RSA vs IBR
In this post, I want to compare the Rational Speech Act approach with the Iterated Best Response approach of Franke (2011). I'm also going to discuss Franke's IBR model of Free Choice, turn it into an RSA model, and explain why I find both unconvincing.
1.The IBR approach
Let's back up a little.
Lewis (1969) argued that linguistic conventions solve a game-theoretic coordination problem.
The coordination problem is easy to see in simple signalling games, where each state of the world calls for a particular action on the part of the hearer, but only the speaker can directly observe that state. The speaker can produce a number of signals, depending on what she observes. The hearer chooses a response based on the signal she receives. Speaker and hearer would like to coordinate their strategies so that the hearer ends up performing the appropriate action for each state.
Human languages are unlike simple signalling games in that there's usually no particular act a hearer is expected (or desired) to perform, in response to a given utterance. Lewis (1969) therefore suggests that our linguistic conventions solve a different kind of coordination problem that arises only among speakers. Lewis was not happy with this conclusion, though. In Lewis (1975), he brings hearers back into the picture, suggesting that their role is to "trust" the speakers.
Franke (2011) adopts a similar perspective. He assumes that human speakers and hearers are, in fact, engaged in a signalling game. The speaker knows the answer to some question (say, whether it is raining) and chooses a signal with the hope of conveying the answer to the hearer. The hearer receives the signal and chooses an interpretation. They might, for example, choose to interpret the signal as saying that it is raining. Speakers and hearers would like to coordinate these strategies, so that they associate the available signals with the same states.
It's not clear to me what kind of act "choosing an interpretation" is meant to be. Perhaps it's a doxastic "act": the speaker comes to believe that it is raining. Or perhaps it's a more indirect act: the speaker decides to act in whatever way would be appropriate if it were raining. Or perhaps it's an act of accepting that it is raining, so that this proposition becomes part of the common ground.
Let's set this issue aside for now.
Signalling games usually have many equilibria. Conventions are supposed to help. But it's not obvious how. What association between signals and states I should use as a speaker depends entirely on what association I think you will use as a hearer, which in turn depends entirely on what association you expect me to use, and so on. Where in this endless loop could a linguistic convention enter the picture?
Franke's answer is that when speakers and hearers replicate each other's reasoning, their replications become increasingly unsophisticated. The iterations terminate in a "level-0" player who simply acts in accordance with the conventional, literal meaning.
This is very similar to (what I take to be) the core idea behind the Rational Speech Act framework. The main difference is that Franke's IBR framework gives an active role to hearers. In the RSA framework, hearers simply update on what they hear. In the IBR framework, hearers choose an interpretation.
2.An IBR model of scalar implicature
Here's an IBR model for the implicature from 'some' to 'not all'.
As usual, the question under discussion is whether some or all students passed. For simplicity, let's assume it is already known that at least one student passed. The speaker knows that not all students passed, and tries to get this across to the hearer. There are two relevant states, ∀ and ∃¬∀, and two relevant utterances, 'some' and 'all'. The literal meaning associates 'some' with both states and 'all' with the ∀ state.
A level-0 speaker would follow the basic convention to utter a (relevant) sentence iff it is true. Knowing that the true state is ∃¬∀, she would utter 'some'. Knowing that the state is ∀, she would randomly choose 'all' or 'some':
Level-0 speaker:
state message ∃¬∀ 'some' ∀ 'some' or 'all'
Now suppose you're a level-1 hearer who models his conversational partner as a (well-informed and cooperative) level-0 speaker. If you hear 'all', you can infer that the state is ∀. If you hear 'some', the state might be ∀ or ∃¬∀. Suppose your prior credence in the two possibilities is 1/2 each. By Bayes' Theorem, your posterior credence in ∀, after hearing 'some', is then 1/3, while your credence in ∃¬∀ is 2/3. But you have to choose an interpretation. Should you choose to interpret 'some' as ∃¬∀ or as ∀? Presumably, you choose the more likely interpretation:
Level-1 hearer:
message interpretation 'all' ∀ 'some' ∃¬∀
Next, consider a "level-2" speaker who models her conversational partner as a level-1 hearer. Evidently, she will use 'all' to convey ∀ and 'some' to convey ∃¬∀:
Level-2 speaker:
state message ∃¬∀ 'some' ∀ 'all'
We've reached equilibrium. A level-3 hearer would choose the same interpretation as the level-1 hearer, and so a level-4 speaker would make the same choice as the level-2 speaker, and so on.
We have derived the fact that people use 'some' to convey 'some and not all', even though the literal meaning of 'some' is compatible with 'all'.
3.Why I prefer RSA
But what's up with that level-1 hearer?
Suppose again that you're a (level-1) hearer who models their conversational partner as a level-0 speaker. You hear them say 'some'. You might become 2/3 confident that the state is ∃¬∀. Why would you then choose to interpret 'some' as ∃¬∀? This makes no sense, on any of the above ideas about what this choice could mean. If you had to bet on the state, you would not bet on ∃¬∀ at all odds. You would not behave as if ∃¬∀ were true. Nor would you be inclined to add ∃¬∀ to the common ground. You would simply be unsure about what the speaker meant, and therefore about the state of the world.
The RSA approach gets this right. In an RSA model of the above scenario, the level-1 hearer would arrive at certain credences about the state, and that's all he has to do. (You might think of this as a "mixed act", if you want.)
Level-1 hearer:
message interpretation 'all' ∀:1 'some' ∃¬∀:2/3, ∀:1/3
A level-2 speaker who wants to maximize the hearer's accuracy in the true state would still choose 'all' to convey ∀ and 'some' to convey ∃¬∀:
Level-2 speaker:
state message ∃¬∀ 'some' ∀ 'all'
A subsequent level-3 hearer could now be certain that 'all' means ∀ and 'some' means ∃¬∀. We reach the same equilibrium, but a little later in the recursion.
I suspect that this generalizes: one can replicate every IBR model by an RSA model in which the "de-probabilification" that occurs on the hearer side in the IBR model occurs at a subsequent speaker stage in the RSA model.
4.Franke's IBR model of Free Choice
Let's now look at Franke's derivation of Free Choice effects – the main topic of Franke (2011).
Suppose it is common knowledge that at least one of A and B is permitted. There are three possible states: A alone is permitted, B alone is permitted, and A and B are both permitted. Let's abbreviate these states as MA, MB, and MAB.
The available messages are '◇A', '◇B', and '◇(A ∨ B)'. The literal meaning of '◇A' is that A is permitted. The literal meaning of '◇B' is that B is permitted. The literal meaning of '◇(A v B)' is that at least one of A and B is permitted. We'd like to predict that '◇(A v B)' can be used to convey that both are permitted.
As before, a level-0 speaker chooses an arbitrary utterance provided that it is true.
Level-0 speaker:
state message MA '◇A' or '◇(A v B)' MB '◇B' or '◇(A v B)' MAB '◇A' or '◇B' or '◇(A v B)'
Now consider a level-1 hearer who models his conversational partner as a level-0 speaker. If the speaker says '◇A', the hearer can infer that the state is either MA or MAB. Given flat priors over the three states, the posterior probability of MA (given '◇A') will be 3/5, that of MAB 2/5. Hearing '◇B' should similarly make the hearer 3/5 confident in MB and 2/5 in MAB. '◇(A v B)' doesn't rule out any of the states, but it favours MA and MB over MAB: the former have posterior probability 3/8 each, the latter probability 2/8.
We're assuming the IBR framework, so the hearer has to choose an interpretation. He will presumably choose to interpret '◇A' as MA, '◇B' as MB, and '◇(A v B)' as either MA or MB:
Level-1 hearer:
message credence interpretation '◇A' MA:3/5, MAB:2/5 MA '◇B' MB:3/5, MAB:2/5 MB '◇(A v B)' MA:3/8, MB:3/8, MAB:2/8 MA or MB
We may note in passing that the choice of interpretation depends on the priors. If the hearer's prior credence in MAB is, say, 0.5 while his credence in MA is 0.3, he will interpret '◇A' as MAB. But let's assume the hearer's priors are flat.
Next we have a level-2 speaker who models her conversational partner as a level-1 hearer. If she knows that the state is MA, and she wants to get this across to the hearer, her best choice is to utter '◇A'. If she knows that the state is MB, her best choice is '◇B'. If she knows that the state is MAB, she has a problem. There's nothing she could say that would make the hearer believe that the state is MAB. Franke assumes that she will randomly choose a message.
level-2 speaker:
state message MA '◇A' MB '◇B' MAB '◇A' or '◇B' or '◇(A v B)'
Now imagine you're a (level-3) hearer who models his conversational partner as a level-2 speaker. If you hear '◇A', you can infer that the state is either MA or MAB. With flat priors, the former is more likely. If you hear '◇B', the most likely state is MB. If you hear '◇(A v B)', the state must be MAB. You'll choose the following interpretation:
level-3 hearer:
message credence interpretation '◇A' MA:3/4, MAB:1/4 MA '◇B' MB:3/4, MAB:1/4 MB '◇(A v B)' MAB:1 MAB
This is what we wanted to predict: '◇A' is interpreted as MA, '◇B' as MB, and '◇(A v B)' as MAB. A level-4 speaker will use the same association to communicate her information. We've reached equilibrium.
5.Converting the IBR model into an RSA model
As above, I would complain that the supposed hearer choices are implausible and unmotivated. Imagine you're the level-1 hearer (with flat priors). You model the speaker as a level-0 speaker. You hear '◇A'. You become 60% confident that the state is MA. In what sense would you choose to interpret '◇A' as MA? Why would a subsequent level-2 speaker care about this choice?
As above, however, we can replicate the IBR model with a slower RSA model in which the hearer does not have to choose an interpretation.
var states = ['MA', 'MB', 'MAB'];
var meanings = {
'◇A': ['MA', 'MAB'],
'◇B': ['MB', 'MAB'],
'◇(A v B)': ['MA', 'MB', 'MAB']
};
var speaker0 = function(observation) {
return Agent({
options: keys(meanings),
credence: update(Indifferent(states), observation),
utility: function(u,s){
return meanings[u].includes(s) ? 1 : -1;
}
});
};
var hearer_prior = Indifferent(states);
// var hearer_prior = Credence({ MA: 0.3, MB: 0.2, MAB: 0.5 });
var hearer1 = Agent({
credence: hearer_prior,
kinematics: function(utterance) {
return function(state) {
var speaker = speaker0(state);
return sample(choice(speaker)) == utterance;
}
}
});
showKinematics(hearer1, keys(meanings));
The level-0 speaker behaves as in the IBR model. The level-1 hearer arrives at the same credence as in the IBR model. He does not choose an interpretation. Here's the output of our simulation in table format:
Level-1 hearer:
message credence '◇A' MA:3/5, MAB:2/5 '◇B' MB:3/5, MAB:2/5 '◇(A v B)' MA:3/8, MB:3/8, MAB:2/8
You can see how the output depend on the priors by uncommenting the line // var hearer_prior = Credence({ MA: 0.3, MB: 0.2, MAB: 0.5 }); in source block #1. (But add the comment slashes back before you run the code blocks below.)
Next, we introduce a level-2 speaker who models the hearer as a level-1 hearer and wants him to have a high degree of belief in the true state.
// continues #1
var speaker2 = function(observation) {
return Agent({
credence: Indifferent([observation]),
options: keys(meanings),
utility: function(u,s) {
return learn(hearer1, u).score(s);
}
});
};
showChoices(speaker2, states);
At this stage, we have the same association between states and messages that we got at level 1 in the IBR model:
Level-2 speaker:
state message MA '◇A' MB '◇B' MAB '◇A' or '◇B'
Now imagine you're a level-3 hearer (with flat priors) who thinks he faces a level-2 speaker. If you hear '◇A', you should become 2/3 confident that the state is MA, and 1/3 that it is MAB. If you hear '◇B', you should become 2/3 confident that the state is MB, and 1/3 that it is MAB. What if you hear '◇(A v B)'? You'll be surprised. A level-2 speaker never utters '◇(A v B)'!
// continues #2
var hearer3 = Agent({
credence: hearer_prior,
kinematics: function(utterance) {
return function(state) {
var speaker = speaker2(state);
return sample(choice(speaker)) == utterance;
}
}
});
showKinematics(hearer3, keys(meanings));
We need to settle how the level-3 hearer updates on '◇(A v B)', so that the speaker at level 4 can decide whether to utter it. Let's assume that if he hears the surprise message '◇(A v B)', the level-3 hearer simply retains his prior credence over states. The following code achieves this.
// continues #2
var hearer3 = Agent({
credence: hearer_prior,
kinematics: function(utterance) {
return function(state) {
var speaker = speaker2(state);
return utterance.includes('v') || sample(choice(speaker)) == utterance;
}
}
});
showKinematics(hearer3, keys(meanings));
Here is the output in table form:
Level-3 hearer:
message credence '◇A' MA:2/3, MAB:1/3 '◇B' MB:2/3, MAB:1/3 '◇(A v B)' MA:1/3, MB:1/3, MAB:1/3
A level-4 speaker should obviously utter '◇A' if the state is MA, and '◇B' if it is MB. If the state is MAB, all options are equally good.
// continues #4
var speaker4 = function(observation) {
return Agent({
credence: Indifferent([observation]),
options: keys(meanings),
utility: function(u,s) {
return learn(hearer3, u).score(s);
}
});
};
showChoices(speaker4, states);
We now have the same association between states and messages that we got at level 2 in the IBR model:
Level-4 speaker:
state message MA '◇A' MB '◇B' MAB '◇A' or '◇B' or '◇(A v B)'
From here on, things go smoothly. A level-5 hearer will take '◇(A v B)' to be a sure sign of MAB. With flat priors, he will be inclined towards MA if he hears '◇A' and towards MB if he hears '◇B'. A level-6 speaker will therefore choose '◇A' in MA, '◇B' in MB, and '◇(A v B)' in MAB.
// continues #5
var hearer5 = Agent({
credence: hearer_prior,
kinematics: function(utterance) {
return function(state) {
var speaker = speaker4(state);
return sample(choice(speaker)) == utterance;
}
}
});
var speaker6 = function(observation) {
return Agent({
credence: Indifferent([observation]),
options: keys(meanings),
utility: function(u,s) {
return learn(hearer5, u).score(s);
}
});
};
showChoices(speaker6, states);
We have replicated Franke's IBR model of Free Choice as an RSA model.
6.Why I'm not convinced
Unfortunately, this derivation of the Free Choice effect depends on many dubious assumptions.
For a start, the derivation only goes through on specific assumptions about the hearer's priors. If you play around with hearer_prior in source block #1 (and then re-run source block #6), you can see that the derivation breaks down whenever the hearer's prior for MA is not exactly equal to that of MB.
The derivation also relies on a very specific assumption about how the level-3 hearer updates on the surprise message '◇(A v B)': that he updates by sticking to his priors. We could alternatively have assumed that the level-3 hearer models the speaker as a soft-maximizer (with possibly very high alpha), so that utterances of '◇(A v B)' are not absolutely impossible. We would then no longer predict the Free Choice effect.
The derivation further relies on the assumption that the level-4 speaker has no preference for simplicity: she is indifferent between '◇A' and '◇B' and '◇(A v B)' in state MAB, even though the last option is needlessly verbose. If she prefers the simpler options, the derivation breaks down.
Analogous problems affect Franke's IBR model. It, too, requires prior indifference between MA and MB. And it requires very specific and peculiar assumptions about the speaker at level 2. Remember that if this speaker is in state MAB, she knows that nothing she could say would get across the true state to the (level-1) hearer. Franke assumes that she randomly chooses between '◇A' and '◇B' and '◇(A v B)'. Any preference for simplicity would break the derivation. So would the availability of a fourth option – say, remaining silent. Why isn't this an option? If the speaker knows that each of the three available messages would cause a false belief, then why would she say anything at all?
On top of these problems, Franke's derivation inherits the general implausibility of IBR models, with their mysterious hearer choices.
As Franke notes, his derivation also only works for cases with exactly two disjuncts. You can confirm this (for the RSA version) by changing the states and meanings in source block #1 to the following and re-running source block #6:
var states = ['MA', 'MB', 'MC', 'MABC'];
var meanings = {
'◇A': ['MA', 'MABC'],
'◇B': ['MB', 'MABC'],
'◇C': ['MC', 'MABC'],
'◇(A v B v C)': ['MA', 'MB', 'MC', 'MABC']
};
Overall, this doesn't look promising. In the next post, I'll try to do better.