfection.
Econometrica, 52:1029–1050.
Rabin, Matthew.
"Communication between rational agents."
--- Journal of Economic Theory, 51.
Rochet, Jean-Charles.
Selection of unique equilibrium payoff for extensive
games with perfect information. Mimeo.
van Rooij, Robert (2009).
"Optimal-theoretic and game-theoretic approaches to
implicature."
In Edward N. Zalta (ed.), Stanford Encyclopedia of Philosophy.
Spring 2009 edition.
Sauerland, Uli (2004).
"Implicatures in complex
sentences."
Linguistics and
Philosophy, 27
Schelling, Thomas C. (1960).
The Strategy of Conflict. Harvard University Press.
Schulz, Katrin and van Rooij, Robert (2006).
Pragmatic meaning and nonmonotonic
reasoning.
Linguistics and Philosophy, 29(2):205–250.
Soames, Scott (1982).
How presuppositions are inherited: A solution to the
projection problem.
Linguistic Inquiry, 13:483–545.
Sobel, Joel (2009).
Signaling games. In Encyclopedia of Complexity and Systems
Science. 8125–8139.
— (2010). Giving and recieving advice. Working Paper UCSD.
Spector, Benjamin (2006).
Aspects de la pragmatique des op´erateurs logiques. Ph.D.
thesis, Universite Paris 7, Denis Diderot.
Spence, Michael (1973).
Job market signaling. The Quarterly Journal of Economics,
87:355–374.
Sperber, D. and D. Wilson, Relevance: Communication and Cognition.
Blackwell.
Stalnaker, Robert
Saying and meaning, cheap talk and credibility.
In
Anton Benz, Gerhard J ¨ager, and Robert van Rooij (eds.), Game Theory and
Pragmatics. Palgrave MacMilla
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